Underdetermined Audio Source Separation Using Laplacian Mixture Modelling
The problem of underdetermined audio source separation has been explored in the literature for many years. The instantaneous \(K\) -sensors, \(L\) -sources mixing scenario (where \(K
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Underdetermined Audio Source Separation Using Laplacian Mixture Modelling Nikolaos Mitianoudis
Abstract The problem of underdetermined audio source separation has been explored in the literature for many years. The instantaneous K -sensors, L-sources mixing scenario (where K < L) has been tackled by many different approaches, provided the sources remain quite distinct in the virtual positioning space spanned by the sensors. In this case, the source separation problem can be solved as a directional clustering problem along the source position angles in the mixture. The use of Laplacian Mixture Models in order to cluster and thus separate sparse sources in underdetermined mixtures will be explained in detail in this chapter. The novel Generalised Directional Laplacian Density will be derived in order to address the problem of modelling multidimensional angular data. The developed scheme demonstrates robust separation performance along with low processing time.
7.1 Introduction Let a set of K sensors x(n) = [x1 (n), . . . , x K (n)]T observe a set of L sound sources s(n) = [s1 (n), . . . , s L (n)]T . We will consider the case of instantaneous mixing, i.e. each sensor captures a scaled version of each signal with no delay in transmission. Moreover, the possible additive noise will be considered negligible. The above instantaneous mixing model can be expressed in mathematical terms, as follows: x(n) = As(n)
(7.1)
where A represents the K × L mixing matrix and n the sample index. The blind source separation problem provides an estimate of the source signals s(n) given N. Mitianoudis (B) Image Processing and Multimedia Lab, Electrical and Computer Engineering Department, Democritus University of Thrace, 67100 Xanthi, Greece e-mail: [email protected] G. R. Naik and W. Wang (eds.), Blind Source Separation, Signals and Communication Technology, DOI: 10.1007/978-3-642-55016-4_7, © Springer-Verlag Berlin Heidelberg 2014
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the sensor signals x(n). Usually, most separation approaches are semi-blind, which implies some knowledge of the source signal’s general statistical structure. A number of algorithms have been proposed to solve the overdetermined and complete source separation problem (K ≥ L) with great success. The additional assumption of statistical independence between the sources led to a group of source separation algorithms, summarised under the general term Independent Component Analysis (ICA). Starting from different interpretations of statistical independence, most algorithms perform source separation with great accuracy. An overview of current ICA and general blind source separation algorithms can be found in tutorial books on ICA by Hyvärinen et al. [27], Cichocki-Amari [11] and Common et al. [12]. The underdetermined source separation problem (K ≤ L) is more challenging, since in this case, the estimation of the mixing matrix A is not sufficient for the estimation of the source signals s(n). This type of mixtures can be encountered in musical audio mixes. A number of solo instrume
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