Structured sampling and fast reconstruction of smooth graph signals

This work concerns sampling of smooth signals on arbitrary graphs. We first study a structured sampling strategy for such smooth graph signals that consists of a random selection of few pre-defined groups of nodes. The number of groups to sample to stably embed the set of $k$-bandlimited signals is driven by a quantity called the \emph{group} graph cumulative coherence. For some optimised sampling distributions, we show that sampling $O(k\log(k))$ groups is always sufficient to stably embed the set of $k$-bandlimited signals but that this number can be smaller -- down to $O(\log(k))$ -- depending on the structure of the groups of nodes. Fast methods to approximate these sampling distributions are detailed. Second, we consider $k$-bandlimited signals that are nearly piecewise constant over pre-defined groups of nodes. We show that it is possible to speed up the reconstruction of such signals by reducing drastically the dimension of the vectors to reconstruct. When combined with the proposed structured sampling procedure, we prove that the method provides stable and accurate reconstruction of the original signal. Finally, we present numerical experiments that illustrate our theoretical results and, as an example, show how to combine these methods for interactive object segmentation in an image using superpixels.