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.

Date Published: 20 September 2007
PDF: 12 pages
Proc. SPIE 6701, Wavelets XII, 67010U (20 September 2007); doi: 10.1117/12.732748

Published in SPIE Proceedings Vol. 6701:
Wavelets XII
Dimitri Van De Ville; Vivek K. Goyal; Manos Papadakis, Editor(s)