Deterministic Constructions of Binary Measurement Matrices from Finite Geometry

Deterministic constructions of measurement matrices in compressed sensing (CS) are considered in this paper. The constructions are inspired by the recent discovery of Dimakis, Smarandache and Vontobel which says that parity-check matrices of good low-density parity-check (LDPC) codes can be used as {provably} good measurement matrices for compressed sensing under $\ell_1$-minimization. The performance of the proposed binary measurement matrices is mainly theoretically analyzed with the help of the analyzing methods and results from (finite geometry) LDPC codes. Particularly, several lower bounds of the spark (i.e., the smallest number of columns that are linearly dependent, which totally characterizes the recovery performance of $\ell_0$-minimization) of general binary matrices and finite geometry matrices are obtained and they improve the previously known results in most cases. Simulation results show that the proposed matrices perform comparably to, sometimes even better than, the corresponding Gaussian random matrices. Moreover, the proposed matrices are sparse, binary, and most of them have cyclic or quasi-cyclic structure, which will make the hardware realization convenient and easy.Comment: 12 pages, 11 figure

Sharp sufficient conditions for stable recovery of block sparse signals by block orthogonal matching pursuit

In this paper, we use the block orthogonal matching pursuit (BOMP) algorithm to recover block sparse signals x from measurements y = Ax + v, where v is an ℓ2-bounded noise vector (i.e., kvk2 ≤ ǫ for some constant ǫ). We investigate some sufficient conditions based on the block restricted isometry property (block-RIP) for exact (when v = 0) and stable (when v , 0) recovery of block sparse signals x. First, on the one hand, we show that if A satisfies the block-RIP with δK+1 1 and √2/2 ≤ δ < 1, the recovery of x may fail in K iterations for a sensingmatrix A which satisfies the block-RIP with δK+1 = δ. Finally, we study some sufficient conditions for partial recovery of block sparse signals. Specifically, if A satisfies the block-RIP with δK+1 < √2/2, then BOMP is guaranteed to recover some blocks of x if these blocks satisfy a sufficient condition. We further show that this condition is also sharp

Pushing towards the Limit of Sampling Rate: Adaptive Chasing Sampling

Measurement samples are often taken in various monitoring applications. To reduce the sensing cost, it is desirable to achieve better sensing quality while using fewer samples. Compressive Sensing (CS) technique finds its role when the signal to be sampled meets certain sparsity requirements. In this paper we investigate the possibility and basic techniques that could further reduce the number of samples involved in conventional CS theory by exploiting learning-based non-uniform adaptive sampling. Based on a typical signal sensing application, we illustrate and evaluate the performance of two of our algorithms, Individual Chasing and Centroid Chasing, for signals of different distribution features. Our proposed learning-based adaptive sampling schemes complement existing efforts in CS fields and do not depend on any specific signal reconstruction technique. Compared to conventional sparse sampling methods, the simulation results demonstrate that our algorithms allow $46\%$ less number of samples for accurate signal reconstruction and achieve up to $57\%$ smaller signal reconstruction error under the same noise condition.Comment: 9 pages, IEEE MASS 201

Nearfield Acoustic Holography using sparsity and compressive sampling principles

Regularization of the inverse problem is a complex issue when using Near-field Acoustic Holography (NAH) techniques to identify the vibrating sources. This paper shows that, for convex homogeneous plates with arbitrary boundary conditions, new regularization schemes can be developed, based on the sparsity of the normal velocity of the plate in a well-designed basis, i.e. the possibility to approximate it as a weighted sum of few elementary basis functions. In particular, these new techniques can handle discontinuities of the velocity field at the boundaries, which can be problematic with standard techniques. This comes at the cost of a higher computational complexity to solve the associated optimization problem, though it remains easily tractable with out-of-the-box software. Furthermore, this sparsity framework allows us to take advantage of the concept of Compressive Sampling: under some conditions on the sampling process (here, the design of a random array, which can be numerically and experimentally validated), it is possible to reconstruct the sparse signals with significantly less measurements (i.e., microphones) than classically required. After introducing the different concepts, this paper presents numerical and experimental results of NAH with two plate geometries, and compares the advantages and limitations of these sparsity-based techniques over standard Tikhonov regularization.Comment: Journal of the Acoustical Society of America (2012