Probability distribution free multi-level sampling pattern design for single pixel cameras

Abstract

Traditional compressed sensing matrices like the Gaussian random matrices and Bernoulli matrices are signal structure agnostic due to their adherence to the Restricted Isometry Property (RIP). However, practical measurement operators like the Walsh measurement matrix do not posses RIP and their reconstruction quality can be improved when the measurements are adapted to the signal structure. Sparsity in natural images decay in an asymptotic manner with dense low spatial frequency subbands and sparser higher frequency subbands. Existing methods use decaying power laws to generate a probability map that mimics this sparsity behaviour. Specifying the subbands to sample and the number of measurements to sample in each subband is difficult in such methods. Also generating frequency selective maps is difficult. We propose a sampling pattern design procedure that adheres to the asymptotic sparsity principle but allows selection of the spatial frequency subbands. Bounds are provided such that maximum measurements are allocated to low spatial frequencies. Through experiments on airborne imagery we show that the proposed method works at-par with existing methods for regular imaging requirements. In an unknown environment the proposed method ensures acquisition of spatial frequencies as- sociated with maximum energy and allows easy and flexible design of sampling patterns for frequency selective imaging in specialized scenarios.