IMAGE DENOISING USING FILTERING AND THRESHOLD TECHNIQUES IN WAVELET DOMAIN

Transcription

1 IMAGE DENOISING USING FILTERING AND THRESHOLD TECHNIQUES IN WAVELET DOMAIN A THESIS Submitted by KALAVATHY.K for the award of the degree of DOCTOR OF PHILOSOPHY DEPARTMENT OF COMPUTER SCIENCE Dr. M.G.R EDUCATIONAL AND RESEARCH INSTITUTE, UNIVERSITY (Declared u/s 3 of the UGC Act, 1956) CHENNAI JULY 2012 CHAPTER 1 1

2 BONAFIDE CERTIFICATE Certified that this thesis titled Image Denoising using Filtering and Threshold Techniques in Wavelet Domain is the bonafide work of Mrs. Kalavathy.K who had carried out the research under my supervision. Certified further that to the best of knowledge the work reported herein does not form part of any other thesis or dissertation on the basis of which a degree or award was conferred on an earlier occasion of this or any candidate. 2

3 DECLARATION This is to certify that the thesis entitled IMAGE DENOISING USING FILTERING AND THRESHOLD TECHNIQUES IN WAVELET DOMAIN, submitted by me to the DR. M.G.R EDUCATIONAL AND RESEARCH INSTITUTE UNIVERSITY for the award of the degree of Doctor of Philosophy is a bonafide record of research work carried out by me under the supervision of Dr.SURESH.R.M. The contents of this thesis, in full or in parts, have not been submitted to any other Institute or University for the award of any degree or diploma. 3

4 ABSTRACT Image Denoising is an important pre-processing task before further processing of image like segmentation, feature extraction, texture analysis etc. The purpose of denoising is to remove the noise while retaining the edges and other detailed features as much as possible. This noise gets introduced during acquisition, transmission & reception and storage & retrieval processes. As a result, there is degradation in visual quality of image. The noises considered in this thesis are Additive White Gaussian Noise (AWGN) and Impulsive Noise. Wavelets play a major role in image compression and image denoising due to the property of sparsity and multiresolution structure. Wavelet Thresholding is another important technique in wavelet domain filtering. Some spatial-domain and transform-domain image filtering algorithms and wavelet thresholding algorithm have been developed in this thesis to suppress Additive White Gaussian noise (AWGN) and Impulse noise. In literature, many efficient image filters are found that perform well under low noise conditions. But their performance is not so good under moderate and high noise conditions. Thus, it is felt that there is sufficient scope to investigate and develop quite efficient but simple algorithms to suppress moderate and high power noise in images. 4

5 The present research work is focused on developing efficient filtering and - thresholding algorithm based on wavelet to suppress AWGN and Impulse Noise under moderate and high noise level quite effectively without yielding much distortion and blurring. The performances of the developed algorithms/methods are compared with the existing methods in terms of Peak Signal to Noise Ratio (PSNR) and Execution Time (TE). The thesis comprises the research contribution for denoising standard images namely 1. Improved wavelet domain algorithm- Neighbourhood Pixel Filter Algorithm (NPFA) 2. Enhancing performance-adaptive Sub-band Thresholding (AST) Technique 3. Modified switching median filter-switching Weighted Adaptive Median (SWAM) Filtering Algorithm 4. Hybrid technique- Filtering and Thresholding Algorithm (FTA)-MRI (Brain Image) The approaches adopted and the novel algorithms designed are summarized here as follows: A filter called Neighbourhood Pixel Filter (NPF) is developed to suppress the Gaussian noise effectively. This filter behaves like a low pass filter in smooth region by decreasing noise variance effectively and giving similar weights to all its neighbouring pixels. This in turn cuts off only high frequency noise signal instead of all noisy signals. After implementing NPFA, it is observed that this method out performs the other wavelet method under moderate and high noise level with an 5

6 average PSNR value of db and minimum execution time TE ms operating on Lena image under various noise conditions. An Adaptive Subband Thresholding (AST) technique is developed based on wavelet coefficients in order to overcome the spatial adaptivity which is not well suited near object edges, where the variance field is not smoothly varied by producing effective results. From simulation and results it is observed that AST technique outperforms the other existing wavelet domain methods under moderate and high noise level by possessing average PSNR value of db and minimum execution time TE ms operating on Lena image under various noise conditions. A new filter called Switching Weighted Adaptive Median Filter (SWAM) is developed for effective suppression of Impulse Noise. This filter is designed by determining the appropriate window length initially based on the width of the Impulse Noise present in the input signal and on the amount of corrupted pixels. Due to this the unwanted filtering of uncorrupted pixels and blurring are reduced even at high density noise. A hybrid method for MRI-Brain Image restoration is developed by combining AST technique based on wavelet coefficient along with NPFA called Filtering and Thresholding Algorithm (FTA). 6

7 ACKNOWLEDGEMENTS As the curtains draw on my journey as a Ph.D student, I look forward to embarking on my career. First and Foremost I would like to thank my supervisor Dr.R.M.Suresh, Principal, Jerusalem College of Engineering without whom this thesis would not have been possible. Words cannot express my gratitude to him for his patience, continuous support and encouragement. I express my sincere thanks to Dean Research, Dr.M.G.R. University and other staff members who provided all the official facilities to me. I am also thankful to my D.C members Dr.Cyril Raj, Professor and Head of CSE and IT Department, Dr.T.Tamilarasi, Professor CSE Department for being a part of my thesis committee and providing useful comments and constructive criticisms that had helped to improve this thesis. It is my duty to remember and acknowledge the encouragement and support provided by (Late) Smt.Manjula Munirathnam, the Chairperson, R.M.D Engineering College Kavarapettai. I express my sincere thanks to Sri.R.S.Munirathinam, the Chairman, Mr.R.M.Kishore, the Vice-Chairman, Mr.Yalamanji Pradeep, the Secretary, and Mr.R.Jodhi Naidu, the Director, R.M.D Engineering College, Kavarapettai for their multi dimensional support and encouragement throughout the period. I would 7

8 like to thank the Principal Dr.K.Sivaram, Dean Dr.K.Dharmalingam, HOD(S&H), Dr.Maria Susai Immanuel, R.M.D Engineering College, Kavarapettai for their valuable suggestions and support in my research throughout my period. I am deeply indebted to Dr.Joice Punitha, Dr.R.Priya for their constant support motivations and encouragement in the progress of my research work. I extend my hearty felt thanks to my friends and colleagues for their prayers, cooperation and support. I want to express my deepest thanks to my husband Mr.R.Sundararajan and my sons Mr.S.Ashwin and Mr.S.Aravind and my parents. I cannot thank them enough for their everlasting love, support, understanding and those are the most precious gift in my life. My Ph.D work will not be possible without their strength and support behind me. Lastly I am thankful to all those who have supported me directly or indirectly during the doctoral research work. KALAVATHY.K 8

9 TABLE OF CONTENTS LIST OF TABLES LIST OF FIGURES LIST OF ABBREVIATIONS xi xiv xix 1 Introduction Preliminaries Performance Evaluation in Image Denoising Noise in Images Sources of Noise Mathematical Representation of Noise Classification of Denoising Methods Spatial Domain Filtering Methods Linear filters Non-Linear Filters Transform Domain Filtering Methods Spatial Frequency Filtering Wavelet Domain Wavelet Transforms Introduction to Wavelet Mathematical Representation of Wavelet Types of Wavelet Transform Discrete Wavelet Transform (DWT) Continuous Wavelet Transform (CWT) Decomposition Process Composition Process Wavelet Thresholding Types of Thresholding Methods Motivation 18 9

10 1.3 Problem Statement Objective and Scope of Study Overview ofthethesis 20 2 Literature Survey Brief History Overview Wavelet Based Techniques Thresholding Algorithm Statistical Model Spatial Domain Filtering Impulse Noise Removal Techniques Color Image and MRI Denoising Technique Conclusion 36 3 Neighbourhood Pixel Filtering Algorithm Introduction Undecimated Algorithm ÀTrous Wavelet Algorithm Determination of Wavelet Coefficient Using ÀTrous Algorithm Neighbourhood Pixel Filter (NPF) Neighbourhood Pixel Filtering Algorithm (NPFA) for Denoising Simulation and Results Conclusion 57 4 Adaptive Subband Thresholding Technique introduction Wavelet Based Image Denoising Visu Shrink Sure Shrink 60 10

11 4.2.3 Baye s Shrink Modified Baye s Shrink Normal Shrink Oracle Shrink and Oracle Thresh Neigh Shrink Denoisingby Wavelet Coefficient Statistical Model Deterministic Modelling of Wavelet Coefficients Statistical Modelling of Wavelet Coefficients Estimation of Noise Variance Optimum Value Threshold and Threshold Selection Threshold Selection Optimum Value Threshold Adaptive Subband Thresholding(AST)Algorithm Simulation and Results Conclusion 92 5 Switching Weighted Adaptive Median Filter introduction Development of SWAM Filter Impulse Noise Model Recursive Weighed Median Filter Recursive Weighted Median Filters-Definition Adaptive Window Size Selection Window Selection Median Controlled Algorithm Impulse Noise Detection Algorithm for Switching Weighted Adaptive Median (SWAM) 100 Filter 5.7 Simulation and Results Conclusion

12 6 Application of Image Denoising in Medical Images introduction Image Denoising in Medical Resonance Images (MRI) Development of Filtering and Thresholding Algorithm (FTA) Filtering and Thresholding Algorithm Analysis of Image Denoising using Wavelet Coefficients 125 and Adaptive AST Technique Analysis of Image Denoising Using Wavelet Coefficients 126 and Adaptive AST Technique Along with NPFA Filters 6.4 Simulation and Results Conclusion Conclusion and Future Work 7.1 Comparative Analysis Removal of AWGN Statistical Analysis Removal of Impulse Noise 7.2Contribution of This Work 7.3 Scope for Future Work REFERENCES 12

13 LIST OF TABLES S.No Table No. Table Name Page No Comparison Table for Existing and Proposed 25 Filtering Algorithms Comparison Table for Existing and Proposed 33 Thresholding Algorithms Comparison Table for PSNR Value for Barbara Image using NPFA and Existing 48 Wavelet Method for Different Noise Level Comparison Table for PSNR Value for Boat Image using NPFA and Existing 48 Wavelet Method for Different Noise Level Comparison Table for PSNR Value for Pepper Image using NPFA and Existing 49 Wavelet Method for Different Noise Level Comparison Table for PSNR Value for House Image using NPFA and Existing 49 Wavelet Method for Different Noise Level Comparison Table for PSNR Value for Lena Image using NPFA and Existing 50 Wavelet Method for Different Noise Level Execution time (milliseconds) taken by Various Filters for Images in the Low Noise Level Range 50 for = Execution time (milliseconds) taken by Various Filters for Images in the Moderate Noise Level Range 51 for = Execution time (milliseconds) taken by Various Filters for Images in the High Noise Level Range 51 for = 70 13

14 Comparison of PSNR Values for Different 79 Noise Standard Deviations for Barbara Image Comparison of PSNR Values for Different 80 Noise Standard Deviations for Lena Image Comparison of PSNR Values for Different 81 Noise Standard Deviations for Boat Image Comparison of PSNR Values for Different 82 Noise Standard Deviations for House Image Comparison of PSNR Values for Different 83 Noise Standard Deviations for Pepper Image Execution Time (Milliseconds) taken by Various Filters for Images in the Low Noise Level 89 Range = Execution Time (Milliseconds) taken by Various Filters for Images in the Moderate Noise Level 90 Range = Execution Time (Milliseconds) taken by Various Filters for Images in the High Noise Level 91 Range = Filter Performance in Terms of PSNR operated on Lena Image for Various Noise Conditions 106 ( varies from 5 to 80) Filter Performance in Terms of PSNR operated on Barbara Image for Various Noise Condition 107 ( varies from 5 to 80) Filter Performance in Terms of PSNR operated on Boat Image for Various Noise Condition 14

15 107 ( varies from 5 to 80) Filter Performance in Terms of PSNR operated on House Image for Various Noise Conditions 107 ( varies from 5 to 80) Filter Performance in Terms of PSNR operated on Pepper Image for Various Noise Conditions 108 ( varies from 5 to 80) Filter Performance in Terms of IEF operated on Lena Image for Various Noise Conditions 108 ( varies from 5 to 80) Filter Performance in Terms of IEF operated on Barbara Image for Various Noise Conditions 108 ( varies from 5 to 80) Filter Performance in Terms of IEF operated on Boat Image for Various Noise Conditions 109 ( varies from 5 to 80) Filter Performance in Terms of IEF operated on House Image for Various Noise Conditions 109 ( varies from 5 to 80) Filter Performance in Terms of IEF operated on Pepper Image for Various Noise Conditions 109 ( varies from 5 to 80) Filter Performance in Terms of PSNR operated on Brain Image 1 for Various Noise Conditions 130 ( varies from 5 to 80) Filter Performance in Terms of PSNR operated on Brain Image 2 for Various Noise Conditions 131 ( varies from 5 to 80) Execution time (milliseconds) taken by 134 Various Filters for Brain Image1, Image 2 15

16 LIST OF FIGURES S.No Figure No. Name of the Figure Page No Illustration of Noise in the Image Representation of Additive White Gaussian 6 Noise Distribution Representation of Salt and Pepper Noise 7 Distribution Representation of Speckle Noise Distribution Classification of Image Denoising Methods Differences between Wave and Wavelet Representation of Wavelet 12 16

17 8 1.8 One Decomposition Step of the Two 14 Dimensional Images One DWT Decomposition Step One Composition Step of the Four Sub Images Subbands of the 2-D Orthogonal Wavelet 16 Transform Noisy Signals in Time Domain and Wavelet 17 Domain Types of Thresholding Denoising Concept Outline of the Research Work (a) Wavelet Frame (b) NPFA Frame Two Label Decomposition of Boat Image 45 before Adding Noise Wavelet Two Label Decomposition of Boat Image 46 after Adding Noise Wavelet Performances of Various Filters for Barbara 46 Image with AWGN = Performances of Various Filters for Boat 47 Image with AWGN = Performances of Various Filters for Pepper 47 Image with AWGN = Performances of Various Filters for House 47 Image with AWGN = Performances of Various Filters for Lena 47 Image with AWGN = 5 17

18 Performance Comparison of Various Filters in Terms of PSNR operated on Barbara for 52 Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in Terms of PSNR operated on Lena for 53 Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in Terms of PSNR operated on Boat for 54 Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in Terms of PSNR operated on Pepper for 55 Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in Terms of PSNR operated on House for 5 6 Various Noise Levels ( varies from 5 to 80) Frame Work of Wavelet Transform based 60 Image Denoising Soft Threshold Characteristics Block Diagram for AST technique Two Label Decomposition of Barbara Image 71 After Adding Noise Wavelet Two Label Decomposition of Barbara Image 72 After Before Noise Wavelet Smooth Noise Wavelet Added Subband Region of the Smooth Noise Performances of Various Filters for Barbara 74 Image with AWGN = Performances of Various Filters for House 75 Image with AWGN = 5 18

19 Performances of Various Filters for Boat 76 Image with AWGN = Performances of Various Filters for Lena 77 Image with AWGN = Performances of Various Filters for Pepper 78 Image with AWGN = Performance Comparison of Various Filters in 84 Terms of PSNR operated on Barbara for Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in 85 Terms of PSNR operated on Lena for Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in 86 Terms of PSNR operated on Boat for Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in 87 Terms of PSNR operated on House for Various Noise Levels ( varies from 5 to 80) Performance Comparison of Various Filters in 88 Terms of PSNR operated on Pepper for Various Noise Levels ( varies from 5 to 80) Block Diagram for Switching Weighted 95 Adaptive Median (SWAM) Filter Block Diagram of Median Controlled 99 Algorithm Noisy Image Input Image-Second Stage Performances of Various Filters for Lena Image 104 with AWGN = 5 19

20 Performances of Various Filters for Barbara 104 Image with AWGN = Performances of Various Filters for Boat Image 105 with AWGN = Performances of Various Filters for House 105 Image with AWGN = Performances of Various Filters for Pepper 106 Image with AWGN = Performance Comparison of Various Filters in Terms of PSNR operated on Lena for Various 110 Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of PSNR operated on Barbara for 111 Various Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of PSNR operated on Boat for Various 112 Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of PSNR operated on House for Various 113 Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of PSNR operated on Pepper for 114 Various Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of IEF operated on Lena for Various 115 Noise Levels ( varies from 10 to 100) 20

21 Performance Comparison of Various Filters in Terms of IEF operated on Barbara for Various 116 Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of IEF operated on Boat for Various 117 Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of IEF operated on House for Various 118 Noise Levels ( varies from 10 to 100) Performance Comparison of Various Filters in Terms of IEF operated on Pepper for Various 119 Noise Levels ( varies from 10 to 100) Block Diagram for FTA Method (Stage 1) Block Diagram for FTA Method (Stage 2) Two Level Decomposition of Brain Image Performances of Various Filters for Brain 128 Image 1 with AWGN = Two Level Decomposition of Brain Image Performances of Various Filters for Brain 129 Image 2 with AWGN = Performance Comparison of Various Filters in Terms of PSNR operated on Brain Image 1 for 132 Various Noise Levels ( varies from 5 to 80) 21

22 Performance Comparison of Various Filters in Terms of PSNR operated on Brain Image 2 for 133 Various Noise Levels ( varies from 5 to 80) LIST OF ABBREVIATIONS AST- Adaptive Subband Thresholding AWGN- Additive White Gaussian Noise CNR- Contrast to Noise Ratio CT-Computerized Tomography CWT- Continuous Wavelet Transform DCT- Discrete Cosine Transform DWT- Discrete Wavelet Transform EQ-Estimation Quantisation FTA-Filtering and Thresholding Algorithm GGD- Generalised Gaussian Distribution GSM-Gaussian Scale Mixture HMM- Hidden Marcov Model HVS- Human Visual System ICA- Independent Component Analysis IEF- Image Enhancement Factor IST- Inter Symbol Interference 22

23 LET-Linear Expansion of Threshold MAP- Maximum a Posterior MBS- Modified Baye s shrink ML-Maximum Likelihood MMSE-Minimum Mean Square Error MRI- Medical Resonance Imaging MSE- Mean Square Error NLM- Non Local Mean NPD-Neighbourhood Pixel Difference NPF- Neighbourhood Pixel Filter NPFA- Neighbourhood Pixel Filtering Algorithm NS- Normal Shrink PDF Probability Density Function PET-Positron Emission Tomography PSNR- Peak Signal to Noise Ratio RVIN- Random- Valued Impulse Noise RWM-Recursive Weighted Median SAM-Switching Adaptive Median SAR- Synthetic Aperture Radar SMF-Standard Median Filter SN- Speckel Noise 23

24 SNR- Signal to Noise Ratio SPN- Salt and Pepper Noise STFT- Short time Fourier Transform SURE-Stein s Unbiased Risk Estimate SVD- Singular Value Decomposition SWAM-Switching Weighted Adaptive Median T E - Execution time TI-Translation Invariant UDWT- Undecimated Discrete Wavelet Transform WFT-Windowed Fourier Transform WMSE- Weighted Mean Square Error WT-Wavelet Transform 24

25 INTRODUCTION Image processing is a field that continues to grow, with new applications being developed at an ever increasing pace. It is a fascinating and exciting area with many applications ranging from the entertainment industry to the space program. One of the most interesting aspects of this information revolution is the ability to send and receive complex data that transcends ordinary written text. Visual information, transmitted in the form of digital images, has become a major method of communication for the 21 st century. Image processing is any form of signal processing for which the input is an image, such as photographs or frames of video and the output of image processing can be either an image or a set of characteristics or parameters related to the image. Most image processing techniques involve treating the image as a two-dimensional signal and applying standard signal-processing techniques to it. There are applications in image processing that require the analysis to be localized in the spatial domain which can be done through Windowed Fourier Transform (WFT). Central idea of windowing is reflected in Short Time Fourier Transform (STFT). The STFT conveys the localized frequency component present in the signal during the short window of time. The same concept can be extended to a twodimensional spatial image where the localized frequency components can be determined from the windowed transform. This is one of the bases of the conceptual understanding of wavelet transforms. Hence, wavelet transforms have been kept as the main consideration in this thesis. Image denoising is a restoration process, where attempts are made to recover an image that has been degraded by using prior knowledge of the degradation process. It is well known that while receiving the input image some aberrations get introduced along with it and hence a noisy image is left with for future processing. The image denoising naturally corrupted by noise is a classical problem in the field of signal or image processing. Images are often corrupted with noise during acquisition, transmission, and retrieval from storage media. For example, many dots can be spotted in a photograph taken with a digital camera under low lighting conditions as shown in Fig

