Saif Basheer Al-Khoja B.Sc Supervised by Prof. Dr. Saleh M. Ali. December 2004

Transcription

1 Republic of Iraq Ministry of Higher Education and Scientific Research University of Baghdad College of Science Computer Science Department A DISSERTATION SUBMITED TO THE COLLEGE OF SCIENCE UNIVERSITY OF BAGHDAD IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER IN COMPUTER SCIENCE By Saif Basheer Al-Khoja B.Sc Supervised by Prof. Dr. Saleh M. Ali December 2004

2 Abstract Abstract In this project the arithmetic coding technique is adopted and used to compress gray tone and colored image. The coding technique is adapted by combining the arithmetic algorithm with the wavelet Tap (3/5) transform. The hybrid technique yields an efficient results; i.e. good quality (high PSNR) with (high Compression Ratio). The presented coding technique converts the lossless arithmetic coding into lossy method to achieve high compression (i.e. without regarding Shannon restriction criterion).

3 Contents Contents Chapter One General Introduction 1.1 IMAGE PROCESSING IMAGE DATA TYPE Binary Images Gray-Scale Images Color Images Multi-Spectral Images IMAGE DATA COMPRESSION METHODOLOGIES METHODOLOGIES OF COMPRESSION TECHNIQUES Lossless Techniques Lossy Compression AIM OF PROJECT PROJECT LAYOUT Chapter Two: The Wavelet Transform 2.1 General Introduction Wavelet Transform Development Stages Wavelet Analysis Continuous Wavelet Transform (CWT) Discrete Wavelet Transform (DWT) Discrete Wavelet Transform and Image Compression Chapter Three: Hybrid Coding Compression System 3.1 Introduction WAVELET TRANSFORM TAP (3/5) FORWARD WAVELET TRANSFORM TAP (3/5) Inverse Wavelet Transform Tap (3/5) ARITHMETIC CODING Arithmetic Encoding Arithmetic De-Coding... 42

4 Contents 3.4 QUANTIZATION DE-QUANTIZATION Chapter Four: Implementation and Experimental Results 4.1 INTRODUCTION PERFORMANCE MEASURES The Compression Factor (CF) The Mean Square Error (MSE) Peak Signal to Noise ratio (PSNR) IMPLEMENTATION Experimental Results Chapter Five: Conclusion and Suggestions for Feature Work 5.1 COMPRESSION USING ARITHMETIC CODING COMPRESSION USING WAVELET TRANSFORM TAP (3/5) COMPRESSION USING HYBRID CODING SYSTEM Suggestions for Feature Work References

5 Chapter One General Introduction General Introduction 1.1 Image Processing Image processing is computer imaging where the application involves a human being in the visual loop. In other word, the images are to be examined and acted upon by people, while computer vision does not involve him. The major topics within the filed of image processing include image restoration, image enhancement, and image compression. Image analysis is often used as preliminary work in the development of image processing algorithms, but the primary distinction between computer vision and image processing is that the output image is to be used by a human being [1]. Image restoration is the process of taking an image with some known, or estimated, degradation, and restoring it to its original appearance. Image restoration is often used in the filed of photography or publishing where an image was somehow degraded but needs to be improved before it can be printed. Image enhancement involves taking an image and improving it visually, typically by taking advantage of the human visual systems response. One of the simplest and often most dramatic enhancement techniques are to simply stretch the contrast of an image. Image compression involves reducing the typically massive amount of data needed to represent an image. This is done by eliminating 1

6 Chapter One General Introduction data that are visually unnecessary and by taking advantage of the redundancy that is inherent in most images. 1.2 Image Representation The most general definition of a digital image is a lattice of sampled measurements of phenomena. The common true color image could be described as a matrix of red, green, and blue reflectance s measured on the projection of the visible electromagnetic radiation from a three-dimensional scene onto the two-dimensional backplane of a camera and sampled in a planar lattice with uniform spatial sampling in two orthogonal directions. Many of the notions of imagery come from present day capabilities to generate image data. The human visual system receives input images as a collection of spatially distributed light energy; this form is called an optical image. Optical images are the types we deal with everyday. We know that these optical images are represented as video information in the form of analog electrical signals and have seen how these are sampled to generate the digital image [1]. 1.3 Image Data Type The digital image f(x,y) is represented as two-dimensional array of data, where each pixel value corresponds to the brightness of the image at the point (x,y). The simplest type of image is the monochrome (one color, this is what we normally refer to as black and white) image data, other types of 2

7 Chapter One General Introduction image data required extension or modification to this model, typically there are multi band images (color, multispectral), and they can be modeled by using different functions f(x,y) each corresponding to single separate band of brightness information. The image may be classified into the following types [2]: Binary Images Binary images are the simplest type of images and can take on two values, typically black and white, or '0' and '1'. A binary images is referred to as a 1-bit/pixel image because it takes only 1 binary digit to represent each pixel. These types of images are most frequently used in computer vision applications where the only information required for the task is general shape, or outline, information. For examples; to position robotic gripper to grasp an object, to check a manufactured object for deformation, for facsimile (FAX) image, or in Optical Character Recognition (OCR) Gray-Scale Images Gray scale images are referred to s monochrome, or one- color, images. They contain rightness information only, no color information. The number of bits used for each pixel determines the number of different brightness levels available. The typical image contains 8 bit/pixel data, which allows us to have 256(0-255) different brightness (gray) levels. 3

