On the Reconstruction of Block-Sparse Signals With an Optimal Number of Measurements

Abstract

Abstract-Letbe an by matrix ( ) which is an instance of a real random Gaussian ensemble. In compressed sensing we are interested in finding the sparsest solution to the system of equations x = y for a given y. In general, whenever the sparsity of x is smaller than half the dimension of y then with overwhelming probability over the sparsest solution is unique and can be found by an exhaustive search over x with an exponential time complexity for any y. The recent work of Candés, Donoho, and Tao shows that minimization of the 1 norm of x subject to x = y results in the sparsest solution provided the sparsity of x, say , is smaller than a certain threshold for a given number of measurements. Specifically, if the dimension of y approaches the dimension of x, the sparsity of x should be 0 239 N. Here, we consider the case where x is block sparse, i.e., x consists of = blocks where each block is of length and is either a zero vector or a nonzero vector (under nonzero vector we consider a vector that can have both, zero and nonzero components). Instead of 1 -norm relaxation, we consider the following relaxation: min x X 1 2 + X 2 2 + + X 2 subject to x = y (*) where X = (x ( 1) +1 x ( 1) +2 . . . x ) for = 1 2 . . . . Our main result is that as, (*) finds the sparsest solution to x = y, with overwhelming probability in , for any x whose sparsity is(1 2) ( ), provided 1 1 , and = (log(1 ) 3 ). The relaxation given in (*) can be solved in polynomial time using semi-definite programming.Index Terms-Compressed sensing, block-sparse signals, semidefinite programming.

Isar Imaging Based on Iterative Reweighted Lp Block Sparse Reconstruction Algorithm

“…We aim to estimate the original signal x with unknown cluster structure. l 2 /l 1 minimization strategy was introduced in [18] to solve the block sparse signal reconstruction problem…”

“…In order to investigate the role of the pulse number, three different amounts of pulses (16,32, and 64 pulses) are implemented. The experimental results are compared to those images obtained by some sparse signal recovery methods including BP method [20], SBL method [18], L 1 L 0 method [12] and S-method [21]. For BIRL2-Lp algorithm, parameter λ is set to 10 −3 .…”

Separation of a subspace-sparse signal: Algorithms and conditions

“…However, the desired solution x now has special structure, as the nonzero coefficients appear in blocks and sparsity within each block (subspace) is no longer important. This property has also been termed block sparsity in the literature [3].…”

“…The separation of a subspace-sparse signal, particularly by convex optimization methods, has been investigated in [3] and [4]. They have provided certain sufficient conditions under which the algorithms are guaranteed to find the correct block-sparse solution.…”

Reference-Driven Compressed Sensing MR Image Reconstruction with Partially Known Support and Group Sparsity Constraints

“…Baraniuk [2] proposed a Model-based CS, which exploits the inherent tree structure of the wavelet coefficients except for sparsity. Stojnic, Parvaresh and Hassibi [10] proposed a block sparsity model for high probability recovery of the block-sparse signals. Since Lustig et al [7] proposed CS-based MR image reconstruction, applying state-of-the-art methods in CS to MRI reconstruction is one of the CS-MRI research trends.…”