26 Figure 1.1 (a) Clean Barbara Image Figure 1.1 (b) Noisy Barbara Image Figure 1.1 Illustration of Noise in the Image Appearance of dots is due to the real signals getting corrupted by noise (unwanted signals). Whereas in television, random black and white snow-like patterns can be seen on the television screens due to loss of reception. Hence noise corrupts both images and videos. In addition, some fine details in the image may be confused with the noise or viceversa. Many image processing algorithms such as an improved non local denoising algorithm, pattern recognition etc. need a clean image to work effectively. 1.1 Preliminaries Performance Evaluation in Image Denoising The quality of an image is examined by objective evaluation as well as subjective evaluation. Objective image quality measures play important roles in various image processing applications. Basically there are two types of objective quality or distortion assessment approaches. The first is mathematically defined measures such as Mean Square Error (MSE), Root Mean Square Error (RMSE) and Peak Signal-Noise Ratio (PSNR) [11, 69]. The second considers Human Visual System (HVS) characteristics in an attempt to incorporate perceptual quality measures. 26

27 In practice, however, the HVS is more tolerant to a certain amount of noise than to a reduced sharpness. Moreover the visual quality is highly subjective and difficult to express objectively [9]. In addition, the HVS is also highly in tolerant artifacts like blips and bumps on the reconstructed image [31]. In the absence of accurate mathematical model for the complete HVS there is no reliable standard measure of image quality that is consistent with human perception and that provides qualitative as well as quantitative measurements. In spite of the lack of such an ideal measure, there are acceptable image quality measures that have been consistently used in the literature. One commonly used image quality measure is known as Root Mean Square Error (RMSE). Although it does not always correlate with human perception it is considered as good measure of fidelity of an image estimate. This is defined as follows: Let the original noise free image, noisy image and the filtered image be represented by,,, respectively. Here represents the discrete spatial coordinate of the digital images. Let the images be of size MxN pixels i.e., = 1,2 = 1,2. The MSE and RMSE is defined as,, = (1.1) = (1.2) Another related image quality measure is Peak Signal-Noise Ratio (PSNR) which is inversely proportional to RMSE; its units are in decibels (db). It is the ratio of Peak Signal Power to Noise Power. It is defined by = 20 (1.3) where 255 is the Maximum Pixel Value for an 8 bits/gray-scale image. MATLAB (MATrix LABoratory) codes have been developed for all the possible combinations separately. The coding for the presented image denoising algorithms has been done using MATLAB R2007B. PSNR is used for comparison of different filters and different wavelet analysis techniques. PSNR analysis uses a standard mathematical model to measure an objective difference between two images. It estimates the quality of reconstructed image with respect to the original image. The basic idea is to compute a single number that reflects the quality 27

28 of reconstructed image. Reconstructed images with lower MSE and higher PSNR are judged better. Execution Time (TE) of a filter is defined as the time taken by a digital computing platform to execute the filtering algorithm when no other software, except the operating system (OS), runs on it. Though TE depends essentially on the computing system s clock time-period, yet it is not necessarily dependant on the clock time alone. Rather, in addition to the clock-period, it depends on the memory-size, the input data size, and the memory access time, etc. The execution time taken by a filter should be low for online and real-time image processing applications. Hence, a filter with lower TE is better than a filter having higher TE value when all other performance-measures are identical Noise in Images In this section various types of noise corrupting an image signal are studied. The sources of noise are discussed and mathematical models for the different types of noise are presented Sources of Noise During acquisition, transmission, storage and retrieval processes an image signal gets contaminated with noise. Acquisition noise is usually Additive White Gaussian Noise (AWGN) with very low variance. In many engineering applications, the acquisition noise is quite negligible. It is mainly due to very high quality sensors. In some applications like remote sensing, biomedical instrumentation, etc., the acquisition noise may be high enough. But in such a system, it is basically due to the fact that the image acquisition system itself comprises of a transmission channel. Hence the researchers are mainly concerned with the noise in a transmission system; usually the transmission channel is linear but dispersive due to a limited bandwidth. The image signal may be transmitted either in the analog form or in digital form. When an analog image signal is transmitted through a linear dispersive channel, the image edges get blurred and image signal gets contaminated with AWGN since no channel is noise free. The noise introduced in the transmission channel of a communication system will be considered in analog form. If the channel is so poor that the noise variances is high 28

29 enough and make the signal excursive to very high positive or high negative value, the thresholding operation which is done at the front end of the receiver will contribute to saturated maximum and minimum values. Such noisy pixels will be seen as white and black spots. Therefore this type of noise is known as Salt and Pepper Noise (SPN). If analog image signal is transmitted the signal gets corrupted with AWGN and SPN as well. Thus there is an effect of mixed noise. If the image signal is transmitted in digital form through a linear dispersive channel, then a noise is introduced due to Bit Error called Inter Symbol Interference (ISI) which takes place along with AGWN which makes the situation worse. Due to ISI and AWGN, it may happen that 1 may be recognized as 0 and vice versa. Under such circumstance, the image pixel values have changed to some random values at random positions in the image frame. Such type of noise is known as Random- Valued Impulse Noise (RVIN). Such kinds of error are taken care by the proposed Switching Weighted Adaptive Median (SWAM) Filter Mathematical Representation of Noise The mathematical representation of AWGN, SPN and RVIN are given below. Gaussian noise is evenly distributing over the signal. This means that each pixel in the noisy image is the sum of true pixel values and random Gaussian distributed noise value is given by, = (1.4) =, +, (1.5) Where a random variable which has a Gaussian probability distribution with bell is shaped probability distribution function given by = / (1.6) Where represents the gray level, is the average or mean of the function and is standard deviation of the noise. Graphically it is represented as shown in Fig.1.2. In Eqn. (1.4) the noisy image is represented as the sum of original uncorrupted image and Gaussian distributed random 29

30 noiseη G. When the variance of the random noise is very low, η G ( x, y ) is zero or very close to zero with many pixel locations. Under such circumstances the noisy image is same or very close to original image at many pixel locations. (x,y). f AWGN Figure 1.2 Representation of Additive White Gaussian Noise Distribution where a random variable which has a Gaussian probability distribution with bell is shaped probability distribution function given by ( g m ) F( g) = e 2σ (1.7) 2πσ 2 Salt and Pepper Noise(SPN) is caused generally due to error in data transmission. It has only two possible values a and b. The probability of each is less than The probability density function for this type of noise is shown in Fig The impulse noise occurs at random locations (x,y) with a probability of. The SPN and RVIN are substitute in nature. An image corrupted with RVIN of density d, ( x, y) is mathematically represented as f RVIN, =, η, = 1 = (1.8) Here η ( x, y) represents a uniformly distributed random variable, ranging from 0 to 1 that replaces the original pixel value f ( x, y). The noise magnitude at any noisy pixel location (x, y) is independent of the original pixel magnitude. 30

31 Figure 1.3 Representations of Salt and Pepper Noise Distribution Another type of noise that may corrupt an image signal is the Speckle Noises corrupt (SN) [43]. This type of noise occurs in almost all coherent imaging systems such as laser, acoustics, SAR (Synthetic Aperture Radar) in bio medical applications like ultrasonic imaging. The SN is a signal dependent noise i.e., if the image pixel magnitude is high, then the noise is also high. Therefore it is also known as multiplicative noise and is given using Eqn. (1.9) and Eqn. (1.10). η ( t) = η( t). s( t) SN f SN ( x, y) = f ( x, y) + η( x, y ). f ( x, y) (1.9) (1.10) Where η(t) is a random variable and s(t) is the magnitude of the signal. The noise is multiplicative since the imaging system transmits a signal to the object and the reflected signal is recorded. Speckle Noise follows a gamma distribution given using Eqn. (1.11). F ( g ) = g α 1 e ( d 1)! a α g a (1.11) Where a α and g is the gray level and is given below in Fig.1.4. The speckle noise is encountered only in a few applications like ultrasonic imaging and SAR, whereas all other types namely AWGN, SPN and RVIN occur in almost all applications. The AWGN is the most common among all. In general, some combinations of AWGN, SPN and RVIN may represent a practical noise. Such type of noise is known as 31

32 Mixed Noise. Some effective schemes are available in the literature for filtration of mixed noise. Figure 1.4 Representation of Speckle Noise Distribution The Neighbourhood Pixel Filter (NPF), Adaptive Subband Thresholding (AST) Technique in subsequent chapters is meant for suppression of AWGN and Switching Weighted Adaptive Median (SWAM) Filter for removal of Impulse noise (SPN, RVIN) Classification of Denoising Methods There are two basic approaches to image denoising, spatial domain filtering methods and transform domain filtering methods as shown in Fig.1.5 [73] Spatial Domain Filtering Methods A traditional way to remove noise from image data is to employ spatial filters. Spatial filters are further classified into linear filters and non linear filters Linear Filters Most classical linear image processing techniques are based on the assumption that image processing applications in which both edge enhancement and noise reduction are desired linear filters too t end to blur sharp edges, destroy lines and other fine image details and perform poorly in the presence of signal dependent noise. 32

33 Non Linear Filters Non linear filters modify the value of each pixel in an image based on the value returned by a non linear filtering function that depends on the neighbouring pixels. Non linear filters are mostly used for noise removal and edge detection. The traditional non linear filters are the median filter. Spatial filters employ a low pass filtering on groups of pixels with the assumption that the noise occupies the higher region of frequency spectrum. Generally spatial filters remove noise to a reasonable extent but at the cost of blurring images which in turn make the edges in pictures invisible. In recent years there are many improved median filters such as Weighted Median Adaptive Filter [125], Centre Weighted Median Adaptive Filter [102] Detail Preserving Median Filters[67], Rank Conditioned Rank Selection[44], The Multilevel Hybrid Median Filter [102] etc have been developed to overcome this drawback Transform Domain Filtering Methods The Transform Domain Filtering methods can be classified according to the choice of the basis or analysis function [104]. The analysis functions can be further classified as Spatial Frequency Filtering and Wavelet domain Spatial Frequency Filtering Spatial Frequency Filtering refers to low pass filters using Fast Fourier Transform (FFT). In frequency smoothing methods [51] the removal of the noise is achieved by designing a frequency domain filter and adapting a cut-off frequency to distinguish the noise components from the useful signal in the frequency domain. These methods are time consuming and depend on the cut-off frequency and the filter function behavior. Furthermore they may produce frequency artifacts in the processed image Wavelet Domain Noise is usually concentrated in high frequency components of the signal which corresponds to small detail size when performing a wavelet analysis. Therefore removing some high frequency (small detail components) which may be distorted by noise is a 33

34 denoising process in the wavelet domain [4, 114]. Filtering operations in wavelet domain can be categorized in to wavelet thresholding, statistical wavelet coefficient model and undecimated. Wavelet domain transform based methods. IMAGE DENOISING METHODS Spatial Domain Transform Domain Linear Non-Linear Non-Data Adaptive Transform Data Adaptive Transform Mean Weiner Median Weighted Median Wavelet Domain Spatial Frequency Domain ICA Linear Filtering Weiner Non-Linear Threshold Filtering Wavelet Coefficient Model Non-Orthogonal Wavelet Transform UDWT Non-Adaptive VISU Shirnk Adaptive SURE Shirnk Deterministic Tree Approximation Statistical SIWPD Multiwavelets Bayes Shrink Cross Shrink Marginal Joint GMM GGD RMF HMM Figure 1.5 Classification of Image Denoising Methods Wavelet Transforms This section discusses the fundamental basics of wavelet. It also contains a brief over view of the types of wavelet transforms used in this work Introduction to Wavelet The concept of wavelet was hidden in the works of mathematicians even more than a century ago. In 1873, Karl Weirstrass mathematically described how a family of functions can be constructed by superimposing scaled versions of a given basis function. The term 34

35 wavelet was originally used in the field of seismology to describe the disturbances that emanate and proceed outward from a sharp seismic impulse [110]. Wavelet means a small wave. The smallness refers to the condition that the window function is of finite length compactly supported [130]. A wave is an oscillating function of time or space and is periodic. In contrast, wavelets are localized waves. They have their energy concentrated in time and are suited to analysis of transient signals. In wavelet analysis, the signal to be analyzed is multiplied with a wavelet function and then the transform is computed for each segment generated. The Wavelet Transform, at high frequencies, gives good time resolution and poor frequency resolution, while at low frequencies; the Wavelet Transform gives good frequency resolution and poor time resolution. An arbitrary signal can be analyzed in terms of scaling and translation of a single mother wavelet function (basis). Wavelets allow both time and frequency analysis of signals simultaneously because of the fact that the energy of wavelets is concentrated in time and still possesses the wave-like (periodic) characteristics. As a result, wavelet representation provides a versatile mathematical tool to analyze transient, time-variant (non stationary) signals that are not statistically predictable especially at the region of discontinuities-a feature that is typical of images having discontinuities at the edges [131]. Figure 1.6 Differences between Wave and Wavelet (a) Wave (b) Wavelet Mathematical Representation of Wavelet Wavelets are functions generated from one single function (basis function) called the Prototype or mother wavelet by dilations (scaling) and translations (shifts) in time (Frequency) domain. If the mother wavelet is denoted by ψ (t), the other wavelets, can be represented as ψ a, b ( t ) = (1 * ψ (( t b) / a)) / a (1.12) Where a and b are two arbitrary real numbers. 35

36 The variables a and b represent the parameters for dilations and translations respectively in the time axis. The mother wavelet can be essentially represented as ψ t) = ψ 1 ( t) (, 0 (1.13) For any arbitrary a=1 and b = 0, ψ a, 0 ( t) = (1 * ψ (( t) / a)) / a (1.14) ψa,0 is the version of mother wavelet function ψ (t) which is time scaled by a and amplitude scaled by a. The parameter a called dilation or scaling parameter causes contraction of ψ (t) in the time axis. When a<1 and expansion or stretching when a> >1. For a<0, the function ψ ( ) results in time reversal with the dilation. Mathematically on substituting t a, b t in Eqn. by (t-b) to cause function ψ ( ) as given in Eqn. (1.12).The parameter b represents the translation in time a, b t a translation or shift in the time axis resulting in the wavelet (shift in frequency) domain as the function ψ a, b( t) is a shift of ψ a, 0( t) in right or left along the time axis by an amount b depending on b>0 or b<0[10]. Figure 1.7 (a) Mother Wavelet ψ (t), b) ψ ( t / α ) : 0 < α < 1c) ψ ( t / α ) : α > 1 Figure 1.7 Representation of Wavelet Types of Wavelet Transform Wavelets capability to give spatial frequency information is the main reason for this investigation. This property promises the possibility for better discrimination between the noise and the data. Successful exploitation of wavelet transform might lessen the blurring 36

37 effect or even overcome it completely. There are mainly two types of wavelet transform namely Continuous Wavelet Transform (CWT) and Discrete Wavelet Transform (DWT) Discrete Wavelet Transform (DWT) DWT of image signals produces a non-redundant image representation, which provides better spatial and spectral localization of image formation compared with other multi scale representation such as Gaussian and Laplacian pyramid. The DWT can be interpreted as signal decomposition in a set of independent spatially oriented frequency channels. The signal S id passed through two complementary filters and emerges a two signals, approximation and details. This is called decomposition or analysis. The components can be associated back into the original signal without loss of information. This process is called reconstruction or synthesis. The mathematical manipulation, which implies analysis and synthesis, is called Discrete Wavelet Transform and Inverse DWT Continuous Wavelet Transform (CWT) CWT is an implementation of the wavelet transform using an arbitrary scales and almost arbitrary wavelets. Non-orthogonal wavelets are used for its development in the data obtained by this transform for highly correlated. CWT works by computing a convolution of the signal with the scaled wavelet and it is implemented in the CWT module that can be accessed with Data Process Integral Transforms CWT Decomposition Process To start with, the image is high and low-pass filtered along the rows and the results of each filter are down- sampled by two. Those two sub-signals correspond to the high and low frequency components along the rows and are each of size N by N/2. Then each of these sub-signals is again high and low-pass filtered, along the column data. The results are again down-sampled by two. As a result the original data is split into four sub-images each of size N/2 by N/2 containing information from different frequency components. Fig.1.8 shows the level one decomposition step of the two dimensional grayscale images. Fig.1.9 shows the four subbands in the typical arrangement. 37

38 Rows Columns L 2 LL L 2 H 2 LH Input L 2 HL H 2 H 2 HH Figure 1.8 One Decomposition Step of the Two Dimensional Images The LL sub-band is the result of low-pass filtering both the rows and columns and it contains a rough description of the image as such. Hence, the LL sub-band is also called the approximation sub-band. The HH sub-band is high-pass filtered in both directions and contains the high-frequency components along the diagonals as well. The HL and LH images are the result of low-pass filtering in one direction and high-pass filtering in another direction. LH contains mostly the vertical detail information that corresponds to horizontal edges. HL represents the horizontal detail information from the vertical edges. All three subbands HL, LH and HH are called the detail subbands, because they add the high-frequency detail to the approximation image. Rows Columns LL HL L H LH HH Figure 1.9 One DWT Decomposition Step Composition Process The inverse process is shown in Fig The information from the four sub-images is up-sampled and then filtered with the corresponding inverse filters along the columns. The two results that belong together are added and then again up-sampled and filtered with 38

39 the corresponding inverse filters. The result of the last step is added together in order to get the original image again. Note that there is no loss of information when the image is decomposed and then composed again at full precision. Columns Rows LL 2 L* LH 2 H* + 2 L* HL 2 L* + Output + 2 H* HH 2 H* Figure 1.10 One Composition Step of the Four Sub Images With DWT, image can be decomposed more than once. Decomposition can be continued until the signal has been entirely decomposed. Another consideration of the wavelet is the subband coding theory or multi resolution analysis. The signal passes through pairs of low pass and high pass filters, the analysis filters, which produces the transform coefficients. These coefficients if passes successively through the synthesis filter may reproduce the initial signal at the decoder s side. In case of 2-D image N-level decomposition can be performed resulting in 3N+1 different frequency bands namely LL (Low frequency or approximation coefficients), LH (vertical details), HL (horizontal details), HH(diagonal details) as shown in Fig In Fig the number next to subband name shows the level. The next level of WT is applied to low frequency subband image LL only. The subbands HH K, HL K, LH K are called the details where K is the level ranging from 1 to J, where J is the largest level. The subband LL J is the low resolution residual. The wavelet thresholding denoising method filters each coefficient from the detail subbands with a threshold function to obtain modified coefficients. The denoised coefficient can be 39

40 estimated by inverse wavelet transform of the modified coefficients. Hence the threshold plays an important role in the denoising process [93]. LL 3 LH 3 HL 3 HH 3 HL 2 LH 2 HH 2 LH 1 1, 2, 3- Decomposition Levels H-High Frequency Bands L-Low Frequency Bands HL 1 HH 1 Figure 1.11 Subbands of the 2-D Orthogonal Wavelet Transform Wavelet Thresholding In wavelet, coefficients with small absolute value are dominated by noise, while coefficients with large absolute value carry more signal information than noise. Replacing noisy coefficients (small coefficient below a certain threshold value) by zero and an inverse wavelet transform may lead to a reconstruction that has lesser noise. The idea of thresholding was motivated based on the following assumptions: The decor relating property of a wavelet transform creates a sparse signal most untouched coefficients are zero or close to zero. Noise is spread out equally along all coefficients. The noise level is not too high so that the signal wavelet coefficients can be distinguished from the noisy ones. This method is a simple and efficient for noise reduction. Further, inserting zeros creates more scarcity in the wavelet domain. 40

41 Figure 1.12 Noisy Signals in Time Domain and Wavelet Domain Types of Thresholding Methods Two thresholding methods are frequently used namely Soft thresholding and Hard thresholding method. Soft thresholding function also a called shrinkage function, takes the argument and shrinks the coefficient towards zeroo by the threshold U. Thresholding operator is defined by Eqn. (1.15) and shown in Fig (a). D( U, λ ) = sgn( u) max(0, u λ) (1.15) Hard thresholding operatorr is defined by Eqn. (1.16), = 0 Hard thresholding is a keep or kill rule whereas soft thresholding shrinks the coefficients above the threshold in absolute value. It is a shrink or kill rule as shown in Fig (b). (1.16) (a) (b) Figure1.13 (a) Hard Thresholding, (b) Soft Thresholding 41