8 Chapter One General Introduction Color Images Color images can be modeled as three band monochrome image data, where each band of data corresponds to different color. The actual information stored in the digital image is the brightness information in each spectral band. When the image is displayed, the corresponding brightness information is displayed on the screen by picture elements that emit light energy corresponding to that particular color. Typical color images are represented as red, green, and blue or RGB images. Using the 8-bits monochrome standard as a model, the corresponding color image would have 24-bits/pixel-8-bits for each of the three-color bands (red, green, and blue) Multi-Spectral Images To extract earth information, a technique of remote sensing is used, it means the observation of, or gathering information about a target by a sensor separated from it by some distance. This information is represented as Multispectral images (i.e. observed by different wavelengths); they are not image in usual sense because some of the information is not directly visible by the human system. 1.4 Image Data Compression Methodologies Image data compression is the process of reducing the number of bits required to represent a digital image data while maintaining an acceptable fidelity or image quality. Due to the advent of multimedia computing, the demand for these images has increased rapidly, their storage and 4

9 Chapter One General Introduction manipulation in their raw form is very expensive, and it significantly increases the transmission time and makes storage costly. Data compression is possible because images are extremely data intensive, and contain a large amount of redundancy which can be removed by accomplishing some kind of transform, with a reversible linear phase to de-correlate the image data pixels. There are two different techniques for image data compression namely; lossless and lossy, the lossless compression techniques are reversible or non destructive compression. It is guaranteed that the decompression image is identical to the original image. This is an important requirement for some applications where high quality is demanded. Whereas, the lossy techniques allows the image quality degradation in each compression and decompression. The performance of any coder is determined by the bit rate achieved and the distortion introduced by the coder. For an image coder, the bit rate is measured as the average number of bits required to express one pixel of the image and the fidelity is measured in terms of PSNR (Peak Signal to Noise Ratio). The applications of image compression include television transmission, video conferencing, facsimile transmission, archiving of graphic images, medical images and drawing, computer communications, data base, etc. In the last few years, using wavelet transforms has become one of the important technologies in image compression research. It involves viewing the image at different resolutions and decomposes the original image into various sub-images [3]. The essential figure of merit for data compression is the Compression ratio. It is the ratio of the original (uncompressed file) to the size of a compressed file [2]: 5

10 Chapter One General Introduction Uncompressed file size Compression ratio (CR) =... (1-1) Compressed file size Another type of compression measurement is the Bit per Pixel (BPP), denoted as the bit rate, it is the average number of bits required to represent each image pixel: Compressed file size Bit per Pixel (BPP) = (1-2) Number of pixels 1.5 Methodologies of Compression Techniques Compression techniques can be divided into two categories, lossless and lossy techniques, as illustrated below: Lossless Techniques Lossless compression, also called noiseless coding, entropy coding or invertible coding, refers to algorithms which allow the original image to be, perfectly, recovered from the digital representation [4]. The problem of these methods is the resultant compression ratios on images are not high; i.e., the maximum compression factor that can achieved is limited by Shannon s theory (which states that the minimum bit rate one can achieve, can not be 6

11 Chapter One General Introduction less than the entropy H of the probability distribution of all gray levels). Therefore, the compression ratio is limited by the entropy rate H of the source (in fact, the source entropy H is a measure of the randomness inherent in the source) [5]. If P 1 is the probability of the occurrence of a gray-level i in an image, the entropy of source is, then, given by [6]: M H= -å i= 1 P i Log (P i ) (1-3) Where i=0 M-1 and M is referred to image gray scale. Lossless compression requires variable-rate coding methods; a varying number of bits are needed for different pixels or pixel patterns. The basic idea of such codes is to use long code words for unlikely inputs and short code words for likely inputs. The main features of the Error free (loss less) compression methods as follows: 1. No data are lost. 2. Original image can be recreated from compressed data. 3. Compression ration. 4. Useful in images archiving (such as legal, satellite images or medical image). There are many types of lossless compression methods based on variable length coding. The most popular Lossless coding methods [7] are the followings: 7

12 Chapter One General Introduction Variable-Length Coding This simplest approach of lossless image compression is to reduce only coding redundancy. Coding redundancy normally is present in any natural binary encoding of the gray levels in an image. In practice, the source symbols may be either color gray levels of an image or the output of mapping operation (likes pixel differences, run-length, and soon) Huffman Coding It is the most popular technique for removing coding redundancy. When the symbols of an information source are coded individually, Huffman coding yields the smallest possible number of code symbols per source symbols. This method is started with a list of the probabilities of the image data elements. Then, take the two least probable elements and make them two nodes with branches (labeled 0 and 1 ) to a common node which represents a new element. The new element has a probability, which is the sum of the two probabilities of the merged elements. The procedure is repeated until the list contains only one element [8] Arithmetic Coding Arithmetic coding is also a kind of statistical coding algorithm similar to Huffman coding. However, it uses a different approach to utilize symbol probabilities, and performs better than Huffman coding. In Huffman coding, optimal codeword length is obtained when the symbol probabilities are of the form (1/2)x, where x is an integer. This is because Huffman coding assigns code with an integral number of bits. This form of symbol 8

13 Chapter One General Introduction probabilities is rare in practice. Arithmetic coding is a statistical coding method that solves this problem. The code form is not restricted to an integral number of bits. It can assign a code as a fraction of a bit. Therefore, when the symbol probabilities are more arbitrary, arithmetic coding has a better compression ratio than Huffman coding. In brief, this is can be considered as grouping input symbols and coding them into one long code. Therefore, different symbols can share a bit from the long code. Although arithmetic coding is more powerful than Huffman coding in compression ratio, arithmetic coding requires more computational power and memory. Huffman coding is more attractive than arithmetic coding when simplicity is the major concern [9] Bit-Plane Coding: - This technique is used for removing coding redundancy and attach inter pixel redundancy; it is based on the concept of decomposing a multilevel (monochrome or color) image into a series of binary images and compressing each binary image via one of several well known binary compression methods Run Length Coding It is one of the simplest forms of data compression, which is some times known as run length limiting (RLL). This coding method is based on representing each row of image or bit Plane by a sequence of lengths that 9

