The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
3. EXTENSION OF ICA 
3.1 Independent Component Analysis 
Independent component analysis (ICA) (Hyvarinen, 1999) is a 
newly developed linear data analysis method to separate blind 
sources, which has been used in some challenging fields of 
medical signals analysis, features extraction and pattern 
recognition. The ICA model is defined as: 
X = A*S (1) 
Where X is observed random vector, S is source random 
vector. The ICA solution for the unmixing problem is to find a 
linear transformation w of dependent sensor signals X, that 
makes the ouputs y as independent as possible, i.e. 
y = W • X (2) 
Where y is an estimate of sources. 
The main task of ICA is to solve the separation matrix W , the 
key of algorithms is to choose the method that measure the 
independence between signals. A large amount of algorithms 
have been developed for performing ICA. One of the best 
methods is the fixed point FastICA algorithm. In the FastICA 
algorithm, negentropy is used as the criterion to estimate y as 
it is a natural measure of the independent between random 
variables. The goal is to maximize their negentropy. In the 
FastICA algorithm, the negentropy is approximated by using 
the contrast function which has the following form: 
Ng(y) = {E[G(y)] - E[G(y gauss )] } 2 (3) 
Where y is random vector, y gauss is a standardized gaussian 
variable. £[•] is mathematics expectation, G[-] is a non 
quadratic function. Here we choose: 
G(u) = -exp(-y-) (4) 
3.2 Extension of Independent Component Analysis 
3.2.1 Topographic Independent Component Analysis: As 
the estimated independent components by standard ICA are not 
completely independent, the residual dependence structure 
could be used to define a topographic order for the components. 
Topographic Independent Component Analysis (TICA) is a 
well-known ICA-based technique, which uses the topographic 
order representation to combines topographic mapping with 
ICA. In contrast to ICA, the components S are no longer 
independent but mutually energy-correlated according to the 
two-layer generative model (Hyvarinen,2001). TICA assumes 
that the variances of sources are dependent on each other 
through neighborhood functions. This idea leads to the 
following representation of the source signals: 
S ; = <7^ (5) 
Where Zj is a random variable having the same distribution as 
Sj, and the variance G| is fixed to unity. The variance CJj is 
further modeled by nonlinearity: 
n 
= ( t>(Z h ( i ’ k ) u k) ( 6 ) 
k=l 
Where U ■ are the higher order independent components used to 
generate the variances, h(i, j) is a neighborhood function, and 
(|) describes some nonlinear. 
Then TICA is given as the following update equation: 
Wy := Wy + aE{z(wj r z)r i } (7) 
where E(u) is the expectation operator, a is the stepsize, and 
r i=Z h ( i ’ k )s(Z h ( k ’jX w I z ) 2 ) w 
k j 
The function g is the derivative of G, such as tanh etc. 
3.2.2 Improved Neighborhood Kernel Function: The 
neighbourhood function eX p resses the proximity 
th 
between the i - th and ^ components. It can be defined in 
the same ways as with the self-organizing map. At the present 
time only the most common square type neighbourhood 
function are used in the standard TICA, i.e. 
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