Graph Spectral Image Processing

Recent advent of graph signal processing (GSP) has spurred intensive studies of signals that live naturally on irregular data kernels described by graphs (e.g., social networks, wireless sensor networks). Though a digital image contains pixels that reside on a regularly sampled 2D grid, if one can design an appropriate underlying graph connecting pixels with weights that reflect the image structure, then one can interpret the image (or image patch) as a signal on a graph, and apply GSP tools for processing and analysis of the signal in graph spectral domain. In this article, we overview recent graph spectral techniques in GSP specifically for image / video processing. The topics covered include image compression, image restoration, image filtering and image segmentation

Steerable Discrete Cosine Transform

In image compression, classical block-based separable transforms tend to be inefficient when image blocks contain arbitrarily shaped discontinuities. For this reason, transforms incorporating directional information are an appealing alternative. In this paper, we propose a new approach to this problem, namely a discrete cosine transform (DCT) that can be steered in any chosen direction. Such transform, called steerable DCT (SDCT), allows to rotate in a flexible way pairs of basis vectors, and enables precise matching of directionality in each image block, achieving improved coding efficiency. The optimal rotation angles for SDCT can be represented as solution of a suitable rate-distortion (RD) problem. We propose iterative methods to search such solution, and we develop a fully fledged image encoder to practically compare our techniques with other competing transforms. Analytical and numerical results prove that SDCT outperforms both DCT and state-of-the-art directional transforms

Design and Optimization of Graph Transform for Image and Video Compression

The main contribution of this thesis is the introduction of new methods for designing adaptive transforms for image and video compression. Exploiting graph signal processing techniques, we develop new graph construction methods targeted for image and video compression applications. In this way, we obtain a graph that is, at the same time, a good representation of the image and easy to transmit to the decoder. To do so, we investigate different research directions. First, we propose a new method for graph construction that employs innovative edge metrics, quantization and edge prediction techniques. Then, we propose to use a graph learning approach and we introduce a new graph learning algorithm targeted for image compression that defines the connectivities between pixels by taking into consideration the coding of the image signal and the graph topology in rate-distortion term. Moreover, we also present a new superpixel-driven graph transform that uses clusters of superpixel as coding blocks and then computes the graph transform inside each region. In the second part of this work, we exploit graphs to design directional transforms. In fact, an efficient representation of the image directional information is extremely important in order to obtain high performance image and video coding. In this thesis, we present a new directional transform, called Steerable Discrete Cosine Transform (SDCT). This new transform can be obtained by steering the 2D-DCT basis in any chosen direction. Moreover, we can also use more complex steering patterns than a single pure rotation. In order to show the advantages of the SDCT, we present a few image and video compression methods based on this new directional transform. The obtained results show that the SDCT can be efficiently applied to image and video compression and it outperforms the classical DCT and other directional transforms. Along the same lines, we present also a new generalization of the DFT, called Steerable DFT (SDFT). Differently from the SDCT, the SDFT can be defined in one or two dimensions. The 1D-SDFT represents a rotation in the complex plane, instead the 2D-SDFT performs a rotation in the 2D Euclidean space

