DIGITAL IMAGE PROCESSING

DIGITAL IMAGE PROCESSING
Instructors: Dr J. Shanbehzadeh M.Gholizadeh ( J.Shanbehzadeh M.Gholizadeh )

Chapter 7 – Wavelet and Multiresolution Processing
DIGITAL IMAGE PROCESSING Chapter 7 – Wavelet and Multiresolution Processing Instructors: Dr J. Shanbehzadeh M.Gholizadeh ( J.Shanbehzadeh M.Gholizadeh )

Road map of chapter 7 Wavelet Transform in Two Dimensions
7.1 7.1 7.2 7.2 7.3 7.7 7.4 7.4 7.5 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 Wavelet Transform in Two Dimensions Wavelet Packets The Fast Wavelet Transform Multi Resolution Expansions Wavelet Transform in One Dimension Background ( J.Shanbehzadeh M.Gholizadeh )

Wavelets and Multi-resolution Processing
Preview What is multi-resolution? - unifies techniques from a variety of disciplines,including subband 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. ( J.Shanbehzadeh M.Gholizadeh )

The difference between Fourier transform and Wavelet transform
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. ( J.Shanbehzadeh M.Gholizadeh )

Background 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) ( J.Shanbehzadeh M.Gholizadeh )

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. Shanbehzadeh M.Gholizadeh )

Background Image Pyramids Image pyramids Subband Coding
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 Image Pyramids Image pyramids Subband Coding The Haar Transform ( J.Shanbehzadeh M.Gholizadeh )

Image Pyramids 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 . 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 )

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 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. ( 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 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.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 )

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 )

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 )

7.4 The Fast Wavelet Transform 7.5 Wavelet Transform in Two Dimensions
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 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.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )

Image Pyramids coarse to fine strategy
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 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 ) ( 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 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 ) ( J.Shanbehzadeh M.Gholizadeh )

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 )

Background Image Pyramids Subband Coding Subband Coding
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 Image Pyramids Subband Coding Subband Coding The Haar Transform ( J.Shanbehzadeh M.Gholizadeh )

Subband Coding Definition :
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 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. ( J.Shanbehzadeh M.Gholizadeh )

filters go(n) and g1(n) combine y0(n) and y1 (n)
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.Shanbehzadeh M.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 approximation vertical detail horizontal detail
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.Shanbehzadeh M.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.Shanbehzadeh M.Gholizadeh )

Subband Coding approximation vertical detail horizontal detail
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.Shanbehzadeh M.Gholizadeh )

Background Image Pyramids Subband Coding The Haar Transform
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 Image Pyramids Subband Coding The Haar Transform ( J.Shanbehzadeh M.Gholizadeh )

The Harr transform 7.1 Background 7.2 Multi Resolution Expansions

The Harr transform 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 or p=0 q= 0≤q≤2^p p≠0 ( J.Shanbehzadeh M.Gholizadeh )

The Harr transform ( 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.Shanbehzadeh M.Gholizadeh )

Problms Problem 7.1 Problem 7.2 Problem 7.7 Problem 7.4 Problem 7.5
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.Shanbehzadeh M.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.Shanbehzadeh M.Gholizadeh )

Ortho v. Non-Ortho Basis

Dual Basis a1-a2 and b1-b2 are biorthogonal Dual Bases
( J.Shanbehzadeh M.Gholizadeh )

Dual Basis (cont) Dual basis may generate different spaces
Verify duality ! Dual basis may generate different spaces Here: a1-a2 and b1-b2 generate two different 2D subspaces in Euclidean 7space. Semiorthogonal: For dual basis that generates the same subspace Orthogonal: Primal and dual are the same bases ( J.Shanbehzadeh M.Gholizadeh )

Multi Resolution Expansions
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 Series Expansions Series Expansions Scaling Functions Wavelet Functions ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )

series expansions closed span of the expansion set expansion set
A signal f(x) can be analyzed as a linear combination of expansion function real-valued expansion functions or basis function real-valued expansion coefficients closed span of the expansion set expansion set Inner product dual functions ( 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 Series Expansions Series Expansions Scaling Functions Wavelet Functions ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )

