## Compressive sensing with local geometric features

##### Author(s)

Gupta, Rishi V.; Indyk, Piotr; Price, Eric C.; Rachlin, Yaron
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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.

##### Date issued

2011-06##### Department

Massachusetts Institute of Technology. Computer Science and Artificial Intelligence Laboratory; Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science##### Journal

Proceedings of the 27th Annual ACM Symposium on Computational Geometry (SoCG '11)

##### Publisher

Association for Computing Machinery (ACM)

##### Citation

Rishi Gupta, Piotr Indyk, Eric Price, and Yaron Rachlin. 2011. Compressive sensing with local geometric features. In Proceedings of the 27th annual ACM symposium on Computational geometry (SoCG '11). ACM, New York, NY, USA, 87-96.

Version: Author's final manuscript

##### ISBN

978-1-4503-0682-9