Performance Bounds For Co-/Sparse Box Constrained Signal Recovery

Abstract

The recovery of structured signals from a few linear measurements is a central point in both compressed sensing (CS) and discrete tomography. In CS the signal structure is described by means of a low complexity model e.g. co-/sparsity. The CS theory shows that any signal/image can be undersampled at a rate dependent on its intrinsic complexity. Moreover, in such undersampling regimes, the signal can be recovered by sparsity promoting convex regularization like 1 -or total variation (TV-) minimization. Precise relations between many low complexity measures and the sufficient number of random measurements are known for many sparsity promoting norms. However, a precise estimate of the undersampling rate for the TV seminorm is still lacking. We address this issue by: a) providing dual certificates testing uniqueness of a given cosparse signal with bounded signal values, b) approximating the undersampling rates via the statistical dimension of the TV descent cone and c) showing empirically that the provided rates also hold for tomographic measurements.

Superiorization vs. Accelerated Convex Optimization: The Superiorized/Regularized Least-Squares Case

“…Each optimization algorithm is guaranteed to reduce both recovery errors (1) and (2), as listed in the beginning of Section 6.2. The subsampling ratio m/n is chosen according to [KP19] so that recovery via (1.2) is stable. As a consequence, also the recovery error (3) in Section 6.2 is reduced by the optimization algorithm.…”

Superiorization vs. accelerated convex optimization: The superiorized / regularized least-squares case

“…The vector representing this test phantom is denoted by x * in the sequel. The subsampling ratio m/n is chosen according to [43] so that recovery error x − x * with x obtained as minimizer of (1.4) is small as illustrated in Figure 5.2. We use the MATLAB routine paralleltomo.m from the AIR Tools II package [38] that implements such a tomographic matrix for a given vector of projection angles.…”

“…the caption of Figure 5 (1) and (2), listed in the beginning of Subsection 5.2. The sub-sampling ratio m/n is chosen according to [43] so that recovery via (1.4) is stable. As a consequence, also the recovery error (3) in Subsection 5.2 is reduced by the optimization algorithm.…”