The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008 
1113 
if\i- j |< m 
otherwise 
(9) 
The constant m defines the width of the neighborhood. 
In fact from the view of neurobiology the feedback intensity 
between the central cell and other neighbor cells has business 
with the distance, so neighborhood should be the function of 
distance. Considering the characteristics of human visual 
system and under our many experiments, this paper introduces 
the gaussian neighborhood series kernel function to express the 
different topology of TIC A, such as: 
1 j-V(i— n ) 2 +(j—m) 2 
M w ij, w „m) = e 2 
(10) 
The new introduced neighborhood functions can obtain the 
image basis with obviously enhanced directionality, which has 
advantages for the coming image analysis task. 
3.2.3 Modified Learning Rule: To resolve the separation 
matrix W, the optimization problem can be induced as follows: 
ÌÌ 
min J(w)= E{G( z(0) 2 )} 
M 
Il II 2 1 
subject to ||wj| 2 = l 
(ll) 
The Lagrange function can be derived as: 
L(w,X) = -E{G(¿h(i,j)(w^z(t)) 2 )} + M||w|! 2 -1) (12> 
i=l 
Finally the batch learning rule can derived as: 
w'— W- Ti(E {g(y)z} - E {g(y)y} w) (i 3) 
The information that remote sensing image represent is the 
reflectivity of different objects in certain band. Each band of 
multi-spectral remote sensing images can be considered as the 
combination of reflectivity of the several independent land 
objects in certain law. Applying ICA to multi-spectral remote 
sensing images, we can obtain the independent component 
bands that concentrate the information of specific land objects, 
resulting in enhancing the degree of separation of different 
objects. 
For single band remote sensing image, most important 
information such as edge features, texture features are nearly 
correlative with high-order statistics. High-order statistics 
reflect the important structure and phase feature of image. 
Image analysis using ICA/TICA with high-order statistics has 
particular advantage, it can realize sparse coding, meanwhile, 
ICA/TICA is excellent edge filter (Zeng,2005). When people 
observe image, a series image patches are picked up firstly and 
then the whole image. Suppose each image patch is denoted by 
x, which can be regarded as a linear combination of the base 
function matrix A, independent component 5 is the statistic 
independent random vector, expressing the coefficients that the 
N 
corresponding basis act on image, i.e. x = '^a j s i , where 
;=l 
A = (a,, a 2 , • • •, a N ) ,column vector a t (i = 1,2, • • •, N) denotes 
a group of N 2 X 1 pixels basis images. Through ICA resolves 
the separation matrix W, one can get the coefficients projected 
in independent component basis by y = Wl, which express 
the image features in ICA domain. Figured are basis matrix A, 
basis vectors have orientation in space domain and localization 
in frequency domain, depict most of the edge features of image. 
Figure.2 illustrates the basis vectors obtained by our improved 
TICA, one can observe the spatial correlation of basis 
introduced by topography, the basis offer a more 
comprehensive representation compared to the general ICA 
model. 
Where TJ is learning rate, here the self-adaptive adjustment Figure 1. ICA basis of natural image data 
method is developed in this paper. 
Through introduction of Lagrange operator to solve the 
optimization of TICA, the method has religious deduction 
procedure and well property of convergence. 
In short, this paper introduces the new topographic kernel 
functions to express the relationships between the independent 
components, which can better satisfy the human vision system 
demand than the former model. Further more, the paper also 
gives the new optimization rule to realize the farther 
development of TICA. The proposed modified TICA is more 
applicable in image fusion. 
Figure 2. Improved TICA basis of natural image data