14 Chapter One General Introduction describe successive runs of 0 s and 1 s values. This coding method takes the advantage of the fact that the neighboring image elements have the same value. The run length coding is in itself an effective method of compressing an image. The most common approaches for determining the value of a run are either to specify the value of the first run of each row or to assume that each row begins with a white / block run, whose run length may in fact be zero, so it can be mapped into sequence of pairs (g 1, l 1 ), (g 2, l 2 ),..(g n, l n ),where (g n ) is the gray level and (l n ) is the length of image pixel [4] Lossy Compression Lossy compression method is based on the concept of compromising the accuracy of the reconstructed image in exchange for increasing compression. If the resulting distortion (which may or may not be visually apparent) can be tolerated, the increase in compression can be significant. The main features of this type of compression is as follows: 1. Allows a loss in the actual image data. 2. Original image cannot be recovered exactly from compressed data. 3. Compression ratio range from 10:1 to 200:1. 4. Useful for broadcast television, video conferencing, and facsimile transmission, etc [4]. 10

15 Chapter One General Introduction Pulse Code Modulation (PCM) Firstly, applied by Goodlin in The basic idea of the PCM is discretizing of the continuous signal into samples or pixels, each sample is then quantized, using a limit number of levels, after that levels are coded by a fixed codeword length which depend on the maximum value of the graylevel. Usually, the brightness is quantized into N levels where N is a power of two (N=2 B ), the commonly values used for B are 7 or 8. The PCM decoder converts the binary codeword into discrete samples, which are lowpass filtered. High compression ratio or, alternatively, the low bit rate is achieved by reducing the number of quantization levels. Acceptable quality image could be obtained with three bits per pixel; therefore an 8/3 to 1 compression ratio can be gained by using this technique without making noticeable error in the reconstructed image [10] Differential Pulse Code Modulation (DPCM) The DPCM technique is a simple and one of the most popular predictive coding methods. Cutler has invented it in It exploits the property that the values of adjacent pixels in an image are often similar and highly correlated. In DPCM, only one previous pixel is required for the prediction. More precisely, the coded signal is the moving difference between the pixel to be coded and the previously coded pixel, just before it. Generally, 3 bits are used to quantize and in code samples of different image sings, it can produce a compression ratio around 2.5 to 1. Several DPCM schemes were developed to improve the result [11]. 11

16 Chapter One General Introduction Fractal Compression The application of fractals in image compression started with M.F. Barnsley and A. Jacquin. Fractal image compression is a process to find a small set of mathematical equations that can describe the image. By ending the parameters of these equations to the decoder, we can reconstruct the original image. In general, the theory of fractal compression is based on the contraction mapping theorem in the mathematics of metric spaces [12] Transform Coding (TC) For the past three decades, the common technique for image compression is the transform coding method. This coding method yields an optimum trade off between the compression ratio and the decoded image quality. A typical transform coder treats the image pixels as a matrix of numbers. A linear transform is applied to this matrix to create a new matrix coefficient, i.e., transform data from spatial domain into frequency domain. To recover the original image, an inverse process must be applied. The transform method has two effects. First, it concentrates the energy of the image so that many of the transformed coefficients are nearly zero. Second, it spectrally decomposes the image into high and low frequencies. However, the aim of an image coding is to produce the best possible trade-off between high compression and low distortion [13]. In most transform image coding, large fraction of the available bits is assigned to encode the high-energy elements while a smaller fraction is devoted to code those of low energy. By these techniques, one can increase the compression ratio by either decreasing 12

17 Chapter One General Introduction the quantization levels or by increasing the number of discarding coefficients Fourier Transform Fourier transform is a powerful tool in linear system analysis. In 1965 a method of computing discrete Fourier transform (DFT) suddenly has become widely known, and has led to the investigation of the Fourier transform in image coding technique. The main applications for (DFT) are the spectrum estimation, convolution, adaptive filtering, and spectral analysis [14,15]. The 1-D, discrete Fourier transform is one of important linear transforms that can be expressed in terms of general relation N T(u) = å - 1 x = 0 f ( x ) g ( x, n ) (1-4) Where, T(u) is the transform of f(x), g(x, u) is the forward transformation Kernel, and u has the range 0,1,.., N-1. Similarly, the inverse transform is the relation. N f(x) = å - 1 u = 0 T ( u ) h ( x, u ) (1-5) Where, h(x,u) is the inverse transformation Kernel, and x has values in the range 0,1,.. N-1. For 2-D arrays the forward and inverse transforms are: W - 1 N -1 T(u, v) = å å x = 0 y = 0 f ( x, y ) g ( x, y, u, v ) (1-6) 13

18 Chapter One General Introduction W 1 N - 1 f(x, y) =å å u = 0 v = 0 T ( u, v ) h ( x, y, u, v ).. (1-7) Where g(x,y,u,v) and h(x,y,u,v) are called the forward and inverse transform Kernels, respectively. The Kernels depend only on the indexes (x, y, u,v) not on the values of f(x,y) or T(u,v), so g(x,y,u,v) and h(x,y,u,v) can be viewed as the basis function of aseries expansion via Equation. (1-6) or (1-7) Discrete Cosine Transform (DCT) The most particular transform coding systems are based on discrete cosine transform (DCT), which provide a good compromise between information packing ability and computational complexity. The properties of this transformation method have provided to be of such particular value that it has become the international standard for transform coding system. Compared to the other input independent transforms, it has the advantages of having been implemented in a single integrated circuit. DCT has the capability of packing the most information into the few coefficients (for most natural images), and minimizing the block like appearance (called blocking artifact, which results when the boundaries between sub images become visible). This last property is particularly important in comparison With the other sinusoidal transforms [2]. The 1-D DCT is mathematically defined as. N é + ù c(u)= å - 1 (2 x 1) u p a ( u ) f ( x ) cos ê ú (1-8) X = 0 ë 2 N û 14

