Compressive sensing with local geometric features

• Main Authors: Gupta, Rishi V. (Contributor), Indyk, Piotr (Contributor), Price, Eric C. (Contributor), Rachlin, Yaron (Author)
• Other Authors: Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory (Contributor), Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science (Contributor)
• Format: Article
• Language: English
• Published: Association for Computing Machinery (ACM), 2012-09-17T18:00:43Z.
• Subjects: Article
• Online Access: Get fulltext

 We propose a framework for compressive sensing of images with local geometric features. Specifically, let x ∈ R[superscript N] be an N-pixel image, where each pixel p has value x[subscript p]. The image is acquired by computing the measurement vector Ax, where A is an m x N measurement matrix for some m l 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 log[subscript k] 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.