We focus on a multidimensional field with uncorrelated spectrum and study the quality of the reconstructed signal when the field samples are irregularly spaced and affected by independent and identically distributed noise. More specifically, we apply linear reconstruction techniques and take the mean-square error (MSE) of the field estimate as a metric to evaluate the signal reconstruction quality. We find that the MSE analysis could be carried out by using the closed-form expression of the eigenvalue distribution of the matrix representing the sampling system. Unfortunately, such distribution is still unknown. Thus, we first derive a closed-form expression of the distribution moments, and we find that the eigenvalue distribution tends to the Marcenko-Pastur distribution as the field dimension goes to infinity. Finally, by using our approach, we derive a tight approximation to the MSE of the reconstructed field.
The goal of this work is to provide an analytical study on the reconstruction quality of a multidimensional physical field, with uncorrelated spectrum. The field samples are (i) irregularly spaced, since they are gathered by a randomly deployed sensor network and (ii) affected by i.i.d. noise. The sink node receives the field samples and runs the reconstruction algorithm in a centralized manner. Our major contributions with respect to previous work are as follows. [1.] Given a @math -dimensional problem formulation, we obtain analytical expressions for the moments of the eigenvalue distribution of the reconstruction matrix. Using the expressions of the moments, we show that the eigenvalue distribution tends to the Mar c enko-Pastur distribution @cite_26 as the field dimension @math . [2.] We apply our results to the study of the quality of a reconstructed field and derive a tight approximation to the MSE of the estimated field.
@cite_26
2571527823
We study the compressed sensing reconstruction problem for a broad class of random, band-diagonal sensing matrices. This construction is inspired by the idea of spatial coupling in coding theory. As demonstrated heuristically and numerically by Krzakala [30], message passing algorithms can effectively solve the reconstruction problem for spatially coupled measurements with undersampling rates close to the fraction of nonzero coordinates. We use an approximate message passing (AMP) algorithm and analyze it through the state evolution method. We give a rigorous proof that this approach is successful as soon as the undersampling rate δ exceeds the (upper) Renyi information dimension of the signal, d(pX). More precisely, for a sequence of signals of diverging dimension n whose empirical distribution converges to pX, reconstruction is with high probability successful from d(pX) n+o(n) measurements taken according to a band diagonal matrix. For sparse signals, i.e., sequences of dimension n and k(n) nonzero entries, this implies reconstruction from k(n)+o(n) measurements. For “discrete” signals, i.e., signals whose coordinates take a fixed finite set of values, this implies reconstruction from o(n) measurements. The result is robust with respect to noise, does not apply uniquely to random signals, but requires the knowledge of the empirical distribution of the signal pX.