SYNTHESIS AND ANALYSIS PRIOR ALGORITHMS FOR JOINT-SPARSE RECOVERY A. Majumdar and R. K. Ward Department of Electrical and Computer Engineering, University of British Columbia {angshulm, rababw}@ece.ubc.ca ABSTRACT This paper proposes a Majorization-Minimization approach for solving the synthesis and analysis prior joint-sparse multiple measurement vector reconstruction problem. The proposed synthesis prior algorithm yielded the same results as the Spectral Projected Gradient (SPG) method. The analysis prior algorithm is the first to be proposed for this problem. It yielded considerably better results than the proposed synthesis prior algorithm. For problems of a given size, the run times for our proposed algorithms are fixed; unlike SPG where the reconstruction time also depends on the support size of the vectors. Index TermsÏ Compressed Sensing, Multiple Measurement Vector, Convex Optimization 1. INTRODUCTION In Compressed Sensing (CS) the problem is to recover a sparse vector from its random low dimensional Fourier projections that may be corrupted by noise. 1 1 1 , n nN N n y H x n N j · · · · ? - > (1) where x is the high dimensional s-sparse vector to be recovered, H is the random/Fourier projection, y is the lower dimensional projection of x cpf さ ku vjg pqkug *cuuwogf vq be Normally distributed). An extension of the Single Measurement Vector (SMV) problem represented by (1) is the Multiple Measurement Vector (MMV) problem. For MMV, a set of vectors with a common sparse support are to be recovered. By common support, it is meant that all the vectors have non-zero coefficients at the same positions. The MMV model is, , 1...r j j j y Hx j j ? - ? (2) where y j is the lower dimensional projection corresponding to x j (all x j Óu have a common support as mentioned earlier). The rest of the symbols have the same meanings as in (1). It is possible to represent (2) in the following compact form, Y HX N ? - (3) where ] _ 1 | ... | r Y y y ? , ] _ 1 | ... | r X x x ? and ] _ 1 | ... | r N j j ? . Since the unknown vectors (xÓu+ jcxg c eqooqp urctug support, the matrix X will be row-sparse. The MMV problem has been studied in the past [1-5]. Optimization based recovery methods recover X by solving the following problem 2 , min subject to p mp F X X Y HX g / ~ (4) where , 1 n p p j mp m j X X › ? ? Â ( j X › is the vector whose entries form the j th row of X), . F denotes the Frobenius norm (l 2 - norm of all the elements) of the matrix and i is a parameter dependent on the variance of noise. Kp ]4_. vjg xcnwgu o?4 cpf rø3 ygtg rtqrqugf0 Ukpeg xcnwgu of p. such as p<1, make the problem non-convex, m=2 and p=1 are used [6]. Thus the inverse problem (3) is solved via, 2 2,1 min subject to F X X Y HX g / ~ (5) The choice of such values for the norms can be understood intuitively. The l 2 -norm over the rows ( j X › Óu+ gphqtegu non-zero values on all the elements of the row vector whereas the summation over the l 2 -norm ( 2 1 r j j X › ? Â ) enforces row-sparsity, i.e. the selection of few rows. MMV problems arise in varied areas of applied signal processing like communication [7], Seismic Imaging [8] and MRI [9, 10]. For example, in multi-echo T1/T2 weighted MR imaging, the same anatomical cross section is imaged by varying certain parameters in order to acquire images with differing tissue contrasts. It has been argued in [9, 10], that such multi-echo images have a common sparse support in the wavelet domain. It is possible to recover the signal by (5) when the signal itself is sparse. However in MRI, the image to be recovered is not sparse, but has a sparse representation in some transform domains (e.g. finite difference). In such a case the synthesis prior formulation (5) is not applicable. Once needs to employ the following analysis prior formulation, 2 2,1 min subject to F X AX Y HX g / ~ (6) where A is the sparsifying transform. The Spectral Projected Gradient L1 algorithm [6] solves the synthesis prior joint-sparse MMV recovery problem (5). There is no existing algorithm to solve the analysis prior problem (6). In this paper we propose algorithms for solving the synthesis prior (5) and the analysis prior problems using the Majorization-Minimization approach. These algorithms 3421 978-1-4673-0046-9/12/$26.00 ©2012 IEEE ICASSP 2012