1. A<br />Project Report on<br />Image Compression Using Wavelet Transform<br />In partial fulfillment of the requirements of<br />Bachelor of Technology (Computer Science & Engineering)<br />Submitted By<br />Saurabh Sharma (Roll No.10506058)<br />Dewakar Prasad(Roll No.10506060)<br />Manish Tripathy(Roll No 10406023)<br />Session: 2008-09<br />Department of Computer Science &Engineering<br />National Institute of Technology<br />Rourkela-769008<br />Orissa<br />A<br />Project Report on<br />Image Compression Using Wavelet Transform<br />In partial fulfillment of the requirements of<br />Bachelor of Technology (Computer Science & Engineering)<br />Submitted By<br /> Saurabh Sharma (Roll No.10506058)<br />Dewakar Prasad(Roll no.10506060)<br />Manish tripathy(Roll No.10406023)<br />Session: 2008-09<br />Under the guidance of<br />Prof. Baliar Singh<br />Department of Computer Science & Engineering<br />National Institute of Technology<br />Rourkela-769008<br />Orissa<br /> National Institute of Technology<br /> Rourkela<br /> CERTIFICATE<br />This is to certify that that the work in this thesis report entitled “Image compression using wavelet transform” submitted by Saurabh Sharma ,Dewakar Prasad and Manish tripathy in partial fulfillment of the requirements for the degree of Bachelor of Technology in Computer Science & Engineering Session 2005-2009 in the department of Computer Science & Engineering, National Institute of Technology Rourkela, This is an authentic work carried out by them under my supervision and guidance.<br />To the best of my knowledge the matter embodied in the thesis has not been submitted to any other University /Institute for the award of any degree.<br />Date: Proff. Baliar Singh<br /> Department of computer science & engineering National Institute of Technology, Rourkela<br /> <br /> <br /> ACKNOWLEDGEMENT<br />I owe a debt of deepest gratitude to my thesis supervisor, Dr.Baliar Singh, Professor, Department of Computer Science & Engineering, for his guidance, support, motivation and encouragement through out the period this work was carried out. His readiness for consultation at all times, his educative comments, his concern and assistance even with practical things have been invaluable. <br /> <br /> <br /> I am grateful to Prof. B Majhi, Head of the Department, Computer Science of Engineering for providing us the necessary opportunities for the completion of our project. I also thank the other staff members of my department for their invaluable help and guidance.<br /> <br /> Saurabh Sharma(10506058)<br /> Dewakar Prasad(10506060) <br /> Manish tripathy(10406023)<br /> B.Tech. Final Yr. CSE.<br /> CONTENTS<br />Chapter Page No.<br />Certificate 3<br />Acknowledgement 4 <br />Abstract 7<br />Chapter 1 Literature Review 8-12<br />1.1 Introduction 9<br />1.2 Why compression is needed 10<br />1.3 Fundamental of image compression technique 11<br />1.4 Objective 12<br />1.5 Organization of Report 12<br />Chapter 2 Image compression methodology 13-25<br />2.1 Overview 14<br />2.2 Different types of transform used for coding 15<br />2.3 Entropy coding 24<br />Chapter 3 Wavelet Transform 23-47<br />3.1 Overview 23<br />3.2 What are basis function 26<br />3.3 Fourier Analysis 30<br />3.4 Similarities between Fourier and wavelet transform 32<br />3.5 Dissimilarities between Fourier and wavelet transform 33<br />3.6 List of Wavelet transform 34<br />Chapter 4 Results and Dicussion 48-58<br />4.2 Results 49<br />4.3 Conclusion 58<br />NOMENCLATURE 59<br />REFERENCES 60-61<br /> <br /> ABSTRACT<br /> <br />Abstract: - Data compression which can be lossy or lossless is required to decrease the storage requirement and better data transfer rate. One of the best image compression techniques is using wavelet transform. It is comparatively new and has many advantages over others.<br />Wavelet transform uses a large variety of wavelets for decomposition of images. The state of the art coding techniques like EZW, SPIHT (set<br />partitioning in hierarchical trees) and EBCOT(embedded block coding with optimized truncation)use the wavelet transform as basic and common step for their own further technical advantages. The wavelet transform results therefore have the importance which is dependent on the type of wavelet used .In our project we have used different wavelets to perform the transform of a test image and the results have been discussed and analyzed. The analysis has been carried out in terms of PSNR (peak signal to noise ratio) obtained and time taken for decomposition and reconstruction. <br /> <br /> CHAPTER 1<br /> <br /> <br /> LITURATURE REVIEW<br /> <br /> CHAPTER 1: LITURATURE REVIEW<br /> 1.1 INTRODUCTION<br /> Uncompressed multimedia (graphics, audio and video) data requires considerable storage capacity and transmission bandwidth. Despite rapid progress in mass-storage density, processor speeds, and digital communication system performance, demand for data storage capacity and data-transmission bandwidth continues to outstrip the capabilities of available technologies. The recent growth of data intensive multimedia-based web applications have not only sustained the need for more efficient ways to encode signals and images but have made compression of such signals central to storage and communication technology.