Transformées basées graphes pour la compression de nouvelles modalités d’image

Due to the large availability of new camera types capturing extra geometrical information, as well as the emergence of new image modalities such as light fields and omni-directional images, a huge amount of high dimensional data has to be stored and delivered. The ever growing streaming and storage requirements of these new image modalities require novel image coding tools that exploit the complex structure of those data. This thesis aims at exploring novel graph based approaches for adapting traditional image transform coding techniques to the emerging data types where the sampled information are lying on irregular structures. In a first contribution, novel local graph based transforms are designed for light field compact representations. By leveraging a careful design of local transform supports and a local basis functions optimization procedure, significant improvements in terms of energy compaction can be obtained. Nevertheless, the locality of the supports did not permit to exploit long term dependencies of the signal. This led to a second contribution where different sampling strategies are investigated. Coupled with novel prediction methods, they led to very prominent results for quasi-lossless compression of light fields. The third part of the thesis focuses on the definition of rate-distortion optimized sub-graphs for the coding of omni-directional content. If we move further and give more degree of freedom to the graphs we wish to use, we can learn or define a model (set of weights on the edges) that might not be entirely reliable for transform design. The last part of the thesis is dedicated to theoretically analyze the effect of the uncertainty on the efficiency of the graph transforms.En raison de la grande disponibilité de nouveaux types de caméras capturant des informations géométriques supplémentaires, ainsi que de l'émergence de nouvelles modalités d'image telles que les champs de lumière et les images omnidirectionnelles, il est nécessaire de stocker et de diffuser une quantité énorme de hautes dimensions. Les exigences croissantes en matière de streaming et de stockage de ces nouvelles modalités d’image nécessitent de nouveaux outils de codage d’images exploitant la structure complexe de ces données. Cette thèse a pour but d'explorer de nouvelles approches basées sur les graphes pour adapter les techniques de codage de transformées d'image aux types de données émergents où les informations échantillonnées reposent sur des structures irrégulières. Dans une première contribution, de nouvelles transformées basées sur des graphes locaux sont conçues pour des représentations compactes des champs de lumière. En tirant parti d’une conception minutieuse des supports de transformées locaux et d’une procédure d’optimisation locale des fonctions de base , il est possible d’améliorer considérablement le compaction d'énergie. Néanmoins, la localisation des supports ne permettait pas d'exploiter les dépendances à long terme du signal. Cela a conduit à une deuxième contribution où différentes stratégies d'échantillonnage sont étudiées. Couplés à de nouvelles méthodes de prédiction, ils ont conduit à des résultats très importants en ce qui concerne la compression quasi sans perte de champs de lumière statiques. La troisième partie de la thèse porte sur la définition de sous-graphes optimisés en distorsion de débit pour le codage de contenu omnidirectionnel. Si nous allons plus loin et donnons plus de liberté aux graphes que nous souhaitons utiliser, nous pouvons apprendre ou définir un modèle (ensemble de poids sur les arêtes) qui pourrait ne pas être entièrement fiable pour la conception de transformées. La dernière partie de la thèse est consacrée à l'analyse théorique de l'effet de l'incertitude sur l'efficacité des transformées basées graphes

Optimized Update/Prediction Assignment for Lifting Transforms on Graphs

Transformations on graphs can provide compact representations of signals with many applications in denoising, feature extraction or compression. In particular, lifting transforms have the advantage of being critically sampled and invertible by construction, but the efficiency of the transform depends on the choice of a good bipartition of the graph into update (U) and prediction (P) nodes. This is the update/prediction (U=P) assignment problem, which is the focus of this paper. We analyze this problem theoretically and derive an optimal U=P assignment under assumptions about signal model and filters. Furthermore, we prove that the best U=P partition is related to the correlation between nodes on the graph and is not the one that minimizes the number of conflicts (connections between nodes of same label) or maximizes the weight of the cut. We also provide experimental results in randomly generated graph signals and real data from image and video signals that validate our theoretical conclusions, demonstrating improved performance over state of the art solutions for this problem.This work was supported in part by NSF under Grant CCF-1018977 and in part by the Spanish Ministry of Economy and Competitiveness under Grants TEC2014-53390-P, TEC2014-52289-R, TEC2016-81900-REDT/AEI and TEC2017-83838-RPublicad

Perfect Reconstruction Two-Channel Wavelet Filter-Banks for Graph Structured Data

In this work we propose the construction of two-channel wavelet filterbanks for analyzing functions defined on the vertices of any arbitrary finite weighted undirected graph. These graph based functions are referred to as graph-signals as we build a framework in which many concepts from the classical signal processing domain, such as Fourier decomposition, signal filtering and downsampling can be extended to graph domain. Especially, we observe a spectral folding phenomenon in bipartite graphs which occurs during downsampling of these graphs and produces aliasing in graph signals. This property of bipartite graphs, allows us to design critically sampled two-channel filterbanks, and we propose quadrature mirror filters (referred to as graph-QMF) for bipartite graph which cancel aliasing and lead to perfect reconstruction. For arbitrary graphs we present a bipartite subgraph decomposition which produces an edge-disjoint collection of bipartite subgraphs. Graph-QMFs are then constructed on each bipartite subgraph leading to "multi-dimensional" separable wavelet filterbanks on graphs. Our proposed filterbanks are critically sampled and we state necessary and sufficient conditions for orthogonality, aliasing cancellation and perfect reconstruction. The filterbanks are realized by Chebychev polynomial approximations.Comment: 32 pages double spaced 12 Figures, to appear in IEEE Transactions of Signal Processin