Smoothness on rank-order path graphs and its use in compressive spectral imaging with side information

This paper proposes a novel reconstruction approach to compressive spectral imaging (CSI) with panchromatic side information, which is based on the notion of approximate rank-order statistics. To that end, we assume that the signal of interest is sufficiently smooth on an unknown graph. When restricted to the family of path graphs, we show that the best path is indeed the rank-order path graph induced by the signal. That is, the path graph whose edge structure is given by the permutation that sorts the entries of the signal in ascending order. Our goal is to show that smoothness on rank-order path graphs inferred from the rank-order statistics of a co-registered panchromatic signal can be used to find accurate spectral image estimates from a compressive snapshot of the scene. We derive theoretical properties of rank-order path graphs and give illustrative examples of their use in signal recovery from undersampled measurements. Our approach leads to solutions with a closed-form, found efficiently by iterative inversion of highly sparse systems of linear equations. We evaluate our method through an experimental demonstration and extensive simulations. Our method performs notably better against a bilateral-filter graph model, adapted to the task, and some traditional and state-of-the-art algorithms.