<br />To enable Modern High Bandwidth required in wireless data services such as mobile multimedia, email, mobile, internet access, mobile commerce, mobile data sensing in sensor networks, Home and Medical Monitoring Services and Mobile Conferencing, there is a growing demand for rich Content Cellular Data Communication, including Voice, Text, Image and Video. <br />One of the major challenges in enabling mobile multimedia data services will be the need to process and wirelessly transmit very large volume of this rich content data. This will impose severe demands on the battery resources of multimedia mobile appliances as well as the bandwidth of the wireless network. While significant improvements in achievable bandwidth are expected with future wireless access technology, improvements in battery technology will lag the rapidly growing energy requirements of the future wireless data services. One approach to mitigate this problem is to reduce the volume of multimedia data transmitted over the wireless channel via data compression technique such as JPEG, JPEG2000 and MPEG . These approaches concentrate on achieving higher compression ratio without sacrificing the quality of the Image. However these Multimedia data Compression Technique ignore the energy consumption during the compression and RF transmission. Here one more factor, which is not considered, is the processing power requirement at both the ends i.e. at the Server/Mobile to Mobile/Server. Thus in this paper we have considered all of these parameters like the processing power required in the mobile handset which is limited and also the processing time considerations at the server/mobile ends which will handle all the loads. <br />Since images will constitute a large part of future wireless data, we focus in this paper on developing energy efficient, computing efficient and adaptive image compression and communication techniques. Based on a popular image compression algorithm, namely, wavelet image compression, we present an Implementation of Advanced Image Compression Algorithm Using Wavelet Transform.<br />1.2 Why Compression is needed? <br />In the last decade, there has been a lot of technological transformation in the way we communicate. This transformation includes the ever present, ever growing internet, the explosive development in mobile communication and ever increasing importance of video communication. <br />Data Compression is one of the technologies for each of the aspect of this multimedia revolution. Cellular phones would not be able to provide communication with increasing clarity without data compression. Data compression is art and science of representing information in compact form. <br />Despite rapid progress in mass-storage density, processor speeds, and digital communication system performance, demand for data storage capacity and data-transmission bandwidth continues to outstrip the capabilities of available technologies. In a distributed environment large image files remain a major bottleneck within systems. <br />Image Compression is an important component of the solutions available for creating image file sizes of manageable and transmittable dimensions. Platform portability and performance are important in the selection of the compression/decompression technique to be employed. <br />Four Stage model of Data Compression <br />Almost all data compression systems can be viewed as comprising four successive stages of data processing arranged as a processing pipeline (though some stages will often be combined with a neighboring stage, performed "off-line," or otherwise made rudimentary). <br />The four stages are <br />(A) Preliminary pre-processing steps. <br />(B) Organization by context. <br />(C) Probability estimation. <br />(D) Length-reducing code. <br />The ubiquitous compression pipeline (A-B-C-D) is what is of interest. <br />With (A) we mean various pre-processing steps that may be appropriate before the final compression engine<br />Lossy compression often follows the same pattern as lossless, but with one or more quantization steps somewhere in (A). Sometimes clever designers may defer the loss until suggested by statistics detected in (C); an example of this would be modern zero tree image coding. <br />• (B) Organization by context often means data reordering, for which a simple but