Simultaneously Sparse Solutions to Linear Inverse Problems with Multiple System Matrices and a Single Observation Vector

A linear inverse problem is proposed that requires the determination of multiple unknown signal vectors. Each unknown vector passes through a different system matrix and the results are added to yield a single observation vector. Given the matrices and lone observation, the objective is to find a simultaneously sparse set of unknown vectors that solves the system. We will refer to this as the multiple-system single-output (MSSO) simultaneous sparsity problem. This manuscript contrasts the MSSO problem with other simultaneous sparsity problems and conducts a thorough initial exploration of algorithms with which to solve it. Seven algorithms are formulated that approximately solve this NP-Hard problem. Three greedy techniques are developed (matching pursuit, orthogonal matching pursuit, and least squares matching pursuit) along with four methods based on a convex relaxation (iteratively reweighted least squares, two forms of iterative shrinkage, and formulation as a second-order cone program). The algorithms are evaluated across three experiments: the first and second involve sparsity profile recovery in noiseless and noisy scenarios, respectively, while the third deals with magnetic resonance imaging radio-frequency excitation pulse design.Comment: 36 pages; manuscript unchanged from July 21, 2008, except for updated references; content appears in September 2008 PhD thesi

Denoising hyperspectral imagery and recovering junk bands using wavelets and sparse approximation

Abstract — In this paper, we present two novel algorithms for denoising hyperspectral data. Each algorithm exploits correlation between bands by enforcing simultaneous sparsity on their wavelet representations. This is done in a non-linear manner using wavelet decompositions and sparse approximation techniques. The first algorithm denoises an entire cube of data. Our experiments show that it outperforms wavelet-based global soft thresholding techniques in both a mean-square error (MSE) and a qualitative visual sense. The second algorithm denoises a set of noisy, user designated bands (“junk bands”) by exploiting correlated information from higher quality bands within the same cube. We prove the utility of our junk band denoising algorithm by denoising ten bands of actual AVIRIS data by a significant amount. Preprocessing data cubes with these algorithms is likely to increase the performance of classifiers that make use of hyperspectral data, especially if the denoised and/or recovered bands contain spectral features useful for discriminating between classes. I

Automatic Cost Minimization for Multiplierless Implementations of Discrete Signal Transforms

The computation of linear DSP transforms consists entirely of additions and multiplications by constants, which, in a hardware realization, can be implemented as a network of wired shifts and additions. Thus, a light weight fixed point implementation that approximates an exact transform can be built from only adders. This paper presents an automatic approach for minimizing the number of additions required for a given transform under the constraint of a particular quality measure. We present an evaluation of our approach. For example, one experiment shows that the IMDCT transform within an MP3 decoder can be reduced from 572 additions to 260 additions while maintaining Limited Accuracy as defined by the MP3 ISO standard