Regularized Modified BPDN for Noisy Sparse Reconstruction with Partial Erroneous Support and Signal Value Knowledge

TL;DR

This paper introduces a regularized modified-BPDN method for noisy sparse signal reconstruction that leverages partial, possibly erroneous prior support and signal value knowledge, providing error bounds and empirical performance comparisons.

Contribution

The paper proposes a novel regularized modified-BPDN algorithm that incorporates prior support and signal estimates, with computable error bounds and empirical validation.

Findings

Provides error bounds without sufficient conditions

Demonstrates improved reconstruction accuracy

Offers empirical comparisons with existing methods

Abstract

We study the problem of sparse reconstruction from noisy undersampled measurements when the following two things are available. (1) We are given partial, and partly erroneous, knowledge of the signal's support, denoted by $T$. (2) We are also given an erroneous estimate of the signal values on $T$, denoted by $(\hat{\mu})_T$. In practice, both these may be available from available prior knowledge. Alternatively, in recursive reconstruction applications, like real-time dynamic MRI, one can use the support estimate and the signal value estimate from the previous time instant as $T$ and $(\hat{\mu})_T$. In this work, we introduce regularized modified-BPDN (reg-mod-BPDN) and obtain computable bounds on its reconstruction error. Reg-mod-BPDN tries to find the signal that is sparsest outside the set $T$, while being "close enough" to $(\hat{\mu})_T$ on $T$ and while satisfying the data constraint. Corresponding results for modified-BPDN and BPDN follow as direct corollaries. A second key contribution is an approach to obtain computable error bounds that hold without any sufficient conditions. This makes it easy to compare the bounds for the various approaches. Empirical reconstruction error comparisons with many existing approaches are also provided.