good example is JPEG's "Zigzag" ordering. The purpose of this step is to improve the estimates found by the next step. <br />• (C) A probability estimate (or its heuristic equivalent) is formed for each token to be encoded. Often the estimation formula will depend on context found by (B) with separate 'bins' of state variables maintained for each conditioned class. <br />(D) Finally, based on its estimated probability, each compressed file token is represented as bits in the compressed file. Ideally, a 12.5%-probable token should be encoded with three bits, but details become complicated <br />Principle behind Image Compression <br />Images have considerably higher storage requirement than text; Audio and Video Data require more demanding properties for data storage. An image stored in an uncompressed file format, such as the popular BMP format, can be huge. An image with a pixel resolution of 640 by 480 pixels and 24-bit colour resolution will take up 640 * 480 * 24/8 = 921,600 bytes in an uncompressed format. <br />The huge amount of storage space is not only the consideration but also the data transmission rates for communication of continuous media are also significantly large. An image, 1024 pixel x 1024 pixel x 24 bit, without compression, would require 3 MB of storage and 7 minutes for transmission, utilizing a high speed, 64 Kbits /s, ISDN line. <br />Image data compression becomes still more important because of the fact that the transfer of uncompressed graphical data requires far more bandwidth and data transfer rate. For example, throughput in a multimedia system can be as high as 140 Mbits/s, which must be transferred between systems. This kind of data transfer rate is not realizable with today’s technology, or in near the future with reasonably priced hardware. <br />1.3 Fundamentals of Image Compression Techniques <br /> <br />A digital image, or "bitmap", consists of a grid of dots, or "pixels", with each pixel defined by a numeric value that gives its colour. The term data compression refers to the process of reducing the amount of data required to represent a given quantity of information. Now, a particular piece of information may contain some portion which is not important and can be comfortably removed. All such data is referred as Redundant Data. Data redundancy is a central issue in digital image compression. Image compression research aims at reducing the number of bits needed to represent an image by removing the spatial and spectral redundancies as much as possible. <br />A common characteristic of most images is that the neighboring pixels are correlated and therefore contain redundant information. The foremost task then is to find less correlated representation of the image. In general, three types of redundancy can be identified: <br />1. Coding Redundancy <br />2. Inter Pixel Redundancy<br />3.PsychovisualRedundancy<br />Coding Redundancy <br />If the gray levels of an image are coded in a way that uses more code symbols than absolutely necessary to represent each gray level, the resulting image is said to contain coding redundancy. It is almost always present when an image’s gray levels are represented with a straight or natural binary code. Let us assume that a random variable r K lying in the interval [0, 1] represents the gray levels of an image and that each r K occurs with probability Pr (r K). <br />Pr (r K) = N k / n where k = 0, 1, 2… L-1 <br />L = No. of gray levels. <br />N k =No. of times that gray appears in that image <br />N = Total no. of pixels in the image <br />If no. of bits used to represent each value of r K is l (r K), the average no. of bits required to represent each pixel is <br />L avg = l (r K) Pr (r K) <br />That is average length of code words assigned to the various gray levels is found by summing the product of the no. of bits used to represent each gray level and the probability that the gray level occurs. Thus the total no. of bits required to code an M×N image is M×N× L avg. <br />Inter Pixel Redundancy <br />The Information of any given pixel can be reasonably predicted from the value of its neighbouring pixel. The information carried by an individual pixel is relatively small. <br />In order to reduce the inter pixel redundancies in an image, the 2-D pixel array normally used for viewing and interpretation must be transformed into a more efficient but usually ‘non visual’ format. For example, the differences between adjacent pixels can be used to represent an image. These types of transformations are referred as mappings. They are called reversible if the original image elements can be reconstructed from the transformed data set. <br />Psycho visual Redundancy <br />Certain information simply has less relative importance than other information in normal visual processing. This information is