This paper introduces the Reed Muller Sieve, a deterministic measurement matrix for compressed sensing. The columns of this matrix are obtained by exponentiating codewords in the quaternary second order Reed Muller code of length N. For k = O(N), the Reed Muller Sieve improves upon prior methods for identifying the support of a k-sparse vector by removing the requirement that the signal entries be independent. The Sieve also enables local detection; an algorithm is presented with complexity N 2  logN that detects the presence or absence of a signal at any given position in the data domain without explicitly reconstructing the entire signal. Reconstruction is shown to be resilient to noise in both the measurement and data domains; the l2/l2 error bounds derived in this paper are tighter than the l2/l1 bounds arising from random ensembles and the l1/l1 bounds arising from expander-based ensembles.
['A. Robert Calderbank', 'Stephen D. Howard', 'Sina Jafarpour']
Sparse reconstruction via the Reed-Muller Sieve