Scaling functions For J=J0 Scaling functions
- Consider the set of expansion functions composed of integer translations and binary scaling of the real, square-integrable function φ(x),this is the set {φj,k(x)}, where k the position of φj,k(x)along the x-axis j width of φj,k(x) 2^(j/2) controls the amplitude of the function. Because the shape of φj,k(x)changes with j, φ(x) is called a scaling function For J=J0 ( J.Shanbehzadeh M.Gholizadeh )

Scaling functions (a-d) :four of the many expansion
functions that can be generated by substituting this pulse-shaped scaling function into e: shows a member of subspace V1. It does not belong to V0, because the V0 expansion functions in(a,b) are too coarse to represent it. f: the decomposition ofΦ0,0 (x) as a sum of V1 expansion functions. ( J.Shanbehzadeh M.Gholizadeh )

Scaling functions four fundamental requirements of multiresolution analysis : 1) The scaling function is orthogonal to its integer translates. for Haar function, it has a value of 1, its integer translates are 0, so that the product of the two is 0 . 2) The subspaces spanned by the scaling function at low scales are nested within those spanned at higher scales. 7) The only function that is common to all Vj is F(X) = 0.If we consider the coarsest possible expansion functions ( j = -∞), the only representable function is the function of no information. 4) Any function can be represented with arbitrary precision. all measurable, square-integrable functions can be represented by the scaling functions in the limit as j∞. ( J.Shanbehzadeh M.Gholizadeh )

Scaling Functions 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 subspaces containing high-resolution functions must also contain all lower resolution functions. ( J.Shanbehzadeh M.Gholizadeh )

Multi resolution expansions
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 Series Expansions Scaling Functions Wavelet Functions Wavelet Functions ( J.Shanbehzadeh M.Gholizadeh )

hψ (n) : wavelet function coefficients
Wavelet functions wavelet function ψ(x) that, together with its integer translates and binary scaling, spans the difference between any two adjacent scaling subspaces. Vj and Vj+1. The set {ψj,k(x)} of wavelets for all k€Z that span the Wj spaces in the figure. As with scaling functions, we write and if f(x)€Wj Since wavelet spaces reside within the spaces spanned by the next higher solution scaling functions , any wavelet function can be expressed as a weighted sum shifted, double-resolution scaling functions. we can write : hψ (n) : wavelet function coefficients ( J.Shanbehzadeh M.Gholizadeh )

Wavelet functions 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 scaling and wavelet function subspaces in Fig are related by union of subspaces ( J.Shanbehzadeh M.Gholizadeh )

Wavelet functions generate the universe of scaled and translated Haar wavelets Waveletψ1,0(x) for space W1 is narrower than ψ0,2(x) for W0; it can be used to represent finer detail. d : shows a function of subspace V1 that is not in subspace V0. (e-f) divide f(x) in a manner similar to a lowpass and highpass filter ( J.Shanbehzadeh M.Gholizadeh )

added to the approximation to provide increasing detail.
The Wavelet series Expansions defining the wavelet series expansion of function f(x) relative to wavelet ψ(x) and scaling functionφ(x) : (j0 is an arbitrary starting scale ) The c j0(k) are normally called approximation and/or scaling coefficients the dj(k) are referred to as detail and/or wavelet coefficients. For each higher scale j≥j0 in the second sum, a finer resolution function —a sum of wavelets-is added to the approximation to provide increasing detail. ( J.Shanbehzadeh M.Gholizadeh )

Problems 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 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.17 7.14 Due date Monday 25/12/88 ( 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 The Wavelet Series Expansions The Wavelet Series Expansions The Discrete Wavelet Transform The Continuous Wavelet Transform ( J.Shanbehzadeh M.Gholizadeh )

The Wavelet series Expansions

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 Wavelet Series Expansions The Discrete Wavelet Transform The Discrete Wavelet Transform The Continuous Wavelet Transform ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )

