# Performance Bounds For Co-/Sparse Box Constrained Signal Recovery

## TL;DR

This paper establishes performance bounds for recovering co-/sparse signals with box constraints using convex regularization, focusing on TV minimization and its application to compressed sensing and tomography.

## Contribution

It introduces dual certificates and statistical dimension analysis to estimate undersampling rates for TV-based recovery, filling a gap in theoretical understanding.

## Key findings

- Derived dual certificates for uniqueness testing.
- Estimated undersampling rates via statistical dimension.
- Validated bounds empirically for tomographic measurements.

## 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 $\ell_1$- 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.