42 Though Hard thresholding seem to be natural, it does not even work with some algorithms where as the pure noise coefficients may pass the hard threshold and appear as annoying blips in the output. Soft thresholding shrinks these false structure thresholding by making algorithms mathematically more tractable. The wavelet thresholding procedure removes noise by thresholding only the wavelet coefficients of the detail subbands, while keeping the low resolution coefficients unaltered. Soft thresholding yield, visually more pleasing images whereas hard thresholding introduce artifacts in the recovered images [31,129]. 1.2 Motivation The basic idea behind this thesis is the estimation of the unsuppressed image from the distorted or noisy image, and is also referred to as image denoising. There are various methods to help restore an image from noisy distortions. Selecting the appropriate method plays a major role in getting the desired image. Image denoising is usually required to be performed before display or further processing like segmentation, feature extraction, object recognition, texture analysis, etc. The purpose of denoising is to suppress the noise quite efficiently while retaining the edges and other detailed features like image smoothening, image sharpening, contrast adjustment as much as possible. For real-time applications like television/photo-phone, etc. it is essential to reduce the noise power as much as possible and to retain the fine details and the edges in the image as well. Moreover, it is very important to have very low computational complexity so that the filtering operation is performed in a short time for online and real-time applications. Therefore, the objective of this research work is to develop an efficient filtering and thresholding algorithm based on wavelet to suppress moderate and high power AWGN and impulse noise under moderate and high noise level quite effectively without yielding much distortion and blurring. The performances of the developed algorithms/methods are compared with the existing methods in terms of Peak Signal to Noise Ratio (PSNR)and execution time. 42

43 1.3 Problem Definition Efficient suppression of noise in an image is a very important issue. Denoising finds extensive applications in many fields of image processing. Image Denoising is an important pre-processing task before further processing of image like segmentation, feature extraction, texture analysis etc. The purpose of Denoising is to remove the noise while retaining the edges and other detailed features as much as possible. In order to quantify the performance of various denoising algorithms, a high quality image is taken and some known noise is added to it. This would then be given as input to the denoising algorithm, which produces an image close to the original high quality image. In case of image denoising methods, the characteristics of the degrading system and the noise are assumed to be known in advance. The image, is blurred by a linear operation and noise, is added to form the degraded image,. This convolutes with the restoration procedure g, to produce the restored image z,. s(x,y) LINEAR OPERATION w(x,y) DENOISING TECHNIQUE g(x,y) z(x,y) n(x,y) Figure 1.14 Denoising Concept The linear operation shown in Fig.1.14 is the addition or multiplication of the noise n( x, y) to the signal S( x, y ).Once the corrupted image w( x, y ) is obtained, it is subjected to denoising technique to get the denoised image z( x, y ). The point of focus in this thesis is comparing and contrasting several denoising techniques. This motivated the author to take up the problem to reduce the noise without corrupting the originality of the image. 43

44 1.4 Objective and Scope of Study The primary objective of the thesis work is to develop an efficient filtering and thresholding algorithm based on wavelet to suppress moderate and high power AWGN and impulse noise quite effectively without yielding much distortion blurring. Calculating and applying thresholds either globally in a level dependent manner or in a sub-band dependent manner may lead to quality image with less blurring and preserving more detailed information. The filtering algorithms are derived to make sure non-corrupted image pixels are left intact, irrespective of noise in image. To develop filtering and thresholding algorithm which gives better performance in both moderate and high noise level. The scope of this thesis is to improve clarity of the resultant image by adding a filter to the wavelet coefficients which gains the effect of both low pass and high pass filter. This in turn cuts off only high frequency noisy signal instead of all noisy signals, for improving detail preservations, edge preservations, wavelet threshold denoising algorithms are essential. The sub-band thresholding technique applied in this thesis work is to maintain the spatial adaptivity near object edges, produces effective results. Implementation of these algorithms in biomedical images like Magnetic Resonance Imaging (MRI) for removal of Gaussian noise yields a significant result is also focused. 1.5 Overview of the Thesis The thesis comprises seven chapters each describing the techniques used in this research work. In the first chapter, an introduction, motivation and problem statement, objective and scope of study were given. The second chapter comprises of history and literature survey to understand the merits and demerits of existing denoising methods. The third chapter deals with the proposed algorithm called Neighbourhood Pixel Filtering Algorithm (NPFA) which is used to remove Gaussian noise from an image. The fourth chapter contains a thresholding technique using wavelet coefficients and threshold formula based on decomposition level for removal of Gaussian noise. The chapter five describes a 44

45 filter called Switching Weighted Adaptive Median (SWAM) filter for suppression of impulse noise. In chapter six, a hybrid method called Filtering and Thresholding Algorithm (FTA) is described for eliminating AWGN in biomedical images like Magnetic Resonance Imaging (MRI) brain images. The concluding chapter describes the strengths and weaknesses of this work and future extension work. Selection of images Addition of Noise AGWN/Impulse Denoiseing Techniques AGWN Impulse Neighbourhood pixel filtering algorithm (NPFA) Subband thresholding Technique with wavelet coefficients and decomposition level Switching Weighted Adaptive Median (SWAM) Filter (Selection of appropriate window length) Implementation in Biomedical Images (MRI) Figure 1.15 Outline of the Research Work 45

46 CHAPTER 2 LITERATURE SURVEY 2.1 Brief History In the 1970s, image denoising was studied by control theorist Nasser Nahi at USC and computer vision pioneers such as S. Zucker and Azriel Rosenfeld. In 1980, J. S. Lee published an important paper titled "Digital image enhancement and noise filtering by use of local statistics" [57]. The invention of wavelet transforms in late 1980s has led to dramatic progress in image denoising in 1990s. The Bayesian view towards image denoising was put forward by Simoncelli & Adelson in 1996 and since then, many wavelet-domain denoising techniques have been proposed [100]. The simple yet elegant Gaussian Scalar Mixture (GSM) algorithm published by Portilla et al. in 2003 [52] and the NonLocal Mean (NLM) algorithm by Buades et al. in 2005 have renewed the interest into this classical inverse problem. In the past three years, many more powerful denoising algorithms have appeared among them the patch-based nonlocal schemes, such as BM3D, have shown outstanding performance and its theoretic interpretation has been given by an expectation-maximization (EM)-based inference on stochastic factor graphs. From a historical point of view, wavelet analysis is a new method, though its mathematical underpinnings date back to the work of Joseph Fourier in the nineteenth century. Fourier laid the foundations with his theories of frequency analysis, which proved to be enormously important and influential. The attention of researchers gradually turned from frequency-based analysis to scale-based analysis when it started to become clear that an approach measuring average fluctuations at different scales might prove less sensitive to noise. In 1909, the first recorded wavelet analysis was mentioned by Alfred Haarin his thesis. In the late nineteen-eighties, when Daubechies and Mallat first explored and popularized the ideas of wavelet transforms, skeptics described this new field as contributing additional useful tools to a growing toolbox of transforms [65]. One particular 46

47 wavelet technique, wavelet denoising, has been hailed as offering all that we may desire of a technique from optimality to generality [14]. The inquiring skeptic, however maybe reluctant to accept these claims based on asymptotic theory without looking at real-world evidence. Fortunately, there is an increasing amount of literature now addressing these concerns that help us appraise of the utility of wavelet shrinkage more realistically. 2.2Overview This section deals with the survey of various research papers that have been contributed in the denoising of images using wavelet transforms. There is a growing demand of image processing in diverse applications such as multimedia computing, secured image data communication, biomedical imaging, biometrics, remote sensing, texture understanding, pattern recognition, content-based image retrieval, compression, and so on. The wavelet transform has been providing a major contribution in all the above mentioned areas since long time. But the quest for betterment never ends. It is very essential to keep the useful data in the exact original form for further processing and wavelet denoising being the latest technique that has proved its command over this issue. The following literature review discusses denoising using wavelet transforms in a wide scenario, i.e. using a number of thresholding techniques for a wide variety of test images Wavelet Based Techniques Wavelet based methods are always a good choice for image denoising and has been discussed widely in literatures for the past two decades [61, 54, 58,95, 27, 71, 22].The problem of image denoising is to recover an image that is cleaner than its noisy observations. M. C. Motwani et.al.analyzed that noise reduction as an important technique in image analysis which is the first step to be taken before the images are considered for further processing. D.L. Donoho and L.M. Johnstone introduced wavelet based denoising scheme, as wavelets give as superior image denoising due to the property of sparsity and multi resolution structure [30]. While applying wavelet based denoising, the noisy wavelet coefficients are modified accordingly M. Vatterili and J. Kovacevic analyzed that soft thresholding is one of the most well known rules due to its effectiveness and simplicity [113]. S. Gauangmin and L. Fudong introduced the main idea of soft thresholding by 47

48 subtracting the threshold values T from all the coefficients larger than T and to set all other coefficients to zero [39]. Wavelets give a superior performance in image denoising due to properties such as sparsity and multiresolution structure. The focus was shifted from the Spatial and Fourier domain to the Wavelet; a different class of methods exploits the decomposition of the data into the wavelet basis and shrinks the wavelet coefficients in order to denoise the data. The wavelet based techniques use wavelets to transform the data into a different basis, where "large" coefficients correspond to the signal, and "small" ones represent mostly noise. The denoised data is obtained by inverse-transforming suitably the threshold or shrunk coefficients. Two dimensional versions of methods were implemented with that which were originally developed for one-dimensional signals and compared with the method proposed for images. Thus, there was a renewed interest in wavelet based denoising techniques since Donoho demonstrated a simple approach to a difficult problem. A wide class of image processing algorithms is based on the DWT. The transform coefficients within the sub-bands can be locally modeled as independent identically distributed (iid) random variables with Generalized Gaussian Distribution (GGD) [65, 74, 90]. This model has been successfully used in image denoising and restoration. It approximates first order statistics of wavelet coefficients fairly well, but does not take higher order statistics into account and thus presents some limitations. The dependency that exists between wavelet coefficients have been studied in many years in the image compression community. Most wavelet models can be loosely classified into 2 categories. Those exploiting interscale dependency and those exploiting intra scale dependencies. These dependencies can be formulated explicitly (e.g., the EQ coder [62]), or implicitly (e.g., the morphological coder [96]) Donoho s wavelet denoising method performs well under a number of applications because wavelet transform has the compaction property of having only a small number of large coefficients. All the rest coefficients are very small. The denoising is done only on the detail coefficients of the wavelet transform. This algorithm offers the advantage of smoothness and adaptation but exhibits visual artifacts. This disadvantage was overcome 48

49 by Coifman and Donoho [24] by proposing a Translation Invariant (TI) denoising scheme to suppress such artifacts by averaging over the denoised signals of all circular shifts. A better denoising scheme using multiwavelet was proposed by Bui and Chen [15] than the TI single wavelet denoising. Cai and Silverman, (2001) proposed a thresholding scheme by taking the immediate neighborhood coefficient by motivating the idea that a large wavelet coefficient will probably have large wavelet coefficients as its neighbors [13]. The problem of wavelet based denoising can be expressed as the estimation of clean coefficients from noisy data with Bayesian estimation techniques, such as the maximum a posterior (MAP) estimator [31]. However, it has a weak model for wavelet coefficients of natural images because they ignore the dependencies between coefficients, and its major problem lies in the difficulties in determining a proper shrinkage function and threshold [39,124]. Tree structures ordering the wavelet coefficients based on their magnitudes, scale and spatial location have been researched. Then the use of the wavelet tree was found to be more efficient [129, 30]. The advantages and disadvantages of the filtering technique that is closest to NPFA are given below. Table 2.1 Comparison Table for Existing and Proposed Filtering Algorithms Filtering Technique Advantage Disadvantage Remarks 1 Wavelet Method Overcomes the Exhibits visual Efficiency and problem of translation artifacts contrast of variant mechanism. Detail resultant Undecimated preservations wavelet images Algorithm. are not uniform are low. Also Omits down-sampling at different the clarity of and up-sampling in scales the resultant forward and inverse Random noise image is weaker transform. Offers the advantage rapidly attenuates with of smoothness and increasing scale adaptation. 2 Neighbourhood Pixel Filtering Algorithm (NPFA) For the removal of AWGN. Gains the effect of both LPF and HPF. Clarity of resultant image is improved Achievement of Improved quality is attributed to exploitation in the correction of NPD value within a small region. As the NPF gains the effect of LPF and HPF it cuts off only high frequency signal instead of all noisy signal. 49

50 The clarity of the resulting image is weaker in the existing trous algorithm because the efficiency of the wavelet images is low and the detailed preservations of images at different scales are not uniform. Also random noise rapidly attenuates with increase in scales. This problem is overcome by NPFA which is implemented, based on the concept that the low pass filter preserves the energy of the signal and attenuates high pass features at discontinuities. By this concept the NPF used in algorithm gains the effect of both LPF and HPF which in turn cuts off only high frequency signal instead of all noisy signals. Due to this the clarity of the resultant image is improved Thresholding Algorithm In recent years there has been a fair amount of research on filtering and wavelet coefficients thresholding, because wavelets provide an appropriate basis for separating noisy signal from the image signal. Simple denoising algorithms that used the wavelet transform consist of the three steps [39]. Step 1. Calculate the wavelet transform of the noisy signal; Step 2. Modify the noisy wavelet coefficients according to a rule; Step 3. Compute the inverse transform using the modified coefficients; One of the most well-known rules for Step 2 is soft thresholding which was analyzed by M. Vatterili, J. Kovacevic [132]. Due to its effectiveness and simplicity, it is frequently used in the literature. The main idea is to subtract the threshold value T from all coefficients larger than T and to set all other coefficients to zero [39]. Generally, these methods use the estimated threshold value to obtain good performance. Hence these wavelets based methods mainly rely on thresholding the Discrete Wavelet Transform (DWT) coefficients, which have been affected by Additive White Gaussian Noise (AWGN). Since [28,30,32,33], there has been a lot of research based on the work of Donoho and Johnston and on the way of defining the threshold levels and their type (i.e. hard or soft threshold),researchers published different ways to compute the parameters for the thresholding of wavelet coefficients. Donoho s concept was not revolutionary; his methods did not require tracking or correlation of the wavelet maxima and minima across the different scales as proposed by Mallat [67]. Data adaptive thresholds [50] were introduced to achieve optimum value of threshold. Later efforts found that substantial improvements in perceptual quality could be obtained by translation invariant methods 50

51 based on thresholding of an undecimated Wavelet Transform [24]. These thresholding techniques were applied to the non orthogonal wavelet coefficients to reduce artifacts. Multi wavelets were also used to achieve similar results. Data adaptive transforms such as Independent Component Analysis (ICA) have been explored for sparse shrinkage. Thus, there was a renewed interest in wavelet based denoising techniques since Donoho [31] demonstrated a simple approach to a difficult problem. Wavelet thresholding is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. The idea is to transform the data into the wavelet basis, in which the large coefficients are mainly the signal and the smaller one represents the noise. By suitably modeling these coefficients the noise can be removed from the data. The classical soft threshold shrinkage function can also be obtained by a Laplacian probability density function (pdf) [95]. Many researchers have proposed the bivariate pdfs for modeling the interscale dependency [95, 106, 87, 1]. Though these pdfs improve the denoising results but they may lead to complicated algorithms. Intrascale dependency states that pdfs using spatial local parameters are able to better capture the statistical properties of wavelets [95, 71]. Mihcak proposes a Gaussian pdf with local variance for denoising and earns impressive results with his simple algorithm [71]. Laplacian pdf with local variance to model the heavy-tailed property, interscale and intrascale dependencies of wavelet coefficients are used in this thesis. This pdf is univariate and the local variance of each coefficients are estimated using its spatial adjacent and its parent s spatial adjacent to incorporate both inter and intrascale dependencies in this estimation. Regardless of the type of employed DWT, denoising is commonly done by wavelet shrinkage. Wavelet shrinkage is a method of removing noise from images by shrinking the empirical wavelet coefficients in the wavelet domain and it is a non linear image denoising procedure to remove the noise. The most straight forward way of distinguishing information from noise in the wavelet domain consists of thresholding the wavelet coefficients. Thresholding method is a common shrinkage approach, which sets the wavelet coefficients with small magnitudes to zero while retaining shrinking in magnitude the remaining ones. Of the various thresholding strategies, soft-thresholding proposed by Donoho and Johnstone is most popular [31]. They have introduced a universal threshold T as given in Eqn. (2.1) 51

52 = 2 log (2.1) where, is the noise variance and N is the number of samples in the signal, also proposed the use of a universal threshold uniformly throughout the entire wavelet decomposition tree [31, 71]. The use of the universal threshold to denoise images in wavelet domain is known as Visu Shrink. Although thresholding with a uniform per subband threshold is attractive due to its simplicity, the performance is limited and the denoising quality is often not satisfactory. Thus wavelet shrinkage methods using separate threshold in each subband have been developed over recent years namely Sure Shrink subband adaptive systems having superior performance. Recently, another data driven subband adaptive technique was proposed by Chang et al., namely Baye s Shrink which outperforms Sure Shrink and Visu Shrink [21]. They have also stated another two shrinkage methods known as Oracle Shrink and Oracle Thresh. By using global thresholding of wavelet coefficients it was observed that the performance in real life images is not sufficiently effective. Later it was observed that the use of wavelet tree was found to be more efficient [129, 32, 30]. Some methods were proposed and investigated methods of selecting thresholds that are adaptive to different spatial characteristics and concluded adaptive approaches have been formed to be more effective than their global counter parts [21, 20, 19]. The basic idea in wavelet shrinkage technique is to model wavelet transform coefficients with priori probability distributions. In 1995, D.L. Donoho expressed the problem as the estimation of clean coefficients using a priori information with Bayesian estimation technique, like Maximum a Posterior (MAP) estimator [31]. In 1989, S.Mallat expressed that the transform coefficients with in subbands can be locally modeled as independent identically distributed (iid) random variables with Generalized Gaussian Distribution (GGD) [65, 74]. Also he expressed that the denoised coefficients may be evaluated by an MMSE (Minimum Mean Square Error) estimator in terms of the noised coefficients and the variances of signal and noise. In 1999 M. K. Mihcak et. al. derived a method in which the denoised coefficients are statistically estimated in small regions for every subband instead of applying a global threshold [72]. In 2000 S. G. Chang et. al. derived a similar spatially adaptive model via wavelet thresholding wavelet image coefficients [21]. 52

53 In the wavelet decomposition, the magnitude of the coefficients varies depending on the decomposition level. If all levels are processed with universal threshold value the processed image may be overly smoothened so that the sufficient information, preservation is not possible and the image get blurry. In order to overcome the subband adaptive system having superior performance which is a data driven system and level dependent methods namely Baye s Shrink, Oracle shrink and Oracle thresh were proposed by Chan et al, (2000) [21]. These methods out performance sure shrink technique which is a sub-banded adaptive and data driven system proposed by D.L. Donoho et al (1995), [32]. Later Iman Elyasi and Sadegh Zarmehi, (2009) [49] proposed different adaptive wavelet threshold methods like Modified Bayes Shrink (MBS) Normal Shrink (NS) for image denoising. They proved that in low noise NS yields the best results for denoising because it has maximum SNR and minimum MSE. In high noise MBS yields the best results. Also NS preserves edges better than noise removal. All these thresholding techniques tend to kill too many wavelet coefficients that might contain useful image information. To overcome this G. Y. Chen et al.(2004),[22] proposed wavelet image thresholding by incorporating neighboring coefficients called Neigh Shrink which is an extension of Cai and silver man s [13] idea of considering the immediate neighbourhood coefficients into account. This method thresholds wavelet coefficient according to the magnitude of the square sum of the entire wavelet coefficient within the neighbourhood window. Due to the suppression of too many detail wavelet coefficients Neigh Shrink produces denoised image with more blurring. This problem was avoided by reducing the value of threshold itself. Hence modified shrinkage factor was introduced in Modi Neigh Shrink method proposed by S.Kother Mohideen et al. (2008), [56]. Tan et al. [108] proposed a wavelet domain denoising algorithm by combining the expectation maximization scheme and the properties of the Gaussian scale mixture models. The algorithm is iterative in nature and the number of iterations depends on the noise variance. For high variance Gaussian noise, the method undergoes many iterations and therefore the method is computational-intensive. In 2002, Shengqian et.al., proposed an adaptive shrinkage denoising scheme by using neighborhood characteristics and claimed that this method produced better results than 53