19 Chapter One General Introduction For u = 0, l, N- 1. Similarly, the inverse DCT is defined as N f(x) =å - = u 1 é (2x + 1) up ù ( u) c( u) cos ê ú ë N û a (1-9) 0 2 For x = 0, 1,,N-1.In both Equation.(1-8) and (1-9) a ( u ) is N a = 1 N For u=0. For u=0. 2 For u=1,2,,n-1. For u=1,2,,n-1.(1-10) The corresponding 2-D DCT pair is c(u,v)= a ( u ) a ( v) N - 1 N -1 å å é (2 x + 1) up ù é (2 N + 1) vp ù f ( x, y ) cos ê ú cos ê ú ë 2 N û ë û X = 0 Y = 0 2 N (1-11) For u, v=0,1,,n-1 and f(x,u)= N - 1 N -1 å å é (2 N + 1) up ù é (2 N + 1) vp ù a ( u ) a ( v) c( u, v) cos ê ú cos ê ú (1-12) ë 2 N û ë û u = 0 v = 0 2 N 15

20 Chapter One General Introduction For x,y=0,1,.,n-1,where a(u) is given by Equation.(1-10) Walsh-Hadamard Transform The Walsh-Hadamard transform differs from the Fourier transform and cosine transforms in that the basis are not sinusoids. The basis functions are based on square or rectangular waves with peaks of ±1. Here the term rectangular wave refers to any function of this form, where the width of the pulse may vary. One primary advantage of a transform with these types of basis functions is that the computations are very simple. When we project the image onto the basis functions, all we need to do is multiply each pixel by ±1, as is seen is the Walsh-Hadamard transform equation (assuming an N N image): 1 WH ( u, v) = N N - 1 N -1 åå r = 0 c= 0 N -1 å i = 0 I ( r, c)( -1) [ b ( r ) p ( u) + b ( c) p ( v)] i i i i Where N=2 n, the exponent on the (-1) is performed in modulo 2 arithmetic, and b i (r) is found by considering r as a binary number and finding the ith bit. P i (u) is found as follows: p 0 (u)=b n-1 (u) p 1 (u)=b n-1 (u)+b n-2 (u) p 2 (u)=b n-2 (u)+b n-3 (u)... p n-1 (u)=b 1 (u)+b 0 (u) Strictly speaking we con not call the Walsh-Hadamard transform a frequency transform because the basis functions do not exhibit the frequency concept in the manner of sinusoidal functions. If we consider the number of 16

21 Chapter One General Introduction zero crossings (or sign changes), we have a measure that is comparable to frequency, and we call this sequence [1]. WH The inverse Walsh-Hadamard transform equation is: N - 1 N -1 å[ bi ( r ) pi ( u) + bi ( c) pi ( v)] -1 1 [ WH ( u, v) ] = I( r, c) = åå N u= 0 v= 0 WH ( u, v)( -1) n-1 i = Wavelet Transform One of the modern methods for data compression is called wavelet encoding which involves a discrete-wavelet transform on image and quantizing the wavelet coefficients using suitable bit allocation scheme. A wavelets-based compression scheme shows much promise for the next generation of image compression methods. The wavelet transform has combined with vector quantization, which has led to the development of compression algorithms with high compression ratios. In wavelet compression, an analysis filter bank followed by down sampling produces the decomposition (analysis) of the image. Any or all of the resulting sub bands may be, further, input to an analysis filter bank and down sampled again. This operation may be continued for as many stages as desired. The final goal of the sub band analysis is to transform the source signal into alternative representation, so that most of the energy is concentrated in a small fraction of sample [16]. 17

22 Chapter One General Introduction 1.6 Aim of Dissertation This project is concerned on compressing an image using the hybrid between wavelet transform and arithmetic coding compression method, to occupy better compression. The details of this process will be clarify through the next chapters. 1.7 Dissertation Layout In chapter 2, the wavelet transform process is evaluated. Chapter 3, shows the compression system components. Chapter 4, explains the system implementation and finally chapter 5, will be cover the results and conclusions. 18

23 Chapter Two The Wavelet Transform The Wavelet Transform 2.1 General Introduction Wavelets are functions that satisfy certain requirements. The wavelet name comes from the requirement that they should integrate to zero, waving above and below the x-axis. The diminutive connotation of wavelet suggests the function has to be well localized. Other requirements are technical and needed mostly to insure quick and easy calculation of the direct and inverse wavelet transforms [17]. So wavelets are mathematical functions that cut up data into different frequency components, and then study each component with a resolution matched to its scale. They have advantages over traditional Fourier methods in analyzing physical situations where the signal contains discontinuities and sharp spikes. Wavelet theory provides a uniform framework for a number of techniques, which has been developed independently in the fields of mathematics, quantum physics, electrical engineering, and seismic geology. Interchanges between these fields during the last ten years have led to many new wavelet applications such as image compression, speech compression, turbulence, human vision, radar, and earthquake prediction [18,20]. In fact, wavelet theory covers quite a large area. It treats both the continuous and the discrete-time cases, it provides very general techniques that can be applied to many tasks in signal processing, and therefore has numerous potential applications [18].

