3.5 Filtering with the 2D Fourier Transform Basic Low Pass and High Pass Filtering using 2D DFT Other Low Pass Filters

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1 Contents Part I Decomposition and Recovery. Images 1 Filter Banks Introduction Filter Banks and Multirate Systems Discrete Fourier Transforms Modulated Filter Banks Decimators and Interpolators The Polyphase Representation Symmetries and Filter Types Linear Phase FIR Filters with Linear Phase Complementary Filters Symmetries in the Frequency Response Orthogonal FIR Filters Mirror FIR Filters Zeros of FIR Filters. Spectral Factorization Two-Channel Filters and Perfect Reconstruction Automatic Aliasing Cancellation, Perfect Reconstruction Design Approaches for Two-Channel Filter Banks with PR Conditions for Filters and Perfect Reconstruction Aspects of Unity Gain Systems Matrices of Interest Allpass Filters Lattice Structure The Case of 2-Channel Filter Banks Tree-Structured Filter Banks ix

2 x Contents 1.7 Uniform M-Channel Filter Banks Basic Equations Paraunitary M-Channel Filter Banks Cosine-Modulated Filter Banks Linear-Phase Filter Banks IIR Filter Banks Orthogonal IIR Filter Banks Linear Phase Orthogonal IIR Filter Banks Implementation Aspects Experiments Perfect Reconstruction, Music JPEG Watermarking Watermarking in Spectral Domain Watermarking in Signal Domain Resources MATLAB Internet References Wavelets Introduction An Important Example: The Haar Wavelets Definitions Multiresolution Analysis Wavelets and Filter Banks The Multiresolution Analysis Equation Solving the MAE Scaling Functions, Wavelets, and Function Expansions Examples Shannon Wavelets Splines Orthogonal Wavelets Meyer Wavelet Battle-Lemarié Wavelet Daubechies Wavelets Biorthogonal Wavelets Daubechies Approach More Ways to Find Biorthogonal Wavelets Continuous Wavelets The Mexican Hat Wavelet The Morlet Wavelet Complex B-Spline Wavelets

3 Contents xi 2.7 Continuous Wavelet Transform (CWT) The Lifting Method and the Second Generation Wavelets Example Decomposition into Lifting Steps Examples More Analysis Flexibility M-Band Wavelets Wavelet Packets Multiwavelets Experiments ECG Analysis Using the Morlet Wavelet Signal Denoising Compression Applications Earth Sciences Medicine, Biology Chemical Industrial The MATLAB Wavelet Toolbox D Continuous Wavelet D Discrete Wavelet Wavelet Packets Lifting Resources MATLAB Internet References Image and 2D Signal Processing Introduction Image Files and Display Image Files Image Display with MATLAB Basic Image Analysis and Filtering Histograms Histogram Equalization Image Adjust D Filtering with Neighbours Gaussian 2D Filters Picture Sharpening D Fourier Transform D Fourier Transform of Edges D Fourier Transform of a Picture

4 xii Contents 3.5 Filtering with the 2D Fourier Transform Basic Low Pass and High Pass Filtering using 2D DFT Other Low Pass Filters Using 2D DFT Other High-Pass Filters Using 2D DFT Edges Thresholding Edges Color Images RGB Example HSV Example YIQ Example Indexed Images Hough Transform and Radon Transform The Sinogram The Hough Transform The Radon Transform, and Computerized Tomography IPT Functions for the Radon Transform Filter Banks and Images Initial Overview Design of 2D Filters Nonequispaced Data and the Fourier Transform Fourier Transform Versions for the Polar Context Nonequispaced Fourier Transform Experiments Capturing Images with a Webcam Backprojection Steps Resources MATLAB Internet References Wavelet Variants for 2D Analysis Introduction Laplacian Pyramid Steerable Filters and Pyramids Steerable Filters Steerable Pyramid Application of Wavelets to Images Application to a Test Image Application to a Photograph Some Wavelet-Based Algorithms for Image Coding and Compression