54 Donoho s methods [98]. Later Sendur and Selesnick, (2002) had proposed bivariate shrinkage functions for denoising and indicated that the estimated wavelet coefficients depend on the parent coefficients, also they observed that the shrinkage is more when parent coefficients are smaller [87, 94]. Chen and Bui, (2003) extended this neighbor wavelet thresholding idea to the multiwavelet denoising [87] Statistical Model Statistical models pretend wavelet coefficients are random variables described by some probability distribution, like Gaussian, Laplacian, GGD, Bayesian methods for image denoising using other distributions had also been proposed and these models ignore the dependencies between coefficients, and they were considered as weak model. E. P. Simoncelli proposed a better model in which the coefficients are statistically dependent based on two properties of wavelet transform [99]. A new framework to capture the statistical dependencies by using wavelet domain Hidden Markov Model (HMT) was developed by M.S.Crouse et. al., in 1998 [26]. In 2001 G.Fan and X.G. Xia introduced improved local contextual hidden Markov models in their work [37]. In 1999 M. K. Mihcak, et.al., proposed a adaptive window-based image denoising algorithm using ML and MAP estimates, which was a powerful low-complexity algorithm exploiting the intrascale dependencies of wavelet coefficients [71]. J. N. Ellinas and D. E. Manolakisderived a model in which the denoised coefficients are statistically estimated in a variable block size frame work resulting in a quad tree decomposition of sub-bands with respect to local variance [35]. P. Kenterlis and D. Salonikidis proposed an effective method for spatial adaptive model by performing MMSE coefficients estimation instead of the classical threshold estimation. E.P. Simoncelli and J.Portilla, 1998 [101] proposed a new model which explicitly combines inter scale and intra scale dependencies of image wavelets coefficients with strong correlation between magnitudes of coefficients across scales. Portilla et al. [85] developed a method for removing noise from digital images based on a statistical model of the coefficients of an over-complete multiscale oriented basis. Neighborhoods of coefficients at adjacent positions and scales are modeled as a product of two independent random variables: a Gaussian vector and a hidden positive scalar 54

55 multiplier. The latter modulates the local variance of the coefficients in the neighborhood, and thus able to account for the empirically observed correlation between the coefficients amplitudes. Under this model, the Bayesian least squares estimate of each coefficient reduces to a weighted average of the local linear estimates over all possible values of the hidden multiplier variable. Also in GGD models the denoised coefficients may be evaluated by an MMSE (Minimum Mean Square Error) estimator, in terms of the noised coefficients and the variances of signal and noise. The signal variance is locally estimated by a ML (Maximum Likelihood) estimator, whereas noise variance is estimated from the first level diagonal details. Therefore, the denoised coefficients are statistically estimated in small regions for every subband instead of applying a global threshold [72]. These methods present efficient results but their spatial adaptivity is not well suited near object edges where the variance field is not smoothly varied. In [21] a similar spatially adaptive model for wavelet image coefficients was used to perform image denoising via wavelet thresholding. In [35] the denoised coefficients are statistically estimated in a variable block size framework resulting in a quad-tree decomposition of subband with respect to local variance Spatial Domain Filtering Spatial filters have long been used as the traditional means of removing noise from images and signals (Weeks 1996) [122]. These filters usually smooth the data to reduce the noise, but, in the process, also blur the data. In the last decade, several new techniques have been developed that improve on spatial filters by removing the noise more effectively while preserving the edges in the data. Some of these techniques borrow ideas from partial differential equations and computational fluid dynamics such as level set methods (Sethian 1999) [97], total variation methods (Chambolle 1998, Chan 2000) [16, 18], non-linear isotropic and anisotropic diffusion (Black 1998, Weickert 1998) [10], and essentially nonoscillatory (ENO) schemes (Chan 1999) [17]. Other techniques combine impulse removal filters with local adaptive filtering in the transform domain to remove not only white and mixed noise, but also their mixtures (Egiazarian 1999) [36]. Linear spatial domain filters such as wiener filter were introduced [41]. The disadvantage of using linear filter is that it introduces blurring effect to the filter image. 55

56 Many efforts have been devoted in reducing this undesired effect [81, 57, 78]. We prevent the blurring across edges but fail to smooth the homogenous regions. An iterative method known as Total Variation [92] is proposed by Rudin et al. to smoothen the homogenous region while retaining edges. It is a constrained optimization type of algorithm where the total variation of the image is minimized subject to constraints involving the statistics of the noise. The constraints are imposed using Lagrange multipliers. The solution is obtained using gradient projection method. This amounts to solving a time dependent partial differential Eqn. on a manifold determined by the constraints. As time increases the solution converges to a steady state which is the denoised image. For a small variance Gaussian noise, large time is required to do more iteration which leads to blurring effect. The method suppresses noise very well but introduces blurring with more iteration. R.C. Gonzalez and R.E. Woods, (2000) [81] proposed a linear spatial domain filter called wiener filter to remove Additive White Gaussian Noise (AWGN).But this filter introduces blurring effect to the filter image. Due to this lots of details of original image had been removed during denoising. In 1990 I.Pitas, J.S. Lee, P. Perona and J. Malik, many efforts were made in creating nonlinear noise filtering by the use of local statistics, edge detection by using anisotropic diffusion [81, 57, 78].All these methods were able to prevent the blurring across edges without smoothening the homogeneous region. An iterative method known as Total Variation was proposed by Rudin et al. (1992) [92]. This method smoothen the homogenous region by retaining edges. It is a constrained optimization type of algorithm where the total variation of the image is minimized subject to constraints involving the statistics of the noise. 56

57 Table 2.2 Comparison Table for Existing and Proposed Thresholding Algorithms Filtering Technique Advantage Disadvantage Remarks 1 Normal Shrink Data driven subband dependent threshold. Introduce mat like structure in the smooth region of the filtered image. Normal shrink preserves edges better than removing noise. 2 Baye s Shrink Performs denoising that is consistent with the Human Visual System. 3 Modified Baye s Shrink Level method. dependent Edges are not preserved properly compared to Normal Shrink. Tends to kill too many wavelet coefficients that might contain useful image information. Bayes shrink performs little denoising in high activity subregions to preserve the sharpness of edges but completely denoise the flat sub-parts of the image. Yields better results for denoising and adapts thresholding strategy. 4 Adaptive Subband Thresholding (AST) Technique Based on Wavelet Coefficient. New approach for suppression of AWGN by fusing the wavelet denoising technique with optimized thresholding function The inclusion of decomposition level makes AST less efficient for higher noise density in greater level of decomposition. AST preserves edges as well as denoise image with better performance. Adaptive Subband Thresholding (AST) technique is developed based on wavelet coefficients in order to overcome the spatial adaptivity which is not well suited near object edges, where the variance field is not smoothly varied by producing effective results. This technique describes a new approach for suppression of AWGN by fusing the wavelet denoising technique with optimized thresholding function to which a multiplying factor is included to make the threshold value dependent on decomposition level. 57

58 2.2.5 Impulse Noise Removal Techniques Images are also corrupted by impulse noise during image acquisition or transmission. The intensity of impulse noise has the tendency of being either relatively high or relatively low. Thus it could severely degrade the image quality and cause great loss of information details. So it is important to eliminate noise in the images before some subsequent processing. Various filtering techniques have been proposed for removing impulse noise in the past. It is well known that linear filtering techniques fail when the noise is non-additive and were not effective in removing impulse noise. As a result non linear filters had been widely exploited due to their much improved filtering performance, in terms of noise attenuation and edge/detail preservation. One of the most popular robust non linear filters is the Standard Median Filter which exploits the rank-order information of pixel intensities with in a window and replaces the center pixel with the median value [80]. The main drawback of SMF is that it is effective only for low noise densities. At high noise densities, SMF s often exhibit blurring for large window size and insufficient noise suppression for small window size (1990,1984) [79, 84]. S.Zhang et al., H.L Eng et al., G.Pok and J.C.Liu had identified and proposed methods that the filtering should be applied to the corrupted pixels while leaving uncorrupted pixels intact. Therefore a noise detection process to discriminate between uncorrupted pixels and corrupted pixel, prior to applying non linear filtering is highly desirable [127, 34, 82]. They had adopted median filtering frame work called Switching Median Filter in their work by showing significant performance improvement. All these filters had all demonstrated excellent performance but at the price of significant computational complexity [107, 38, 23, 116, 83, 48]. To overcome this problem a switching based adaptive median filtering scheme was proposed by Mamta Juneja and Rajni Mohana [68]. In this method noises attenuated by estimating the values of the noisy pixels with a switching based median filter applied exclusively to the uncorrupted neighborhood pixels and also the size of filtering window is adaptive in nature and it depends on the number of noise free pixels in current filtering window. Later G. Arce and J. Paredes proposed Recursive Weighted Median filter (RWM) which produces better results when compared with other median type filter (2000) [5]. In this filter window length are selected based on 58

59 the amount of noise present in the input signal. Ho-Ming Lin and Alan proposed Adaptive Length Median Filter for removal of impulse noise in images. As the weights are chosen in accordance window length, their algorithm had high complexity by producing less efficient output (1988) [46]. The proposed SWAM filter is designed where the window length is determined appropriately based on the width of the impulsive noise presented in the input signal and the uncorrupted pixel is not filtered. Also the weights of the filter are calculated by using the median controlled algorithm. Due to this the results are very effective and the resulting image, will have less blurring in the output signal Color Image and MRI Denoising Technique Many color image denoising techniques [45, 64, 60, 53, 63, 59] are available in the literature for suppression of AWGN. Lian et al. [60] proposed an edge preserving image denoising via optimal color space projection method. SURE-LET multichannel image denoising is proposed by F. Luiser and T. Blu [63] where the denoising algorithm is parameterized as a linear expansion of thresholds (LET) and optimized using Stein s unbiased risk estimate (SURE). A non-redundant, orthonormal wavelet transform is first applied to the noisy data, followed by the (subband-dependent) vector-valued thresholding of individual multi-channel wavelet coefficients which are finally brought back to the image domain by inverse wavelet transform. Lian et. al. [59] proposed a color image denoising technique in wavelet domain for suppression of AWGN. This method is based on minimum cut algorithm where the interscale and intra-scale correlations of wavelet coefficients are exploited to suppress the additive noise. Some blind techniques using independent component analysis (ICA) [128, 126] for image denoising are available in the literature. But, none of the filters available in literature is able to achieve perfect restoration. Further, there is a need to reduce computational complexity of a filtering algorithm for its use in real-time applications. Lee and Tsai discuss the use of wavelets to enhance MR images [112]. They used a mapping function to manipulate the transform coefficients before reconstruction. The mapping function was chosen such that the low frequency coefficients are not affected which prevents distortion. The coefficients with larger absolute values contain more information while the high frequency coefficients contain important edge information. 59

60 Hence, coefficients belonging to either of these classes were heavily weighted compared to other coefficients. Zadeh et. al. compare various filters (ratio, log ratio and angle image filters) to enhance MR images in [77]. In 2002, R. Archibald and A. Gelb [6] have looked at noise suppression in MR images using Fourier spectral methods. FIR filters along with wavelet decomposition for image enhancement was used specifically for edge enhancement and edge detection [86]. From the above review of research papers, it is quite clear that wavelet has provided a very handsome amount of contribution in image denoising. Hence, it is concluded that there is enough scope to develop better filtering and thresholding schemes with very low computational complexity that may yield high noise reduction as well as preservation of edges and fine details in an image. 2.3 Conclusion This chapter investigates noise models and includes an in-depth literature survey of denoising based on wavelets. Desirable features and complexities of denoising algorithms are discussed. In addition, it explains common mechanisms used to evaluate the performance of denoising algorithms. According to the current literature, denoising algorithms based on wavelet transform are the best choice for achieving the desired denoising performance. However, the computational complexity must also be considered. Thresholding techniques used with the Discrete Wavelet Transform are the simplest to implement. A universal denoising algorithm is a dream of researchers, although there are no universal method, in this study, the denoised results of the proposed algorithms and existing algorithms are compared under different noise models and variances by means of the evaluation methods introduced above. 60

61 CHAPTER 3 NEIGHBOURHOOD PIXEL FILTERINGALGORITHM 3.1 Introduction In this chapter a new filtering algorithm is developed in order to improve the clarity of the resultant image. This filtering algorithm is called Neighbourhood Pixel Filtering Algorithm (NPFA) which is based on the concept that the low pass filter preserves the energy of the signal and attenuates the high pass features at the discontinuities. Byenhancing the lower frequencies and removing higher frequencies, many denoising techniques are developed with low pass filtering method. This algorithm is one such technique, because this filter behaves like a low pass filter in smooth region by decreasing noise variance effectively and giving similar weights to all neighbouring pixels to improve the clarity of the images. Trous algorithm is one of the simplest wavelet algorithms, used to overcome the problem of translation variant mechanism existing on DWT. This algorithm up samples low pass filter by inserting zeros between the filter coefficient at each level and accordingly low pass and high pass filter coefficients are modified. The efficiency of wavelet images by using this algorithm low because the details preservations of images at different scales are not uniform. Also random noise rapidly attenuates with increasing scales. Due to this the contrast of resultant image is weaker. In order to improve the clarity of the image NPF algorithm is added along with the Trous algorithm. 3.2 Undecimated Algorithm In this section, the development of Trous algorithm is explained to understand the development of NPFA. Translations of original signal lead to different wavelet coefficients. In order to overcome this and to get more complete characteristic of the 61

62 analyzed signal the Undecimated Discrete Wavelet Transform (UDWT) are proposed. The general idea behind it was that it did not decimate the signal. Thus it produces more precise information for the frequency localization. From the computational point of view the undecimated wavelet transform has larger storage space requirements and involve more computations. This algorithm is based on the idea of no decimation. It applies the wavelet transform and omits both down-sampling in the forward and up sampling in the inverse transform. More precisely, it applies the transform at each point of the image and saves the detail coefficients and uses the low-frequency coefficients for the next level. The sizes of the coefficients array do not diminish from level to level. By using all coefficients at each level, very well allocated high-frequency information is obtained. From level to level there is a difference in the width of the scaling filter. For example, instead of 8 pixels at the third level of DWT here the width difference is 5 pixels, which is not a power of 2 but a sum of 2. Generally, the step is not a power of 2 but a sum with 2. This property is good for noise removal because the noise is usually spread over small number of neighboring pixels. With this transform the number of pixels involved in computing a given coefficient grows slower and so the relation between the frequency and spatial information is more precise. In the ideal case the noise can be removed only at the places that it really exists, without affecting the neighboring pixels. Trous is one of the undecimated algorithms which approach the problem from different ways [47] ÀTrous Wavelet Algorithm ÀTrous algorithm computes the redundant wavelet transform in which low pass and high pass filters are modified at each consecutive level. The up samples of low pass filters are done by inserting zeros between the filter coefficients at each consecutive level. The difference between low pass images from two consecutive levels calculates the detail coefficients. By adding all levels of high pass coefficients to the final low resolution image, an inverse transform are computed [109] Determination of Wavelet Coefficient Using ÀTrous Algorithm By applying À Trous algorithm the resultant images obtained will be of the same dimension as the original one. In this algorithm an image gets decomposed into an 62

63 approximate and a detail image at a scale. The detailed image is the same as the original one in dimension called wavelet plane. The mathematical derivation for determining Wavelet coefficient using À Trous algorithm is given below. Let { } c ( ) 0 k be the sampled data at each pixel k of the function f(x) with a scaling function φ (x) which corresponds to a low pass filter. Then { h( k )} is the magnified scale obtained by performing the first filter twice [123]. The difference { c ( k) } { c ( k) } 0 1 is called signal difference which contains the information between these two scales and is the discrete set associated with the wavelet transforms corresponding to φ (x). The associated wavelet ψ (x) is defined by the Eqn. (3.1). 1 x 1 x ψ = φ( x) φ (3.1) The distance between samples increasing by a factor 2 from the scale (i - 1) (i> 0) to the next one, c i (k) is given by transform w i (k) by Eqn. (3.2). i i 1 i c k h l c k l and the discrete wavelet i 1 i ( ) = ( ) ( i 1) ( + 2 ) l w ( k) = c ( k) c ( k) (3.2) The coefficients { h( k) } can be derived from φ (x) 1 x φ = h( l) φ( x l) (3.3) 2 2 l The last smoothed array c np is added to all the differences w i : c0 ( k) = cn ( k) wj ( k). 1 x if x [ 1,1] Choosing the linear interpolation for the scaling functionφ( x) =. The 0 if x [ 1,1] wavelet coefficients at the scale j are 1 j 1 1 j c j+ 1( k) = c j ( k 2 ) + c j ( k) c j ( k + 2 ) (3.4) p np j= 1 63

64 3.3 Neighbourhood Pixel Filter (NPF) A filter called Neighbourhood Pixel Filter (NPF) is developed to suppress the Gaussian noise effectively. In this filter first find Neighbourhood Pixel Difference (NPD) by subtracting the neighbouring pixel values from its current noisy pixel value. Also calculate weight of each pixel which depends on this NPD. A filtered value is assigned for each current pixel in order to approximate the original pixel value of that pixel. This filtered value is generated by minimizing NPD and Weighted Mean Square Error (WMSE) using method of least square. A reduction in noise pixel is observed on replacing the optimal weight namely NPFA filter solution with the noisy value of the current pixel. Due to this NPFA filter gain the effect of both high pass and low pass filter. This filter behaves like a low pass filter in smooth region by decreasing noise variance effectively and giving similar weights to all its neighbouring pixels. This in turn cuts off only high frequency noise signal instead of all noisy signals. The resultant image thus obtained is observed to have much less blurring effect compared to the other wavelet method Neighbourhood Pixel Filtering Algorithm (NPFA) for Denoising The first step in the algorithm is to detect whether the noisy pixel value input B ij itself is corrupted by a noise or not. The deduction of noise is made by applying the denoising model as shown in Fig.3.1 (a) and Fig. 3.1 (b). After comparing the pixel value of the current pixel B ij to that of the neighboring pixels B i-1,j, B i+1,j, B i,j-1, B i,j+1 and colocated pixel P ij in the previous wavelet frame. Find the Neighbouring Pixel Difference (NPD) by subtracting the neighbouring pixel values namely B i-1,j, B i+1,j, B i,j-1, B i,j+1 and current pixel P ij from the pixel value B ij. If all the differences are greater than a specified threshold T then the pixel B ij is corrupted by a noise. Otherwise it is not corrupted by a noise. After noise detection in the first step, a filtered value f ij would be assigned for each corrupted noisy pixel as the weighted average of its neighbouring pixel values in the NPFA frame given by Eqn. (3.5) f ij Bi 1, j + B i + 1, j + B i, j B i, j + 1 = (3.5) 4 64

65 B ij-1 P ij B i-1j Bij B i+1j B ij+1 Figure 3.1(a) Wavelet Frame Figure 3.1 (b) NPFA Frame The correlation between the noisy pixel value B ij and its original pixel value would be reflected by NPD. For corrupted pixels this correlation between the pixel differences is small and a filtered value f ij is assigned to each current pixel in order to approximate the original pixel value of current pixel. While for each uncorrupted pixels the noisy pixel value is highly correlated with the original pixel value. In this case the filtered value f ij will be generated by minimizing NPD and Weighted Mean Square Error (WMSE) which is given in Eqn. (3.9) using method of Least Squares. For which the weight W ij of each pixel is required which depends on NPD and calculated using Eqn. (3.6) and (3.7) W M N i= 1 j= 1 ij = M N g( i) P B * f i= 1 j= 1 ij ij ij g( i) Pij B ij (3.6) where ( T /8 i /8 2 ) i < T g( i) = 0 else (3.7) where P ij is the current pixel value, B ij is the noisy pixel value and f ij is the filtered value. On replacing the optimal weight namely NPFA filter solution obtained from Eqn. (3.10) with the noisy value of current pixel a reduction in noise pixel is observed. Due to this, NPFA filter gain the effect of both low pass filter and high pass filter. In smooth areas this filter decreases noise variance effectively by giving similar weights to all its neighbouring pixels. Hence in this area the proposed filter behaves like a low pass filter and preserves edge information. In high texture areas, this filter exploits the edge 65