24 Chapter Two The Wavelet Transform Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Approximation by using superposition of functions was existed, since the early 1800's, when Joseph Fourier discovered that he could superpose sins and cosines to represent other functions. However, in wavelet analysis, the scale that we use to look at data plays a special role. Wavelet algorithms process data at different scales or resolutions. If we look at a signal with a large window, we would notice gross features. Similarly, if we look at a signal with a small window, we would notice small features. The result in wavelet analysis is to see both the forest and the trees, so to speak [21]. This makes wavelets interesting and useful. For many decades, scientists have wanted more appropriate functions than the sins and cosines, which comprise the bases of Fourier analysis, to approximate choppy signals. By their definition, these functions are non-local (and stretch out to infinity). They therefore do a very poor job in approximating sharp spikes. But with wavelet analysis, we can use approximating functions that are contained neatly infinite domains. Wavelets are well suited for approximating data with sharp discontinuities [19,20]. The fundamental idea behind wavelets is to analyze according to scale. The wavelet analysis procedure is to adopt a wavelet prototype function called an analyzing wavelet or mother wavelet. Any signal can then be represented by translated and scaled versions of the mother wavelet. Wavelet analysis is capable of revealing aspects of data that other signal analysis techniques (such as Fourier analysis) miss aspects like trends, breakdown points, and discontinuities in higher derivatives, and selfsimilarity. Furthermore, because it affords a different view of data than those

25 Chapter Two The Wavelet Transform presented by traditional techniques, it can compress or de-noise a signal without appreciable degradation [17]. Wavelet transforms are a new area in mathematics and have many different applications. The wavelet transform comes in two different forms, the first form is the Continuous Wavelet Transform (CWT) which is dealing with continuous input signal, the time and scale parameters can be continuous. The second form is the Discrete Wavelet Transform (DWT) which can be used for representing the discrete-time signals [19]. Actually, there are many kinds of wavelets. One can choose between smooth wavelets, compactly supported wavelets, wavelets with simple mathematical expressions, wavelets with simple associated filters, etc. The most simple is the Haar wavelet. Examples of some wavelets are given in Figure (2.1). Scale Time Scale Time Figure (2.1) some examples of wavelet families

26 Chapter Two The Wavelet Transform Using wavelets primarily links the idea behind signal compression by using wavelets to the relative scarceness of the wavelet domain representation for the signal. 2.2 Wavelet Transform Development Stages The main branch of mathematics leading to wavelets began with Joseph Fourier (1807) with his theories of frequency analysis, now often referred to as Fourier synthesis. But Wavelet appeared for the first time in an appendix of the thesis of Alfred Haar (1909) [17]. In the history of mathematics, wavelet analysis has many different origins. Much of the work was performed in the 1930s, and at that time, the separate efforts did not appear to be parts of a coherent theory. Since 1930 several groups worked independently a lot of research work concerned with the representation of the wavelet function using scale varying basis functions was accomplished. But these bases had several weak points [22]. In 1946 Gabor introduced a family of non-orthogonal wavelets with unbounded support. These wavelets are based on translated Gaussians. Between 1960 and 1980, the mathematicians Weiss and Coifman studied the simplest elements of a function space, called atoms. Their objectives were to find atoms for a common function and finding the "assembly rules" that allow the reconstruction of all the elements of the function space using these atoms. In 1976 Croisier, Esteban and Galand introduced filter banks for the decomposition and reconstruction of a signal. Flanagan, Crochiere and

27 Chapter Two The Wavelet Transform Weber introduced roughly the same idea in speech acoustics, called sub band coding. In 1980, Grossman and Morlet, a physicist and an engineer, broadly defined wavelets in the context of quantum physics. In 1985, Stephane Mallat gave wavelets an additional jump-start through his work in digital signal processing. He discovered some relationships between quadrate mirror filters, pyramid algorithms, and orthonormal wavelet bases. Inspired in part by these results, Y. Meyer constructed the first non-trivial wavelets. Unlike the Haar wavelets, the Meyer wavelets are continuously differentiable; however they do not have compact support. A couple of years later, exactly in 1988, Ingrid Daubechies used Mallat's work to construct a set of wavelet orthonormal basis functions that are perhaps the most elegant, and have become the cornerstone of wavelet applications today. In 1996 Sweldens, Calderbank and Daubechies proposed the lifting scheme for developing second-generation wavelets that can be derived from FIR filters and map integers to integers. Figure (2.2) illustrates the important stages in the historical review of Wavelet Transform [23,24].

28 Chapter Two The Wavelet Transform Alfred Haar Wavelet Analysis Grossman and Morlet 1980 Stephane Mallat 1985 Ingrid Daubechies 1988 Figure (2.2) Important stages in wavelet history We have two types of analysis: the continuous and discrete time analysis. The distinction among the various types of wavelet transform (WT) depends on the way in which the scale and shift parameters are discredited. In this section we will look closer to the tree type of these possibilities [22] Continuous Wavelet Transform (CWT) The continuous wavelet transform was developed as an alternative approach to the short time Fourier transforms (STFT) to overcome the resolution problem. The wavelet analysis is done in a similar way to the STFT analysis, in the sense that the signal is multiplied with a function, similar to the window function in the STFT, and the transform is computed separately for different segments of the time-domain signal [25]. The continuous wavelet transform is defined as follows

29 Chapter Two The Wavelet Transform CWT y x y 1 æ t - T ö ( T, S) = Ys ( T, S) = ò x( t) y * ç dt...(2.1) S è S ø Where: T Represents the translation parameter. S Represents the scale parameter. Y(t) Represents the transforming function. Y(t) Is also called the mother wavelet. The term mother wavelet gets its name due to two important properties of the wavelet analysis as will be explained below. The term wavelet means a small wave. The smallness refers to the condition that this (window) function is of finite length (compactly supported). The wave refers to the condition that this function is oscillatory. The term mother implies that the functions with different region of support that used in the transformation process are derived from one main function, which is called the mother wavelet. In other words, the mother wavelet is a prototype for generating the other window functions [23,25]. An example of continuous wavelet transform is shown in figure (2.3).

