Mathematics in Independent Component Analysis

Chapter 15. Neurocomputing, 69:1485-1501, 2006 213
SNR Enhancement [dB]
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Fig. 2. Comparison between LPCA and LICA based denoising. Here the mean square
error of two signals x, y with L samples is 1 �
L ||xi − yi|| 2 . For all noise levels a
complete parameter estimation was done in the sets {10, 15, . . . , 60} for M and
{20, 30, . . . , 80} for K.
4.1.3 LICA denoising with multi-dimensional data sets
A generalization of the LICA algorithm to multidimensional data sets like
images where pixel intensities depend on two coordinates is desirable. A simple
generalization would be to look at delayed coordinates of vectors instead
of scalars. However, this appears impractical due to the prohibitive computational
effort. More importantly, this direct approach reduces the number
of available samples significantly. This leads to far less accurate estimators
of important aspects like the MDL estimation of the dimension of the signal
subspace or the estimation of the kurtosis criterion in the LICA case.
Another approach could be to convert the data to a 1D string by choosing some
path through the data and concatenating the pixel intensities accordingly. But
this can easily create unwanted artifacts along the chosen path. Further, local
correlations are broken up, hence not all the available information is used.
But a more sophisticated and, depending on the nature of the signal, very effective
alternative approach can be envisaged. Instead of converting the multidimensional
data into 1D data strings prior to applying the LICA algorithm,
we can use a modified delay transformation using shifts along all available
dimensions. This concept is similar to the multidimensional auto-covariances
used in the Multi Dimensional SOBI (mdSOBI) algorithm introduced in [31].
In the 2D case, for example, consider an n × n image represented by a matrix
P = (aij)i,j=1...n. Then the transformed data set consists of copies of P which
are shifted either along columns or rows or both. For instance, a translation
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