Graph adjacency matrix learning for irregularly sampled Markovian natural images

The boost of signal processing on graph has recently solicited research on the problem of identifying (learning) the graph underlying the observed signal values according to given criteria, such as graph smoothness or graph sparsity. This paper proposes a procedure for learning the adjacency matrix of a graph providing support to a set of irregularly sampled image values. Our approach to the graph adjacency matrix learning takes into account both the image luminance and the spatial samples' distances, and leads to a flexible and computationally light parametric procedure. We show that, under mild conditions, the proposed procedure identifies a near optimal graph for Markovian fields; specifically, the links identified by the learning procedure minimize the potential energy of the Markov random field for the signal samples under concern. We also show, by numerical simulations, that the learned adjacency matrix leads to a higly compact spectral wavelet graph transform of the so obtained signal on graph and favourably compares to state-of-the-art graph learning procedures, definetly matching the intrinsic signal structure.