said to be Psycho visually redundant, it can be eliminated without significantly impairing the quality of image perception. <br />In general, an observer searches for distinguishing features such as edges or textual regions and mentally combines them in recognizable groupings. The brain then correlates these groupings with prior knowledge in order to complete the image interpretation process. <br />The elimination of psycho visually redundant data results in loss of quantitative information; it is commonly referred as quantization. As this is an irreversible process i.e. visual information is lost, thus it results in Lossy Data Compression. An image reconstructed following Lossy compression contains degradation relative to the original. Often this is because the compression scheme completely discards redundant information. <br />Image Compression Model<br /> <br />As figure shows a compression system consists of two distinct structural blocks: an encoder and a decoder. An input image f(x, y) is fed into the encoder, which creates a set of symbols from the input data. <br />Image Compression Techniques <br />There are basically two methods of Image Compression: <br />1. Lossless Coding Techniques <br />2. Lossy Coding Techniques <br />Lossless Coding Techniques: <br />In Lossless Compression schemes, the reconstructed image, after compression, is numerically identical to the original image. However Lossless Compression can achieve a modest amount of Compression. <br />Lossless coding guaranties that the decompressed image is absolutely identical to the image before compression. This is an important requirement for some application domains, e.g. Medical Imaging, where not only high quality is in the demand, but unaltered archiving is a legal requirement. Lossless techniques can also be used for the compression of other data types where loss of information is not acceptable, e.g. text documents and program executables. Lossless compression algorithms can be used to squeeze down images and then restore them again for viewing completely unchanged. <br />Lossless Coding Techniques are as follows: Source Encoder Input Image F(x, y) <br /> 1. Run Length Encoding <br /> 2. Huffman Encoding <br /> 3. Entropy Encoding <br /> 4. Area Encoding <br /> Lossy Coding Techniques: <br />Lossy techniques cause image quality degradation in each Compression / De-compression step. Careful consideration of the Human Visual perception ensures that the degradation is often unrecognizable, though this depends on the selected compression ratio. An image reconstructed following Lossy compression contains degradation relative to the original. Often this is because the compression schemes are capable of achieving much higher compression. Under normal viewing conditions, no visible loss is perceived (visually Lossless). <br />Lossy Image Coding Techniques normally have three Components: <br />1. Image Modeling: <br />It is aimed at the exploitation of statistical characteristics of the image (i.e. high correlation, redundancy). It defines such things as the transformation to be applied to the Image. <br /> 2. Parameter Quantization: <br />The aim of Quantization is to reduce the amount of data used to represent the information within the new domain. <br /> 3. Encoding: <br />Here a code is generated by associating appropriate code words to the raw produced by the Quantizer. Encoding is usually error free. It optimizes the representation of the information and may introduce some error detection codes. <br /> Measurement of Image Quality <br />The design of an imaging system should begin with an analysis of the physical characteristics of the originals and the means through which the images may be generated. For example, one might examine a representative sample of the originals and determine the level of detail that must be preserved, the depth of field that must be captured, whether they can be placed on a glass platen or require a custom book-edge scanner, whether they can tolerate exposure to high light intensity, and whether specular reflections must be captured or minimized. A detailed examination of some of the originals, perhaps with a magnifier or microscope, may be necessary to determine the level of detail within the original that might be meaningful for a researcher or scholar. For example, in drawings or paintings it may be important to preserve stippling or other techniques characteristic <br /> CHAPTER 1: LITURATURE REVIEW<br />1.4 OBJECTIVE <br /> The objective of this project is to compress an image using haar wavelet transform.