CoE4TN3 Image Processing Image Pyramids Wavelet and Multiresolution Processing 4 Introduction Unlie Fourier transform, whose basis functions are sinusoids, wavelet transforms are based on small waves, called wavelets, of limited duration. Fourier transform provides only frequency information, but wavelet transform provides time-frequency information. Wavelets lead to a multiresolution analysis of signals. Multiresolution analysis: representation of a signal (e.g., an images) in more than one resolution/scale. Features that might go undetected at one resolution may be easy to spot in another. Image pyramids At each level we have an approximation image and a residual image. The original image (which is at the base of pyramid) and its P approximation form the approximation pyramid. The residual outputs form the residual pyramid. Approximation and residual pyramids are computed in an iterative fashion. A P+ level pyramid is build by executing the operations in the bloc diagram P times. 5 Multiresolution Image pyramids During the first iteration, the original J x J image is applied as the input image. This produces the level J- approximate and level J prediction residual results For iterations j=j-, J-,, J-p+, the previous iteration s level j- approximation output is used as the input. 3 6
Image pyramids Each iteration is composed of three sequential steps:. Compute a reduced resolution approximation of the input image. This is done by filtering the input and downsampling (subsampling) the filtered result by a factor of. Filter: neighborhood averaging, Gaussian filtering The quality of the generated approximation is a function of the filter selected Subband coding In subband coding, an image is decomposed into a set of bandlimited components, called subbands. Since the bandwidth of the resulting subbands is smaller than that of the original image, the subbands can be downsampled without loss of information. 7 0 Image pyramids. Upsample output of the previous step by a factor of and filter the result. This creates a prediction image with the same resolution as the input. By interpolating intensities between the pixels of step, the interpolation filter determines how accurately the prediction approximates the input to step. 3. Compute the difference between the prediction of step and the input to step. This difference can be later used to reconstruct progressively the original image 8 Perfect Reconstruction Filter Z transform: ˆ X ( z ) = [ H ] 0( z) G0( z) + H( z) G( z) X( z) + [ H0( zg ) 0( z) + H( zg ) ( z) ] X( z) Goal: find H 0, H, G 0 and G so that x( n) = xˆ ( n) ( ie.. X ( z ) = Xˆ ( z ) ) Perfect Reconstruction Filter: Conditions If Then H0( z) G0( z) + H( z) G( z) = 0 H0( z) G0( z) + H( z) G( z) = X( z) = Xˆ ( z) 9
Perfect Reconstruction Filter Families QMF: quadrature mirror filters CQF: conjugate mirror filters 3 6 -D The Haar Transform Haar proposed the Haar Transform in 90, more than 70 years before the wavelet theory was born. Actually, Haar Transform employs the Haar wavelet filters but is expressed in a matrix form. Haar wavelet is the oldest and simplest wavelet basis. Haar wavelet is the only one wavelet basis, which holds the properties of orthogonal, (anti-)symmetric and compactly supported. 4 7 Example of Filters The Haar Wavelet Filters h 0 = {, } g 0 = {, } h = {, } g 0 = {, } 5 8 3
Multiresolution Expansions Scaling functions Integer translations and dyadic scalings of a scaling function j x x j/ j, ( ) = ( ) Express f as the combination of j ( ) 0, x f = α j0, 9 Multiresolution Expansions Series Expansions A function can be expressed as f = α where α, f * f dx = = Dual function of Complex conjugate operation * 0 0 x < = otherwise 0, =, + (, x ) + f = 0.5,0 + 0.5,,4 0 3 Multiresolution Expansions Series Expansions Orthonormal basis = 0 j j, = j = biorthogonal j, = 0 j 0 j j, = j = Multiresolution Expansions Scaling functions Dilation equation for scaling function = h ( n) ( x n) n h ( n ) are called scaling function coefficients i Example: Haar wavelet, h (0) = h () = = (x ) + 4 4
Multiresolution Expansions Wavelet functions ψ = h ( n) ( x n) n ψ h are called wavelet function coefficients ψ ( n) Translation and scaling of ψ ψ j x ψ x j/ j, ( ) = ( ) condition for orthogonal wavelets n h ( n) = ( ) h ( n) ψ Wavelet Transform: -D Wavelet series expansion f = c + d ( ) ψ where j0 j0, j j, j= j0 c ( ) f, f dx = = j0 j0, j0, d ( ) f, ψ f ψ dx = = j j, j, 5 8 yx ( ) = 0,0 3 ψ 0,0 4 ψ,0 ( x ) 3 3 ψ, 3 +... 6 9 Haar Wavelet 0 x < 0.5 = 0.5 x< 0 elsewhere Wavelet Transform: -D Discrete Wavelet Transform f = W ( j0, ) j0, M + Wψ ( j, ) ψ j, M j = j0 where Approximation W ( j0, ) = f j0, coefficients M x Detail Wψ ( j, ) = f ψ j, coefficients M x 7 30 5
Fast Wavelet Transform: Decomposition Fast Wavelet Transform: Reconstruction W ( j, ) = h ( n) W ( j+, n) n ψ ψ =, 0 W ( j+, ) = h ( ) W ( j, ) + h ( ) W ( j, ) up up ψ ψ 0 W ( j, ) = h ( n) W ( j+, n) n =, 0 3 34 Fast Wavelet Transform: Decomposition Fast Wavelet Transform: Reconstruction 3 35 Example: Haar Wavelet n = 0 / n = 0, h ( n) = hψ ( n) = n= 0 otherwise 0 otherwise Fast Wavelet Transform: Reconstruction 33 36 6
Wavelet Transform vs. Fourier Transform -D Wavelet Transform: Reconstruction 37 40 Wavelet Transform: -D Scaling function: ( x, y) = ( y) Wavelet functions: ψ H ( x, y) = ψ ( y) Horizontal direction ψ V ( x, y) ψ( y) = Vertical direction ψ D ( x, y) ψ ψ( y) = Diagonal direction 38 4 -D Wavelet Transform: Decomposition 39 4 7
Fig. 7.4 (g) Wavelet Transform based Denoising Three Steps: Decompose the image into several scales. For each wavelet coefficient y: y y t Hard thresholding: y = 0 y < t sign( y) i( y t) y t Soft thresholding: y = 0 y < t Reconstruct the image with the altered wavelet coefficients. 43 46 Image Processing by Wavelet Transform Three Steps: Decompose the image into wavelet domain Alter the wavelet coefficients, according to your applications such as denoising, compression, edge enhancement, etc. Reconstruct the image with the altered wavelet coefficients. 44 47 Assignment Get familiar with the Matlab Wavelet Toolbox. By using the Wavelet Toolbox functions, write a program to realize the softthresholding denoising on a noisy MRI image. 45 48 8
End of the lecture 49 9