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## DIGITAL IMAGE PROCESSING

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**DIGITAL IMAGE PROCESSING**Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh mhdgholizadeh@gmail.com ( J.Shanbehzadeh M.Gholizadeh )**DIGITAL IMAGE PROCESSING**Chapter 7 – Wavelet and Multiresolution Processing Instructors: Dr J. Shanbehzadeh Shanbehzadeh@gmail.com M.Gholizadeh mhdgholizadeh@gmail.com ( J.ShanbehzadehM.Gholizadeh )**Road map of chapter 7**7.5 7.7 7.3 7.4 7.1 7.1 7.2 7.2 7.4 7.5 7.6 7.6 • 7.1 Background • 7.2 Multi Resolution Expansions • 7.3 Wavelet Transform in One Dimension • 7.4 The Fast Wavelet Transform • 7.5 Wavelet Transform in Two Dimensions • 7.6 Wavelet Packets • The Fast Wavelet Transform • Wavelet Transform in Two Dimensions • Wavelet Packets • Wavelet Transform in One Dimension • Multi Resolution Expansions • Background ( J.Shanbehzadeh M.Gholizadeh )**Wavelets and Multi-resolution Processing**( J.Shanbehzadeh M.Gholizadeh ) • Preview • What is multi-resolution? - unifies techniques from a variety of disciplines,includingsubband coding from signal processing, quadrature mirror filtering from digital speech recognition, and pyramidal image processing. - features that might go undetected at one resolution may be easy to detect at another.**The difference between Fourier transform and Wavelet**transform ( J.Shanbehzadeh M.Gholizadeh ) 1) Fourier transform’ s basis functions are sinusoids, wavelet transforms are based on small waves, called wavelets, of varying frequency and limited duration. 2) Fourier transforms, provide only frequency information and temporal information is lost in the transformation process.**Background**( J.ShanbehzadehM.Gholizadeh ) If both small and large objects, or low and high contrast objects are present need multiresolution Examine an object --Depending on the size or contrast of the object choose the resolution(high , low) Local histogram variations (Fig. 7.1)**background**local histograms can vary from one part of an image to another making statistical modeling over the span of an entire image is a difficult, or impossible task. ( J. ShanbehzadehM.Gholizadeh )**Image pyramids**The Haar Transform • Image Pyramids Background Subband Coding 7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J. ShanbehzadehM.Gholizadeh ) What is an image pyramid? A powerful , simple structure for representing images at more than one resolution. an image pyramid is a collection of decreasing resolution images arranged in the shape of a pyramid .**Image Pyramids**• 7.1 Background • 7.2 Multi Resolution • Expansions • 7.3 Wavelet Transform in • One Dimension • 7.4 The Fast Wavelet Transform • 7.5 Wavelet Transform in Two Dimensions • 7.6 Wavelet Packets ( J. ShanbehzadehM.Gholizadeh ) • : The base of the pyramid contains a high-resolution representation of the image being Processed; the apex contains a low-resolution approximation . As you move up the pyramid, both size and resolution decrease.**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets provides the images needed to build an approximation pyramid is used to build a complementary prediction residual pyramid. prediction residual pyramids contain only one reduced-resolution approximation of the input image ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J. ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J. ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**7.1 Background**7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh ) • A P+I level pyramid is built by executing the operations in the block diagram P times • first iteration produces the level J-1 approximation and level J residual results • each pass is composed of three steps (Fig. 7.2(b)) • Step 1: compute a reduced-resolution approximation of the input image:filtering and down-sampling • Mean pyramid, low-pass Gaussian filter based on Gaussian pyramid, no filtering (i.e.sub-sampling pyramid) • If we compute without filtering, alias can become pronounced • Step 2 1. up-sample the o/p of the step (a)-again by a factor of 2. filter--interpolate intensities between the pixels of the step 1 • Create a prediction image • Determines how accurately approximate the input by using interpolation • If we delete interpolation filter, blocky effect is inevitable ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) • Step3 : compute the difference between the prediction of step2 and the input to step 1 (prediction residual) • Predict residual of level J • Can be used to reconstruct the original image • Can be used to generate the corresponding approximation pyramid including the original image without quantization error • level j-1 approximation can be used to populate the approximation pyramid • coarse to fine strategy • High resolution pyramid—used for analysis of large structure or overall image context • Low resolution pyramid —analyzing individual object characteristics ( J.Shanbehzadeh M.