<br /> <br /> <br /> CHAPTER 2<br /> <br /> Image Compression Methodology<br /> <br /> CHAPTER 2: Image compression methodology<br />2.1 Overview<br />The storage requirements for the video of a typical<br />Angiogram procedure is of the order of several hundred Mbytes <br />*Transmission of this data over a low bandwidth network results in very high latency <br />* Lossless compression methods can achieve compression ratios of ~2:1 <br />* We consider lossy techniques operating at much higher compression ratios (~10:1) <br />* Key issues: <br />- High quality reconstruction required <br />- Angiogram data contains considerable high-frequency spatial texture <br />* Proposed method applies a texture-modelling scheme to the high-frequency texture of some regions of the image <br />* This allows more bandwidth allocation to important areas of the image <br />2.2 Different types of Transforms used for coding are:<br /> <br />1. FT (Fourier Transform) <br />2. DCT (Discrete Cosine Transform) <br />3. DWT (Discrete Wavelet Transform) <br /> <br />2.2.2 The Discrete Cosine Transform (DCT):<br /> <br />The discrete cosine transform (DCT) helps separate the image into parts (or spectral sub-bands) of differing importance (with respect to the image's visual quality). The DCT is similar to the discrete Fourier transform: it transforms a signal or image from the spatial domain to the frequency domain. <br /> <br />2.2.3 Discrete Wavelet Transform (DWT): <br /> <br />The discrete wavelet transform (DWT) refers to wavelet transforms for which the wavelets are discretely sampled. A transform which localizes a function both in space and scaling and has some desirable properties compared to the Fourier transform. The transform is based on a wavelet matrix, which can be computed more quickly than the analogous Fourier matrix. Most notably, the discrete wavelet transform is used for signal coding, where the properties of the transform are exploited to represent a discrete signal in a more redundant form, often as a preconditioning for data compression. The discrete wavelet transform has a huge number of applications in Science, Engineering, Mathematics and Computer Science. <br />Wavelet compression is a form of data compression well suited for image compression (sometimes also video compression and audio compression). The goal is to store image data in as little space as possible in a file. A certain loss of quality is accepted (lossy compression). <br />Using a wavelet transform, the wavelet compression methods are better at representing transients, such as percussion sounds in audio, or high-frequency components in two-dimensional images, for example an image of stars on a night sky. This means that the transient elements of a data.<br />signal can be represented by a smaller amount of information than would be the case if some other transform, such as the more widespread discrete cosine transform, had been used. <br />First a wavelet transform is applied. This produces as many coefficients as there are pixels in the image (i.e.: there is no compression yet since it is only a transform). These coefficients can then be compressed more easily because the information is statistically concentrated in just a few coefficients. This principle is called transform coding. After that, the coefficients are quantized and the quantized values are entropy encoded and/or run length encoded. <br />Examples for Wavelet Compressions: <br />· JPEG 2000 <br />· Ogg<br />· Tarkin <br />· SPIHT <br />· MrSID <br />2.3 Quantization: <br />Quantization involved in image processing. Quantization techniques generally compress by compressing a range of values to a single quantum value. By reducing the number of discrete symbols in a given stream, the stream becomes more compressible. For example seeking to reduce the number of colors required to represent an image. Another widely used example — DCT data quantization in JPEG and DWT data quantization in JPEG 2000. <br />Quantization in image compression <br />The human eye is fairly good at seeing small differences in brightness over a relatively large area, but not so good at distinguishing the exact strength of a high frequency brightness variation. This fact allows one to get away with greatly reducing the amount of information in the high frequency components. This is done by simply dividing each component in the frequency domain by a constant for that component, and then rounding to the nearest integer. This is the main lossy operation in the whole process. As a result of this, it is typically the case that many of the higher frequency components are rounded to zero, and many of the rest become small positive or negative numbers. <br />2.3 Entropy Encoding <br />An entropy encoding is a coding scheme that assigns codes to symbols so as to match code lengths with the probabilities of the symbols. Typically, entropy encoders are used to compress data by replacing symbols represented by equal-length codes with symbols represented by codes proportional to the negative logarithm of the