۱- Introduction

۲- Background

۳- Recovery with multiple prior information

۴- Bounds for weighted n-ℓ۱ minimization

۵- Experimental results

۶- Conclusion

References

Abstract

We address the problem of reconstructing a sparse signal from compressive measurements with the aid of multiple known correlated signals. We propose a reconstruction algorithm with multiple side information signals (RAMSI), which solves an n-`1 minimization problem by weighting adaptively the multiple side information signals at every iteration. In addition, we establish theoretical bounds on the number of measurements required to guarantee successful reconstruction of the sparse signal via weighted n-`1 minimization. The analysis of the derived bounds reveals that weighted n-`1 minimization can achieve sharper bounds and significant performance improvements compared to classical compressed sensing (CS). We evaluate experimentally the proposed RAMSI algorithm and the established bounds using numerical sparse signals. The results show that the proposed algorithm outperforms state-of-the-art algorithms—including classical CS, `1-`1 minimization, Modified-CS, regularized Modified-CS, and weighted `1 minimization—in terms of both the theoretical bounds and the practical performance.

Introduction

Compressed sensing (CS) [1–۱۵] states that sparse signals can be recovered in a computationally tractable manner from a limited set of measurements by minimizing the `1-norm. The CS performance can be improved by replacing the `1-norm with a weighted `1-norm [8, 9, 16–۱۸]. The studies in [11, 12] provide bounds on the number of measurements required for successful signal recovery based on convex optimization. Furthermore, distributed compressed sensing [13, 14] allows a correlated ensemble of sparse signals to be jointly recovered by exploiting the intra- and inter-signal correlations. We consider the problem of reconstructing a signal given side or prior information, gleaned from a set of known correlated signals. Initially, this problem was studied in [16, 19–۲۸], where the modified CS method [19, 21] considered that a part of the support is available from prior knowledge and tried to find the signal that satisfies the measurement constraint and is sparsest outside the known support. Prior information on the sparsity pattern of the data was also considered in [23] and informationtheoretic guarantees were presented. The studies in [24, 25] introduced weights into the `1 minimization framework that depend on the partitioning of the source signal into two sets, with the entries of each set having a specific probability of being nonzero.