Multichannel Blind Deconvolution Research Articles

Overview of blind signal separation, part I: Criteria and algorithms

In blind signal separation, multiple independent source signals are separated from multiple linear mixtures of these signals without specific knowledge of either the source signal characteristics or the mixing conditions. This talk provides an introduction to the blind signal separation task. Three different problem formulations—signal separation of instantaneous mixtures, signal separation of convolutive mixtures, and multichannel blind deconvolution—are described, and their similarities and differences are highlighted. An overview of both information-theoretic and contrast-based separation criteria is then given. Natural gradient optimization procedures, when combined with such criteria, yield simple and useful blind signal separation algorithms. An example of speech separation of real room recordings illustrates the capabilities and limitations of one such approach.

Transpose properties in the stability and performance of the classic adaptive algorithms for blind source separation and deconvolution

This paper presents a tutorial review of the problem of Blind Source Separation (BSS) and the properties of the classic adaptive algorithms when either the score-function or a general (nonscore) nonlinearity is employed in the algorithm. In new findings it is shown that the separating solution for both sub- and super-Gaussian signals can be stabilized by an algorithm employing any given nonlinearity. For these separating solutions the steady-state error levels are also given in terms of the nonlinearity and the pdfs of the source signals. These results show that a transpose symmetry exists between the nonlinear algorithms for sub- and super-Gaussian signals. The behavior of the algorithm is then detailed when the ideal score-function nonlinearity is replaced by a general (hard saturation or u 3) nonlinearity. The phases of convergence to decorrelated output signals and then to recovery of the source signals are explained. The results are then extended to single- and multi-channel deconvolution and shown by analysis and extensive simulation to hold for mixed and convolved source signals. The results allow the design of stable algorithms for multichannel blind deconvolution with a general nonlinearity when sub- and super-Gaussian source signals are present.

A unifying information-theoretic framework for independent component analysis

We show that different theories recently proposed for independent component analysis (ICA) lead to the same iterative learning algorithm for blind separation of mixed independent sources. We review those theories and suggest that information theory can be used to unify several lines of research. Pearlmutter and Parra [1] and Cardoso [2] showed that the infomax approach of Bell and Sejnowski [3] and the maximum likelihood estimation approach are equivalent. We show that negentropy maximization also has equivalent properties, and therefore, all three approaches yield the same learning rule for a fixed nonlinearity. Girolami and Fyfe [4] have shown that the nonlinear principal component analysis (PCA) algorithm of Karhunen and Joutsensalo [5] and Oja [6] can also be viewed from information-theoretic principles since it minimizes the sum of squares of the fourth-order marginal cumulants, and therefore, approximately minimizes the mutual information [7]. Lambert [8] has proposed different Bussgang cost functions for multichannel blind deconvolution. We show how the Bussgang property relates to the infomax principle. Finally, we discuss convergence and stability as well as future research issues in blind source separation.

Super-exponential algorithms for multichannel blind deconvolution

Multichannel blind deconvolution has been receiving increasing attention. Shalvi and Weinstein proposed an attractive approach to single-channel blind deconvolution called the super-exponential methods. The objective of this correspondence is to extend the Shalvi and Weinstein (1993, 1994) approach to the multichannel case and present super-exponential algorithms for multichannel blind deconvolution. We propose three approaches to multichannel blind deconvolution. In the first one, we present a multichannel super-exponential algorithm. In the second one, we present a super-exponential deflation algorithm. In the third one, we present a two-stage super-exponential algorithm.

The stability of nonlinear least squares problems and the Cramer-Rao bound

A number of problems of interest in signal processing can be reduced to nonlinear parameter estimation problems. The traditional approach to studying the stability of these estimation problems is to demonstrate finiteness of the Cramer-Rao bound (CRB) for a given noise distribution. We review an alternate, deterministic notion of stability for the associated nonlinear least squares (NLS) problem from the realm of nonlinear programming (i.e., that the global minimizer of the least squares problem exists and varies smoothly with the noise). Furthermore, we show that under mild conditions, identifiability of the parameters along with a finite CRB for the case of Gaussian noise is equivalent to the deterministic stability of the NLS problem. Finally, we demonstrate the application of our result, which is general, to the problems of multichannel blind deconvolution and sinusoid retrieval to generate new stability results for these problems with little additional effort.

Iterative algorithms based on multistage criteria for multichannel blind deconvolution

Blind deconvolution and blind equalization have been important interesting topics in diverse fields including data communication, image processing, and geophysical data processing. Recently, Inouye and Habe introduced a multistage criterion for attaining blind deconvolution of multiple-input multiple-output (MIMO) linear time-invariant (LTI) systems. In this correspondence, based on their criterion, we present iterative algorithms for solving the blind deconvolution problem of MIMO LTI systems. However, their criterion should be subjected to several constraints of equations. Therefore, they proposed a new constraint-free multistage criterion for accomplishing the blind deconvolution of MIMO LTI systems. Based on their unconstrained criterion, we show iterative algorithms for solving the blind deconvolution of multichannel LTI systems.