30 Chapter Two The Wavelet Transform 2 D 3 D Figure (2.3) Example of continuous wavelet transform. In the top figure the signal is decomposed. While in the bottom figure the corresponding wavelet coefficients are depicted Discrete Wavelet Transform (DWT) The Discrete Wavelet Transform (DWT) involves choosing scales and positions based on powers of two, so called dyadic scales and positions. The mother wavelet is rescaled or dilated, by powers of two and translated by integers [26]. The CWT provides highly redundant as far as the reconstruction of the signal is concerned. This redundancy, on the other hand, requires a significant amount of computation time and resources. The

31 Chapter Two The Wavelet Transform discrete wavelet transform (DWT), on the other hand, provides sufficient information both for analysis and synthesis of the original signal, with a significant reduction in the computation time. The DWT is considerably easier to implement when compared to the CWT. The basic concepts of the DWT will be introduced in the next section. Nowadays, the wavelet transform becomes one of the most exciting developments in the signal processing field during the past decades. This is especially true when it is utilized in compressing images [27,28,29] 2.4 Discrete Wavelet Transform and Image Compression In the discrete case, filters of different cutoff frequencies are used to analyze the signal at different scales. The signal is passed through a series of high pass filters to analyze the high frequencies, and it is passed through a series of low pass filters to analyze the low frequencies [30]. The resolution of the signal, which is a measure of the amount of detail information in the signal, is changed by the filtering operations, and the scale is changed by up sampling and down sampling (sub-sampling) operations. Sub-sampling a signal corresponds to reducing the sampling rate, or removing some of the samples of the signal. For example, sub-sampling by two refers to dropping every other sample of the signal. Sub-sampling by a factor n reduces the number of samples in the signal n times [21,30]. In DWT, an analysis filter bank followed by down sampling produces the decomposition (analysis) of the image [31]. The image is decomposed into sub bands correspond to higher image frequencies and other sub bands

32 Chapter Two The Wavelet Transform correspond to lower image frequencies where most of the image energy is concentrated. This is why we can expect the detail coefficients become smaller as we move from high to low levels [32]. Only the lowest frequency sub band is fed to the next level of decomposition. This operation may continue for as many stages as desired, figure (2.4) illustrate the two-stage wavelet decomposition tree. LLLL LL LLLH LH LLHL I (X,Y) LLHH HL HH Figure (2.4) Two stage wavelet decomposition tree

33 Chapter Two The Wavelet Transform The goal of the sub banding analysis is to transform the source image into alternative representation so that most of the energy is concentrated in the lowest frequency sub band and in a few coefficients, to reduce the correlation and provide a useful data structure. The sub band structure is the basis of all the image compression methods based on wavelet transforms. The image reconstruction is obtained by applying an inverse operation to that of the decomposition, as shown in figure (2.5). LLLL LLLH LL LH LLHL IË(X,Y) LLHH HL HH Figure (2.5) Two stage wavelet reconstruction tree

34 Chapter Two The Wavelet Transform Since that most of the image energy is concentrated in the lowest frequency sub band. Therefore, the quality of reconstruction using this sub band has a great influence on the quality of the fully reconstructed image, for this reason this sub-band has to be coded with relatively high fidelity images[33 ]. Figure 2.6 illustrates the image decomposition, defining level and sub band conventions. Figure 2.6 Image Decomposition Using Wavelet Transform After decomposition, the wavelet coefficients are quantized and coded. To reconstruct the image, the code words are decoded to give back the quantized coefficients, which are then de-quantized back to give the approximated values of the transform coefficients, which are then dequantized back to give the approximated values of the transform coefficients. Then, the inverse transform applied to produce the

35 Chapter Two The Wavelet Transform reconstructed image. Numerous filters can be used to implement the wavelet transform, two of the most commonly used filters are the Haar, Daubechies filters [5]. To improve compression performance we have used zero tree coding scheme which is very helpful to progressive coding. Zero tree method implies that the descendants four childs of insignificant coefficients are most probably are insignificant and, therefore, they can be discarded. A variety of approaches to progressive image transmission were proposed and investigated. They which generally fall into three categories: pyramidal, transform based, and iterative encoding. In the transform-based approach, the image first undergoes a block transform and the transformed coefficients are transmitted progressively in some order, usually from low to high sun-bands [34]. In summary, the wavelet packet has a number of useful properties: 1. It can represent smooth functions. 2. It has unconditional basis function; the choice of wavelet basis should usually be reasonable choice. 3. The potential of wavelet packet lies in its capacity to offer a rich menu of basis functions that satisfy perfect reconstruction conditions, from which the best one can be chosen for a given application [35,36].

36 Chapter Three Hybrid Coding Compression System Hybrid Coding Compression System 3.1 Introduction The image compression scheme using wavelet transform (WT) consists of several stages, figure (3.1) shows the block diagram of the hybrid compression system, while figure (3.2) shows the block diagram of hybrid de-compression system. The suggested compression scheme consists of the following steps: 1. Loading Image Data (BMP file 24-bit color). 2. Forward Wavelet Transform (Version Tap (3/5)). 3. Perform Arithmetic Coding on LLPF. 4. Perform quantization process. While the suggested de -compression scheme consists of the following steps: 1. De-Quantization process. 2. Perform Arithmetic De-Coding process to reconstruct LLPF. 3. Inverse Wavelet Transform by setting zeros on the remaining filters (i.e. LH, HL, HH). 4. Display the de-compressed image.

