Estimation algorithms with noisy frame coefficients

Alexander M. Powell

doi:10.1117/12.732748

20 September 2007 Estimation algorithms with noisy frame coefficients

Alexander M. Powell

Proceedings Volume 6701, Wavelets XII; 67010U (2007) https://doi.org/10.1117/12.732748
Event: Optical Engineering + Applications, 2007, San Diego, California, United States

Abstract

The Rangan-Goyal (RG) algorithm is a recursive method for constructing an estimate x_N ∈ R^d of a signal x ∈ R^d, given N ≥ d frame coefficient measurements of x that have been corrupted by uniform noise. Rangan and Goyal proved that the RG-algorithm is constrained by the Bayesian lower bound: lim inf_N→∞N² E||x − x_N||² > 0. As a positive counterpart to this, they also proved that for every p < 1 and x ∈ R^d, the RG-algorithm satisfies lim_N→∞ N^p||x − x_N|| = 0 almost surely. One consequence of the existing results is that one "almost" has mean square error E||x − x_N||² of order 1/N² for random choices of frames. It is proven here that the RG-algorithm achieves mean square error of the optimal order 1/N², and the applicability of such error estimates is also extended to deterministic frames where ordering issues play an important role. Approximation error estimates for consistent reconstruction are also proven.

Citation Download Citation

Alexander M. Powell "Estimation algorithms with noisy frame coefficients", Proc. SPIE 6701, Wavelets XII, 67010U (20 September 2007); https://doi.org/10.1117/12.732748

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