IMAGE DE-NOISING IN WAVELET DOMAIN Aaditya Verma a, Shrey Agarwal a a Department of Civil Engineering, Indian Institute of Technology, Kanpur, India - (aaditya, ashrey)@iitk.ac.in KEY WORDS: Wavelets, Denoising, SAR images ABSTRACT: Wavelets are a widely used tool in many applications of signal and image processing. In image processing, wavelets make it possible to recover better quality images from raw images containing noise. This paper explores image denoising using wavelets with the help of Wavelet Toolbox in MatLab. To understand the procedure, TIFF images from Resourcesat-1 satellite were introduced with a set of known noises and then denoised. The same procedure was repeated to denoise SAR images with unknown noise using haar wavelet. 1. INTRODUCTION Satellite images may be affected by various kinds of noise during capturing and transmission. Noise types vary from Gaussian noise in natural images to multiplicative speckle noise in synthetic aperture radar (SAR) images. So, image denoising is an important step in image processing and remote sensing applications[1]. Image denoising is an important image processing task, both as a process itself, and as a component in other processes. There are many ways to denoise an image or a data set. The main properties of a good image denoising model are that it will remove noise while preserving edges. Traditionally, linear models have been used. One common approach is to use a Gaussian filter. An advantage of linear noise removal models is the speed. But a drawback of the linear models is that they are not able to preserve edges in a good manner. Edges, which are recognized as discontinuities in the image, are smeared out. Wavelet denoising attempts to remove the noise present in the image while preserving its characteristics. It is different from smoothing as smoothing only removes high frequency while retaining the lower ones[2]. The major advantage in wavelet transform is the thresholding of small coefficients without affecting significant features of the image as the small coefficients are more likely due to noise and large coefficients due to important signal features. Thresholding is a simple technique, which operates on one wavelet coefficient at a time. In its most basic form, each coefficient is thresholded by comparing against a threshold, if the coefficient is smaller than threshold, it is set to zero; otherwise it is kept or modified. Replacing the small noisy coefficients by zero and inverse wavelet transform on the result may lead to reconstruction with the essential signal characteristics and with less noise. In this paper, image denoising of SAR images is done using Mat- Lab Wavelet Toolbox and comparison of denoising methods is done on the basis of wavelet and thresholding type. The organization of this paper is as follows. A brief review of Corresponding author. Tel. No. - +91-8765696423. e-mail: ashrey@iitk.ac.in various types of noises is done in Section 2 followed by theory on wavelets and signal to noise ratio in Section 3 and Section 4 respectively. The methodology used for denoising is explained in Section 5. In Section 6 results of the above denoising methods are compared. Section 7 discusses about the results and conclusions drawn from them. 2. IMAGE NOISES Image noise is random variation of brightness or color information in images and is usually an aspect of electronic noise. Digital images are prone to a variety of noises. Noise is the result of errors in the image acquisition process that result in pixel values that do not reflect the true intensities of the real scene. The principle sources of noise in digital images arise during image acquisition and/or transmission. The performance of imaging sensors are affected by a variety of factors during acquisition, such as: Environmental conditions during the acquisition Light levels (low light conditions require high gain amplification) Sensor temperature (higher temperature implies more amplification noise) Images can also be corrupted during transmission due to interference in the channel because of phenomenon like lightening or atmospheric disturbances. 2.1 Gaussian Noise Gaussian noise is statistical noise having a probability density function (PDF) equal to that of the normal distribution, which is also known as the Gaussian distribution. In other words, the values that the noise can take on are Gaussian-distributed. The probability density function p of a Gaussian random variable z is given by: p G(z) = 1 σ 2π e (z µ) 2 2σ 2 (1) where, z represents the grey level, µ the mean value and σ the standard deviation. Principal sources of Gaussian noise in digital images arise during acquisition e.g. sensor noise caused by poor illumination and/or high temperature, and/or transmission e.g. electronic circuit noise.