An unsupervised hybrid network for blind separation of independent non-Gaussian source signals in multipath environment

This paper is concerned with the problem of recovering multiple source signals that are transmitted through a linear Multiple Input Multiple Output (MIMO) system, without resorting to any prior knowledge. Source signals are assumed to be spatially independent and temporally i.i.d. non-Gaussian sequences. We present an unsupervised hybrid network (a linear feedback network with FIR synapses followed by a linear memoryless feedforward network) which is able to recover multiple source signals blindly. A simple criterion for multichannel blind deconvolution and an associated learning algorithm are presented. Extensive computer simulation results confirm the validity and high performance of the proposed method.

Perfect blind restoration of images blurred by multiple filters: theory and efficient algorithms

We address the problem of restoring an image from its noisy convolutions with two or more unknown finite impulse response (FIR) filters. We develop theoretical results about the existence and uniqueness of solutions, and show that under some generically true assumptions, both the filters and the image can be determined exactly in the absence of noise, and stably estimated in its presence. We present efficient algorithms to estimate the blur functions and their sizes. These algorithms are of two types, subspace-based and likelihood-based, and are extensions of techniques proposed for the solution of the multichannel blind deconvolution problem in one dimension. We present memory and computation-efficient techniques to handle the very large matrices arising in the two-dimensional (2-D) case. Once the blur functions are determined, they are used in a multichannel deconvolution step to reconstruct the unknown image. The theoretical and practical implications of edge effects, and "weakly exciting" images are examined. Finally, the algorithms are demonstrated on synthetic and real data.

Criteria for blind deconvolution of multichannel linear time-invariant systems

Blind deconvolution and equalization of linear time-invariant systems have widely received attention in various fields such as data communication, image processing and geophysical data processing. Shalvi and Weinstein (1990) proposed certain criteria for blind deconvolution of nonminimum-phase linear time-invariant systems. Their work is, however, restricted to the single-channel (or scalar) case. This correspondence extends the Shalvi-Weinstein approach to the multichannel case. We propose a multistage maximization criterion and a single-stage maximization criterion for attaining multichannel blind deconvolution. Under the normalized condition specified later, they are shown to be equivalent.

FIR perfect signal reconstruction from multiple convolutions: minimum deconvolver orders

Given p co-prime finite impulse response (FIR) filters h/sub i//spl isin/R(m/sub h/), it well known that there exist q/sub i//spl isin/R(m/sub q/) such that /spl Sigma//sub i/h/sub i/*q/sub i/=/spl delta/. Importantly, this enables signal recovery from its convolutions with p/spl ges/2 co-prime FIR filters by FIR filtering. We show that such q/sub i/ exist almost surely if and only if m/sub q//spl ges/[(m/sub h/-1)/(p-1)], where [x] is the smallest integer greater than or equal to x. The results also provide conditions for full rank of certain key matrices arising in the blind multichannel deconvolution problem.

Generalized contrasts for multichannel blind deconvolution of linear systems

Two contrasts for the problem of multichannel blind deconvolution have been given and theoretically studied by Comon [1996]. The maximization of these criteria allows us to solve the problem of multi-input/multi-output (MIMO) blind deconvolution. In this paper, we show that many other contrast functions may be considered. The two aforementioned criteria are proved to be included in the wide class of contrast functions, which is here defined through simple conditions.

Multichannel blind signal separation and reconstruction

The separation of multiple signals from their superposition recorded at several sensors is addressed. The methods employ polyspectra of the sensor data in order to extract the unknown signals and estimate the finite impulse response (FIR) coupling systems via a linear equation based algorithm. The procedure is useful for multichannel blind deconvolution of colored input signals with (possibly) overlapping spectra. An extension of the main algorithm, which can be applied for quasiperiodic signal separation, is also given. Simulation results corroborate the applicability of the algorithm.

Contrasts for multichannel blind deconvolution

A class of optimization criteria is proposed whose maximization allows us to carry out blind multichannel deconvolution in the presence of additive noise. Contrasts presented in the paper encompass those related to source separation and independent component analysis problems.

EVAM: an eigenvector-based algorithm for multichannel blind deconvolution of input colored signals

A new algorithm is proposed for the deconvolution of an unknown, possibly colored, Gaussian or nonstationary signal that is observed through two or more unknown channels described by rational system transfer functions. More specifically, not only the root (pole and zero) locations but also the orders of the channel transfer functions are unknown. It is assumed that the channel orders may be overestimated. The proposed algorithm estimates the orders and root locations of the channel transfer functions, therefore it can also be used in multichannel system identification problems. The input signal is allowed to be nonstationary and the channel transfer functions may be a nonminimum phase as well as noncausal, hence the proposed algorithm is particularly suitable for applications such as dereverberation of speech signals recorded through multiple microphones. Several experimental results indicate improvement compared to the existing methods in the literature. >