66 information by using the pixel value difference between the current pixel and its neighbouring pixels (NPD). Because of this advantage, NPFA filter attenuates high pass features at the discontinuities and maintains the sharpness of edges by assigning small weight for that pixel. Also this filters cuts off only high frequency signal instead of all noisy signals. Quantitatively the measure of image is given by its PSNR value obtained from formula given by Eqn. (3.8) where the minimized WMSE is used. Hence this filter yields a significant PSNR value to that of other wavelet method. PSNR 255 db WMSE 2 = 10 log10 (3.8) M N ij ij ij (3.9) i= 1 j= 1 Minimise ( ) 2 WMSE = W B Y Y ij is the NPFA filter solution given by the following Eqn.: Y M N i = 1 j = 1 ij = M N i = 1 j = 1 W B ij W ij ij (3.10) Qualitatively the resultant images obtained on using this filter will have less blurring effect compared to other wavelet method. Neighbourhood Pixel Filtering Algorithm (NPFA) Input: Pixel value B ij Step 1 Compare the current pixel value B ij to that of the neighbouring pixels B i-1,j,b i+1,j,b i,j- 1, B i,j+1 and co-located pixel P ij in the previous wavelet frame. Step 2Computethreshold, T 2 = σ 2(log n ) where σ is the noise variance of the corrupted ( X ij ) Median 2 image which can be calculated usingσ =, B i j belongs to each subband. Step 3 If all the differences between B ij and B i-1,j, B i+1,j, B i,j-1, B i,j+1 respectively are greater than T, then mark B ij as noise corrupted pixel. 66

67 Step 4 Compute a filtered value f ij to each corrupted pixel B ij, defined by: f ij B + B + B + B = 4 i 1, j i+ 1, j i, j+ 1 i, j+ 1 Step 5 Correlate between the noisy pixel value B ij and its original pixel value. If it is small then assign its corresponding filtered value f ij. Step 6 Calculate the weight W ij of each pixel by using where M and N are width and height of the image B ij and W M N i= 1 j= 1 ij = M N g( i) P B * f i= 1 j= 1 ij ij ij g( i) Pij B ij ( T /8 i /8 2 ) i < T g( i) = 0 else which depends on value. Here B is the noisy pixel value, ij P ij is the current pixel value, f ij and is the filtered value. Step 7 Minimize NPD value for generating the filtered value f ij using Least Square method. Step 8 Minimize WMSE for generating the filtered value, where M N i= 1 j= 1 ( ) 2 WMSE = W B Y ij ij ij Step 9 Compute NPFA filter solution (optimal solution) by using the Eqn. (3.11) Y = M N i= 1 j= 1 ij M N i= 1 j= 1 W B ij W ij ij and PSNR 255 db WMSE (3.11) 2 = 10log10 Output: Denoised pixel B ij 3.4 Simulation and Results As the ideally image is normally unknown at the receiver end, finding quantitative performance is not an easy task in practical application. Hence experiments are performed by adding Gaussian noise with different variance to the original image. This corrupted image is transformed into original domain on applying wavelet transform. The detail wavelet coefficients are modified according to the NPFA algorithm and the denoised image is finally reconstructed by taking inverse wavelet transform. 67

68 In this method various experiments are carried out with different parameters on several gray level images with dimensions 256x256 and 512x512. Among them Lena, Boat, Barbara (512x512) images and House and Pepper (256x256) images are used for comparison purpose. We assume that the noise model is Gaussian additive white noise 0,. A noise is introduced into the selected image. On applying the wavelet transform, the noise is reduced to some extent but the efficiency of wavelet image is low. Whereas on applying NPFA the efficiency of the picture gets considerably increased and give optimum results. Hence, to add more efficiency to the image, we apply NPFA filter and the image is compared to the one without filter application. This is illustrated in Fig. 3.3to Fig. 3.9 and the comparison of PSNR performance of different filters for various noise levels operated on standard images are illustrated in Fig to Fig

69 Figure 3.2 Two Label Decomposition of Boat Image before Adding Noise Wavelet 69

70 Figure 3.3 Two Label Decomposition of Boat Image after Adding Noise Wavelet Figure 3.4 Performances of Various Filters for Barbara Image with AWGN = 70

71 Figure 3.5 Performances of Various Filters for Boat Image with AWGN = Figure 3.6 Performances of Various Filters for Pepper Image with AWGN = Figure 3.7 Performances of Various Filters for House Image with AWGN = Figure 3.8 Performances of Various Filters for Lena Image with AWGN = 71

72 Table 3.1 Comparison Table for PSNR Value for Barbara image using NPFA and Existing Wavelet Method for Different Noise Level Noise level Noisy Wavelet NPFA Table 3.2 Comparison Table for PSNR Value for Boat image using NPFA and Existing Wavelet Method for Different Noise Level Noise level Noisy Wavelet NPFA

73 Table 3.3 Comparison Table for PSNR Value for Pepper image using NPFA and Existing Wavelet Method with Different Noise Level Noise level Noisy Wavelet NPFA Table 3.4 Comparison Table for PSNR Value for House image using NPFA and Existing Wavelet Method for Different Noise Level Noise level Noisy Wavelet NPFA

74 Table 3.5 Comparison Table for PSNR Value for Lena image using NPFA and Existing Wavelet Method for Different Noise Level Noise level Noisy Wavelet NPFA Table 3.6 Execution Time (Milliseconds) Taken by Various Filters for Images in the Low Noise Level Range for σ=5 Method System Number Barbara Lena Pepper Boat House Wavelet System (À Trous Algorithm) System NPFA System System

75 Table 3.7 Execution Time (Milliseconds) Taken by Various Filters For Images In The Moderate Noise Level Range for = Method System Number Barbara Lena Pepper Boat House Wavelet System (À Trous Algorithm) System NPFA System System Table 3.8 Execution Time (Milliseconds) Taken by Various Filters For Images In The High Noise Level Range for = Method System Number Barbara Lena Pepper Boat House Wavelet System (À Trous Algorithm) System NPFA System System

76 70 60 Performance of NPFA for Barbara image Noise Image Wavelet Image NPFA 50 PSNR Total Noise Standard Deviation Figure 3.9Performance Comparison of Various Filters in Terms of PSNR Operated on Barbara for Various Noise Levels ( varies from 5 to 80) 76

77 70 60 Performance of NPFA for Lena image Noise Image Wavelet Image NPFA 50 PSNR Total Noise Standard Deviation Figure 3.10Performance Comparison of Various Filters in Terms of PSNR Operated on Lena for Various Noise Levels ( varies from 5 to 80) 77

78 70 60 Performance of NPFA for Boat image Noise Image Wavelet Image NPFA 50 PSNR Total Noise Standard Deviation Figure 3.11Performance Comparison of Various Filters in Terms of PSNR Operated on Boat for Various Noise Levels ( varies from 5 to 80) 78

79 70 60 Performance of NPFA for Pepper image Noise Image Wavelet Image NPFA 50 PSNR Total Noise Standard Deviation Figure 3.12 Performance Comparison of Various Filters in Terms of PSNR Operated on Pepper for Various Noise Levels ( varies from 5 to 80) 79

80 70 60 Performance of NPFA for House image Noise Image Wavelet Image NPFA 50 PSNR Total Noise Standard Deviation Figure 3.13Performance Comparison of Various Filters in Terms of PSNR Operated on House for Various Noise Levels ( varies from 5 to 80) From the Tables 3.1 to 3.5, it is observed that NPFA possess high PSNR value. Also from Fig. 3.9 to 3.13 it is observed that as the noise level increases the PSNR decreases because NPFA performs like low pass filter and removes only high frequency noise signals. Execution time is another important image metric to compare the performance of filter. These wavelet domain filters are simulated using MATLAB R2007b platform on two different systems having different operating systems, one has(system 1) 64-bit 80

81 operating system Windows7 Home Basic having Intel(R) Core(TM) i and having RAM of 4GB. Another system (system 2) 64-bit operating system Microsoft Windows XP Professional having Intel(R) Core(TM)2 CPU and having RAM of 504MB. The Matlab function tic-toc and command called Flops are used in each program to calculated the execution time. It is the time taken by the system to run the Matlab program in order to calculate PSNR value for different noise level and for different images. From the Table 3.6, Table 3.7 and Table 3.8 it is observed that at low, moderate and high level of noise NPFA has minimum execution time. As NPFA possess high PSNR value and minimum execution time this method out performs the other wavelet method and the clarity of the image is improved. 3.5 Conclusion Á Trous algorithm is applied to DWT to overcome the shift invariance issue. Samples of the images are taken and noise is included to each of the image. The detection of noise is made by modifying the filter coefficients only. Using ÀTrous algorithm the clarity of the output image is not good, as the contrast of the result image is weaker. To improve the clarity of images an efficient NPF is added along with à Trous algorithm. As the concept low pass filter preserving the energy of signal and attenuating the high pass features at the discontinuities is used this filter gains the effect of both high pass and low pass filter. This filter uses NPD and cuts off only high frequency signal instead of all noisy signals. After implementing and using the results of the NPF a significant improvement in clarity and sharpness of the image is observed. Hence, the achievement of improved quality is attributed to exploitation in the corrections of NPD values within a small region and taking average weights of them. Also, WMSE between the filtered value and the pixel values are minimized using Least Square Method. 81

82 CHAPTER 4 ADAPTIVE SUBBAND THRESHOLDING TECHNIQUE 4.1 Introduction Image denoising is a common procedure in digital image processing aiming at the suppression of additive white Gaussian noise (AWGN) that might have corrupted an image during its acquisition or transmission. This procedure is traditionally performed in the spatial-domain or transform-domain by filtering. In spatial-domain filtering, the filtering operation is performed on image pixels directly. The main idea behind the spatial-domain filtering is to convolve a mask with the whole image. The mask is a small sub-image of any arbitrary size (e.g., 3 3, 5 5, 7 7, etc.). Other common names for mask are: window, template and kernel. An alternative way to suppress additive noise is to perform filtering process in the transform-domain. In order to do this, the image must be transformed into the frequency domains using a 2-D image transform. Various image transforms such as Discrete Cosine Transform (DCT) [51, 89, 2], Singular Value Decomposition (SVD) Transform [51], Discrete Wavelet Transform (DWT) [66, 105, 42, 70, 27, 89, 115, 103, 25] etc. are used. The Discrete Wavelet Transform (DWT) is a powerful tool for image denoising. Image denoising using wavelet techniques is effective because of its ability to capture most of the energy of a signal in a few significant transform coefficients when image is corrupted with Gaussian noise. One method that has received considerable attention in recent years is wavelet thresholding or shrinkage: an idea of killing coefficients of low magnitude relative to some threshold. The various thresholding or shrinkage techniques proposed in the literature are Visu Shrink [31], Sure Shrink [31], Baye s Shrink [21], Neigh Shrink [22], Oracle Shrink [21], Oracle Thresh [21] etc. The wavelet domain methods are suitable in retaining the detail structures but they introduce mat like structure in the smooth region of the filtered image. 82

83 The performances of various thresholding or shrinkage techniques are not quite effective as they are not spatially adaptive. Some other methods evaluate the denoised coefficients by an MMSE (Minimum Mean Square Error) estimator, in terms of the noised coefficients and the variances of signal and noise. The signal variance is locally estimated by a ML (Maximum Likelihood) estimator in small regions for every subband where variance is assumed practically constant. These methods present effective results but their spatial adaptivity is not well suited near object edges where the variance field is not smoothly varied. Further, these methods introduce artifacts in the smooth regions of the output image. In this chapter an Adaptive Subband Thresholding (AST) technique is developed based on wavelet coefficients in order to overcome the spatial adaptivity which is not well suited near object edges, where the variance field is not smoothly varied by producing effective results. This method describes a new way for suppression of Gaussian noise in image by fusing the wavelet denoising technique with optimized thresholding function to which a multiplying factor is included to make the threshold value dependent on decomposition level. The performance of this new method is compared with exisisting wavelet domain filters in terms of objective and subjective evaluations to demonstrate the effectiveness of the developed method. 4.2 Wavelet Based Image Denoising Image denoising algorithms attempts to remove the noise from the image. Ideally the resulting denoised image will not contain any noise or added artifacts. Denoising of natural images corrupted by Gaussian noise using wavelet techniques is very efficient because of its ability to capture the energy of a signal in few energy transform values. The basic frame work of wavelet transform based image denoising is shown in Fig. 4.1.Some efficient wavelet domain methods used as the methods of threshold selection for image denoising based on wavelet transform are discussed in the subsequent sub sections Visu Shrink Visu Shrink is thresholding by applying universal threshold [31] proposed by Dohono and Johnston. This threshold is given by Eqn. (4.1). 83

84 = 2 (4.1) where, is the noise variance of AWGN and L is the total number of pixels in an image. Original Image s(x,y) + Noise n(x,y) Pre filter Noisy Image N(x,y) Forward Wavelet Transform Thresholding/ Shrinkage Inverse Wavelet Transform Post Filter Figure 4.1 Frame Work of Wavelet Transform based Image Denoising Sure Shrink Sure Shrink is an adaptive thresholding method where the wavelet coefficients are treated in level-by-level fashion. It is used for suppression of additive noise in waveletdomain where a threshold is employed for denoising. The threshold parameter is expressed in Eqn. (4.2). = arg ; (4.2) ; is defined as in Eqn. (4.3). ; = 2. #{ : min, (4.3) where, is the noise variance of AWGN, L is the total number of coefficients in a particular subband, is a wavelet coefficient in the particular subband, 0,, is Donoho universal threshold [31]. 84

85 4.2.3 Baye s Shrink Baye s Shrink is an adaptive data-driven threshold is used for image denoising [21]. The wavelet coefficients in a sub-band of a natural image can be represented effectively by a generalized Gaussian distribution (GGD). The Bayesian framework is derived in Eqn. (4.4). = (4.4) where, is the estimated noise variance of AWGN by robust median estimator and is the estimated signal standard deviation in wavelet-domain.the robust median estimator is stated as in Eqn. (4.5). =., (4.5) This estimator is used when there is no a priori knowledge about the noise variance. The estimated signal standard deviation is calculated using Eqn. (4.6). =, 0 (4.6) where, is the variance of Y. Since Y is modeled as zero-mean, foundempirically by Eqn. (4.7). can be =,, (4.7) Incase, will become 0. That is, becomes. Hence, for this case = Modified Baye s Shrink Modified Baye s shrink remove noise better than Baye s shrink [21]. It performs its processing using threshold values that are different for each subband coefficient the threshold T can be determined using the formula Eqn. (4.8) and Eqn. (4.9). = (4.8) = (4.9) 85

86 M is the total of coefficients of wavelet. J is the wavelet decomposition level present in the subband coefficients under scrutiny Normal Shrink The value of threshold by normal shrink [21] can be calculated by using Eqn. (4.10) and Eqn. (4.11). = (4.10) = log (4.11) is length of the subband at the k th scale. J is the total number of decompositions Oracle Shrink and Oracle Thresh Oracle Shrink and Oracle Thresh are two wavelet thresholding methods used for image denoising [21]. These methods are implemented with the assumption that the wavelet coefficients of original decomposed image are known. The Oracle Shrink and Oracle Thresh employ two different thresholds denoted as respectively. Mathematically they are represented by Eqn. (4.12) and Eqn. (4.13). = arg, (4.12) = arg, (4.13) where, is the wavelet coefficient of original decomposed image, and are soft-thresholding and hard-thresholding functions defined by Eqn. (4.14) and Eqn.(4.15). =., 0 (4.14) =. 1{ (4.15) which keeps the input if it is larger than the threshold T; otherwise, it is set to zero Neigh Shrink Chen et al. proposed a wavelet-domain image thresholding scheme by incorporating neighboring coefficients, namely Neigh Shrink [22]. The method Neigh Shrink thresholds the wavelet coefficients according to the magnitude of the squared sum of all the wavelet coefficients. The shrinkage function for Neigh Shrink of any arbitrary 3 3 window centered at (i,j) is expressed by Eqn. (4.16) 86

87 = 1 (4.16) where, is the universal threshold and is the squared sum of all waveletcoefficients in the respective 3 3 window given by Eqn. (4.17). =, (4.17) where, + sign at the end of the formula means to keep the positive values while setting it to zero when it is negative. The estimated center wavelet coefficient is then calculated from its noisycounterpart as in Eqn. (4.18). = Γ (4.18) 4.3 Denoising By Wavelet Coefficient-Statistical Model Considering the advantages and limitations of the statistical model sub-band thresholding method is developed based on the wavelet coefficients. Wavelet coefficients with large magnitudes are representation of edges or some textures. While those with small magnitudes are associated with smooth region such as the background. The coefficient in each subband except the first fine scale is partioned into two classes based on the magnitude of their parents, namely significant classes and insignificant classes in the corresponding region. The significant classes represent the high activity region and insignificant classes correspond to smooth region. The size of the two classes is controlled by significant threshold T. If the magnitude of parent is larger than T then the coefficient is included in the significant classes otherwise it is included in insignificant classes. The two classes have different statistics; the histogram of the coefficient in insignificant classes is highly concentrated around zero while that of significant classes is more spread out. The proposed approach focuses on exploiting the multiresolution properties of wavelet transform. The derived technique identifies close correlation of signal at different resolution by observing the signal across multiple resolutions. The modeling of the wavelet coefficient can neither be deterministic or statistical. 87

88 4.3.1 Deterministic Modeling of Wavelet Coefficients The deterministic method of modeling involves developing a tree-structure of wavelet coefficients in the every level of a tree representing each scale of transformation where nodes represent wavelet coefficients. The optimum tree approximation displaces a hierarchical interpretation of wavelet decomposition Statistical Modeling of Wavelet Coefficients The proposed approach focuses on some appealing properties of the wavelet transform such as multi scale correlation between the wavelet coefficients and local correlations between the neighborhood coefficients. This approach has an inherent goal of perfecting the exact modeling of image data with the use of wavelet transform. Marginal probabilistic model and joint probabilistic model are the two techniques which exploit the statistical properties of the wavelet coefficients based on a probabilistic model. Marginal distribution of wavelet coefficients are highly kurtotic, and usually have a marked peak at zero and heavy tails. The Gaussian Mixture Model (GMM) and the Generalized Gaussian Distribution (GDD) are commonly used to model the wavelet coefficients distribution. In these methods the wavelet coefficients are assumed to be conditionally independent zeromean Gaussian random variables with variances modeled as identically distributed, highly correlated random variables. Also another model in which an approximate Maximum A Posterior (MAP) probabilistic rule is used to estimate marginal prior distribution of wavelet coefficients. In the developed AST techniques the coefficients in significant classes are modeled as independently identically distributed (iid) Laplacian with zero-mean for the coefficients in insignificant model which corresponds to homogeneous regions, the usage of intrascale model in Estimation Quantization [EQ] coder is appropriate [61]. It provides a good fit for the first order statistics of wavelet coefficients and well models the non stationary nature of low-activity regions. The observation model is expressed as follows Y = X + V, where Y is the wavelet transform of the degraded image, X is the wavelet transform of the original image, V denotes the wavelet transform of the noise components following the Gaussian distribution 88

89 N σ Since X and V are mutually independent, the variances of Y, X and V are given 2 (0, V ) by σ = σ + σ (4.19) Y X V 4.4 Estimation of Noise Variance The following steps are used to evaluate the variance estimate for each wavelet coefficients depending on the subband. 2 Step 1 The noise variance σ can be accurately estimated from the first decomposition V level (i.e) diagonal subband HH 1 by the robust and accurate median estimator [49]. m ed ia n y ( V ) 2 σ V = Where y ( V ) represents the coefficients HH 1 subband. (4.20) Step 2 The coarse subbands are not processed because the coarse subband has very high SNR. These coefficients are considered reliable. Step 3 For each of the three subbands (horizontal, vertical and diagonal orientations), coefficients within the subband are modeled as identically independently distributed with zero mean and variance σ (where j indicates the subband). The variance estimate is 2 x, j computed from the noisy coefficients in subband j as in Eqn. (4.21). 2 2 σ = max{0, var{ y, } } x, j i i subbandj σ v (4.21) Using MAP, estimation of x is obtained by applying a soft threshold λ as given in Eqn. (4.22) to each noisy coefficient. λ = 2 σv / σ, j subband x j (4.22) 2 2, Step 4 In each of the other high sub-bands, coefficients are assigned either to significant or insignificant classes depending on the magnitude of their estimated parent relative to the significance threshold T, where = 2 log (4.23) (i) Coefficients in significant classes are modeled as iid Laplacian with zero mean and 2 their variance σ is estimated from the noisy coefficients as mentioned in step 3. x, i n s ig Again the MAP estimator is a simple soft thresholding scheme where its threshold value is adjusted to the signal variance. 89