caused by statistical quantum fluctuations, i.e., variation in the number of photons sensed at a given exposure level. It follows Poisson distribution. The number of photons N measured by a given sensor element over a a time interval t is described by the following discrete probability distribution: p p(n = k) = e λt k! where, λ is the expected number of photons per unit time interval. (5) Figure 1: Gaussian probability density function p G(z) 2.2 Salt-and-Pepper Noise Salt-and-pepper noise is a form of noise that presents itself as sparsely occurring white and black pixels. An image containing salt-and-pepper noise will have dark pixels in bright regions and bright pixels in dark regions. This type of noise can be caused by analog-to-digital converter errors, bit errors in transmission, etc. A simple model is the following: Let f(x, y) be the original image and q(x, y) be the image after it has been altered: p r(q = f) = 1 α (2) p r(q = MAX) = α/2 (3) p r(q = MIN) = α/2 (4) where MAX and MIN are the maximum and minimum image values respectively. For 8 bit images, MIN = 0 and MAX = 255. The idea is that with probability 1 α the pixels are unaltered; with probability α the pixels are changed to largest to smallest values. The altered pixels look like black and white dots sprinkled over the image. Impulse noise takes place in situations where high transients (faulty switching) occurs. The following traditional assumptions about the relationship between signal and noise do not hold for shot noise: Shot noise is not independent of the signal Shot noise is not additive. 2.4 Speckle Noise Speckle is a granular noise that inherently exists in and degrades the quality of the active radar and synthetic aperture radar (SAR) images. It is also known as texture. Generalized model of the speckle is represented as, g(n, m) = f(n, m) u(n, m) + ξ(n, m) (6) where, g(n, m) is the observed image, u(n, m) is the multiplicative component and ξ(n, m) is the additive component of the speckle noise. Speckle noise in SAR is generally serious, causing difficulties for image interpretation. It is caused by coherent processing of backscattered signals from multiple distributed targets. In SAR images, speckle noise is generated in the form of a random pixelto-pixel variation with statistical properties similar to those of thermal noise. Due to its granular appearance in an image, speckle noise makes it very difficult to visually and automatically interpret SAR data[4]. 3. WAVELET THEORY A wavelet is a wave-like oscillation with an amplitude that begins at zero, increases, and then decreases back to zero. As a mathematical tool, wavelets can be used to extract information from different kinds of data. A wavelet is a mathematical function useful in digital signal processing and image compression. A wavelet must satisfy the following two conditions: It must be oscillatory. 2.3 Poisson Noise Figure 2: Example of Salt and Pepper Noise Poisson noise, also known as shot noise, is a basic form of uncertainty associated with the measurement of light. It is typically Its amplitudes are non-zero only during a short interval. Wavelet transform is often compared with Fourier transform, in which signals are represented as a sum of sinusoids. Fourier transform can be viewed as a special case of continuous wavelet transform. The main difference in general is that wavelets are localized in both time and frequency whereas the standard Fourier transform is only localized in frequency.
3.1 Wavelet Denoising The basis of denoising using wavelet analysis is to reduce the noise in the wavelet transform domain. Suppose we have a length- N noisy observation w = [w 1, w 2,...w N ]: w = f + ɛ, (7) where, f = [f 1, f 2,...f N ] is the desired noise-free signal and ɛ = [ɛ 1, ɛ 2,...ɛ N ] is the additive noise. Because a discrete wavelet transform is a linear operator, it yields an additive noise model in the transform domain: y = DW T (w) = DW T (f) + DW T (ɛ) = x + n, (8) where, DW T stands for discrete wavelet transform and IDW T will be used ahead to denote the inverse transform. The main scheme for recovering f from w using wavelet denoising techniques is summarized in the following three steps: 1. Perform a 2-D Wavelet transform: y = DW T (w) 2. Manipulate the wavelet coefficients: ˆx = f(y, σ n) 3. Perform a 2-D wavelet inverse transform: f = IDW T (ˆx) 3.2 Thresholding Technique Thresholding involves a binary decision. Denoising based on thresholding in the wavelet domain was proposed by Donoho and Johnstone[3]. Two thresholding techniques are soft and hard thresholding. For each wavelet coefficient, if its amplitude is smaller than a predefined threshold, it will be set to zero, otherwise it will be kept unchanged (hard thresholding), or shrunk in the absolute value by an amount of the threshold (soft thresholding). { x, if x > T Hard thresholding : y = (9) 0, if x < T Soft thresholding : y = sign(x)( x T ) (10) where x is the input signal, y is the output signal and T is the threshold. 