<p><b>Introduction to Graph Spectral Image Processing </b><b>xi<br /></b><i>Gene CHEUNG and Enrico MAGLI</i></p> <p><b>Part 1. Fundamentals of Graph Signal Processing </b><b>1</b></p> <p><b>Chapter 1. Graph Spectral Filtering </b><b>3<br /></b><i>Yuichi TANAKA</i></p> <p>1.1. Introduction 3</p> <p>1.2. Review: filtering of time-domain signals 4</p> <p>1.3. Filtering of graph signals 5</p> <p>1.3.1. Vertex domain filtering 6</p> <p>1.3.2. Spectral domain filtering 8</p> <p>1.3.3. Relationship between graph spectral filtering and classical filtering 10</p> <p>1.4. Edge-preserving smoothing of images as graph spectral filters 11</p> <p>1.4.1. Early works 11</p> <p>1.4.2. Edge-preserving smoothing 12</p> <p>1.5. Multiple graph filters: graph filter banks 15</p> <p>1.5.1. Framework 16</p> <p>1.5.2. Perfect reconstruction condition 17</p> <p>1.6. Fast computation 20</p> <p>1.6.1. Subdivision 20</p> <p>1.6.2. Downsampling 21</p> <p>1.6.3. Precomputing GFT 22</p> <p>1.6.4. Partial eigendecomposition 22</p> <p>1.6.5. Polynomial approximation 23</p> <p>1.6.6. Krylov subspace method 26</p> <p>1.7. Conclusion 26</p> <p>1.8. References 26</p> <p><b>Chapter 2. Graph Learning </b><b>31<br /></b><i>Xiaowen DONG, Dorina THANOU, Michael RABBAT and Pascal FROSSARD</i></p> <p>2.1. Introduction 31</p> <p>2.2. Literature review 33</p> <p>2.2.1. Statistical models 33</p> <p>2.2.2. Physically motivated models 35</p> <p>2.3. Graph learning: a signal representation perspective 36</p> <p>2.3.1. Models based on signal smoothness 38</p> <p>2.3.2. Models based on spectral filtering of graph signals 43</p> <p>2.3.3. Models based on causal dependencies on graphs 48</p> <p>2.3.4. Connections with the broader literature 50</p> <p>2.4. Applications of graph learning in image processing 52</p> <p>2.5. Concluding remarks and future directions 55</p> <p>2.6. References 57</p> <p><b>Chapter 3. Graph Neural Networks </b><b>63<br /></b><i>Giulia FRACASTORO and Diego VALSESIA</i></p> <p>3.1. Introduction 63</p> <p>3.2. Spectral graph-convolutional layers 64</p> <p>3.3. Spatial graph-convolutional layers 66</p> <p>3.4. Concluding remarks 71</p> <p>3.5. References 72</p> <p><b>Part 2. Imaging Applications of Graph Signal Processing </b><b>73</b></p> <p><b>Chapter 4. Graph Spectral Image and Video Compression </b><b>75<br /></b><i>Hilmi E. EGILMEZ, Yung-Hsuan CHAO and Antonio ORTEGA</i></p> <p>4.1. Introduction 75</p> <p>4.1.1. Basics of image and video compression 77</p> <p>4.1.2. Literature review 78</p> <p>4.1.3. Outline of the chapter 79</p> <p>4.2. Graph-based models for image and video signals 79</p> <p>4.2.1. Graph-based models for residuals of predicted signals 81</p> <p>4.2.2. DCT/DSTs as GFTs and their relation to 1D models 87</p> <p>4.2.3. Interpretation of graph weights for predictive transform coding 88</p> <p>4.3. Graph spectral methods for compression 89</p> <p>4.3.1. GL-GFT design 89</p> <p>4.3.2. EA-GFT design 92</p> <p>4.3.3. Empirical evaluation of GL-GFT and EA-GFT 97</p> <p>4.4. Conclusion and potential future work 100</p> <p>4.5. References 101</p> <p><b>Chapter 5. Graph Spectral 3D Image Compression </b><b>105<br /></b><i>Thomas MAUGEY, Mira RIZKALLAH, Navid MAHMOUDIAN BIDGOLI, Aline ROUMY and Christine GUILLEMOT</i></p> <p>5.1. Introduction to 3D images 106</p> <p>5.1.1. 3D image definition 106</p> <p>5.1.2. Point clouds and meshes 106</p> <p>5.1.3. Omnidirectional images 107</p> <p>5.1.4. Light field images 109</p> <p>5.1.5. Stereo/multi-view images 110</p> <p>5.2. Graph-based 3D image coding: overview 110</p> <p>5.3. Graph construction 115</p> <p>5.3.1. Geometry-based approaches 117</p> <p>5.3.2. Joint geometry and color-based approaches 121</p> <p>5.3.3. Separable transforms 125</p> <p>5.4. Concluding remarks 126</p> <p>5.5. References 128</p> <p><b>Chapter 6. Graph Spectral Image Restoration </b><b>133<br /></b><i>Jiahao PANG and Jin