90 (ii) Coefficients in insignificant classes which has small magnitude representing smooth areas, σ is estimated using ML estimator in order to have an estimate for a local 2 x, i n s i g 2 neighborhood σ where variance is assumed to be constant. The estimate of the classes x coefficient variance is given by the Eqn. (4.24). σ 1 ( ( ) ) M 2 M 2 2 = y V σ x, insig V = 1 V (4.24) where M represents the number of wavelet coefficients residing in local neighborhood N. Considering the coefficients belonging to a insignificant classes inside the window are used by excluding the one which belong to significant classes, the MAP estimator is given by the Eqn. (4.25). x = σ σ 2 x, in sig σ x, in sig V y i (4.25) Thus the coefficient of estimates corresponding to the high subband are obtained by repeating the above steps from parent to child subband, starting from the coarse scale and terminating in the highest subband. 4.5 Optimum Value Threshold and Threshold Selection Wavelet thresholding [32, 4, 31] is a signal estimation technique that exploits the capabilities of wavelet transform for signal denoising. It removes noise by killing coefficients that are insignificant relative to some threshold and turns out to be simple and effective which depends heavily on the choice of a thresholding parameter. The choice of this threshold determines the efficacy of denoising to a great extent Threshold Selection Finding an optimum value thresholding is not an easy task. A small threshold may yield a result close to the input, but the result may still be noisy. A large threshold on the other hand, produces a signal with a large number of zero coefficients. This leads to a smooth signal. Paying too much attention to smoothness destroys details and it may cause blur and artifacts in image processing. 90

91 Soft thresholding method is used to analyze the performance of denoising system for different levels of DWT decomposition, as it results in better denoising performance than other denoising methods. Also it leads to less severe distortion of the object of interest than other thresholding methods [7]. Several approaches have been suggested for setting the threshold for each band of the wavelet decomposition. A common approach is to compute the sample variance 2 σ of the coefficients in each band and set the threshold to any multiple of standard deviation σ for that band [88]. Thus, to implement a soft threshold of the DWT coefficients for a particular wavelet band, the coefficients of that band should be threshold as shown in Fig The soft thresholding is generally represented by Eqn. (4.26). d sign( d )( d λ ) ifd > λ * * soft ik ik ik ik = * 0ifdik λ (4.26) Figure 4.2 (a) Soft Threshold Characteristics With λ = aσ Figure 4.2(b) Soft Threshold Characteristics With λ = 1 Figure 4.2 Soft Threshold Characteristics 91

92 4.5.2 Optimum Value Threshold An adaptive thresholding is proposed by fixing the optimum thresholding value depending on the decomposition level. At every decomposition level, four frequency subbands are obtained namely LL, LH, HL, HH. The next level should be applied to the low frequency subband LL only. This process is continued until a pre specified level (level 2) is reached. In wavelet domain, as the level of subbands increases its coefficients becomes smoother. That is, subband HL 2 is smoother than the corresponding subband in the first level (HL 1 ) and so the threshold value of HL 2 should be smaller than that of HL 1. In the wavelet decomposition, the magnitude of the coefficient varies depending on the decomposition level. Therefore, if all levels are processed with one threshold value, the processed image may be overly smoothened so that sufficient information preservation is not possible and the image gets blurred. To overcome this problem and to obtain a significantly superior quality image, the multiplying factor is included in the threshold formula to get better PSNR value by preserving edges where L K α = 2 log M (4.27) L is the number of wavelet decomposition level, K is the level at which the subband is available, M is the total number of wavelet coefficients. Using this multiplication factor α the optimum threshold formula for the proposed technique is given by λ * = α λ (4.28) Where λ, σ are calculated using Eqn. (4.22) and Eqn. (4.27) respectively. 4.6 Adaptive Subband Thresholding (AST) Algorithm The block diagram of the developed method is shown in Fig Wavelet based denoising method relies on the fact that noise commonly manifests itself as fine- grained structure in the image and DWT provides a scale based decomposition. Thus, most of the noise tends to be represented by wavelet coefficient at the finer scales. Discarding these coefficients would result in a natural filtering of the noise on the basis of scale. As the coefficients at such scales also tend to be primary carriers of edge information, the DWT noisy coefficients can be made zero if their values are below its optimum threshold value. 92

93 On the other hand, the edge relating coefficients are usually above the threshold. The inverse DWT of the threshold coefficients is the denoised image. Noisy Image Apply DWT and Decomposition Wavelet Estimator Apply Threshold Technique Denoised Image Estimation of Variance MAP Estimator Figure 4.3 Block Diagram forast Technique Adaptive Sub-band Thresholding (AST) Algorithm The complete algorithm of the developed wavelet based denoising technique is explained in the following steps: Input: Noisy image Step 1 Perform Multiscale decomposition of the image corrupted by Gaussian noise using wavelet transform. 2 Step 2 Estimate the noise variance σ using Eqn. (4.20) for each scale and compute the V scale parameter. Step 3 For each of the three subbands variance estimate is computed from the noisy coefficient in subband j using Eqn. (4.21). Step 4 In each of the other high subbands the estimates of the classes coefficient variance are estimated using Eqn. (4.21) and (4.24). Step 5 Calculate threshold value using optimum value threshold formula as given in Eqn. (4.28) after finding the multiplying factor σ for each subband using the relation given in (4.27). After computing threshold for each subband except the low pass or approximation subband, apply soft thresholding to each wavelet coefficient using threshold given in Eqn. (4.26), by substituting the threshold value obtained in Step 5. Step 6 Invert the multiple decomposition to reconstruct the denoised image. 93

94 Output: Denoised image 4.7 Simulation and Results The AST algorithm has been applied on several natural gray scale test images like Lena, Barbara and Pepper at different Gaussian Noise level. Daubechius (Db4), the least asymmetric compactly supported wavelet at two levels of decomposition are used. Performance of noise reduction algorithm is measured using quantitative performance measure such as Peak Signal to Noise Ratio (PSNR), Execution Time and in terms of visual quality of image. To evaluate the performance of the AST technique, it is compared with wavelet domain method like the Baye s shrink, Modified Baye s shrink, Oracle Shrink, Neigh Shrink, Visu Shrink and Normal shrink using PSNR and Execution Time. PSNR is calculated using the Eqn. (4.29). = 10 log (4.29) This method finds application in denoising images those are corrupted during transmission which are normally random image. 94

95 Figure 4.4 Two Label Decomposition of Barbara Image after Adding Noise Wavelet 95

96 Figure 4.5 Two Label Decomposition of Barbara Image before Noise Wavelet 96

97 Figure 4.6 Smooth Noise Wavelet Added Figure 4.7 Subband Region of the Smooth Noise 97

98 Figure 4.8 Performances of Various Filters for Barbara Image with AWGN = 98

99 Figure 4.9 Performances of Various Filters for House Image with AWGN = 99

100 Figure 4.10 Performances of Various Filters for Boat Image with AWGN = 100

101 Figure 4.11 Performances of Various Filters for Lena Image with AWGN = 101

102 Figure 4.12 Performances of Various Filters for Pepper Image with AWGN = 102

103 Table 4.1 Comparison of PSNR Values for Different Noise Standard Deviations for Barbara Image Noise Level Noisy Image Baye s Shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST

104 Table 4.2 Comparison of PSNR Values for Different Noise Standard Deviations for Lena Image Noise Level Noisy Image Baye s Shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST

105 Table 4.3 Comparison ofpsnr Values for Different Noise Standard Deviations for Boat Image Noise Level Noisy Image Baye s Shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST

106 Table 4.4 Comparison of PSNR Values for Different Noise Standard Deviations for House Image Noise Level Noisy Image Baye s Shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST

107 Table 4.5 Comparison of PSNR Values for Different Noise Standard Deviations for Pepper Image Noise Level Noisy Image Baye s Shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink Adaptive Subband

108 PSNR Performance of Barbara Image Baye s Shrink Modified Bayes Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST PSNR Total Noise Standard Deviation Figure 4.13 Performance Comparison of Various Filters in Terms of PSNR Operated on Barbara for Various Noise Levels ( varies from 5 to 80) 108

109 PSNR Performance of Lena Image Baye s Shrink Modified Bayes Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST PSNR Total Noise Standard Deviation Figure 4.14 Performance Comparison of Various Filters in Terms of PSNR Operated on Lena for Various Noise Levels ( varies from 5 to 80) 109

110 PSNR Performance of Boat Image Baye s Shrink Modified Bayes Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST PSNR Total Noise Standard Deviation Figure 4.15 Performance Comparison of Various Filters in Terms of PSNR Operated on Boat for Various Noise Levels ( varies from 5 to 80) 110

111 PSNR Performance of House Image Baye s Shrink Modified Bayes Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST PSNR Total Noise Standard Deviation Figure 4.16 Performance Comparison of Various Filters in Terms of PSNR Operated on House for Various Noise Levels ( varies from 5 to 80) 111

112 PSNR Performance of Pepper Image Baye s Shrink Modified Bayes Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink AST PSNR Total Noise Standard Deviation Figure 4.17 Performance Comparison of Various Filters in Terms of PSNR Operated on Pepper for Various Noise Levels ( varies from 5 to 80) 112

113 Table 4.6 Execution Time (Milliseconds) Taken By Various Filters for Images in the Low Noise Level Range = Method Baye s shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink Adaptive Subband System Number Barbara Lena Pepper Boat House System System System System System System System System System System System System System System

114 Table 4.7 Execution Time (Milliseconds) Taken By Various Filters for Images in the Moderate Noise Level Range = Method Baye s shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink Adaptive Subband System Number Barbara Lena Pepper Boat House System System System System System System System System System System System System System System

115 Table 4.8 Execution Time (Milliseconds) Taken By Various Filters for Images in the High Noise Level Range = Method Baye s shrink Modified Baye s Shrink Visu Shrink Neigh Shrink Oracle Shrink Normal Shrink Adaptive Subband System Number Barbara Lena Pepper Boat House System System System System System System System System System System System System System System The performances of various wavelet domain filters in terms of PSNR values are given in Tables 4.1 to 4.5. From which it is observed that the filters Modified Baye s Shrink, Normal Shrink and AST technique perform better in terms of PSNR. Under Moderate and High noise level conditions, AST works well by possessing high PSNR value. The comparison of the PSNR performance of different methods for various noise levels operated on the standard images is graphically represented in Figures 4.13 to From the graph it is observed that Modified Baye s Shrink, Normal Shrink and Neigh Shrink perform better in both Moderate and High noise level. In high activity sub region Baye s Shrink and Neigh Shrink performs little denoising, the sharpness of the edges are 115

116 not maintained and flat subparts of the image are completely denoised. Normal Shrink preserves edges better than noise removal. Modified Baye s Shrink yields better results for denoising and also adapts thresholding strategy. The AST technique gives better performance when compared with the other wavelet domain method by preserving the edges as well as removing the noise due to the advantages of subband thresholding and inclusion multiplying factor in the optimum value threshold formula. The execution time of different filters for Low, Moderate and High noise level conditions are given in table It is observed from the table that the AST technique possess minimum execution time when compared with the other existing methods. Hence under Moderate and High noise levels the AST techniques out performs other methods by possessing high PSNR and minimum execution time. 4.8 Conclusion The developed AST technique is based on the analysis of statistical parameters like standard deviation, variance of the sub-band coefficients using ML or MAP estimator which is more sub-bands adaptive. Since the decomposition level dependent is included as a multiplying factor α in the optimum value threshold formula along with sub-band variance estimation, the proposed technique yields significantly superior image quality by preserving edges and a better PSNR value. It is also observed that the images corrupted with less noise densities, single level of decomposition is sufficient. While for images corrupted with higher noise density second level of decomposition is required irrespective of the images. 116

117 CHAPTER 5 SWITCHING WEIGHTED ADAPTIVE MEDIAN FILTER 5.1 Introduction Images could be contaminated by impulse noises during image acquisition or transmission. The intensity of the impulse noise has the tendency of being either relatively high or relatively low. Thus, it could severely degrade the image quality and cause great loss of information details. So it is important to eliminate such noise in the images before some subsequent processing, such as edge reduction, image segmentation and object recognisation. Various filtering techniques have been proposed for removing impulse noise in the past. The linear filters could produce serious image blurring and as a result non linear filters have been widely exploited due to their mechanism improved filtering performance in terms of impulse noise attenuation and edge/details preservations. Some of the popular and robust non-linear filters are Standard Median Filter (SMF) Weighted Median Filter (WMF) and Center Weighted Median Filter (CWMF) [80, 12, 55]. These filters apply the median operation to each pixel unconditionally whether it is corrupted or uncorrupted and as a result even the uncorrupted pixels are filtered which causes image quality degradation. In order to overcome this problem, is to implement an impulse noise detection mechanism prior to filtering: hence only those pixels identified as corrupted would undergo filtering process while other uncorrupted pixels would remain intact. By incorporating such noise detection mechanism into the median filtering frame work a new filter called Switching Median Filter exists and had shown significant performance improvement. These median type filters exhibit blurring for large window size and insufficient noise suppression for small window size. Hence selection of appropriate window size plays an important role for noise detection. Lin and Huang proposed adaptive length median filter for removal of 117

118 impulse noise in images. The results are not better as the window size is selected based on threshold values [67]. In this chapter a new filter called Switching Weighted Adaptive Median Filter (SWAM) is developed for effective suppression of impulse noise which is used to incorporate the Recursive Weighted Median (RWM) filters and the Switching Adaptive Median (SAM) filter. The adaptive window size is selected using RWM and the output image produced by this filter with least mean square error is considered as input image to SAM filter where impulse detection mechanism is adapted. Due to this the unwanted filtering of uncorrupted pixels and blurring are reduced even at high density noise. The performance of the developed filter algorithm is given in terms of Image Enhancement Factor (IEF) and Peak Signal Noise to Ratio (PSNR) and it is compared with Switching Median Filter (SMF), Adaptive Median Filter (AMF), Weighted Median Filter (WMF), 2D-Median filter. 5.2 Development of SWAM Filter In this paper, a novel Switching Weighted Adaptive Median Filter (SWAM) is proposed that employs the switching scheme with two stages based on impulse detection mechanism as shown in Fig The objective of the proposed filter is to utilize the RWM filter and SAM filter to define more general operators [115, 68]. In the first stage, the size of the adaptive window is selected by RWM and the output image produced by this filter is considered as the input image for the second stage where impulse noise detection mechanism is implemented. 5.3 Impulse Noise Model The impulse detection is based on the assumption that a noise pixel in the filtering window takes a gray value which is substantially different from the neighbouring pixels, whereas noise-free regions in the image have locally smooth varying gray levels separated by edges [3]. In the switching median filter, the difference of the median value of pixels in the filtering window and the current pixel value is compared with a threshold to decide about the presence of the impulse. We assume that the image is of size M N having 8-bit 118

119 x gray scale pixel resolution of I [0, 255]. Now a large Window W i, j is taken whose central pixel is x( i, j ) Input Noisy Image Weight Calculation by Median Controlled Algorithm Selection of Window Size & RWM Filtering Operation Output/Reference Image Iterative Median Based Noise for Reference Image Switch No Filtering Filtered Output Switching Mechanism Based on Noise Threshold SWAM Filter Figure 5.1 Block Diagram for Switching Weighted Adaptive Median (SWAM) Filter In the conventional switching median filter, the output of the filter is given by x x mi, j; if mi, j x( i, j) > threshold y( i, j) = x( i, j); otherwise (5.1) x Where m i, j represents the median value of the pixels inside the filtering window. When the above scheme is applied for impulse detection, a binary flag image { f ( i, j)} is constructed such that f ( i, j ) = 1 if the pixel x( i, j) is noisy and f ( i, j ) = 0 if the pixel x( i, j ) is noiseless. Now during filtering operation, the noisy pixels are replaced by the median of the noise-free pixels. 119

120 5.4 Recursive Weighed Median Filter The success of the median filter in the image processing is based on two intrinsic properties: edge preservation and efficient attenuation of the impulsive noise properties not shared by any traditional filters. The application of weighted median filter however has not significantly spread beyond image processing applications. When a median type filter filters a signal, some characteristic(s) will change. But impulse noise will be reduced significantly. In general changes are more profound nearer edges than homogenous region. Thus the median filter can be understood as a simple detector of impulses and edges. It is a highly data dependent filter, by which weights have been given to the samples according to the changes by the low pass filter. The recursive weighted median filter detects and remove the impulses in the images. The general structure of linear IIR filters is denoted by the Eqn. (5.2). = 1 + (5.2) where the output is formed not only from the input but also from previously computed outputs. The filtered weight consists of two sets: feedback coefficients{ and the feedforward coefficients{, N coefficients are needed to denote the recursive difference Eqn.. For WM filters the summation operation is replaced with the median operation, and the multiplication weighting is replaced by signed replication as in Eqn. (5.3). = sgn A l Y n 1 N M L=1 B k B k X n k 2 k= M1 (5.3) Recursive Weighted Median Filters-Definition Given a set of N real valued feed-back coefficients and a set of M+1 realvalued feed-forward coefficients, the M+N+1 recursive WM filter output is defined as Eqn. (5.4). Y ( n) = median( A sgn( A ) y( n N),... A sgn( A ) N N l N y( n N), B sgn( B ) X, B sgn( B ) X ( n + M )) M M n (5.4) Where < (A N,...A 1,B 0,B 1,...B M ) > the coefficients of recursive are weighted median filter. 120

121 5.4.2 Adaptive Window Size Selection A noise free sample can be estimated with better accuracy from a large number of noisy samples. In order to estimate a true pixel in a particular region from a noisy 2-D image, a large number of pixels in the neighbourhood of the noisy pixel are required. In other words a large sized window surrounding the pixel can be considered for better estimation. A smooth or flat region called as homogenous region is said to be very less complex as compared to an edge region. The region containing edges and textures are treated as highly complex region where the window size is increased for smooth region and also for images with high noise power. In a homogenous region, the correlation amongst the pixels is high. Hence a larger sized window can be taken, if the pixels to be filtered belong to a homogenous region. On the other hand a pixel that belongs to non homogenous region or a region containing edges having less number of correlated pixels in its neighbourhood. In such cases a small sized window length can be taken for denoising a pixel belonging to a non-homogenous region. However, a little bit of noise will still remain in the edges of the non-homogeneous region. Hence a variable sized window may be the right choice of efficient image denoising. Generally in the fixed small window size filters the amount of noise density filtered will be very less, for filtering high density noise the window size of the noise may increase. This may lead to blurring in the output images. In order to overcome this adaptive window length filters are designed for filtering high density noises. In the developed SWAM filter, the window size is selected based on the intensity value and the amount of corrupted pixels but not based on the threshold values. The intensity values of the pixel are used to determine whether the pixel is corrupted or uncorrupted. The window size is reduced or increased based on the amount of corrupted pixel. Due to this the unwanted filtering of uncorrupted pixels and blurring is reduced even at high density noise Window Selection The selection of window is based on the level of noise present in the noisy image. If the noise level is unknown, a robust median estimator [21] may be applied to predict the level of noise. When the noise level is low 0 20, 121

122 i) A 3 3 window is selected for filtering the noisy pixels belonging to homogenous regions; ii) The pixel is unaltered if the noisy pixels belong to edges. When the noise level is moderate 20 50, i) A 5 5 window is chosen for filtration of noisy pixels in the flat regions; ii) The window size is 3 3 if the noisy pixels to be filtered are identified as edge pixels. When the noise level is high 50 i) A 7 7 window is used for filtration of noisy pixels in the flat regions; ii) If the noisy pixels to be filtered are identified as edge pixels then the window size to be used is Median Controlled Algorithm The weight calculation for the Recursive WM filter is performed by threshold decomposition technique, optimal weights by MAE technique for real weight calculation, Complex weights calculation and negative weights calculation. The above method is complex and has high computation. Mean square error and mean absolute error are used for calculating the weights for the RWM filters in optimization techniques. The original image is needed for the calculation of MSE and MAE and also, the weights calculated by the optimization technique may be zero and negative. In case of median controlled algorithm, the selection of weights is simple and also the filter gives small weights for the impulse. For example, for each window, those input samples which are closer to the output of the first filtering operation can be exponentially weighted more. Let the difference of the sample and the result of the low pass filtering at the same position is. Weight values can be obtained by using the Eqn.(5.5), = {,, (5.5) where 0.The output of the first iteration of the median controlled filter is obtained as a weighted sum of the samples inside the moving window of the filter. This moving window need not be the same window that is used in the calculation of weights. The general weighted median filter structure with weights as =,,,. )and the Inputs =(,,,. )is given by Eqn. (5.6) 122