37 Get LLPF Only Arithmetic Encoding Quantization Input Image Forward Wavelet Transform Tap(3/5) HL LH HH LLHL LLLH LLHH... Ignore The Remaining Filters Compressed Image Figure (3.1): The hybrid compression system 33

38 De-Quantization Arithmetic De-Coding LLPF Inverse Wavelet Transform Reconstruct Image Data Compressed File Set Zeros on the Remaining Filters Figure (3.2): The hybrid de-compression system 34

39 Chapter Three Hybrid Coding Compression System 3.2 Wavelet Transform Tap (3/5) This type of biorthogonal wavelet transforms, which was recently recommended by the standard JPEG The low and high pass filters of this kind of transform can be divided into two parts forward and backward [39] Forward Wavelet Transform Tap (3/5) Forward Tap (3/5) transform process three data elements from the image data and the calculations are separate into two steps according to the length of height, or width: A. Odd length processing. For high pass filter H ë( X 2m X 2 2 ) m = X 2 m m+ The calculation process on the last element is the only / 2û difference between the odd and even length: H m = Xn - Xn-1 For low pass filter The first element is calculated by: L + ë 0 = X 0 H 0 While L / 2 û the other elements: ë( H + m - 1 H 1) = + m X 2 m m + / 4û

40 Chapter Three Hybrid Coding Compression System B. Even length processing. For high pass filter H ë( X 2m X 2 2 ) m = X 2 m m+ For low pass filter / 2û The first element is calculated by: L + ë 0 = X 0 H 0 / 2 û While other elements are calculated by: L ë( H + m- 1 H 1) = + m X 2 m m+ / 4û The low and high filters are processed on the height and then on width. The LL image part will be passed as input to the arithmetic coding process Inverse Wavelet Transform Tap (3/5) The following equations are used to perform the inverse wavelet transform based on the biorthogonal filters Tap (3/5): A. Odd length processing. For low pass filter The first element is calculated by: X + ë 0 = L0 H 0 While X / 2û the other elements are calculated by: ë( H + m - 1 H 1) = + 2 m Lm m + / 4û

41 Chapter Three Hybrid Coding Compression System For high pass filter X ë( X 2m X 2 2 ) 2 m+ 1 = Lm m+ / 2û The calculation process on the last element: X n = H m - X n-1 B. Even length processing. For low pass filter The first element is calculated by: X + ë 0 = L0 H0 While X / 2û the other elements are calculated by: ë( H + m- 1 H 1) = + 2 m Lm m+ For high pass filter X ë( X 2m X 2 2 ) 2 m+ 1 = Lm m+ / 4û / 2û From the previous equations, the backward Tap (3/5) works on five coefficients to produce two elements that represent the reconstructed image data. 3.3 Arithmetic Coding Only in the last fifteen years has a respectable candidate to replace Huffman coding been successfully demonstrated: arithmetic coding. Arithmetic coding bypasses the idea of replacing an input symbol with a specific code. It replaces a stream of input symbols with a single floatingpoint output number. More bits are needed in the output number for longer, complex messages. This concept has been known for some time, but only

42 Chapter Three Hybrid Coding Compression System recently were practical methods found to implement arithmetic coding on computers with fixed-sized registers [37,38] Arithmetic Encoding The output from an arithmetic coding process is a single number less than 1 and greater than or equal to 0. This single number can be uniquely decoded to create the exact stream of symbols that went into its construction. To construct the output number, the symbols are assigned a set probability. The message BILL GATES, for example, would have a probability distribution illustrated in table 3.1. Table 3.1 Probabilities of the message BILL GATES Character Probability SPACE 1/10 A 1/10 B 1/10 E 1/10 G 1/10 I 1/10 L 2/10 S 1/10 T 1/10

43 Chapter Three Hybrid Coding Compression System Once character probabilities are known, individual symbols need to be assigned a range along a probability line, nominally 0 to 1. It doesn t matter which characters are assigned which segment of the range, as long as it is done in the same manner by both the encoder and the decoder. The ninecharacter symbol set used here as shown in the table 3.2. Table 3.2 Range of each probability Character Probability Range SPACE 1/ [gte] r > 0.10 A 1/ [gte] r > 0.20 B 1/ [gte] r > 0.30 E 1/ [gte] r > 0.40 G 1/ [gte] r > 0.50 I 1/ [gte] r > 0.60 L 2/ [gte] r > 0.80 S 1/ [gte] r > 0.90 T 1/ [gte] r > 1.00 Each character is assigned the portion of the 0 to 1 range that corresponds to its probability of appearance. Note that the character owns everything up to, but not including, the higher number. So the letter T in fact has the range.90 to.9999

44 Chapter Three Hybrid Coding Compression System The most significant portion of an arithmetic-coded message belongs to the first symbols or B, in the message BILL GATES. To decode the first character properly, the final coded message has to be a number greater than or equal to.20 and less than.30. To encode this number, track the range it could fall in. After the first character is encoded, the low end for this range is.20 and the high end is.30. During the rest of the encoding process, each new symbol will further restrict the possible range of the output number. The next character to be encoded, the letter I, owns the range.50 to.60 in the new sub range of.2 to.3. So the new encoded number will fall somewhere in the 50th to 60th percentile of the currently established range. Applying this logic will further restrict our number to.25 to.26. here: The algorithm to accomplish this for a message of any length is shown low = 0.0; high = 1.0; while ( ( c = getc( input ) )!= EOF ) { range = high - low; high = low + range * high_range( c ); low = low + range * low_range( c ); } output ( low ); Following this process to its natural conclusion with our message results in table 3.3.