probability. Therefore, the most common symbols use the shortest codes. <br />According to Shannon's source coding theorem, the optimal code length for a symbol is −logbP, where b is the number of symbols used to make output codes and P is the probability of the input symbol. <br />Three of the most common entropy encoding techniques are Huffman coding, range encoding, and arithmetic coding. If the approximate entropy characteristics of a data stream are known in advance (especially for signal compression), a simpler static code such as unary coding, Elias gamma coding, Fibonacci coding, Golomb coding, or Rice coding may be useful. <br />There are three main techniques for achieving entropy coding: <br />• Huffman Coding - one of the simplest variable length coding schemes. <br />• Run-length Coding (RLC) - very useful for binary data containing long runs of ones of zeros. <br />• Arithmetic Coding - a relatively new variable length coding scheme that can combine the best features of Huffman and run-length coding, and also adapt to data with non-stationary statistics. <br />We shall concentrate on the Huffman and RLC methods for simplicity. Interested readers may find out more about Arithmetic Coding in chapters 12 and 13 of the JPEG Book. <br />First we consider the change in compression performance if simple Huffman Coding is used to code the subimages of the 4-level Haar transform.<br />This is an example DCT coefficient matrix: <br />A common quantization matrix is: <br />Using this quantization matrix with the DCT coefficient matrix from above results in: <br />For example, using −415 (the DC coefficient) and rounding to the nearest integer <br /> <br /><ul><li>
2. Chapter 3</li></ul> WAVELET TRANSFORM <br /> CHAPTER 3: NUMERICAL MODELING<br />3.1 OVERVIEW <br />The fundamental idea behind wavelets is to analyze according to scale. Indeed, some researchers in the wavelet field feel that, by using wavelets, one is adopting a whole new mindset or perspective in processing data. <br />Wavelets are functions that satisfy certain mathematical requirements and are used in representing data or other functions. This idea is not new. Approximation using superposition of functions has existed since the early 1800's, when Joseph Fourier discovered that he could superpose sines 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. <br />This makes wavelets interesting and useful. For many decades, scientists have wanted more appropriate functions than the sines 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 in finite domains. Wavelets are well-suited for approximating data with sharp discontinuities. <br />The wavelet analysis procedure is to adopt a wavelet prototype function, called an analyzing wavelet or mother wavelet. Temporal analysis is performed with a contracted, high-frequency version of the prototype wavelet, while frequency analysis is performed with a dilated, low-frequency version of the same wavelet. Because the original signal or function can be represented in terms of a wavelet expansion (using coefficients in a linear combination of the wavelet functions), data operations can be performed using just the corresponding wavelet coefficients. And if you further choose the best wavelets adapted to your data, or truncate the coefficients below a threshold, your data is sparsely represented. This sparse coding makes wavelets an excellent tool in the field of data compression. <br />Other applied fields that are making use of wavelets include astronomy, acoustics, nuclear engineering, sub-band coding, signal and image processing, neurophysiology, music, magnetic resonance imaging, speech discrimination, optics, fractals, turbulence, earthquake-prediction, radar, human vision, and pure mathematics applications such as solving partial differential equations. <br />3.2 What are Basis Functions?<br />It is simpler to explain a basis function if we move out of the realm of analog (functions) and into the realm of digital (vectors) (*). Every two-dimensional vector (x,y) is a combination of the vector (1,0) and (0,1). These two vectors are the basis vectors for (x,y). Why? Notice that x multiplied by (1,0) is the vector (x,0), and y multiplied by (0,1) is the vector (0,y). The sum is (x,y). <br />The best basis vectors have the valuable extra property that the vectors are perpendicular, or orthogonal to each other. For the basis (1,0) and (0,1), this criteria is satisfied. <br />Now let's go back to the analog world, and see how to relate these concepts to basis functions. Instead of the vector (x,y), we have a function f(x). Imagine that f(x) is a musical tone, say the note A in a particular octave. We can construct A by adding sines and cosines using combinations of amplitudes and frequencies. The sines and cosines are the basis functions in this example, and the elements of Fourier synthesis. For the sines and cosines chosen, we can set the additional requirement that they be orthogonal. How? By choosing the appropriate combination of sine and cosine function terms whose inner product add up to zero. The particular set of functions that are orthogonal and that construct f(x) are our orthogonal basis functions for this problem. <br /> What are Scale-Varying Basis Functions?