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) • the level j prediction residual outputs are placed in the prediction residual pyramid • Ex. Fig. 7.7 (P=7) • Approximation pyramid--Gaussian pyramid (5x5 low-pass Gaussian kernel) • Prediction residual--Laplacian pyramid • 64x64 Laplacian pyramid predict the Gaussian pyramid’s level 7 prediction residual • First order statistics of the pyramid are highly peaked around zero ( J.Shanbehzadeh M.Gholizadeh )**Image Pyramids**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets the lower-resolution levels of a pyramid can be used for the analysis of large structures or overall image context ( J.Shanbehzadeh M.Gholizadeh )**Image Pyramids**The Haar Transform Background Subband Coding 7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets • Subband Coding ( J.ShanbehzadehM.Gholizadeh )**Subband Coding**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh ) Definition : in subband coding :an image is decomposed into a set of band limited components, called subbands. The decomposition is performed so that the subbands can be reassembled to reconstruct the original image without error. • A filter bank is a collection of two or more filters.**Subband Coding**The goal in subband coding is to select h0(n),h1(n),g0(n),g1(n) so that x(n) = x’(n) . 7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets filters go(n) and g1(n) combine y0(n) and y1 (n) to produce x’(n). ( J.ShanbehzadehM.Gholizadeh )**Subband Coding**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets • An image is decomposed into a set of band-limited component sub-bands, which can be reassemble to reconstruct the original image • Each sub-band is generated by band-pass filtering its I/p • the sub-band can be down sampled without loss of information • Reconstruction of the original image is accomplished by sampling, filtering, and summing the individual sub-band • The principal components of a two-band sub-band coding and decoding system (Fig. 7.4) • The output sequence is formed through the decomposition of x(n) into y0(n) and y1(n) via analysis filter h0(n) and h1(n),and subsequent recombination via synthesis filters g0(n) and g1(n) ( J.Shanbehzadeh M.Gholizadeh )**Subband Coding**For perfect reconstruction, the impulse responses of the synthesis and analysis filters must be related in one of the following two ways: Bio-orthogonal- filter bank satisfying the conditions Filter response of two-band, real coefficient, perfect reconstruction filter bank are subject to bio-orthogonality constraints Orthonormal**Subband Coding**• 1-D orthonormal and biorthogonal filters can be used as 2-D separable • filters for the processing of images. approximation vertical detail horizontal detail diagonal detail • the separable filters are first applied in one dimension (e.g., vertically) and then in the • other(e.g..horizontally) . • Down sampling is performed in two stages-once before the second filtering operation to • reduce the overall number of computations . ( J.ShanbehzadehM.Gholizadeh )**Subband Coding**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets It is easy to show numerically that the filters are both biorthogonal and orthonormal. As a result, it supports error-free reconstruction of the decomposed input. ( J.ShanbehzadehM.Gholizadeh )**Subband Coding**vertical detail approximation • visual effects of aliasing that are present in Figs. 7.7(b) and c. • The wavy lines in the window area are due to the down-sampling of a barely discernable window screen in Fig. 7.1. • Despite the aliasing, the original image can be reconstructed from the subbands in Fig. 7.7 without • error. horizontal detail diagonal detail ( J.ShanbehzadehM.Gholizadeh )**Image Pyramids**Background Subband Coding 7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets The Haar Transform ( J.Shanbehzadeh M.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**The Harr transform**( J.Shanbehzadeh M.Gholizadeh ) Basic functions are the oldest and simplest known orthnormal wavelet Separable and symmetric and can be expressed in matrix form T=HFH where F is an N * N image matrix, H is an N X N Haar transformation matrix, and T is the resulting N X N transform The Harr basic functions are : z€[0 1],k=0,1,2,…,N,N=2^n , k=2^p+q-1,0≤p≤n-1 0 or 1 p=0 q= 0≤q≤2^p p≠0**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.ShanbehzadehM.Gholizadeh )**The Harr transform**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.ShanbehzadehM.Gholizadeh )**The Harr transform**( J.ShanbehzadehM.Gholizadeh )**The Harr transform**( J.ShanbehzadehM.Gholizadeh )**7.1 Background**7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )**7.1 Background**7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh )**7.1 Background**7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )**Problms**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets • Problem 7.1 • Problem 7.2 • Problem 7.7 • Problem 7.4 • Problem 7.5 • Due Date Friday 21/12/88 ( J.ShanbehzadehM.Gholizadeh )**Why is orthogonality useful**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets • Orthonormal bases further simplify the computation ( J.Shanbehzadeh M.Gholizadeh ) ( J.ShanbehzadehM.Gholizadeh )**Ortho v. Non-Ortho Basis**7.1 Background 7.2 Multi Resolution Expansions 7.3 Wavelet Transform in One Dimension 7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions 7.6 Wavelet Packets ( J.ShanbehzadehM.Gholizadeh ) ( J.ShanbehzadehM.Gholizadeh )**Dual Bases**Dual Basis a1-a2 and b1-b2 are biorthogonal ( J.ShanbehzadehM.Gholizadeh )