Compound Markov random field model of signals on graph: an application to graph learning

In this work we address the problem of Signal on Graph (SoG) modeling, which can provide a powerful image processing tool for suitable SoG construction. We propose a novel SoG Markovian model suited to jointly characterizing the graph signal values and the graph edge processes. Specifically, we resort to the compound MRF called pixel-edge model formerly introduced in natural images modeling and we reformulate it to frame SoG modeling. We derive the Maximum A Posteriori Laplacian estimator associated to the compound MRF, and we show that it encompasses simpler state-of-the-art estimators for proper parameter settings. Numerical simulations show that the Maximum A Priori Laplacian estimator based on the proposed model outperforms state-of-the-art competitors under different respects. The Spectral Graph Wavelet Transform basis associated to the Maximum A Priori Laplacian estimation guarantees excellent compression of the given SoG. These results show that the compound MRF represents a powerful theoretical tool to characterize the strong and rich interactions that can be found between the signal values and the graph structures, and pave the way to its application to various SoG problems.