Compressive sensing with local geometric features Citation

We propose a framework for compressive sensing of images with local geometric features. Specifically, let x ∈ R be an N -pixel image, where each pixel p has value xp. The image is acquired by computing the measurement vector Ax, where A is an m×N measurement matrix for some m N . The goal is then to design the matrix A and recovery algorithm which, given Ax, returns an approximation to x. In this paper we investigate this problem for the case where x consists of a small number (k) of “local geometric objects” (e.g., stars in an image of a sky), plus noise. We construct a matrix A and recovery algorithm with the following features: (i) the number of measurements m is O(k logk N), which undercuts currently known schemes that achieve m = O(k log(N/k)) (ii) the matrix A is ultra-sparse, which is important for hardware considerations (iii) the recovery algorithm is fast and runs in time sub-linear in N . We also present a comprehensive study of an application of our algorithm to a problem in satellite navigation.