Digital Image Processing using Local Segmentation Torsten Seemann B. Sc (Hons) School of Computer Science and Software Engineering Faculty of Information Technology Monash University Australia. Submission for the degree of Doctor of Philosophy April 2002 Contents 1 Introduction 1 2 Notation and Terminology 7 2.1 Digital images . 7 2.2 Image statistics . 9 2.2.1 The histogram . 9 2.2.2 The mean . 9 2.2.3 The variance . 10 2.2.4 The entropy . 11 2.3 Image algebra . 11 2.3.1 Image-scalar operations . 11 2.3.2 Image-image operations . 12 2.4 Image acquisition . 13 2.5 Types of noise . 14 2.5.1 Additive noise . 14 2.5.2 Multiplicative noise . 15 2.5.3 Impulse noise . 16 iii iv CONTENTS 2.5.4 Quantization noise . 16 2.5.5 The noise function . 16 2.6 Segmentation . 17 2.7 Local windows . 18 3 Local Segmentation in Image Processing 19 3.1 Introduction . 19 3.2 Global segmentation . 21 3.2.1 Clustering . 25 3.2.2 Spatial segmentation . 32 3.3 Local segmentation . 37 3.3.1 The facet model . 37 3.3.2 Block truncation coding . 38 3.3.3 SUSAN . 39 3.4 Denoising . 41 3.4.1 Temporal filtering . 41 3.4.2 Spatial filtering . 42 3.4.3 Fixed linear filters . 43 3.4.4 Rank filters . 45 3.4.5 Locally adaptive filters . 49 3.4.6 Filtering based on global segmentation . 59 3.5 Conclusions . 60 CONTENTS v 4 Denoising with Local Segmentation 63 4.1 Introduction . 63 4.2 Global image models . 64 4.2.1 The facet model . 65 4.2.2 Generalizing the facet model . 66 4.2.3 Image sampling . 67 4.2.4 Edges and lines . 68 4.2.5 A useful noise model . 69 4.2.6 Pixel quantization . 71 4.2.7 A suitable image model . 72 4.2.8 Image margins . 73 4.3 Measuring image similarity . 76 4.3.1 Visual inspection . 77 4.3.2 Traditional quantitative measures . 78 4.3.3 Difference images . 80 4.4 Test images . 82 4.4.1 Square . 82 4.4.2 Lenna . 82 4.4.3 Montage . 83 4.5 A one segment model . 84 4.6 A two segment model . 86 4.6.1 Application to denoising . 89 vi CONTENTS 4.7 Discriminating between two models . 91 4.7.1 A simple model selection criterion . 92 4.7.2 Student's -test . 96 4.8 Iterative reapplication of the filter . 102 4.9 Rejecting poor local models . 107 4.10 Different local windows . 111 4.11 Multiple overlapping local approximations . 113 4.11.1 Choosing the weights . 115 4.12 Larger numbers of segments . 121 4.12.1 Extending the clustering algorithm . 122 4.12.2 Extending the model order selection technique . 122 4.12.3 Choosing the initial means . 123 4.12.4 Quantitative results for multiple classes . 126 4.13 Estimation of the noise variance . 132 4.14 Comparison to other denoising algorithms . 138 4.14.1 The montage image . 139 4.14.2 The lenna image . 143 4.14.3 The barb2 image . 145 4.15 Conclusions . 147 4.16 Related work . 151 CONTENTS vii 5 Information Theoretic Local Segmentation 153 5.1 Introduction . 153 5.2 Statistical model selection . 154 5.2.1 Maximum likelihood . 155 5.2.2 Interchangeability of probability and code length . 157 5.2.3 Penalized likelihood . 158 5.2.4 Bayesianism . 159 5.2.5 MDL : Minimum Description Length . 160 5.2.6 MML : Minimum Message Length . 161 5.3 Describing segmentations . 163 5.3.1 Possible segmentations . 164 5.3.2 Canonical segmentations . 164 5.3.3 Valid segmentations . 165 5.3.4 Segmentation parameters . 166 5.4 Using MML for local segmentation . 167 5.4.1 A message format . 167 5.4.2 A uniform prior . 169 5.4.3 A worked example . 169 5.4.4 Posterior probability . 171 5.5 Application to denoising . 172 5.6 A better segment map prior . 173 5.6.1 Results . 174 viii CONTENTS 5.7 Optimal quantization of segment means . 174 5.7.1 Results . 178 5.8 Posterior blending of models . 179 5.8.1 Results . 180 5.9 Incorporating “Do No Harm” . 181 5.9.1 The FUELS approach to DNH . 181 5.9.2 Information based DNH . 182 5.10 Combining overlapping estimates . 186 5.10.1 Equal weighting . 187 5.10.2 Message length weightings . 187 5.11 Estimating the noise level . 190 5.12 Evolution of the message length . 191 5.13 Learning from the data . 192 5.13.1 Learning a prior for the segment maps . 194 5.13.2 Learning the prior for DNH models . 196 5.14 Incorporating more models . 199 5.15 Improving the running time . 200 5.16 Results . ..