Machine Recognition in Complex Domain - High Dimensional Neurocomputing: Growth, Appraisal and Applications - page 144

7. Maximizing joint entropy of output Y gives independent basis images in U. It also
minimize the mutual information between the individual outputs (basis images).
Thus, ICA algorithm produces transformation matrix W t

= W OPT × W w such

W t E T . That will let us know how much the extracted signals in U are
close to being independent.
8. Let R be the M by M matrix of PC representation of the images in X, R =
X E , also approximation of X = R E T . Assumption that W is invertible, we get
E T

that U =

W − 1
t

RW − 1
t

= U . Hence X = U .

RW − t contains the coefficients for the linear combina-
tion of statistically independent basis images in U; X

9. Each row of matrix B =

BU . This X comprises
the images in its rows, X is the reconstruction of the original data. Thus, sta-
tistically independent feature vectors (IC representation) of images have been
obtained.

=

7.2.6 Feature Extraction with C ICA

Independent component analysis in complex domain ( C ICA ) has been used for source
separation of complex-valued data such as fMRI [ 10 ], EEG [ 9 ] and communication
data, yet this concept is not well developed hence demanding more applications. The
most important application of C ICA , in machine recognition is still untouched, there-
fore it is worthwhile to develop the feature extraction algorithm with basic concepts
of ICA. This chapter is devoted to build C ICA algorithm for image processing and
vision applications. It is observed in Chap. 4 that ANN in a complex domain gives
a far better performance in the real-valued problems. It will be fruitful to investigate
the principle of optimal information transfer through complex-valued neurons, incor-
porating nonanalytic activation function, for feature extraction from image database.
A bounded but nonanalytic activation function f C given in Eq. ( 3.3 ) has performed
well with neurons in complex domain [ 56 ]. The motivation of using this function in
C ICA algorithm is that it can approximate roughly well the joint cdf ( F

)of
source distribution [ 57 , 58 ]. The apparent problem in this complex function comes
from the fact that it is real-valued and therefore is not complex differentiable unless
it is a constant. The differentiation of f C can be conveniently done [ 59 , 60 ] with-
out separating real and imaginary parts with following complex differential (partial)
operator:

( u

,

u

)

∂

∂ f C ( z ) 1

2

f C ( z )

j ∂

f C ( z )

f C ( z ) = = − (7.18)

∂ z ∂ z ∂ z

C ICA algorithm is derived by maximizing the entropy of the output from a sin-
gle layered complex-valued neural network. The update equations in unsupervised
learning involve first- and second-order derivatives of the nonlinearity. The complex-
valued sigmoid function is flexible enough to obey the joint cdf. The Infomax algo-
rithm [ 9 , 43 , 58 , 61 ] in complex domain setup is modified for feature extraction in