45 Chapter Three Hybrid Coding Compression System Table 3.3 Arithmetic encoding process New Character Low value High Value B I L L SPACE G A T E S So the final low value, , will uniquely encode the message BILL GATES using our present coding scheme.

46 Chapter Three Hybrid Coding Compression System Arithmetic De-Coding Given the previous encoding scheme, it is relatively easy to see how the decoding process operates. Find the first symbol in the message by seeing which symbol owns the space our encoded message falls in. Since falls between.2 and.3, the first character must be B. Then remove B from the encoded number. Since we know the low and high ranges of B, remove their effects by reversing the process that put them in. First, subtract the low value of B, giving Then divide by the width of the range of B, or.1. This gives a value of Then calculate where that lands, which is in the range of the next letter, I. The algorithm for decoding the incoming number is shown next: number = input_code(); for ( ; ; ) { symbol = find_symbol_straddling_this_range( number ); putc( symbol ); range = high_range( symbol ) - low_range( symbol ); number = number - low_range( symbol ); number = number / range; }

47 Chapter Three Hybrid Coding Compression System as shown: The decoding algorithm for the BILL GATES message will proceed Table 3.4 Arithmetic decoding process Encoded Number Output Symbol Low High Range B I L L SPACE G A T E S In summary, the encoding process is simply one of narrowing the range of possible numbers with every new symbol. The new range is proportional to the predefined probability attached to that symbol. Decoding is the inverse procedure, in which the range is expanded in proportion to the probability of each symbol as it is extracted.

48 Chapter Three Hybrid Coding Compression System 3.4 Quantization Quantization is a non-reversible mapping process, it return an approximated values. By this process, the wide range of real numbers is mapped to a small set of integers which require less number of bits in representation (i.e. in storage or transmission). In this Project the quantization process was implemented as follows: Code[i]= Round (Code[i]*Pow) Where: Code is a real number that obtained from calculating the arithmetic encoding process, Pow, is 2 to the power number of bits. 3.5 De-Quantization This step is to reconstruct the real number of Code values. In this Project the de-quantization process was implemented as follows: Code[i]=Code[i]/Pow

49 Chapter Four Implementation and Experimental Results Implementation and Experimental Results 4.1 Introduction The hybrid coding compression system was implemented as a program using Borland Delphi 7 programming language. The main form page of hybrid coding compression system is shone in Figure 4.1. In this form the forward wavelet transform Tap (3/5) was applied using one level. Figure 4.1 Forward wavelet transforms Tap (3/5) 45

50 Chapter Four Implementation and Experimental Results 4.2 Performance Measures A number of quantitative parameters have been used to control or evaluate the performance of the suggested wavelet based image compression scheme [1] The Compression Factor (CF) This parameter is used to calculate the how much that the size of the tested image file was compressed; the compression ratio is defined as follows: Compressio n factor( CF) = size of the original image,...(4.1) size of the compresseddataimage CF>1 means positive compression, while CF<1 means the output stream is bigger than the input stream (negative compression). Whenever this factor is big it indicates that the compression is better, otherwise the compression performance is week The Mean Square Error (MSE) This parameter is a fidelity parameter which is used to measure the error level caused by the compression system, Mean square error can be defined as: MSE= H * W H W - åå y= 0 x= 0 / 2 ( f ( x, y) - f ( x, y) ),...(4.2) Where: f(x,y) is the value of the original image at row (y) and column (x). f / (x,y) is the corresponding values of the reconstructed image. 46

51 Chapter Four Implementation and Experimental Results H is the height of the image. W is the width of the image. The small result of MSE means that there is a small overall error in the reconstructed version of image caused by the compression system; this indicates that the objective quality of the reconstructed data is acceptable. When the value of MSE is high it will indicate that the compression system cause a significant error Peak Signal to Noise ratio (PSNR) It is also a fidelity parameter used to measure the distortion level caused by the compression system, the error in the reconstructed image is considered as a noise, whose relative ratio is determine. Peak Signal to Noise Ratio (PSNR) can be defined as: 2 æ (255) 10log10 ö PSNR= ç,...(4.3) è MSE ø Where: MSE is the mean square error. In this case the large results means that there is a small noise in the compression system and the quality of the reconstructed image is better. When the value of this parameter is small it means that the compression performance is weak. For colored images, the PSNR is computed as an average value of the Red, Green, and Blue bands. 4.3 Implementation After applying wavelet transform, the LLPF captured from wavelet coefficients will implement arithmetic coding on it to get the final 47

52 Chapter Four Implementation and Experimental Results compressed image file. To reconstruct the image, the compressed image file must go thru arithmetic de-coding, finally applying backward wavelet transform Tap (3/5) to get the final image result. 4.4 Experimental Results The previous sequence of procedures was implemented partially. Figure 4.2 and Figure 4.3 illustrates, the resulted images of the different implementations accompanied with there arithmetic measurements. Original Image1 Size 192 KB 256 X 256 pixels Hybrid Compression System MSE PSNR Db C.R Compressed Size 48 KB Vector length 4 One level wavelet transform Figure 4.2 illustrate experimental results on image1 48

53 Chapter Four Implementation and Experimental Results Original Image2 Binary Image Size 786 KB 512 X 512 pixels Compressed by Arithmetic Coding MSE 0 PSNR Infinity db (i.e. Lossless) C.R Vector length 6 Compressed Size 512 KB Compressed by Wavelet MSE PSNR db C.R 4 One level wavelet transform Compressed Size 384 KB Hybrid Compression System MSE 1364 PSNR db C.R Vector length 6 One level wavelet transform Compressed Size 128 KB Figure 4.3 illustrate experimental 49 results on image2