Signaux stationnaires sur graphe : étude d'un cas réel

National audienceBased on a real geographical dataset, we apply the stationarity characterisation of a graph signal, through the analysis of its spectral decomposition. In the course, we identify possible sources of non-stationarity and we elaborate on the impact of the graph used to model the structural coherence of the data.Sur un jeu de données géographiques réelles, nous appliquons la caractérisation de la propriété de stationnarité d'un signal sur graphe via l'analyse de ses coefficients spectraux. Nous identifions différentes sources possibles de non-stationnarité et isolons l'influence qu'a le graphe sous-jacent sur la cohérence structurelle des données

On localisation and uncertainty measures on graphs

Due to the appearance of data on networks such as internet or Facebook, the number of applications of signal on weighted graph is increasing. Unfortunately, because of the irregular structure of this data, classical signal processing techniques are not applicable. In this paper, we examine the windowed graph Fourier transform (WGFT) and propose ambiguity functions to analyze the spread of the window in the vertex-frequency plane. We then observe through examples that there is a trade-off between the vertex and frequency resolution. This matches our intuition form classical signal processing. Finally, we demonstrate an uncertainty principle for the spread of the ambiguity function. We verify with examples that this principle is sharp for the extreme values of and emphasize the difference between the generalized graph ambiguity function and the classical one. We finish with demonstration of some Young and Hausdorff-Young like inequalities for graphs

Learning parametric dictionaries for graph signals

In sparse signal representation, the choice of a dictionary often involves a tradeoff between two desirable properties -- the ability to adapt to specific signal data and a fast implementation of the dictionary. To sparsely represent signals residing on weighted graphs, an additional design challenge is to incorporate the intrinsic geometric structure of the irregular data domain into the atoms of the dictionary. In this work, we propose a parametric dictionary learning algorithm to design data-adapted, structured dictionaries that sparsely represent graph signals. In particular, we model graph signals as combinations of overlapping local patterns. We impose the constraint that each dictionary is a concatenation of subdictionaries, with each subdictionary being a polynomial of the graph Laplacian matrix, representing a single pattern translated to different areas of the graph. The learning algorithm adapts the patterns to a training set of graph signals. Experimental results on both synthetic and real datasets demonstrate that the dictionaries learned by the proposed algorithm are competitive with and often better than unstructured dictionaries learned by state-of-the-art numerical learning algorithms in terms of sparse approximation of graph signals. In contrast to the unstructured dictionaries, however, the dictionaries learned by the proposed algorithm feature localized atoms and can be implemented in a computationally efficient manner in signal processing tasks such as compression, denoising, and classification