123 h (,,,. )= (,,,. ) (5.6) Where is the replication operator defined as =(,,,. ) times[5]. By selecting the output of the first iteration to be the reference signal, computing the new weights by comparing the new reference signal to the original signal, and computing the output again using the new weight can continue the procedure. This procedure is repeated until the number of the iterations is reached. Thus the Median controlled Recursive Weighted Median filter is obtained. By using Recursive weighted median filter with weights first Reference signal calculation can be changed. This gives more freedom for the designer. Furthermore, one can completely reject potential outliers by letting the weights be zero when the difference between the filtered signal and the original signal exceeds a certain level. The block diagram of Median Controlled Algorithm is shown in Fig.5.2. Steps involved in the Median controlled algorithm are as follows 1. Get the median filtered image using the window sliding W and store the result in reference image. 2. Calculate the weight as ( ) = α ( ) ( ) Weight i, j exp{ original i, j Reference i, j } 3. Using the above weights, perform the Recursive weighted median operation and store the output as reference image. 4. The process is done iteratively, so that output image is produced with least mean square error. This output image which is produced with least mean square error is considered as an input image for the second stage where impulse noise detection mechanism is implemented. Median Operation Weight Calculation RWM Operation Figure 5.2 Block Diagram of Median Controlled Algorithm 123

124 5.5 Impulse Noise Detection The impulse detection is usually based on the following two assumptions: 1) a noisefree image consisting of locally smoothly varying areas separated by edges and 2) a noisy pixel having very high or very low gray value compared to its neighbors. During the impulse reduction procedure two image sequences are generated. The first is the sequence of gray scale image { x, x, x,... x } where the initial image x (0) (1) (2) ( n) ( i, j) ( i, j) ( i, j ) ( i, j) (0) ( i, j) is noisy image itself, ( i, j ) is position of pixel in image, where1 i M,1 j N, M and N are number of pixels in horizontal and vertical directions respectively and x ( n) ( i, j) is image after nth iteration. The second is the binary flag image sequence { f, f, f,... f } where (0) (1) (2) ( n) ( i, j) ( i, j ) ( i, j) ( i, j) the binary flag f is used to indicate whether the pixel at (i, j) in noisy image detected as ( n) ( i, j) noisy or noise-free after n th iteration. If f = means pixel at ( i, j ) has been found as (0) ( i, j) 0 noise-free after n th iteration and if f = means pixel at ( i, j ) has been found as noisy ( n) ( i, j) 1 after n th iteration. For the selected window size the value of x is modified and given by ( n) ( i, j) Eqn. (5.7), x m if f f ( n 1) ( n) ( n 1) ( n) ( i, j) ( i, j) ( i, j) ( i, j) = ( n 1) ( n) ( n 1) x( i, j) if f( i, j) = f( i, j) (5.7) After noise detection, only binary flag image is required for noise filtering process. The elements of this image give information about whether the pixel is corrupted or not at location ( i, j ) in noisy image x (0). If ( i, j ) th image has detected as a noise then it will go ( i, j) through median filtering process otherwise it will remain the same which is called Switching based Median Filter. 5.6 Algorithm for the Switching Weighted Adaptive Median (SWAM) Filter The SWAM filtering technique has two stages. In the first stage the adaptive window size for the output image y( i, j) as defined in Eqn. (5.1) is obtained by using RWM filter. Also a weight adjustment is made to central pixel x( i, j) within the size of the sliding window and it is obtained on using median controlled algorithm. As RWM filter uses the 124

125 intensity value of the pixels the unwanted filtering of uncorrupted pixels as well as blurring is reduced even at high noise density level. In second stage for the output image which was produced by RWM filter with least mean square error is considered as input image for the detection mechanism. Algorithm: Input: Noisy image (Reference Image of first stage) Step 1 To detect the impulse noise. The detection of noisy and noise free pixels is decided by checking whether the value of a processed fixed element I ( x, y ) lies in the range I max I or not, as the impulse noise pixel can take a maximum and minimum value in min the dynamic range (0,255). If the value lies within this range then it is uncorrupted pixel and left unchanged. Otherwise, it is a noisy pixel and is replaced by the median value of the window or by its neighborhood. Step 2 On applying adaptive median filtering to the corrupted image yields a filtered image and a binary flag image { f ( i, j)} and is given by Eqn. (5.8) f ( n ) ( i, j ) ( n 1) ( n 1) ( n 1) f ( i, j ) if X ij m ij < T = 1, oth erw ise (5.8) where T is pre-defined threshold value. The impulse detection scheme detects noise even at high corruption level setting flag matrix value as 1 wherever noise exists. Step 3 Find on how many pixels are detected as noise free in current filtering window with respect to the corresponding binary flag window. Step 4 Extend window size outward by one pixel on all the four sides of the window if the number of uncorrupted pixels is less than ¼ th of the total number of pixels within filtering window. Repeat the above steps until the end of the image is reached. Step 5 The pixels that are classified as noise free in filtering window will continue in median filtering process and the other pixels which are noisy cannot continue in filtering process. This will yield a better filtering result with less blurring and distortion. Output Denoised Filtered Image. 125

126 5.7 Simulation and Results The performance of the developed algorithm is tested with different gray scale images (Lena) and with their dynamic range of values [0,255]. In the simulation, images will be corrupted by impulse noise with equal probability. The noise levels are varied from 10% to 90% with increments of 10%. The performance evaluations of the filtering operation are quantified by the PSNR and Image Enhancement Factor (IEF) given by Eqn. (5.7) and Eqn. (5.9). PSNR 2 = 10log MSE (5.9) 1 M S E = ( I ˆ ij I ij ) M N where IEF = ij ij I i j, Iˆ i j ( η I ) ij ij 2 ij ( Iˆ I ) ij 2 ij where ηij is corrupted image. 2 are the original and filtered images, MΧN size of the image. (5.10) (5.11) The restoration performance is assessed according to the noise density of the corrupted pixels in the standard test images. Both PSNR, IEF measure the difference in the intensity values of a pixel in original and enhanced images. These values are calculated for the developed algorithm and a comparison performance with various filters SMF, AMF, WMF, 2D-MF are shown in Tables 5.1 to

127 Figure 5.3 Noisy Image Figure 5.4 Input Image-Second Stage 127

128 Figure 5.5 Performances of Various Filters for Lena Image with AWGN = Figure 5.6 Performances of Various Filters for Barbara Image with AWGN = 128

129 Figure 5.7 Performances of Various Filters for Boat Image with AWGN = Figure 5.8 Performances of Various Filters for House Image with AWGN = 129

130 Figure 5.9 Performances of Various Filters for Pepper Image with AWGN = Table 5.1 Filter Performance in Terms of PSNR Operated on Lena Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF

131 Table 5.2 Filter Performance in Terms of PSNR Operated on Barbara for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF Table 5.3 Filter Performance in Terms of PSNR Operated on Boat Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF Table 5.4 Filter Performance in Terms of PSNR Operated on House Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF

132 Table 5.5 Filter Performance in Terms of PSNR Operated on Pepper Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF Table 5.6 Filter Performance in Terms of IEF Operated on Lena Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF Table 5.7 Filter Performance in Terms of IEF Operated on Barbara Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF

133 Table 5.8 Filter Performance in Terms of IEF Operated on Boat Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF Table 5.9 Filter Performance in Terms of IEF Operated on House Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF Table 5.10 Filter Performance in Terms of IEF Operated on Pepper Image for Various Noise Conditions ( varies from 5 to 80) Noise SWAM DMF AMF SMF WMF

134 Performance of variours filters in terms of PSNR for Lena image SWAM DMF AMF SMF WMF 36 PSNR Noise Percentage Figure 5.10 Performance Comparison of Various Filters in Terms of PSNR Operated on Lena for Various Noise Levels ( varies from 10 to 100) 134

135 38 36 Performance of variours filters in terms of PSNR for Barbara image SWAM DMF AMF SMF WMF 34 PSNR Noise Percentage Figure 5.11 Performance Comparison of Various Filters in Terms of PSNR Operated on Barbara for Various Noise Levels ( varies from 10 to 100) 135

136 40 38 Performance of variours filters in terms of PSNR for Boat image SWAM DMF AMF SMF WMF PSNR Noise Percentage Figure 5.12 Performance Comparison of Various Filters in Terms of PSNR Operated on Boat for Various Noise Levels ( varies from 10 to 100) 136

137 Performance of variours filters in terms of PSNR for House image SWAM DMF AMF SMF WMF 36 PSNR Noise Percentage Figure 5.13 Performance Comparison of Various Filters in Terms of PSNR Operated on House for Various Noise Levels ( varies from 10 to 100) 137

138 40 38 Performance of variours filters in terms of PSNR for Pepper image SWAM DMF AMF SMF WMF PSNR Noise Percentage Figure 5.14 Performance Comparison of Various Filters in Terms of PSNR Operated on Pepper for Various Noise Levels ( varies from 10 to 100) 138

139 60 50 Performance of variours filters in terms of IEF for Lena image SWAM DMF AMF SMF WMF 40 IEF Noise Percentage Figure 5.15 Performance Comparison of Various Filters in Terms of IEF Operated on Lena for Various Noise Levels ( varies from 10 to 100) 139

140 60 50 Performance of variours filters in terms of IEF for Barbara image SWAM DMF AMF SMF WMF 40 IEF Noise Percentage Figure 5.16 Performance Comparison of Various Filters in Terms of IEF Operated on Barbara for Various Noise Levels ( varies from 10 to 100) 140

141 Performance of variours filters in terms of IEF for Boat image SWAM DMF AMF SMF WMF 25 IEF Noise Percentage Figure 5.17 Performance Comparison of Various Filters in Terms of IEF Operated on Boat for Various Noise Levels ( varies from 10 to 100) 141

142 Performance of variours filters in terms of IEF for House image SWAM DMF AMF SMF WMF 25 IEF Noise Percentage Figure 5.18 Performance Comparison of Various Filters in Terms of IEF Operated on House for Various Noise Levels ( varies from 10 to 100) 142

143 Performance of variours filters in terms of IEF for Pepper image SWAM DMF AMF SMF WMF 10 IEF Noise Percentage Figure 5.19 Performance Comparison of Various Filters in Terms of IEF Operated on Pepper for Various Noise Levels ( varies from 10 to 100) The visual quality clearly shows that SWAM method s performance is best when compared with other filters. Tables 5.1 to 5.10 shows the comparison of PSNR,IEF for standard test images like Lena, Barbara, Boat, House, Pepper corrupted by different noise densities applied to different noise filters. Figures 5.10 to 5.19 respectively show the graphical comparison of the PSNR, IEF parameter. From visual quality, comparison tables and graphs, it is observed that SWAM produce better results compared to other existing methods. 143

144 Figures 5.5 to 5.9 show that the developed algorithm removes very high density impulse noise. An extensive experimental result shows that the developed filtering algorithm performs much better than the standard non-linear median based filters. It also works even for low and medium density noise, also preserves all fine details of the image. For lower noise density up to 30% almost all algorithms perform equally well in removing the noise completely with edge preservation. For noise density above 50% the standard algorithms such as SMF, WMF, and AMF fails to remove the noise completely. In case of high density noise, the performance of these methods is very poor in terms of noise cleaning and edge detail preservation. From the results it is observed that the 2-D median filter removes noise at high densities. But they produce streaking effect and not suitable for above 60% noise densities. Also it is observed that the developed method restores the original image much better than standard non-linear based filters. The developed filter requires less computation time when compared with other adaptive methods. 5.8 Conclusion To demonstrate the performance of the developed method extensive experiments have been conducted on a standard test image to compare our method with many other well known techniques. The developed filter is designed where the window length is determined appropriately based on the width of the impulsive noise presented in the input signal and the uncorrupted pixel is not filtered. Also the weights of the filter are calculated by using the median controlled algorithm. Due to this the results are very effective the resulting image, will have less blurring in the output signal. The results reveal that the developed SWAM filter exhibits better performance in terms of PSNR and IEF. The SWAM filter also shows consistent and stable performance across a wide range of noise densities varying from 10% to 90% densities. 144

145 CHAPTER6 APPLICATION OF IMAGE DENOISING IN MEDICAL IMAGES 6.1 Introduction Medical image processing is the use of algorithms and procedures for operations such as image enhancement, image compression, image analysis, mapping, geo referencing etc. The rapid development in medical research produces a continuous stream of new knowledge about disease processes, new therapeutic targets and the complex relationship between a person s genome and his/her related risk for disease. The visual distortion might arise due to various factors such as time of exposure, lighting, and movement of patient, sensitivity of the imaging devices affecting images in terms of contrast, distortion, artifacts being introduced, blur and contrast sensitivity. The influence and impact of digital images on modern society is tremendous, and image processing is now a critical component in science and technology. The rapid progress in computerized medical image reconstruction, and the associated developments in analysis methods and computer-aided diagnosis, has propelled medical imaging into one of the most important sub-fields in scientific imaging [8]. The arrival of digital medical imaging technologies such as positron emission tomography (PET), magnetic resonance imaging (MRI), computerized tomography (CT) and ultrasound Imaging has revolutionized modern medicine [75]. Today, many patients no longer need to go through invasive and often dangerous procedures to diagnose a wide variety of illnesses. With the widespread use of digital imaging in medicine today the quality of digital medical images becomes an important issue. To achieve the best possible diagnosis it is important that medical images to be sharp, clear, and free of noise While the technologies for acquiring digital medical images continue to improve, resulting in images of higher and higher resolution and quality, removing noise in these digital images 145

146 remains one of the major challenges in the study of medical imaging, because they could mask and blur important subtle features in the images, Many proposed de-noising techniques have their own problems. Image de-noising still remains a challenge for researchers because noise removal introduces artifacts and causes blurring of the images [117]. Medical images which are acquired using different devices are affected by a distortion metric called Noise. Noise in images, in particular, medical images have two disadvantages. They are (1) degradation of the image quality and (2) obscuring important information required for accurate diagnosis. As both these points have serious impact, all medical imaging devices need some denoising algorithm to enhance the image under consideration. MR images are typically corrupted with noise, which hinder the medical diagnosis. There has been substantial interest in the problem of denoising of images in general. Tools from traditional image processing field have been applied to denoised MR images. However, the process of noise suppression must not appreciably degrade the useful features in an image. In particular, edges are important features for MR images and thus the denoising must be balanced with edge preservation Magnetic Resonance Imaging (MRI) is a powerful diagnostic technique. However, the incorporated noise during image acquisition degrades the human interpretation, or computer-aided analysis of the images. Time averaging of image sequences aimed at improving the signal-to-noise ratio (SNR) would result in additional acquisition time and reduce the temporal resolution. Therefore, denoising should be performed to improve the image quality for more accurate diagnosis [77]. As MR magnitude images suffer from a contrast-reducing signal-dependent bias image denoising has become an essential exercise in medical imaging especially the Magnetic Resonance Imaging (MRI).Also the noise is often assumed to be white, however a widely used acquisition technique to decrease the acquisition time gives rise to correlated noise. In this chapter hybrid method for MRI restoration is developed by combining AST technique based on wavelet coefficient along with Neighbourhood Pixel Filtering Algorithm (NPFA) called Filtering and Thresholding Algorithm (FTA). Different positions of brain images with Gaussian noise have been obtained from Sri Ramachandra Medical College Hospital, Chennai. The denoising filtering algorithm developed in the MATRIX 146

147 LABORATORY (MATLAB) environment have been tested in more than 200 images of the brain in different position of ten patients. The quantitative performance of the input image obtained for different noise level has been estimated by using PSNR value and execution time. To confirm the efficiency this is further compared with Median filter, Weiner Filter, AST technique along with NPFA filter (FTA). 6.2 Image Denoising in Medical Resonance Images (MRI) Medical images such as magnetic resonance imaging (MRI) have been widely exploited for more truthful pathological changes as well as diagnosis. However it suffer from a number of shortcomings and these includes: acquisition noise from the equipment, ambient noise from the environment, the presence of background tissue, other organs and anatomical influences such as body fat, and breathing motion. Therefore, noise reduction is very important, as various types of noise generated limits the effectiveness of medical image diagnosis. The amount of the noise has the tendency of being either relatively high or low. Thus, it could harshly degrade the image quality and cause some loss of image information details. Magnetic Resonance Imaging (MRI) is a notable medical imaging technique that has proven to be particularly valuable for examination of the soft tissues in the body. MRI is an imaging technique that makes use of the phenomenon of nuclear spin resonance. Since the discovery of MRI, this technology has been used for many medical applications. Because of the resolution of MRI and the technology being essentially harmless it has emerged as the most accurate and desirable imaging technology. MRI is primarily used to demonstrate pathological or other physiological alterations of living tissues and is a commonly used form of medical imaging. Despite significant improvements in recent years, magnetic resonance (MR) images often suffer from low SNR or Contrast-to-Noise Ratio (CNR), especially in cardiac and brain imaging. This is problematic for further tasks such as segmentation of important features, three-dimensional image reconstruction, and registration. Therefore, noise reduction techniques are of great interest in MR imaging as well as in other imaging modalities [120]. 147

148 Recently, many of the popular de-noising algorithms suggested are based on wavelet thresholding [121, 76]. These approaches attempt to separate significant features/signals from noise in the frequency domain and simultaneously preserve them while removing noise. If the wavelet transform is applied on MR magnitude data directly, both the wavelet and the scaling coefficients of a noisy MR image become biased estimates of their noisefree counterparts [118, 119]. Therefore, it was suggested [76] that the application of the wavelet transform on squared MR magnitude image data (which is non central chi-square distributed) would result in the wavelet coefficients no longer being biased estimates of their noise-free counterparts. Although the bias ill remains in the scaling coefficients, it is not signal-dependent and can therefore be easily removed [76, 119]. The difficulty with wavelet or anisotropic diffusion algorithms is again the risk of over-smoothing fine details particularly in low SNR images [111]. From the above discussion, it is understood that all the algorithms have the drawback of over smoothing fine details. It is stated that oscillatory functions or oriented textures have a significantly sparser expansion in wave atoms than in other fixed standard representations like Gabor filters, wavelets and curve lets [40]. Due to the signal dependent mean of the Rician noise, one can overcome this problem by filtering the square of the noisy MR magnitude image in the transformed coefficients [76, 91]. 6.3 Development of Filtering and Thresholding Algorithm (FTA) FTA involves with a statistical model to estimate the noise variance for each coefficient based on the subband using Maximum Likelihood (ML) estimator or a Maximum a Posterior (MAP) estimator. Also this algorithm describes a new method for suppression of noise by fusing the wavelet denoising technique with optimized thresholding function. This is achieved by including a multiplying factor (α) to make the threshold value dependent on decomposition level. By finding Neighbourhood Pixel Difference (NPD) and adding NPFA along with subband thresholding, the clarity of the image is improved. The filtered value is generated by minimizing NPD and Weighted Mean Square Error (WMSE) using method of least square. A reduction in noise pixel is well observed on replacing the optimal weight namely NPFA filter solution with the noisy value of the current pixel. Due to this NPFA filter gains the effect of both high pass and low pass filter. This filter behaves like a low pass filter in smooth region by decreasing noise variance effectively and giving similar weights to all its neighboring pixels. Also it 148

149 attenuates the high pass features at the discontinuities by maintaining the sharpness of edges and gives small weight for that pixel. After computing threshold, apply soft thresholding to each noisy coefficient. By inverting the multi scale decomposition, the resultant quality image with less blurring and preserving more detail information is reconstructed hence the developed technique yields significantly superior image quality by preserving the edges, producing a better PSNR value with low execution time Filtering and Thresholding Algorithm There are two stages in this algorithm by which the denoised image of MRI can be obtained. In the first stage the image gets denoised by using AST technique without adding NPFA filter. The block diagram of Stage 1 of the algorithm is given in Fig The qualitative and quantitative measures are given in Table 6.1 and Fig.6.7 based on the following steps involved in this stage of the algorithm Analysis of Image Denoising using Wavelet Coefficient and AST Technique Noisy Image Apply DWT & Decompose Apply Threshold Denoised Image Estimate Noise Variance Algorithm: Figure 6.1 Block Diagram forfta Method (Stage 1) The complete algorithm of the proposed wavelet based denoising technique is explained in the following steps: Input: Noisy image Step 1 Perform Multiscale decomposition of the image corrupted by Gaussian noise using wavelet transform. 2 Step 2 Estimate the noise variance σ using Eqn. (4.20) for each scale and compute the V scale parameter. 149