The Discrete Wavelet Transform
Like the Fourier series expansion, the wavelet series expansion maps a function of a continuous variable into a sequence of coefficients. If the function being expanded is discrete ,the resulting coefficients are called the discrete wavelet transform (DWT) . the series expansion becomes the DWT transform pair: (x=0,….,M-1) TheWφ(jo, k) and Wψ (j, k) correspond to the cj0(k) and dj(k) of the wavelet series expansion in the previous section. Inverse DWT 1/√M is normalizing factor ( J.Shanbehzadeh M.Gholizadeh )

( J.Shanbehzadeh M.Gholizadeh )

EXAMPLE7.8 ( 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 Series Expansions Scaling Functions Wavelet Functions Wavelet Functions ( J.Shanbehzadeh M.Gholizadeh )

The Continuous Wavelet 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 The natural extension of the discrete wavelet transform Transforms a continuous function into a highly redundant function of two continuous variables(translation and scale ) Inverse CWT ( J.Shanbehzadeh M.Gholizadeh )

Fourier spectrum Mexican hat
b:reveals the close connection between scaled wavelets and Fourier frequency bands. The spectrum contains Two peaks that correspond two Gaussian-like perturbations of the function. c:a portion of the CWT of(a) relative to the Mexican hat wavelet.Unlike(b), it provides both spatial and frequency information d: the absolute value of the transform |Wψ(s,τ ) | is displayed as intensities between black and white ( J.Shanbehzadeh M.Gholizadeh )

t 7.4 the fast wavelet transform

Multiresolution Refinement
scaling x by 2^j. translating it by k, and letting m = 2k + n gives ( J.Shanbehzadeh M.Gholizadeh )

Multiresolution Refinement
interchanging the sum and integral If j0 =j+1 Like above finally we have ( J.Shanbehzadeh M.Gholizadeh )

FWT a computationally efficient implementation of
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 FWT a computationally efficient implementation of the discrete wavelet transform (DWT) . the filter bank can be "iterated" to create multistage structures for computing DWT coefficients at two or more successive scales. ( J.Shanbehzadeh M.Gholizadeh )

splits the original function into a lowpass, approximation component
and a highpass, detail component splits the spectrum and subspace, the lower half-band, into quarter-band subspaces. a two-stage filter bank for generating the coefficients at the two highest scales of the transform.The highest scale coefficients are assumed to be samples of the function itself. ( 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.Shanbehzadeh M.Gholizadeh )

Fast wavelet Transform synthesis filter bank
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 Is identical to the synthesis portion of the two-band subband coding and decoding system in Fig. 7.4(a). ( J.Shanbehzadeh M.Gholizadeh )

and quarantees perfect reconstruction of sequence f(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 a two-scale structure for computing the final two scales of a (FWT)^(-1) reconstruction is depicted. This can be extended to any number of scales and quarantees perfect reconstruction of sequence f(n). ( J.Shanbehzadeh M.Gholizadeh )

Negative indices n<0
shows the sequences that result from the required FWT convolutions and downsamplings. Function f(n) itself is the scaling (approximation) input to the leftmost filter bank. ( J.Shanbehzadeh M.Gholizadeh )

illustrates the process for the sequence considered in Example 7.10.
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 Negative indices n<0 illustrates the process for the sequence considered in Example 7.10. ( J.Shanbehzadeh M.Gholizadeh )

an impulse function basis
sinusoidal (FFT) basis FWT basis low frequencies the tiles are shorter but are wider high frequencies tile width is smaller height is greater . A: events occur but provides no frequency information . Thus, to represent a single frequency sinusoid as an expansion using impulse basis functions, every basis function is required. B: the frequencies in events that occur over long periods but provides no time resolution . Thus, the single frequency sinusoid that was represented by an infinite number of impulse basis functions can be represented as an expansion involving one sinusoidal basis function. The time and frequency resolution of the FWT tiles in (c) vary, but the area of each tile (rectangle) is the same. Thus,the FWT basis functions provide a compromise between the two limiting cases(a) and (b). ( J.Shanbehzadeh M.Gholizadeh )