ZENG</i></p> <p>6.1. Introduction 133</p> <p>6.1.1. A simple image degradation model 133</p> <p>6.1.2. Restoration with signal priors 135</p> <p>6.1.3. Restoration via filtering 137</p> <p>6.1.4. GSP for image restoration 140</p> <p>6.2. Discrete-domain methods 141</p> <p>6.2.1. Non-local graph-based transform for depth image denoising 141</p> <p>6.2.2. Doubly stochastic graph Laplacian 142</p> <p>6.2.3. Reweighted graph total variation prior 145</p> <p>6.2.4. Left eigenvectors of random walk graph Laplacian 150</p> <p>6.2.5. Graph-based image filtering 155</p> <p>6.3. Continuous-domain methods 155</p> <p>6.3.1. Continuous-domain analysis of graph Laplacian regularization 156</p> <p>6.3.2. Low-dimensional manifold model for image restoration 163</p> <p>6.3.3. LDMM as graph Laplacian regularization 165</p> <p>6.4. Learning-based methods 167</p> <p>6.4.1. CNN with GLR 169</p> <p>6.4.2. CNN with graph wavelet filter 171</p> <p>6.5. Concluding remarks 172</p> <p>6.6. References 173</p> <p><b>Chapter 7. Graph Spectral Point Cloud Processing </b><b>181<br /></b><i>Wei HU, Siheng CHEN and Dong TIAN</i></p> <p>7.1. Introduction 181</p> <p>7.2. Graph and graph-signals in point cloud processing 183</p> <p>7.3. Graph spectral methodologies for point cloud processing 185</p> <p>7.3.1. Spectral-domain graph filtering for point clouds 185</p> <p>7.3.2. Nodal-domain graph filtering for point clouds 188</p> <p>7.3.3. Learning-based graph spectral methods for point clouds 189</p> <p>7.4. Low-level point cloud processing 190</p> <p>7.4.1. Point cloud denoising 191</p> <p>7.4.2. Point cloud resampling 193</p> <p>7.4.3. Datasets and evaluation metrics 198</p> <p>7.5. High-level point cloud understanding 199</p> <p>7.5.1. Data auto-encoding for point clouds 199</p> <p>7.5.2. Transformation auto-encoding for point clouds 206</p> <p>7.5.3. Applications of GraphTER in point clouds 211</p> <p>7.5.4. Datasets and evaluation metrics 211</p> <p>7.6. Summary and further reading 213</p> <p>7.7. References 214</p> <p><b>Chapter 8. Graph Spectral Image Segmentation </b><b>221<br /></b><i>Michael NG</i></p> <p>8.1. Introduction 221</p> <p>8.2. Pixel membership functions 222</p> <p>8.2.1. Two-class problems 222</p> <p>8.2.2. Multiple-class problems 226</p> <p>8.2.3. Multiple images 227</p> <p>8.3. Matrix properties 230</p> <p>8.4. Graph cuts 232</p> <p>8.4.1. The Mumford-Shah model 234</p> <p>8.4.2. Graph cuts minimization 235</p> <p>8.5. Summary 237</p> <p>8.6. References 237</p> <p><b>Chapter 9. Graph Spectral Image Classification 241<br /></b><i>Minxiang YE, Vladimir STANKOVIC, Lina STANKOVIC and Gene CHEUNG</i></p> <p>9.1. Formulation of graph-based classification problems 243</p> <p>9.1.1. Graph spectral classifiers with noiseless labels 243</p> <p>9.1.2. Graph spectral classifiers with noisy labels 246</p> <p>9.2. Toward practical graph classifier implementation 247</p> <p>9.2.1. Graph construction 247</p> <p>9.2.2. Experimental setup and analysis 249</p> <p>9.3. Feature learning via deep neural network 255</p> <p>9.3.1. Deep feature learning for graph construction 258</p> <p>9.3.2. Iterative graph construction 260</p> <p>9.3.3. Toward practical implementation of deep feature learning 262</p> <p>9.3.4. Analysis on iterative graph construction for robust classification 267</p> <p>9.3.5. Graph spectrum visualization 269</p> <p>9.3.6. Classification error rate comparison using insufficient training data 270</p> <p>9.3.7. Classification error rate comparison using sufficient training data with label noise 270</p> <p>9.4. Conclusion 271</p> <p>9.5. References 272</p> <p><b>Chapter 10. Graph Neural Networks for Image Processing </b><b>277<br /></b><i>Giulia FRACASTORO and Diego VALSESIA</i></p> <p>10.1. Introduction 277</p> <p>10.2. Supervised learning problems 278</p> <p>10.2.1. Point cloud classification 278</p> <p>10.2.2. Po.

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