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

A unified framework for spectral clustering in sparse graphs

This article considers spectral community detection in the regime of sparse networks with heterogeneous degree distributions, for which we devise an algorithm to efficiently retrieve communities. Specifically, we demonstrate that a conveniently parametrized form of regularized Laplacian matrix can be used to perform spectral clustering in sparse networks, without suffering from its degree heterogeneity. Besides, we exhibit important connections between this proposed matrix and the now popular non-backtracking matrix, the Bethe-Hessian matrix, as well as the standard Laplacian matrix. Interestingly, as opposed to competitive methods, our proposed improved parametrization inherently accounts for the hardness of the classification problem. These findings are summarized under the form of an algorithm capable of both estimating the number of communities and achieving high-quality community reconstruction

Approximating Spectral Clustering via Sampling: a Review

International audienceSpectral clustering refers to a family of well-known unsupervised learning algorithms. Rather than attempting to cluster points in their native domain, one constructs a (usually sparse) similarity graph and computes the principal eigenvec-tors of its Laplacian. The eigenvectors are then interpreted as transformed points and fed into a k-means clustering algorithm. As a result of this non-linear transformation , it becomes possible to use a simple centroid-based algorithm in order to identify non-convex clusters, something that was otherwise impossible. Unfortunately , what makes spectral clustering so successful is also its Achilles heel: forming a graph and computing its dominant eigenvectors can be computationally prohibitive when dealing with more that a few tens of thousands of points. In this chapter, we review the principal research efforts aiming to reduce this computational cost. We focus on methods that come with a theoretical control on the clustering performance and incorporate some form of sampling in their operation. Such methods abound in the machine learning, numerical linear algebra, and graph signal processing literature and, amongst others, include Nyström-approximation, landmarks, coarsening, coresets, and compressive spectral clustering. We present the approximation guarantees available for each and discuss practical merits and limitations. Surprisingly, despite the breadth of the literature explored, we conclude that there is still a gap between theory and practice: the most scalable methods are only intuitively motivated or loosely controlled, whereas those that come with end-to-end guarantees rely on strong assumptions or enable a limited gain of computation time

Approximating Spectral Clustering via Sampling: a Review

Spectral clustering refers to a family of unsupervised learning algorithms that compute a spectral embedding of the original data based on the eigenvectors of a similarity graph. This non-linear transformation of the data is both the key of these algorithms' success and their Achilles heel: forming a graph and computing its dominant eigenvectors can indeed be computationally prohibitive when dealing with more that a few tens of thousands of points. In this paper, we review the principal research efforts aiming to reduce this computational cost. We focus on methods that come with a theoretical control on the clustering performance and incorporate some form of sampling in their operation. Such methods abound in the machine learning, numerical linear algebra, and graph signal processing literature and, amongst others, include Nystr\"om-approximation, landmarks, coarsening, coresets, and compressive spectral clustering. We present the approximation guarantees available for each and discuss practical merits and limitations. Surprisingly, despite the breadth of the literature explored, we conclude that there is still a gap between theory and practice: the most scalable methods are only intuitively motivated or loosely controlled, whereas those that come with end-to-end guarantees rely on strong assumptions or enable a limited gain of computation time

Data-Driven Shape Analysis and Processing

Data-driven methods play an increasingly important role in discovering geometric, structural, and semantic relationships between 3D shapes in collections, and applying this analysis to support intelligent modeling, editing, and visualization of geometric data. In contrast to traditional approaches, a key feature of data-driven approaches is that they aggregate information from a collection of shapes to improve the analysis and processing of individual shapes. In addition, they are able to learn models that reason about properties and relationships of shapes without relying on hard-coded rules or explicitly programmed instructions. We provide an overview of the main concepts and components of these techniques, and discuss their application to shape classification, segmentation, matching, reconstruction, modeling and exploration, as well as scene analysis and synthesis, through reviewing the literature and relating the existing works with both qualitative and numerical comparisons. We conclude our report with ideas that can inspire future research in data-driven shape analysis and processing.Comment: 10 pages, 19 figure

Courbure discrète : théorie et applications

International audienceThe present volume contains the proceedings of the 2013 Meeting on discrete curvature, held at CIRM, Luminy, France. The aim of this meeting was to bring together researchers from various backgrounds, ranging from mathematics to computer science, with a focus on both theory and applications. With 27 invited talks and 8 posters, the conference attracted 70 researchers from all over the world. The challenge of finding a common ground on the topic of discrete curvature was met with success, and these proceedings are a testimony of this wor

From Dynamics to Structure of Complex Networks: Exploiting Heterogeneity in the Sakaguchi-Kuramoto Model