<br />A basis function varies in scale by chopping up the same function or data space using different scale sizes. For example, imagine we have a signal over the domain from 0 to 1. We can divide the signal with two step functions that range from 0 to 1/2 and 1/2 to 1. Then we can divide the original signal again using four step functions from 0 to 1/4, 1/4 to 1/2, 1/2 to 3/4, and 3/4 to 1. And so on. Each set of representations code the original signal with a particular resolution or scale. <br />3.3 Fourier analysis<br />FOURIER TRANSFORM<br />The Fourier transform's utility lies in its ability to analyze a signal in the time domain for its frequency content. The transform works by first translating a function in the time domain into a function in the frequency domain. The signal can then be analyzed for its frequency content because the Fourier coefficients of the transformed function represent the contribution of each sine and cosine function at each frequency. An inverse Fourier transform does just what you'd expect, transform data from the frequency domain into the time domain. <br />DISCRETE FOURIER TRANSFORM<br />The discrete Fourier transform (DFT) estimates the Fourier transform of a function from a finite number of its sampled points. The sampled points are supposed to be typical of what the signal looks like at all other times. <br />The DFT has symmetry properties almost exactly the same as the continuous Fourier transform. In addition, the formula for the inverse discrete Fourier transform is easily calculated using the one for the discrete Fourier transform because the two formulas are almost identical. <br />WINDOWED FOURIER TRANSFORM<br />If f(t) is a nonperiodic signal, the summation of the periodic functions, sine and cosine, does not accurately represent the signal. You could artificially extend the signal to make it periodic but it would require additional continuity at the endpoints. The windowed Fourier transform (WFT) is one solution to the problem of better representing the non periodic signal. The WFT can be used to give information about signals simultaneously in the time domain and in the frequency domain. <br />With the WFT, the input signal f(t) is chopped up into sections, and each section is analyzed for its frequency content separately. If the signal has sharp transitions, we window the input data so that the sections converge to zero at the endpoint. This windowing is accomplished via a weight function that places less emphasis near the interval's endpoints than in the middle. The effect of the window is to localize the signal in time. <br />FAST FOURIER TRANSFORM<br />To approximate a function by samples, and to approximate the Fourier integral by the discrete Fourier transform, requires applying a matrix whose order is the number sample points n. Since multiplying an matrix by a vector costs on the order of arithmetic operations, the problem gets quickly worse as the number of sample points increases. However, if the samples are uniformly spaced, then the Fourier matrix can be factored into a product of just a few sparse matrices, and the resulting factors can be applied to a vector in a total of order arithmetic operations. This is the so-called fast Fourier transform or FFT.<br /> 3.4 SIMILARITIES BETWEEN FOURIER AND WAVELET TRANSFORM<br />The fast Fourier transform (FFT) and the discrete wavelet transform (DWT) are both linear operations that generate a data structure that contains segments of various lengths, usually filling and transforming it into a different data vector of length . <br />The mathematical properties of the matrices involved in the transforms are similar as well. The inverse transform matrix for both the FFT and the DWT is the transpose of the original. As a result, both transforms can be viewed as a rotation in function space to a different domain. For the FFT, this new domain contains basis functions that are sines and cosines. For the wavelet transform, this new domain contains more complicated basis functions called wavelets, mother wavelets, or analyzing wavelets. <br />Both transforms have another similarity. The basis functions are localized in frequency, making mathematical tools such as power spectra (how much power is contained in a frequency interval) and scale grams (to be defined later) useful at picking out frequencies and calculating power distributions. <br /> 3.5 DISSIMILARITIES BETWEEN FOURIER AND WAVELET TRANSFORM<br /> <br />The most interesting dissimilarity between these two kinds of transforms is that individual wavelet functions are localized in space. Fourier sine and cosine functions are not. This localization feature, along with wavelets' localization of frequency, makes many functions and operators using wavelets "sparse" when transformed into the wavelet domain. This sparseness, in turn, results in a number of useful applications such as data compression, detecting features in images, and removing noise from time series.