and 
7 Chapter 6Figure 6.1. Several representative image restoration problems. The bottom-middle image is the original image. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.2. A simple grid graph . This figure has only showed a part of the graph. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.3. Denoising results of different approaches, with depth map Teddy corrupted by AWGN(σ=10). © 2013 IEEE. Reprinted, with permission, from Hu et al.(2013)Figure 6.4. Illustrations of different kinds of images.(a) A true natural image,(b) a blurry image,(c) a skeleton image, and(d),(e) and(f) are patches in the green squares of(a),(b) and(c), respectively. © 2018 IEEE. Reprinted, with permission, from Bai et al.(2019). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.5. Edge weight distribution around image edges.(a) A true natural patch,(b) a blurry patch, and(c) a skeleton patch. © 2018 IEEE. Reprinted, with permission, from Bai et al.(2019). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.6. Deblurring results comparison.(a) Blurry image.(b) Sun et al.(2013).(c) Michaeli and Irani(2014).(d) Lai et al.(2015).(e) Pan et al.(2016).(f) RGTV. The blur kernel is shown at the lower left corner. © 2018 IEEE. Reprinted, with permission, from Bai et al.(2019). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.7. (a) A patch being optimized encloses a smaller code block. Boundary discontinuity is removed by averaging overlapping patches.(b) The relationship between the dictionary size and the restoration performance. © 2016 IEEE. Reprinted, with permission, from Liu et al.(2017). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.8. Comparison with competing schemes on Butterfly at QF = 5. The corresponding PSNR and SSIM values are shown © 2016 IEEE. Reprinted, with permission, from Liu et al.(2017)Figure 6.9. Sampling the exemplar function fn at pixel locations in domain Ω. © 2017 IEEE. Reprinted, with permission, from Pang and Cheung(2017). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.10. (a) –(c) Different scenarios of using the metric norm as a “pointwise” regularizer.(d) The ideal metric space. The red dots mark the ground-truth gradient. © 2017 IEEE. Reprinted, with permission, from Pang and Cheung(2017). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.11. Denoising of the image Lena, where the original image is corrupted by AWGN with σI = 40. Two cropped fragments of each image are presented for comparison. © 2017 IEEE. Reprinted, with permission, from Pang and Cheung(2017)Figure 6.12. Inpainting of the image Peppers with the low-dimensional manifold model(LDMM), where the corrupted image only keeps 10% of the pixels from the original imageFigure 6.13. Results of real image denoising.(a) Noise clinic(model-based)(Lebrun et al. 2015);(b) CDnCNN(data-driven)(Zhang et al. 2017);(c) DeepGLR. CDnCNN and DeepGLR are trained for Gaussian denoising. ©2019 IEEE. Reprinted, with permission, from Zeng et al.(2019). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.14. Block diagram of the proposed GLRNet that employs a graph Laplacian regularization layer for image denoising. © 2019 IEEE. Reprinted, with permission, from Zeng et al.(2019)Figure 6.15. Block diagram of the overall DeepGLR framework. © 2019 IEEE. Reprinted, with permission, from Zeng et al.(2019)Figure 6.16. DeepAGF framework. Top: Block diagram of the AGFNet, using analytical graph filter for image denoising. Bottom: Block diagram of the N stacks DeepAGF framework. © 2020 IEEE. Reprinted, with permission, from Su et al.(2020). For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 6.17. Denoising result comparison on the Starfish image with input noisy level σ = 70(a) Original,(b) DnCNN(Zhang et al. 2017) and(c) DeepAGF. DnCNN and DeepAGF are trained with noise level σ = 50. © 2020 IEEE. Reprinted, with permission, from Su et al.(2020). For a color version of this figure, see www.iste.co.uk/cheung/graph.zip
8 Chapter 7Figure 7.1. Examples of 3D point clouds. The 3D point cloud in plot shows (a) a 3D Bunny model obtained by range scanners with some postprocessing and (b) one LiDAR sweep directly collected by Velodyne HDL-64E for autonomous drivingFigure 7.2. Graph and graph signals for 3D point clouds. A K-nearest-neighbor graph is constructed to capture the pairwise spatial relationships among 3D points. The values of graph signals are reflected via color. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 7.3. Low-pass approximation of the point cloud Bunny. Plot (a) is the original sampled point cloud with 1,797 points. Plots (b)–(d) show the low-pass approximations with 10, 100 and 1,000 graph frequenciesFigure 7.4. Graph filtering for 3D point clouds. Low-pass graph filtering smooths the sharp transition in a graph signal, while high-pass graph filtering highlights the sharp transition. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 7.5. A synthetic noisy point cloud with Gaussian noise σ = 0.04 for Quasimoto and one denoised result: (a) the ground truth; (b) the noisy point cloud; (c) the denoised result by Hu et al. (2020a)Figure 7.6. Dual problems of 3D point cloud downsampling and upsampling. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 7.7. Local variation based downsampling enhances the contour information (Chen et al. 2018). For a color version of this figure, see Figure 7.8. Two auto-encoder frameworks for 3D point clouds. For a color version of this figure, see Figure 7.9. Graph topology learning and filtering improves the reconstruction of a 3D point cloud (Chen et al. 2019)Figure 7.10. An illustration of the GraphTER model for unsupervised feature learning (Gao et al. 2020). For a color version of this figure, see Figure 7.11. The architecture of the unsupervised feature learning in GraphTER. The representation encoder and transformation decoder are jointly trained by minimizing equation [7.33]. For a color version of this figure, see Figure 7.12. Visual comparison (Gao et al. 2020) of point cloud segmentation between GraphTER and MAP-VAE. For a color version of this figure, see
9 Chapter 8Figure 8.1. An example of the matrix W. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 8.2. (a) The original 480 × 640 image with initial scribbles for three regions (blue, red and green). (b)–(d) The regions viewed against a uniform blue background, respectively. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 8.3. An example of multiple images: two images of cells of benign and malignant types. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 8.4. A one-dimensional example of five grid points for three regions segmentation (blue circle: image pixel; blue arrow : an edge between two grid point vertices; brown arrow: an edge from the source vertex to a grid point vertex; green arrow: an edge from a grid point vertex to the sink vertex; red arrow: an edge from one region to another region. For a color version of this figure, see www.iste.co.uk/cheung/graph.zip
10 Chapter 9Figure 9.1. Classification error rate (%) as a function of node degree k for the two datasets. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 9.2. Classification error rate (%) as a function of labeling ratio for the two datasets. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 9.3. Classification