5 Contents xiii 4.5 New Wavelets for Images Perspective Wedgelets Ridgelets and First Generation Curvelets Curvelets (Second Generation) Contourlets Bandelets Shearlets Other Wavelet Variants Complex Wavelets Implementation Issues D Application Experiments Level Haar Decomposition of the Image Fine Noise Is Added. Denoising Is Applied Patched Noise Is Added. Denoising Is Applied Display of LL Regions, No Noise Applications Denoising Compression Image Registration Seismic Signals Other Applications Resources MATLAB Internet References Part II Data-Based Actions: Adaptive Filtering, Modelling, Analysis, and Classification 5 Adaptive Filters and Observers Introduction The Wiener Filter Problem Statement. Transfer Function Versions of the Filter Spectral Factorization The Error Surface A Simple Example of Batch Mode and Recursive Mode Recursive Estimation of Filter Coefficients The RLS Method Search-Based Methods

6 xiv Contents 5.4 Adaptive Filters System Identification Inverse System Identification Noise Cancellation Linear Prediction Image Deblurring Motion blur More Adaptive Filters and Some Mathematical Aspects LMS Variants Other Adaptive Filters Mathematical Aspects Unifying Perspective Bayesian Estimation: Application to Images Introduction to Image Restoration Uniform Out-of-Focus Blur Atmospheric Turbulence Blur Linear Motion Blur The Lucy-Richardson Algorithm (RLA) Other Aspects of the Topic Observers The Luenberger Observer Noises Experiments Eigenvalues of Signals Water Fetal Heart Rate Monitoring Some Motivating Applications Resources MATLAB Internet References Experimental Modelling Introduction Data Fitting Coherence. Delays Coherence Between Two Signals Delays Basic Experimental Transfer Function Modelling Two Simple Transfer Function Examples Obtaining a Transfer Function Model from Impulse Response Obtaining a Transfer Function Model from Sine Sweep Obtaining a Transfer Function Model from Response to Noise

7 Contents xv 6.5 The Case of Transfer Functions with Delay Two Simple Examples Responses of Case 1d Responses of Case 2d Detecting the Delay Getting Strange Models Methods for Frequency-Domain Modelling The Levi s Approximation The SK Iterative Weighted Approach The Vector Fitting (VF) Approach Methods for Time-Series Modelling Basic Identification Methods Variants of Recursive Parameter Estimation Experiments AR Model Identification of Canadian Lynx Data Model Order Introduction to the MATLAB System Identification Toolbox Identification Steps Using the Toolbox Functions Using the GUI Resources MATLAB Internet References Data Analysis and Classification Introduction A Basic Idea of Component Analysis Principal Component Analysis (PCA) Mathematical Aspects Principal Components Application Examples Independent Component Analysis (ICA) Blind Source Separation and the Cocktail Party Problem PCA and ICA Whitening Determination of Non-Gaussianity Assumptions of the ICA Method. Independence Contrast Functions Optimization Algorithms Application Examples Clusters. Discrimination Discrimination Clustering

8 xvi Contents Kernels Other Approaches Classification and Probabilities The Expectation-Maximization Algorithm (EM) Naïve Bayes Classifier Quadratic Discriminant Analysis (QDA) Logistic Discriminantion Bayesian Linear Regression. Prediction Sets of Random Variables. Kriging Gaussian Processes (GP) Entropy, Divergence, and Related Aspects Entropy Divergence Jensen s Inequality Variational Bayes Methodology Neurons The Perceptron The Adaline Multilayer Neural Networks Experiments Face Detection Color Reduction Using K-Means Some Pointers to Related Topics Resources MATLAB Internet References Appendix: Long Programs Index

Digital Image Processing

Digital Image Processing Third Edition Rafael C. Gonzalez University of Tennessee Richard E. Woods MedData Interactive PEARSON Prentice Hall Pearson Education International Contents Preface xv Acknowledgments