Robust Sparse Component Analysis Based on a Generalized Hough Transform
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Research Article Robust Sparse Component Analysis Based on a Generalized Hough Transform Fabian J. Theis,1 Pando Georgiev,2 and Andrzej Cichocki3, 4 1 Institute
of Biophysics, University of Regensburg, 93040 Regensburg, Germany Department and Department of Mathematical Sciences, University of Cincinnati, Cincinnati, OH 45221, USA 3 BSI RIKEN, Laboratory for Advanced Brain Signal Processing, 2-1, Hirosawa, Wako, Saitama 351-0198, Japan 4 Faculty of Electrical Engineering, Warsaw University of Technology, Pl. Politechniki 1, 00-661 Warsaw, Poland 2 ECECS
Received 21 October 2005; Revised 11 April 2006; Accepted 11 June 2006 Recommended by Frank Ehlers An algorithm called Hough SCA is presented for recovering the matrix A in x(t) = As(t), where x(t) is a multivariate observed signal, possibly is of lower dimension than the unknown sources s(t). They are assumed to be sparse in the sense that at every time instant t, s(t) has fewer nonzero elements than the dimension of x(t). The presented algorithm performs a global search for hyperplane clusters within the mixture space by gathering possible hyperplane parameters within a Hough accumulator tensor. This renders the algorithm immune to the many local minima typically exhibited by the corresponding cost function. In contrast to previous approaches, Hough SCA is linear in the sample number and independent of the source dimension as well as robust against noise and outliers. Experiments demonstrate the flexibility of the proposed algorithm. Copyright © 2007 Fabian J. Theis et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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INTRODUCTION
One goal of multichannel signal analysis lies in the detection of underlying sources within some given set of observations. If both the mixture process and the sources are unknown, this is denoted as blind source separation (BSS). BSS can be applied in many different fields such as medical and biological data analysis, broadcasting systems, and audio and image processing. In order to decompose the data set, different assumptions on the sources have to be made. The most common assumption currently used is statistical independence of the sources, which leads to the task of independent component analysis (ICA); see, for instance, [1, 2] and references therein. ICA very successfully separates data in the linear complete case, when as many signals as underlying sources are observed, and in this case the mixing matrix and the sources are identifiable except for permutation and scaling [3, 4]. In the overcomplete or underdetermined case, fewer observations than sources are given. It can be shown that the mixing matrix can still be recovered [5], but source identifiability does not hold. In order to approximately detect the sources, additional requirements have to be made, usually sparsity of the sources [6– 8].
Recently, we have introduced a novel measure for sparsity and
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