<br />3.6 LIST OF WAVELET RELATED TRANSFORM<br /> 1. Continuous wavelet transform<br />A continuous wavelet transform is used to divide a continuous-time function into wavelets. Unlike Fourier transform, the continuous wavelet transform possesses the ability to construct a time frequency represented of a signal that offers very good time and frequency localization.<br /> 2 .Multiresolution analysis<br />A multiresolution analysis (MRA) or multiscale approximation (MSA) is the design methods of most of the practically relevant discrete wavelet transform (DWT) and the justification for the algorithm of the fast Fourier wavelet transform (FWT)<br />3. Discrete wavelet transform<br />In numerical analysis and functional analysis, a discrete wavelet transform (DWT) is any wavelet transform for which the wavelets are discretely sampled. As with other wavelet transforms, a key advantage it has over Fourier transforms is temporal resolution: it captures both frequency and location information.<br />4. Fast wavelet transform<br />The Fast Wavelet Transform is a mathematical algorithm designed to turn a waveform or signal in the time domain into a sequence of coefficients based on an orthogonal basis of small finite waves, or wavelets. The transform can be easily extended to multidimensional signals, such as images, where the time domain is replaced with the space domain<br />3.2 HAAR WAVELET<br /> In mathematics, the Haar wavelet is a certain sequence of functions. It is now recognised as the first known wavelet.<br />This sequence was proposed in 1909 by Alfred Haar. Haar used these functions to give an example of a countable orthonormal system for the space of square integrable functions on the real line. The study of wavelets, and even the term "wavelet", did not come until much later. <br />The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continous , and therefore not differentiable.<br />The Haar wavelet's mother wavelet function ψ(t) can be described as<br />and its scaling function φ(t) can be described as<br />Wavelets are mathematical functions that were developed by scientists working in several different fields for the purpose of sorting data by frequency. Translated data can then be sorted at a resolution which matches its scale. Studying data at different levels allows for the development of a more complete picture. Both small features and large features are discernable because they are studied separately. Unlike the discrete cosine transform, the wavelet transform is not Fourier-based and therefore wavelets do a better job of handling discontinuities in data.<br />The Haar wavelet operates on data by calculating the sums and differences of adjacent elements. The Haar wavelet operates first on adjacent horizontal elements and then on adjacent vertical elements. The Haar transform is computed using:<br /> <br /> <br /> <br /> CHAPTER 4 <br /> <br /> RESULTS AND DISCUSSION<br /> CHAPTER 4 : RESULTS AND DISCUSSION <br />4.1 RESULTS<br />The image on the left is the original image and the image on the right is the compressed one <br />(The point is that the image on the left you are right now viewing is compressed using Haar wavelet method and the<br />loss of quality is not visible. Of course, image compression using Haar Wavelet is one of the simplest ways.)<br />Original image compressed image<br />4.2 CONCLUSION<br />Haar wavelet transform for image compression is simple and crudest algorithm.as compared to other algorithms it is more effective.The quality of compressed image is also maintained <br />BIBILOGRAPHY :- [1<br />[1] Aldroubi, Akram and Unser, Michael (editors), Wavelets in Medicine and Biology, CRC Press, Boca Raton FL, 1996.<br />[2] Benedetto, John J. and Frazier, Michael (editors), Wavelets; Mathematics and Applications, CRC Press, Boca Raton<br />FL, 1996.<br />[3] Brislawn, Christopher M., Fingerprints go digital," AMS Notices 42(1995), 1278{1283.<br />[4] Chui, Charles, An Introduction to Wavelets, Academic Press, San Diego CA, 1992.<br />[5] Daubechies, Ingrid, Ten Lectures on Wavelets, CBMS 61, SIAM Press, Philadelphia PA, 1992.<br />[6] Glassner, Andrew S., Principles of Digital Image Synthesis, Morgan Kaufmann, San Francisco CA, 1995.<br />
Be the first to comment