Gene Cheung received his PhD in Electrical Engineering and Computer Science from the University of California, Berkeley, USA. He is Associate Professor at York University, Canada, and an IEEE fellow. His research interests include image and graph signal processing.

Enrico Magli is Full Professor at Politecnico di Torino, Italy, and is an IEEE fellow. His research interests are within the field of graph signal processing and deep learning for image and video analysis.

Introduction to Graph Spectral Image Processing xi
Gene CHEUNG and Enrico MAGLI

Part 1. Fundamentals of Graph Signal Processing 1

Chapter 1. Graph Spectral Filtering 3
Yuichi TANAKA

1.1. Introduction 3

1.2. Review: filtering of time-domain signals 4

1.3. Filtering of graph signals 5

1.3.1. Vertex domain filtering 6

1.3.2. Spectral domain filtering 8

1.3.3. Relationship between graph spectral filtering and classical filtering 10

1.4. Edge-preserving smoothing of images as graph spectral filters 11

1.4.1. Early works 11

1.4.2. Edge-preserving smoothing 12

1.5. Multiple graph filters: graph filter banks 15

1.5.1. Framework 16

1.5.2. Perfect reconstruction condition 17

1.6. Fast computation 20

1.6.1. Subdivision 20

1.6.2. Downsampling 21

1.6.3. Precomputing GFT 22

1.6.4. Partial eigendecomposition 22

1.6.5. Polynomial approximation 23

1.6.6. Krylov subspace method 26

1.7. Conclusion 26

1.8. References 26

Chapter 2. Graph Learning 31
Xiaowen DONG, Dorina THANOU, Michael RABBAT and Pascal FROSSARD

2.1. Introduction 31

2.2. Literature review 33

2.2.1. Statistical models 33

2.2.2. Physically motivated models 35

2.3. Graph learning: a signal representation perspective 36

2.3.1. Models based on signal smoothness 38

2.3.2. Models based on spectral filtering of graph signals 43

2.3.3. Models based on causal dependencies on graphs 48

2.3.4. Connections with the broader literature 50

2.4. Applications of graph learning in image processing 52

2.5. Concluding remarks and future directions 55

2.6. References 57

Chapter 3. Graph Neural Networks 63
Giulia FRACASTORO and Diego VALSESIA

3.1. Introduction 63

3.2. Spectral graph-convolutional layers 64

3.3. Spatial graph-convolutional layers 66

3.4. Concluding remarks 71

3.5. References 72

Part 2. Imaging Applications of Graph Signal Processing 73

Chapter 4. Graph Spectral Image and Video Compression 75
Hilmi E. EGILMEZ, Yung-Hsuan CHAO and Antonio ORTEGA

4.1. Introduction 75

4.1.1. Basics of image and video compression 77

4.1.2. Literature review 78

4.1.3. Outline of the chapter 79

4.2. Graph-based models for image and video signals 79

4.2.1. Graph-based models for residuals of predicted signals 81

4.2.2. DCT/DSTs as GFTs and their relation to 1D models 87

4.2.3. Interpretation of graph weights for predictive transform coding 88

4.3. Graph spectral methods for compression 89

4.3.1. GL-GFT design 89

4.3.2. EA-GFT design 92

4.3.3. Empirical evaluation of GL-GFT and EA-GFT 97

4.4. Conclusion and potential future work 100

4.5. References 101

Chapter 5. Graph Spectral 3D Image Compression 105
Thomas MAUGEY, Mira RIZKALLAH, Navid MAHMOUDIAN BIDGOLI, Aline ROUMY and Christine GUILLEMOT

5.1. Introduction to 3D images 106

5.1.1. 3D image definition 106

5.1.2. Point clouds and meshes 106

5.1.3. Omnidirectional images 107

5.1.4. Light field images 109

5.1.5. Stereo/multi-view images 110

5.2. Graph-based 3D image coding: overview 110

5.3. Graph construction 115

5.3.1. Geometry-based approaches 117

5.3.2. Joint geometry and color-based approaches 121

5.3.3. Separable transforms 125

5.4. Concluding remarks 126

5.5. References 128

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