7.5Wavelet Transforms in Two Dimensions

Wavelet Transforms in Two Dimensions
a two-dimensional scaling function, φ(x, y), and three two dimensional wavelets, ψH (x, y),ψV(x,y),ψD(x,y), are required. Each is the product of two one-dimensional functions. the separable scaling function measures variations along columns (for example, horizontal edges) responds to variations along rows (like vertical edges) corresponds to variations along diagonals ( J.Shanbehzadeh M.Gholizadeh )

Wavelet Transforms in Two Dimensions
scaled and translated basis functions: The discrete wavelet transform of image f(x. y) of size M X N is then coefficients define an approximation of f(x, y) at scale j0 coefficients add horizontal, vertical, and diagonal details for scales j≥j0 Inverse DWT ( J.Shanbehzadeh M.Gholizadeh )

with vertical orientation;
a:Convolving its rows with hφ (-n) and hψ (-n) and downsampling its columns, we get two subimages whose horizontal resolutions are reduced by a factor of 2. Both subimages are then filtered columnwise and downsampled to yield four quarter-size output subimages . The highpass or detail component  the image's high-frequency information with vertical orientation; The lowpass, approximation component  low-frequency, vertical information. b:These subimages, are the inner products of f(x, y) and the two-dimensional scaling and wavelet functions followed by downsampling by two in each dimension. c:the synthesis filter bank reverses the process. At each iteration, four scale j approximation and detail subimages are upsampled and convolved with two one-dimensional filters one operating on the subimages' columns and the other on its rows. Addition of the results yields the scale j + 1approximation, and the process is repeated until the original image is reconstructed. ( J.Shanbehzadeh M.Gholizadeh )

computer-generated image consisting of
Each pass through the filter bank produced four quarter-size output images that were substituted for the input from which they were derived. 2-Dfilter bank of 7.24(a) and the decomposition filters shown in Figs. 7.26(a,b) were used to generate all three results. d: is the three-scale FWT that resulted when the subimage from the upper-left-hand corner of(c) was used as the filter bank input . A similar process for generating the two-scale FWT in (c), but the input to the filter bank was changed to the quarter-size approximation subimage from the upper-left-hand corner of (b). (b) To compute this transform, the original image was used as the input to the filter bank of 7.24(a) . The four resulting quarter-size decomposition outputs (i.e.,on the mechanics of the the approximation and horizontal, vertical, and diagonal details)were then arranged in accordance with Fig. 7.24(b) to produce the image in 7.25(b). computer-generated image consisting of 2-D sine-like pulses on a black background ( J.Shanbehzadeh M.Gholizadeh )

Next example The decomposition filters usd in the preceding example are part of a well known family of wavelets called symlets, short for "symmetrical wavelets.“ (e) and (f) show the fourth-ordervalues. 1-D symlets (wavelet and scaling functions) 7.26(a) through (d) show the corresponding decomposition and reconstruction filters. Figure 7.26(g), a low-resolution graphic depiction of wavelet ψV(x, y), is provided as an illustration of how a one-dimensional scaling and wavelet function can combine to form a separable, two-dimensional wavelet . ( J.Shanbehzadeh M.Gholizadeh )

with vertical orientation;
a:Convolving its rows with hφ (-n) and hψ (-n) and downsampling its columns, we get two subimages whose horizontal resolutions are reduced by a factor of 2. Both subimages are then filtered columnwise and downsampled to yield four quarter-size output subimages . The highpass or detail component  the image's high-frequency information with vertical orientation; The lowpass, approximation component  low-frequency, vertical information. b:These subimages, are the inner products of f(x, y) and the two-dimensional scaling and wavelet functions followed by downsampling by two in each dimension. c:the synthesis filter bank reverses the process. At each iteration, four scale j approximation and detail subimages are upsampled and convolved with two one-dimensional filters one operating on the subimages' columns and the other on its rows. Addition of the results yields the scale j + 1approximation, and the process is repeated until the original image is reconstructed. ( J.Shanbehzadeh M.Gholizadeh )