4. SIGNAL-TO-NOISE RATIO Signal-to-noise ratio (SNR) is a physical measure of the sensitivity of an imaging system. It is defined as the ratio of the noise-free signal f to the mean-squared error (MSE) between the noise-free image and the denoised image ˆf. var(f) SNR = 10log 10 1 N ΣN i=1 (fi ˆf (11) i) 2 5. METHODOLOGY 5.1 Denoising GeoTiff Images: In order to denoise the given SAR images, the procedure of denoising was first applied on a color composite derived from given GeoTiff images from Resourcesat-1 satellite which uses LISS-III sensor. The given GeoTiff band images are noise-free. So, noises mentioned in the previous section are first added to the color composite to generate a noised image. The Matlab code to denoise the images (as explained in next section) is used to generate denoised image. The noised image is denoised using both hard and soft thresholding techniques. The final image is used to calculate the Signal-to-Noise Ratio (SNR) and analyse the success of the denoising technique applied. Higher the value of SNR, better is the quality of the denoised image. 5.2 Denoising SAR Images The above denoising procedure is applied to the SAR images using MatLab and the denoised images are generated for the same. Finally, the SNR is calculated and the success of the denoising techniques applied is compared. 5.3 Algorithm Used The algorithm used for denoising the images includes two Mat- Lab functions. The first function returns the threshold value to be used for the image denoising. The second function returns the denoised matrix of the input image matrix obtained by wavelet coefficients thresholding using the above positive threshold. Wavelet decomposition is performed at the specified level and according to the specified wavelet type. In this paper, only haar wavelet has been used for image denoising. 6. RESULTS Initially, the following four kinds of noises are added to color composite generated from given GeoTiff images: Figure 3: Threshold types: (a)original Signal (b)hard (c)soft The major step in thresholding is the selection of an approriate threshold. If the threshold value is too small, the recovered image will remain noisy. Otherwise, if the value is too strong, important image details will be smoothed out. Donoho proposed universal threshold T = 2log(N)σ n where N is the sample size and σ n is the noise standard deviation. Gaussian noise Salt & Pepper noise Poisson noise Speckle noise Then, the generated noised images are denoised using hard and soft thresholding and haar wavelet. The results including noised and denoised images and Signal-to- Noise ratios are shown in Table 1. Soft and hard thresholding is applied on the SAR images using haar wavelet and results obtained are shown in Table 2 and Table 3.
Noise Type Thresholding Type Noised Image Denoised Image Signal-to-Noise Ratio Hard 5.0201 Gaussian Soft 5.0005
Noise Type Thresholding Type Noised Image Denoised Image Signal-to-Noise Ratio Hard 22.6835 Salt & Pepper Soft 14.6010
Noise Type Thresholding Type Noised Image Denoised Image Signal-to-Noise Ratio Hard 10.7225 Poisson Soft 10.3833
Noise Type Thresholding Type Noised Image Denoised Image Signal-to-Noise Ratio Hard 5.4752 Speckle Soft 5.3927 Table 1: Results of GeoTiff denoising
Thresholding Type Noised Image Denoised Image Signal-to-Noise Ratio Hard 4.9999 Soft 4.6885 Table 2: Results of denoising for SAR image 1
Thresholding Type Noised Image Denoised Image Signal-to-Noise Ratio Hard 3.4221 Soft 3.3837 Table : Results of denoising for SAR image 2
7. CONCLUSIONS In this paper, simple hard and soft thresholding techniques using haar wavelet have been used to recover two SAR images from noise contamination effectively. The whole procedure from determining the threshold to producing denoised images has been implemented in MatLab. The same procedure can be repeated with waveletes other than haar wavelet. In addition to denoising of the images, a comparison between hard and soft thresholding techniques has been made. The effectiveness of these techniques has been compared by comparing the signal-to-noise ratios of the denoised images by these techniques. It is observed that hard thresholding produces better results in terms of signal-to-noise ratio in all the cases. 8. REFERENCES [1] Nasri M. and Nezamabadi-pour H., 2008. Image denoising in the wavelet domain using a new adaptive thresholding function Neurocomputing, 72(2009), pp. 1012-1025. [2] Rangarajan, R., et. al., 2002. Image Denoising Using Wavelets. pp. 3-5 [3] Donoho D.L., and Johnstone I. M., 1994. Ideal spatial adaptation via wavelet shrinkage Biometrika, vol. 81, pp. 425-455 [4] Chen, C. H., 2003. Frontiers of Remote Sensing Information Processing. World Scientific, Singapore, pp. 168-170.