Graph Spectral Image Processing. Gene Cheung
of this figure, see www.iste.co.uk/cheung/graph.zipFigure 5.8. Graph topology for point clouds. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 5.9. Graph topology for omnidirectional images based on two different sampling approaches. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 5.10. Example of a rate-distortion optimized graph partition in an omnidirectional image. Left: in the equirectangular domain; right: in the sphere domain. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 5.11. Illustration of the output of the optimization process for a super-ray in four views of a light field. The first row corresponds to a super-ray across four views of the light field. The signal on the vertices corresponds to the color values lying on super-pixels corresponding to the same super-ray and the blue lines denote the correspondences based on the geometrical information in hand. The second to fourth rows are illustrations of basis functions before and after optimization. The signals on the vertices are the eigenvectors values. For a color version of this figure, see www.iste.co.uk/cheung/graph.zip
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 error rate (%) as a function of γ for the two datasets.Figure 9.4. Classification error rate (%) as a function of μ for the two datasets. For a color version of this figure, see www.iste.co.uk/cheung/graph.zipFigure 9.5. Classification error rate (%) as a function of label noise and the threshold τ for (a) MNIST dataset and (b) CIFAR10 dataset. We show classification error rate in color space as improvement when using two iterations, relative to the case with only one iteration;