The coefficients of lowpass reconstruction filter go(n) = hφ(n) for 0 < n < 7 .
The coefficients of the remaining orthonormal filters are obtained using

The coefficients of the remaining orthonormal filters are obtained using ( J.Shanbehzadeh M.Gholizadeh )

As in the Fourier domain, the basic approach is to
Step 1. Compute a 2-D wavelet transform of an image. Step 2. Alter the transform. Step 7. Compute the inverse transform. a: the lowest scale approximation component of the discrete wavelet transform shown in Fig. 7.25(c) has been eliminated by setting its values to zero. 7.27(b) shows, the net effect of computing the inverse wavelet transform using these modified coefficients is edge enhancement. Note how well the transitions between signal and background are delineated, despite the fact that they are relatively soft, sinusoidal transitions. By zeroing the horizontal details as well—see Figs. 7.27(c) and (d)—we can isolate the vertical edges . ( J.Shanbehzadeh M.Gholizadeh )

The coefficients of the remaining orthonormal filters are obtained using ( J.Shanbehzadeh M.Gholizadeh )

Next Example As in the Fourier domain, the basic approach is to
Step 1. Compute a 2-D wavelet transform of an image. Step 2. Alter the transform. Step 7. Compute the inverse transform. a: the lowest scale approximation component of the discrete wavelet transform shown in Fig. 7.25(c) has been eliminated by setting its values to zero. 7.27(b) shows, the net effect of computing the inverse wavelet transform using these modified coefficients is edge enhancement. Note how well the transitions between signal and background are delineated, despite the fact that they are relatively soft, sinusoidal transitions. By zeroing the horizontal details as well—see Figs. 7.27(c) and (d)—we can isolate the vertical edges . ( J.Shanbehzadeh M.Gholizadeh )

Next Example Thresholding
general wavelet-based procedure for denoising the image : Step 1. Choose a wavelet (Haar. symlet ) and number of levels (scales), P, for the decomposition. Then compute the FWT of the noisyimage. Step 2. Threshold the detail coefficients. That is, select and apply a threshold to the detail coefficients .Soft thresholding eliminates the discontinuity (at the threshold) that is inherent in hard thresholding. Step 7. Compute the inverse wavelet transform (i.e., perform a wavelet reconstruction) using the original approximation coefficients at level J - P and the modified detail coefficients for levels J — 1 to J — P. Thresholding hard thresholding,means setting to zero the elements whose absolute values are lower than the threshold soft thresholding, involves first setting to zero the elements whose absolute values are lower than the threshold and then scaling the nonzero coefficients toward zero ( J.Shanbehzadeh M.Gholizadeh )

(b) :result of performing these
operations with fourth-order symlets, two scales (P = 2), and a global threshold(determined interactively) the reduction in noise and blurring of image edges. This loss of edge detail is reduced significantly in (c) f) shows the information that is lost. note the increase in edge information in(f) and the corresponding decrease in edge detail in (e). generated by simply zeroing the highest-resolution detail coefficients and reconstructing e) Reconstruction of the DWT in which the details at both levels of the two-scale transform have been zeroed; shows the information that is lost in the process. which was generated by computing the inverse FWT of the two-scale transform with all but the highest-resolution detail coefficients zeroed ( J.Shanbehzadeh M.Gholizadeh )

7.6 Wavelet Packets

Wavelet Packets Wavelet Packets
If we want greater control over the partitioning of the time-frequency plane , the FWT must be generalized to yield a more flexible decomposition . The cost increase in computational complexity from O(M) for the FWT to O(Mlog M) for a wavelet packet.

approximation that is ultimately filtered to become two leaf nodes .
Figure 7.29(a) links the appropriate FWT scaling and wavelet coefficients to its nodes. The root node is assigned the highest-scale approximation coefficients, which are samples of the function itself, while the leaves inherit the transform‘s approximation and detail coefficient outputs. The lone intermediate node, is a filter bank approximation that is ultimately filtered to become two leaf nodes . replace the generating coefficients in Fig. 7.29(a) by the corresponding subspace. The result is the subspace analysis tree of Fig. 7.29( b). ( J.Shanbehzadeh M.Gholizadeh )

