4 Open Issues, Future Research Directions, and Conclusions
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Chapter 7
On Independent Component Analysis for
Multimedia Signals
Lars Kai Hansen, Jan Larsen, and Thomas Kolenda
7.1
Background
Blind reconstruction of statistically independent source signals from linear mixtures is relevant to many signal processing contexts [1, 6, 8, 9, 22, 24, 36]. With reference to principal
component analysis (PCA), the problem is often referred to as independent component analysis
(ICA).1
The source separation problem can be formulated as a likelihood formulation (see, e.g.,
[7, 32, 35, 37]). The likelihood formulation is attractive for several reasons. First, it allows
a principled discussion of the inevitable priors implicit in any separation scheme. The prior
distribution of the source signals can take many forms and factorizes in the source index
expressing the fact that we look for independent sources. Second, the likelihood approach
allows for direct adaptation of the plethora of powerful schemes for parameter optimization,
regularization, and evaluation of supervised learning algorithms. Finally, for the case of linear
mixtures without noise, the likelihood approach is equivalent to another popular approach
based on information maximization [1, 6, 27].
The source separation problem can be analyzed under the assumption that the sources either
are time independent or possess a more general time-dependence structure. The separation
problem for autocorrelated sequences was studied by Molgedey and Schuster [33]. They
proposed a source separation scheme based on assumed nonvanishing temporal autocorrelation
functions of the independent source sequences evaluated at a specific time lag. Their analysis
was developed for sources mixed by square, nonsingular matrices. Attias and Schreiner derived
a likelihood-based algorithm for separation of correlated sequences with a frequency domain
implementation [2]–[4]. The approach of Molgedey and Schuster is particularly interesting as
regards computational complexity because it forms a noniterative, constructive solution.
Belouchrani and Cardoso presented a general likelihood approach allowing for additive
noise and nonsquare mixing matrices. They applied the method to separation of sources taking
discrete values [7], estimating the mixing matrix using an estimate–maximize (EM) approach
with both a deterministic and a stochastic formulation. Moulines et al. generalized the EM
approach to separation of autocorrelated sequences in the presence of noise, and they explored
a family of flexible source priors based on Gaussian mixtures [34]. The difficult problem
1 There are a number of very useful ICA Web pages providing links to theoretical analysis, implementations, and
applications. Follow links from the page http://eivind.imm.dtu.dk/staff/lkhansen/ica.html.
© 2001 CRC Press LLC
of noisy, overcomplete source models (i.e., more sources than acquired mixture signals) was
recently analyzed by Lewicki and Sejnowski within the likelihood framework [28, 31].
In this chapter we study the likelihood approach and entertain two different approaches
to the problem: a modified version of the Molgedey–Schuster scheme [15], based on time
correlations, and a novel iterative scheme generalizing the mixing problem to separation of
noisy mixtures of time-independent white sources [16]. The Molgedey–Schuster scheme
is extended to the undercomplete case (i.e., more acquired mixture signals than sources),
and further inherent erroneous complex number results are alleviated. In the noisy mixture
problem we find a maximum posterior estimate for the sources that, interestingly, turns out
to be nonlinear in the observed signal. The specific model investigated here is a special case
of the general framework proposed by Belouchrani and Cardoso [7]; however, we formulate
the parameter estimation problem in terms of the Boltzmann learning rule, which allows for a
particular transparent derivation of the mixing matrix estimate.
The methods are applied within several multimedia applications: separation of sound, image
sequences, and text.
7.2
Principal and Independent Component Analysis
PCA is a very popular tool for analysis of correlated data, such as temporal correlated image
databases. With PCA the image database is decomposed in terms of “eigenimages” that often
lend themselves to direct interpretation. A most striking example is face recognition, where
so-called eigenfaces are used as orthogonal preprocessing projection directions for pattern
recognition. The principal components (the sequence of projections of the image data onto the
eigenimages) are also uncorrelated and, hence, perhaps the simplest example of independent
components [9]. The basic tool for PCA is singular value decomposition (SVD).
Define the observed M × N signal matrix, representing a multichannel signal, by
X = {Xm,n } = {xm (n)} = [x(1), x(2), . . . , x(N )]
(7.1)
where M is the number of measurements and N is the number of samples. xm (n), n =
1, 2, . . . , N is the mth signal and x(n) = [x1 (n), x2 (n), · · · , xM (n)] . In the case of image
sequences, M is the number of pixels.
For the fixed choice of P ≤ M, the SVD of X reads2
P
X = U DV
=
i=1
P
ui Di,i vi ,
Xm,n =
Um,i Di,i Vn,i
(7.2)
i=1
where M × P matrix U = {Um,i } = [u1 , u2 , . . . , uP ] and N × P matrix V = {Vn,i } =
[v1 , v2 , . . . , vP ] represent the orthonormal basis vectors (i.e., eigenvectors of the symmetric
matrices XX and X X, respectively). D = {Di,i } is a P × P diagonal matrix of singular
values. In terms of independent sources, SVD can identify a set of uncorrelated time sequences,
the principal components: Di,i vi , enumerated by the source index i = 1, 2, . . . , P . That is,
we can write the observed signal as a weighted sum of fixed eigenvectors (eigenimages) ui .
However, considering the likelihood for the time-correlated source density, we are often
interested in a slightly more general separation of image sources that are independent in time
2 Usually, SVD expresses X = U D V
where U is M × M, D is M × N , and V is N × N. U is the first P columns
of U , D is the P × P upper-left submatrix of D, and V is the first P columns of V .
© 2001 CRC Press LLC