150 Step 3 For each of the three sub bands variance estimate is computed from the noisy coefficient in subband j using Eqn. (4.21). Step 4 In each of the other high sub bands the estimates of the class coefficient variance are estimated using Eqn. (4.22) and Eqn. (4.23). Step 5 Calculate threshold value using optimum value threshold formula as given in Eqn. (4.28) after finding the multiplying factor α for each subband using the relation given in Eqn. (4.27). After computing threshold for each subband except the low pass or approximation subband, apply soft thresholding to each wavelet coefficient using threshold given in Eqn. (4.26), by substituting the threshold value obtained in Step 5. Step6 Invert the multiple decomposition to reconstruct the denoised image. Output: Denoised image In the second stage NPFA filter is added before applying soft thresholding to the Subband. By adding this filter to each wavelet coefficients in the entire subband except the LL 1 subband reduction in noise is observed on replacing the optimal weight namely NPFA filter solution with the noisy values of the current pixel. The block diagram of Stage 2 of the FTA is given in Fig The qualitative and quantitative measures are given in Table 6.2 and Fig. 6.8 based on the following steps involved in this stage of the algorithm Analysis of Image Denoising Using Wavelet Coefficients and AST Technique along with NPFA Filters Noisy Image Apply DWT & Decompose NPFA Filter Apply Threshold Denoised Image Estimate Noise Variance Figure 6.2 Block Diagram forfta Method (Stage 2) Algorithm: Input: Noisy image Step 1 Perform Multiscale decomposition of the image corrupted by Gaussian noise using wavelet transform. 150

151 2 Step 2 Estimate the noise variance σ using Eqn. (4.20) for each scale and compute the V scale parameter. Step 3For each of the three sub bands variance estimate is computed from the noisy coefficient in subband j using Eqn. (4.21). Step 4 In each of the other high sub bands the estimates of the class coefficient variance are estimated using Eqn. (4.21) and Eqn. (4.24). Step 5Add NPFA filter to each coefficient in all the three sub bands except the low pass or approximation subband. Step 6Find optimal NPFA filter solution for each coefficient using the Eqn. (3.10). Step7Calculate threshold value using optimum value threshold formula as given in Eqn.(4.27) after finding the multiplying factor α for each subband using the relation given in Eqn. (4.26). Step 8 After computing threshold for each subband except the low pass or approximation subband, apply soft thresholding to each wavelet coefficient using threshold given in Eqn. (4.26), by substituting the threshold value obtained in Step 7. Step 9 Invert the multiple decomposition to reconstruct the denoised image. Output: Denoised image 6.4 Simulation and Results The medical image noise reduction algorithm has been implemented in the MATLAB environment has been tested in more than 200 MR brain images. The algorithm was tested with Weiner Filtering, Median Filtering; AST and FTA in wavelet domain. To estimate the performance of the reduction algorithm is carried out in two stages. In the first stage without adding NPFA filter, apply soft thresholding to each wavelet coefficient except in low or approximation band and the algorithm is implemented. In the second stage of the algorithm before applying soft thresholding NPFA filter is added to each coefficient in three bands except the low or approximation band and the steps in the algorithm have been implemented. To estimate the filter performance the quantitative measure such as PSNR and the execution time of the different filtering algorithms on two different MR images of brain have been used, which is calculated by using the Eqn. (4.29) for the first stage of algorithm and by Eqn. (3.8) for the second stage of algorithm. 151

152 The performance metrics calculated from the denoised image after implementing FTA for two different MR Brain images namely Image 1 and Image 2 is given in Table 6.5 and Table 6.6 respectively. Also graphical representation based on the performance of various filters with the corresponding PSNR value is given in Fig.6.8. Figure 6.3 Two Level Decomposition of Brain Image 1 Figure 6.4 Performances of Various Filters for Brain Image 1 with AWGN = 152

153 Figure 6.5 Two Level Decomposition of Brain Image 2 Figure 6.6 Performances of Various Filters for Brain Image 2 with AWGN = 153

154 Table 6.1 Filter Performance In Terms Of PSNR Operated on Brain Image 1 for Various Noise Conditions ( varies from 5 to 80) Noise Level Noisy Image Weiner Filter Median Subband Adaptive FTA

155 Table 6.2 Filter Performance In Terms Of PSNR Operated on Brain Image 2 for Various Noise Conditions ( varies from 5 to 80) Noise Level Noisy Image Weiner Filter Median Subband Adaptive FTA

156 Noise Image Wiener Filter FTA Median Subband Adaptive PSNR Total Noise Standard Deviation Figure 6.7 Performance Comparison of Various Filters in Terms of PSNR Operated on Brain Image 1 for Various Noise Levels ( varies from 5 to 80) 156

157 Noise Image Wiener Filter FTA Median Subband Adaptive PSNR Total Noise Standard Deviation Figure 6.8 Performance Comparison of Various Filters in Terms of PSNR Operated on Brain Image 2 for Various Noise Levels ( varies from 5 to 80) 157

158 Table 6.3 Execution time (milliseconds) taken by various filters for Brain Image1 and Image 2 Method System Brain Image 1 Brain Image 2 Number = 5 = 30 = 70 = 5 = 30 = 70 Median System Filter System Weiner System Filter System Subband Adaptive FTA System System System System From Fig. 6.4 and Fig. 6.6 and from Table 6.1 and Table 6.2 it is observed that PSNR value obtained for FTA is more. Qualitatively also it is observed that the images are having more clarity without loss of much detailed information. This is due to the addition of NPFA filter which gains the effect of low pass filter and high pass filter which in turn cuts off only high frequency noise signal instead of all noise signals. Also the decomposition level dependent is included as a multiplying factor in the optimum value threshold formula along with sub band variance estimation. From Fig. 6.7 and Fig. 6.8 it is observed that Weiner filter performs little denoising in high activity sub regions to preserve the sharpness of the edges but completely denoise the flat subparts of the image. Median filter preserve edges better than noise removal method using wiener filter. AST yields better results for denoising and also adopt thresholding strategy by preserving edges better than Wiener and median filter. The FTA gives better performance than other spatial domain filter like Wiener, Median, and AST Wavelet in terms PSNR value for Image 1 in all noise levels, where as for Image 2 in moderate and high noise level. In low noise level for Image 2 the filter AST technique gives better performance in terms of PSNR value. Further FTA out performs the 158

159 performance of the above mentioned filtering algorithm by preserving the edges as well as removing the noise, due to the advantages of using the multiplying factor included in the optimum value threshold formula and subband thresholding, and addition of NPFA filter. Execution time is another important image metric to compare the performance of filter. These wavelet domain filters are simulated using MATLAB R2007b platform on two different systems having different operating systems, one has(system 1) 64-bit operating system Windows7 Home Basic having Intel(R) Core(TM) i and having RAM of 4GB. Another system (system 2) 64-bit operating system Microsoft Windows XP Professional having Intel(R) Core(TM)2 CPU and having RAM of 504MB.From the Table 6.3 it is observed that at low level of noise all the filters except Median filter take same execution time. Hence in this level Weiner Filter, AST and FTA are better techniques for Image 1 and Image 2 in terms of execution time. At moderate and high noise level for Image 1the execution time of FTA is more compared to other filters but the PSNR performance is far better to other filters and for Image 2 at moderate noise level the execution time is same for all filters, where as in high noise level the execution time for FTA s slightly greater but its PSNR performance is better. Hence FTA out performs the other compared Linear and Non-Linear filters both in terms of PSNR and Execution Time. 6.5 Conclusion The FTA is based on the analysis of statistical parameters like standard deviation, variance of the sub band coefficients using ML or MAP estimator which is more sub bands adaptive. Since the decomposition level dependent is included as a multiplying factor α in the optimum value threshold formula along with sub band variance estimation, the proposed technique yields significantly superior image quality by preserving edges and a better PSNR value. After implementing and using the results of the FTA a significant improvement in clarity and sharpness of the image is observed. Comparative studies have been made between Median filter Wiener filter, AST, FTA. The outcome of the study reveals that FTA filter out performs all the other filters in terms of PSNR, Execution Time and Visual Quality. 159

160 CHAPTER 7 CONCLUSION AND FUTURE WORK In this thesis various wavelet thresholding and filtering algorithm for suppression of AWGN and Impulse noise are studied and their performances are analyzed. Considering the limitations of the existing algorithms, efforts have been made to develop wavelet based filtering and thresholding algorithm namely a) NPFA b) AST Technique for suppression of AWGN. Also a new filter SWAM is implemented for effective suppression of Impulse noise. The implementation of NPFA and AST technique are made effectively in the application of image denoising for medical images (MR Brain Image). The performance of these implemented algorithms and filter are compared with existing wavelet based domain filters and algorithms. The objective evaluation metrics: PSNR and TE are considered for comparing the filtering and thresholding performance. 7.1 Comparative Analysis Removal of AWGN The implemented filtering and thresholding algorithms are simulated on tested images: Barbara, Pepper, House, Boat and Lena of sizes 512x512, 256x256 each corrupted with different level AWGN. The noise level of AWGN is categorized as Low ( 20), Moderate (20 50), High (50 80). The implemented algorithms are compared with the existing wavelet domain methods: trous algorithm, Visu Shrink, Normal Shrink, Baye s Shrink, Modified Baye s Shrink, Neigh Shrink and Oracle Shrink to check the filtering performance of the implemented as well as existing algorithms, the performance measures: PSNR and TE are evaluated for all cases. To give a concise presentation of all simulation results so as to have a precise comparative study, the performance measures of the implemented and existing methods are shown in Table

161 only for 3 case noise levels =18,38,78 to give an insight of Low, Moderate and High noise level conditions respectively. Here Lena image is taken as test image. Table 7.1 Filtering Performances of Various Methods In Terms of PSNR (db) and TE (ms) Operated On Lena Image under Various Noise Conditions Denoising Methods Low Level =18 Moderate Level =38 High Level =78 PSNR TE PSNR TE PSNR TE Trous Algorithm Visu Shrink Normal Existing Shrink Methods Baye s Shrink Modified Baye s Shrink Neigh Shrink Oracle Shrink Implemented NPFA Methods AST After implementing NPFA it is observed that the above method possess high PSNR value also as noise level increases the PSNR value decreases and TE is minimum at Low, moderate and high noise levels. As NPFA possess high PSNR and minimum TE this method out performs the other wavelet method under moderate and high noise level and the clarity of the resultant image is improved as shown in Figures 3.4 to 3.8. AST yields significantly superior image quality by preserving edges as well as removing noise when compared to other wavelet domain methods shown in Figures 4.8 to From the simulation results it is clear that at moderate and high noise level, this technique possess a better PSNR value and minimum TE. Hence under moderate and high noise level AST technique outperforms the other methods both qualitatively and quantitatively. Also it is observed that single level of decomposition is sufficient for the 161

162 images corrupted with less noise densities. While for images corrupted with high noise density second level of decomposition is required irrespective of the image Statistical Analysis Statistical analysis is also made to show that the implemented methods give significant results when compared with the existing methods using Student s t-test. Consider the PSNR value obtained for Barbara Image of different noise level using our implemented technique and Modified Baye s Shrink method as two samples X 1 and X 2 of sizes n 1 and 2 n n1 n2 ( = = 10) respectively. The significant differences between these two samples were tested using Student s t-test: Two samples assuming equal variance. By using the test statistic t = s x x n n 1 2 (~ ( + 2)) d.f (degrees of freedom) for the above sample it is seen that the calculated value of t is greater than the critical value of t at 5% level of significance for 18 d.f. Due to this the null hypothesis H 0 : µ 1 = µ 2 is rejected and : > (1 tailed test) is accepted. By this significant test the PSNR value obtained for Barbara image of different noise level by our implemented methods is significant than by using Modified Baye s Shrink method. A similar test is also done by considering the other sample as trous algorithm, Visu Shrink, Normal Shrink, Baye s Shrink, Neigh Shrink and Oracle Shrink. In these cases also it is observed that the implemented method yields a significant value. Hence NPFA and AST technique outperform the other wavelet method under Moderate and High noise level by possessing high PSNR value and minimum execution time. These methods finds application in denoising images those are corrupted during transmission which is normally random in nature Removal of Impulse Noise The implemented filtering algorithm is simulated on tested images: Barbara, Pepper, House, Boat and Lena of sizes 512x512, 256x256 each corrupted with different level Impulse noise. This algorithm is compared with the existing filtering algorithm: 2DMF, WMF AWMF, SMF to check the filtering performance of the implemented as well as existing algorithms, the performance measures: PSNR and IEF are evaluated for different 162

163 noise levels. To give a concise presentation of all simulation results so as to have a precise comparative study, the performance measures of the implemented and existing methods are shown in Table 7.2. Here Lena image is taken as test image. Table 7.2 Filtering Performances of Various Methods In Terms of PSNR (db) and TE (ms) Operated On Lena Image under Various Noise Conditions Performance Noise Level in % Denoising Methods Metrics Existing Methods Implemented Method 2DMF WMF AMF SMF SWAM PSNR IEF PSNR IEF PSNR IEF PSNR IEF PSNR IEF The standard non-linear based median filters, RWM filters are designed only for the fixed window length which causes blurring in the output samples. In the fixed window length, the filtering operation is done for the original samples which cause the blurring in the output. The selected window size ensures the high correlation between the pixels; this provides more edge details, leading to better edge preservation. In the implemented SWAM filter the window length is determined based on the intensity value of the pixel and the amount of corrupted pixels due to this there is no chance of filtering the uncorrupted pixel which reduces the blurring in the output sample by using the region of interest the window size is minimized. The implemented filter employs the switching scheme in two stages based on impulse noise deduction. In the first stage, the size of the window is selected by RWM and the output image produced by this filter is considered as the input image for the second stage. Hence in the second stage partially denoised images with least mean square error are 163

164 considered for impulse noise deduction. Also the weights of the pixel are calculated by the median control algorithm. Due to this the results are very effective by preserving fine details and edges, and the resulting image will have less blurring. The proposed filter also shows effective filtering performance across wide range of noise density varying from 10% to 90%. 7.2 Contribution of This Work In this work two filtering algorithms and one thresholding algorithm are implemented for the removal of AWGN and impulse noise. Salient points of the thesis, highlighting the contribution at each stage are presented below. Neighbourhood Pixel Filtering Algorithm (NPFA) is implemented for the removal of AWGN. The clarity of the resulting image is weaker in the existing trous algorithm because the efficiency of the wavelet images is low and the detailed preservations of images at different scales are not uniform. Also random noise rapidly attenuates with increase in scales. This problem is overcome by NPFA which is implemented, based on the concept that the low pass filter preserves the energy of the signal and attenuates high pass features at discontinuities. By this concept the NPF used in algorithm gains the effect of both LPF and HPF which in turn cuts off only high frequency signal instead of all noisy signals. Due to this the clarity of the resultant image is improved. Adaptive Subband Thresholding Algorithm (AST) is implemented for removal of AWGN. There are some methods in which the denoised coefficients are evaluated by MMSE estimator, ML estimator in terms of the noise coefficients. These methods produce effective results, but their spatial adaptively is not well suited near object edges, where variance field is not smoothly varied. Also these methods introduce artifacts in smooth regions of the output image. To overcome this problem AST technique is implemented based on the wavelet coefficients. This technique describes a new approach for suppression of AWGN by fusing the wavelet denoising technique with optimized thresholding function to which a multiplying factor is included to make the threshold value dependent on decomposition level. 164

165 Switching Weighted Adaptive Median Filtering Algorithm (SWAM) is developed for suppression of impulse noise. The existing switching median filter filters only identified corrupted pixels by leaving the uncorrupted pixels. Even though these types of filters show significant improvement in their performance, they exhibit blurring for large window size and insufficient noise suppression for small window size. To overcome this problem SWAM filter is derived in which the selection of appropriate window length is made initially based on the intensity value of the pixel, the amount of corrupted pixels. Due to this selection the unwanted filtering of uncorrupted pixels are reduced which in turn reduces the blurring even at high density noise. Hence this filter shows consistent and stable performance across a wide range of noise densities varying from 10% to 90% densities. Filtering and Thresholding Algorithm (FTA), an hybrid method is implemented for MRI restoration by combining AST and NPFA. This algorithm is implemented on MRIbrain images for removal of AWGN The FTA is based on the analysis of statistical parameters like standard deviation, variance of the sub band coefficients using ML or MAP estimator which is more sub bands adaptive. After implementing and using the results of the FTA a significant improvement in clarity and sharpness of the image is observed. 7.3 Scope for Future Work The field of image processing has been growing at a very fast pace. The day to day emerging technology requires more and more revolution and evolution in the image processing field. The well known saying A picture says a thousand words can be taken as the main motive behind the need of image processing. The work proposed in this thesis also portrays a small contribution in this regard. The proposed denoising technique can provide a good platform for further research work in this respect. This work can be further enhanced to denoise the other type of images, as well, like RGB, Indexed and Binary images. It will provide a good add on to the already existing denoising techniques used for denoising these images. Moreover, for future work the implemented algorithms can be trained using various AI techniques like fuzzy logic or neural network, in order to attain the best output without performing calculations for each and every combination. Use of AI techniques will lead to the optimal solution directly, with more efficiency and less tedious 165

166 work. The widow size of different filters can be made adaptive for efficient denoising. The shape of the window can also be varied and made adaptive to develop very effective filters. Future work may be done by considering the technique of incorporating neighbour wavelet coefficients in the thresholding process for multiwavelet image denoising. It is further suggested that the proposed algorithm may be extended to various metrics for describing and quantifying the qualities of an image as well those of a filtering process the color images and video framework which may further improve the video denoising. 166

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177 PRESENTATIONS IN NATIONAL CONFERENCES [1] S. Kalavathy, R.M. Suresh, Image Denoising Techniques In Wavelet Domain, National Conference on Applied Mathematics (NACM 2010), January 28-29, B.S Abdur Rahman University Chennai-48, ISBN: [2] S. Kalavathy, R. M. Suresh, Image Denoising Based On Wavelet Coefficient And Adaptive Subband Technique, 3 rd National Conference on Mathematical Modelling in Global Perspectives April 21, 2011, Velammal Engineering College, Surapet, Chennai-66, ISBN: PRESENTATIONS IN INTERNATIONAL CONFERENCES [1] S. Kalavathy, R. M. Suresh, Image Denoising Improved Semantic Approximation Algorithm In Wavelet Domain Markov Model, 3 rd IEEE International Conference on Signal and Image Processing (ICSIP 2010), January 15-17, 2010, R.M.D Engineering College, Kavarapettai IEEE Catalog Number: CFP1015L-DVD. ISBN: INTERNATIONAL JOURNAL PUBLICATIONS [1] S. Kalavathy, R. M. Suresh, Analysis Of Image Denoising Using Wavelet Coefficient And Adaptive Subband Thresholding Technique, IJCSI International Journal Of Computer Science Issues, Vol. 8, Issue 4, pp , July Impact Factor:0.242 [2] S. Kalavathy, R. M. Suresh, A Switching Weighted Adaptive Median Filter for Impulse Noise Removal, International Journal Of Computer Applications ( ), Vol. 28, No.9, pp.8-13, August Impact Factor:0.835 [3] S. Kalavathy, R. M. Suresh, An Efficient Neighbourhood Pixel Filtering Algorithm for Wavelet Based Image Denoising, International Journal Of Computer Application, Vol. 34, Issue 2, April Impact Factor:1.76 [4] S. Kalavathy, R. M. Suresh, Application of Subband Adaptive Thresholding Technique with Neighbourhood Pixel Filtering for Denoising 177

178 MRI Images, International Journal on Engineering Science and Technology, Vol. 4, No.02, February Mrs.K.Kalavathy M.Sc.,M.Phil.,(Ph.D.) Professor Mrs.K.Kalavathy is the Professor in Mathematics of the Science and Humanities Department in R.M.D.Engineering College. She Passed B.Sc. Mathematics from Madras University in She passed M.Sc. in Applied Mathematics from Anna University Guindy in the year Also she had received her M.Phil, degree in Mathematics from Bharathidasan University in the year At present she is persuing her Ph.D. degree from Dr. M.G.R Educational and Research Institute- University, Maduravoyal, Chennai in the field of Image processing. She is in teaching profession for the past 25 years. She handled almost all engineering mathematics subjects for various discipline in UG and PG courses MBA/MCA. She is a life member of ISTE. She published 4papers in International Journals and presented papers in refereed national & International Conferences. She is an author of 4 books namely Operations Research with C language, Numerical Methods, Laplace transform and Differential equations, Engineering Mathematics I. 178