[eng] Most of the real-world complex systems are best described as complex networks and can be mathematically described as oscillatory systems, coupled with the neighbours through the connections of the network. The flashing of fireflies, the neuronal brain signals or the energy flow through the power grid are some examples. Yoshiki Kuramoto came up with a tractable mathematical model that could capture the phenomenology of collective synchronization by suggesting that oscillators were coupled by a sinusoidal function of their phase differences. Later, Yoshiki Kuramoto together with Hidetsugu Sakaguchi presented a generalization of the previous limit-cycle set of oscillators Kuramoto’s model which incorporated a constant phase lag between oscillators. Subsequent studies of the model included the network structure within the model together with the global shift. For a wide range of the phase lag values, the system becomes synchronized to a resulting frequency, i.e., the dynamics reaches a stationary state. In the original work of Kuramoto and Sakaguchi and in most of the consequent later studies, a uniform distribution of phase lag parameters is customarily assumed. However, the intrinsic properties of nodes – that assuredly represent the constituents of real systems – do not need be identical but distributed non-homogeneously among the population. This thesis contributes to the understanding of the Kuramoto-Sakaguchi model with a generalization for nonhomogeneous phase lag parameter distribution. Considering different scenarios concerning the distribution of the frustration parameter among the oscillators represents a major step towards the extension of the original model and provides significant novel insights into the structure and function of the considered network. The first setting that the present thesis considers consists in perturbing the stationary state of the system by introducing a non-zero phase lag shift into the dynamics of a single node. The aim of this work is to sort the nodes by their potential effect on the whole network when a change on their individual dynamics spreads over the entire oscillatory system by disrupting the otherwise synchronized state. In particular, we define functionability, a novel centrality measure that addresses the question of which are the nodes that, when individually perturbed, are best able to move the system away from the fully synchronized state. This issue may be relevant for the identification of critical nodes that are either beneficial – by enabling access to a broader spectrum of states – or harmful – by destroying the overall synchronization. The second scenario that the present thesis addresses considers a more general configuration in which the phase lag parameter is an intrinsic property of each node, not necessarily zero, and hence exploring the potential heterogeneity of the frustration among oscillators. We obtain the analytical solution of frustration parameters so as to achieve any phase configuration, by linearizing the most general model. We also address the fact that the question ’among all the possible solutions, which is the one that makes the system achieve a particular phase configuration with the minimum required cost?’ is of particular relevance when we consider the plausible real nature of the system. Finally, the homogenous distribution of phase lag parameters is revisited in the last scenario. As studied in the literature, a certain degree of symmetry is an attribute of real-world networks. Nevertheless, beyond structural or topological symmetry, one should consider the fact that real- world networks are exposed to fluctuations or errors, as well as mistaken insertions or removals. In the present thesis, we provide an alternative notion to approximate symmetries, which we call ‘Quasi-Symmetries’ and are defined such that they remain free to impose any invariance of a particular network property and are obtained from the stationary state of the Kuramoto-Sakaguchi model with a homogeneous phase lag distribution. A first contribution is exploring the distributions of structural similarity among all pairs of nodes. Secondly, we define the ‘dual network’, a weighted network –and its corresponding binarized counterpart– that effectively encloses all the information of quasi-symmetries in the original one.[cat] La major part dels sistemes complexos presents en la natura i la societat es poden descriure com a xarxes complexes. Molts d’aquests sistemes es poden modelitzar matemàticament com un sistema oscil·latori, on les unitats queden acoblades amb els components veïns a través de les connexions de la xarxa. Yoshiki Kuramoto i Hidetsugu Sakaguchi van presentar la generalització del ben conegut model d’oscil·ladors de Kuramoto, on s’incorporava un terme de desfasament entre parelles d’oscil·ladors. Aquesta tesi contribueix en la comprensió d’aquest model, tot considerant una distribució no homogènia d’aquest paràmetre de desfasament o frustració. S’han considerat tres escenaris diferents, tots ells donant lloc a resultats que permeten una millor descripció de l’estructura i funció de la xarxa que s’està considerant. Una primera configuració consisteix en pertorbar l’estat estacionari tot introduint un desfasament en la dinàmica d’un node de manera aïllada. Seguidament, definim la funcionabilitat, una mesura de centralitat única que respon a la pregunta de, quins nodes, quan són pertorbats individualment, són més capaços d’allunyar el sistema de l’estat sincronitzat. Aquest fet podria suposar un comportament beneficiós o perjudicial per sistemes reals. El segon escenari considera la configuració més flexible, explorant la potencial heterogeneïtat dels paràmetres de frustració dels diferents nodes. Obtenim la solució analítica d’aquesta distribució per tal d’assolir qualsevol configuració de les fases dels oscil·ladors, a través de la linearització del model. També contestem a la pregunta: “de totes les possibles solucions, quina és la que implica un menor cost per tal d’assolir una configuració en particular?”. Finalment, en l’últim escenari, proporcionem una definició alternativa al concepte de simetria aproximada d’una xarxa, i que anomenem “Quasi simetries”. Aquestes són definides sense imposar invariàncies en les propietats del sistema, sinó que s’obtenen de l’estat estacionari del model de Kuramoto-Sakaguchi model, tot considerant una distribució homogènia dels paràmetres de frustració

Data-driven shape analysis and processing

Data-driven methods serve an increasingly important role in discovering geometric, structural, and semantic relationships between shapes. In contrast to traditional approaches that process shapes in isolation of each other, data-driven methods aggregate information from 3D model collections to improve the analysis, modeling and editing of shapes. Through reviewing the literature, we provide an overview of the main concepts and components of these methods, as well as discuss their application to classification, segmentation, matching, reconstruction, modeling and exploration, as well as scene analysis and synthesis. We conclude our report with ideas that can inspire future research in data-driven shape analysis and processing