Chapter 1 Introduction 1
1.1 Motivation 2
1.2 Outline of Book 5
1.3 Original Contributions 7
Chapter 2 Wavelet and Hidden Markov Model(HMM) 10
2.1 Wavelet Analysis 11
2.1.1 Integral Wavelet Transform 11
2.1.2 Discrete Wavelet Transform 13
2.1.3 Multiresolution Analysis(MRA)of L2(R) 14
2.1.4 Wavelet Packet 22
2.1.5 Two Dimensional Wavelets 23
2.2 Hidden Markov Model 28
2.2.1 From Markov Process to Markov Model 28
2.2.2 Elements of an Hidden Markov Model 32
2.2.3 The Three Basic Problems for HMM 34
2.2.4 Solution to Problem 1：Probability Evaluation 36
2.2.5 Solution to Problem 2："Optimal"State Sequence 38
2.2.6 Solution to Problem 3：Parameter Estimation 40
2.2.7 Continuous Observation Densities in HMMs 45
Chapter 3 Image Denoising on Wavelet Domain 48
3.1 Introduction 48
3.2 Important Statistical Preparations 51
3.2.1 Prior and Posterior Distribution 52
3.2.2 Markov Random Field 54
3.2.3 Maximum Likelihood Estimate 56
3.2.4 Expectation-Maximization(EM) 58
3.3 Description of Image Denoising 60
3.3.1 Gaussian White Noise 61
3.3.2 Signal and Noise Ratios 73
3.3.3 Criteria 77
3.4 Wavelet Thresholding 78
3.4.1 Hard and Soft Thresholding 79
3.4.2 Improvements of Wavelet Thresholding 80
3.5 Least Square Estimation 85
3.6 Spatial Image Denoising 90
3.6.1 New Frameworks 93
3.6.2 Denoising Results 96
3.7 Summary 110
Chapter 4 Spatial Wavelet Domain Hidden Markov Model 111
4.1 Preparations 112
4.1.1 Statistical Models and Wavelet 112
4.1.2 Gaussian Mixture Model 115
4.1.3 EM Algorithm 120
4.1.4 Least Square Estimation on Wavelet Domain 125
4.2 Wavelet Domain-Hidden Markov Models 126
4.2.1 Independent Mixture(IM)Model 126
4.2.2 Hidden Markov Tree(HMT)Model 127
4.2.3 Contextual Hidden Markov Model(CHMM) 133
4.3 Spatial Wavelet Domain-Hidden Markov Models 137
4.3.1 Gaussian Markov Field 138
4.3.2 Block 139
4.3.3 Template 144
4.3.4 EM Algorithm of the Block HMM 148
4.3.5 EM Algorithm of the Template HMM 150
4.3.6 Some Views about Improved WD-HMM 151
Chapter 5 Signal Denoising Using the Block HMM 154
5.1 Signal Denoising 156
5.2 The Signal Denoising Algorithm Using the Block HMM 159
5.3 Experimental Results and Discussion 165
5.4 Conclusions 171
Chapter 6 Image Denoising Using the Template HMM 173
6.1 Image Denoising 175
6.2 The Image Denoising Algorithm Using the Template HMM 180
6.3 Experimental Results and Discussion 186
6.4 Conclusions 193
Chapter 7 Conclusions and Perspectives 195
Bibliography 199