A non-convex regularization approach for compressive sensing
- PDF / 2,128,752 Bytes
- 26 Pages / 439.642 x 666.49 pts Page_size
- 51 Downloads / 203 Views
A non-convex regularization approach for compressive sensing Ya-Ru Fan1,2 · Alessandro Buccini3
· Marco Donatelli4 · Ting-Zhu Huang5
Received: 19 May 2017 / Accepted: 30 July 2018 / © Springer Science+Business Media, LLC, part of Springer Nature 2018
Abstract Compressive sensing (CS) aims at reconstructing high dimensional data from a small number of samples or measurements. In this paper, we propose the minimization of a non-convex functional for the solution of the CS problem. The considered functional incorporates information on the self-similarity of the image by measuring the rank of some appropriately constructed matrices of fairly small dimensions. However, since the rank minimization is a NP hard problem, we consider, as a surrogate function for the rank, a non-convex, but smooth function. We provide a theoretical analysis of the proposed functional and develop an iterative algorithm to compute one of its stationary points. We prove the convergence of such algorithm and show, with some selected numerical experiments, that the proposed approach achieves good performances, even when compared with the state of the art. Keywords Compressive sensing · Non-convex low-rank regularization · Smoothed rank function Mathematics Subject Classification (2010) 90C26 · 65K10 · 94A12 Communicated by: Gitta Kutyniok Alessandro Buccini
[email protected] Ya-Ru Fan [email protected] Marco Donatelli [email protected] Ting-Zhu Huang [email protected]
Extended author information available on the last page of the article.
Y.-R. Fan et al.
1 Introduction The theory of compressive sensing (CS) [11, 17] asserts that sparse signals can be exactly reconstructed from a very small number of samples or measurements. A key condition behind the exact recovery for CS is sparsity. In practice, many signals can be represented in some domain with most generating coefficients close to or equal to zero. In recent years, CS has received considerable attention, thanks to the rapid growth of new applications which required the development of several reliable and computationally efficient numerical methods. A few of the potential CS-based applications are medical image reconstruction [26, 27], image acquisition [20, 31], and sensor networks [30]. The CS recovery problem seeks the perfect reconstruction of a signal x ∈ RN from M linear measurements y = x, where ∈ RM×N is the measure matrix. We assume that y is attainable, i.e., that y lies in the range of , and that M N. Under these assumptions, it holds that there exists more than one x that satisfies the linear equation y = x. The standard CS recovery technique aims at recovering the unknown signal by finding the sparsest x such that y = x, i.e., it looks for the solution of the minimization problem arg min x0 x
s.t. y = x,
(1)
where ·0 denotes the so-called 0 -norm. We recall that the 0 -norm is not a norm and x0 is defined as the number of nonzero elements of x. Since 0 -norm minimization is a NP-hard and non-convex problem, it is difficult to computat
Data Loading...
