Abstract:The blind source separation problem is to extract the underlying source signals from a set of linear mixtures, where the mixing matrix is unknown. This situation is common in acoustics, radio, medical signal and image processing, hyperspectral imaging, and other areas. We suggest a twostage separation process: a priori selection of a possibly overcomplete signal dictionary (for instance, a wavelet frame or a learned dictionary) in which the sources are assumed to be sparsely representable, followed by unmixing… Show more
“…• Empirical work, showing that combined representations such as wavelets with curvelets or wavelets with sinusoids often gave very compelling separations of real signals and images, see, for instance, [1,10,25,26,35,42,40,41,43,44,30,47].…”
Section: Minimum ℓ 1 Decomposition and Perfect Separationmentioning
Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically 'pure' images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem, but recent empirical work achieved highly persuasive separations.We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the ℓ 1 norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by ℓ 1 minimization; microlocal phase space helps organize the calculations that cluster coherence requires.
“…• Empirical work, showing that combined representations such as wavelets with curvelets or wavelets with sinusoids often gave very compelling separations of real signals and images, see, for instance, [1,10,25,26,35,42,40,41,43,44,30,47].…”
Section: Minimum ℓ 1 Decomposition and Perfect Separationmentioning
Image data are often composed of two or more geometrically distinct constituents; in galaxy catalogs, for instance, one sees a mixture of pointlike structures (galaxy superclusters) and curvelike structures (filaments). It would be ideal to process a single image and extract two geometrically 'pure' images, each one containing features from only one of the two geometric constituents. This seems to be a seriously underdetermined problem, but recent empirical work achieved highly persuasive separations.We present a theoretical analysis showing that accurate geometric separation of point and curve singularities can be achieved by minimizing the ℓ 1 norm of the representing coefficients in two geometrically complementary frames: wavelets and curvelets. Driving our analysis is a specific property of the ideal (but unachievable) representation where each content type is expanded in the frame best adapted to it. This ideal representation has the property that important coefficients are clustered geometrically in phase space, and that at fine scales, there is very little coherence between a cluster of elements in one frame expansion and individual elements in the complementary frame. We formally introduce notions of cluster coherence and clustered sparsity and use this machinery to show that the underdetermined systems of linear equations can be stably solved by ℓ 1 minimization; microlocal phase space helps organize the calculations that cluster coherence requires.
“…It has been shown that when sources are sparse, they can be easily recovered from their linear mixtures using simple geometrical methods [2], [5]. This is based on the observation that whenever sources are sparse, there is a high probability that each data point in each mixture will result from the contribution of only one source.…”
Abstract. ICA and similar techniques have been previously applied to either one-dimensional signals or still images. We consider the problem of blind separation of dynamic sources, i.e. functions of both time and two spatial variables. We extend the Sparse ICA (SPICA) approach and apply it to a sliding data cube, defined by the two dimensions of the visual scene and the extent in time over which the mixing problem can be considered to be stationary and linear. This framework and formalism are applied to two special problems encountered in two different fields: The first deals with separation of dynamic reflections from a desired moving visual scene, without having any a priori knowledge on the structure of the images and/or their statistics. The second problem concerns blind separation of 'neural cliques' from the background firing activity of a neural network. The approach is generic in that it is applicable to any linearly mixed dynamic sources.
“…In the audio domain, sparse prior distributions are usually a Laplacian [1], a generalized Gaussian [2], a Student-t [3], or a mixture of two Gaussians [4].…”
Section: Introductionmentioning
confidence: 99%
“…In each time-frequency point, the maximum number of nonzero sources is usually assumed to be limited to the number M of channels [1,2]. 2.…”
Abstract. The underdetermined blind audio source separation problem is often addressed in the time-frequency domain by assuming that each time-frequency point is an independently distributed random variable. Other approaches which are not blind assume a more structured model, like the Spectral Gaussian Mixture Models (Spectral-GMMs), thus exploiting statistical diversity of audio sources in the separation process. However, in this last approach, Spectral-GMMs are supposed to be learned from some training signals. In this paper, we propose a new approach for learning Spectral-GMMs of the sources without the need of using training signals. The proposed blind method significantly outperforms state-of-the-art approaches on stereophonic instantaneous music mixtures.
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