Ihe output of the upper-left filter and subsampler is. to be accurate,
the block diagram of (a) is labeled to resemble the analysis tree in (b). while Ihe output of the upper-left filter and subsampler is. to be accurate, Wψ(J - 1, n), it has been labeled WJ-1. This subspace corresponds to the upper-right leaf of the associated analysis tree, as well as the right most (widest bandwidth) segment of the corresponding frequency spectrum. Analysis trees provide a compact and informative way of representing multiscale wavelet transforms . ( J.Shanbehzadeh M.Gholizadeh ) ( J.Shanbehzadeh M.Gholizadeh )

Note the additional subscripting :
the three-scale FWT analysis tree of 70(b) becomes the three-scale wavelet packet tree of 71. Note the additional subscripting : The first subscript of a double-subscripted node identifies the scale of the FWT parent node from which it descended. The second(a variable length string of As and Ds) encodes the path from the parent to the node. An A designates approximation filtering, while a D indicates detail filtering. ( J.Shanbehzadeh M.Gholizadeh )

7.72(a-b) are the filter bank and spectrum splitting characteristics of the analysis tree in7.71. Note that the "naturally ordered“ output of the filter bank in (a) have been reordered based on frequency content in (b) ( J.Shanbehzadeh M.Gholizadeh )

consider the two-dimensional, four-band filter bank of Fig. 7. 24(a)
consider the two-dimensional, four-band filter bank of Fig. 7.24(a). As in the one dimensional case, it can be "iterated" to generate P scale transforms for scales j = J- 1,... ,J- P, with Wφ(J.m. n) = f(m, n). The spectrum resulting from the first iteration is shown in 7.74(a). Note that it divides the frequency plane into four equal areas.The low-frequency quarter- band in the center of the plane coincides with transform coefficients Wφ(J – 1,m, n) and scaling space Vj-1 b: shows the resulting four-band.single-scale quaternary FWT analysis tree.the superscripts that link the wavelet subspace designations to their transform coefficient counterparts. ( J.Shanbehzadeh M.Gholizadeh )

conventional FWT detail node is the scale of that parent detail node.
Like its one-dimensional , the first subscript of every node that is a descendant of a conventional FWT detail node is the scale of that parent detail node. The second subscript(a variable length string of As, Hs,Vs,Ds)encodes the path from the parent to the node under consideration. ( J.Shanbehzadeh M.Gholizadeh )

efficient algorithm for finding optimal decompositions with
respect to application specific criteria select the "best“ tree-scale wavelet packet decomposition problem : reducing the amount of data needed to represent e fingerprint image in 7.76(a). Using three-scale wavelet packet trees, there are 87,522 potential decompositions. Figure 7.76(b) shows one of them. One reasonable criterion for selecting a decomposition for the compression the image of 7.76(a) is the energy content ,include the dimensional function ( J.Shanbehzadeh M.Gholizadeh )

level by level to the leaves:
For each node of the analysis tree, beginning with the root and proceeding level by level to the leaves: Step 1. Compute both the energy of the node, denoted E (for parent energy), and the energy of its four offspring. Step 2. If the combined energy of the offspring is less than the energy of the Parent include the offspring in the analysis tree. If the combined energy of the offspring is greater than or equal to that of the parent, prune the offspring, keeping only the parent. It is a leaf of the optimized analysis tree. ( J.Shanbehzadeh M.Gholizadeh )

many of the original full packet decomposition's 64 subbands in Fig. 7
many of the original full packet decomposition's 64 subbands in Fig. 7.76(b) have been eliminated. In addition, the subimages that are not split (further decomposed) in Fig are relatively smooth and composed of pixels that are middle gray in value. ( J.Shanbehzadeh M.Gholizadeh )

DIGITAL IMAGE PROCESSING

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