y — Wv (3) 
Our target is to estimate the independent components from 
all observed signals alone, which is equivalent to find the 
optimum estimate matrix W in (3), which makes y , the 
estimate of variables s, as statistically independent as possible. 
Because a linear mixture of Gaussian variables is still a 
Gaussian variable, at most one source in the mixture model can 
be allowed to be a Gaussian type. 
In detail, the algorithm falls into three steps[6]. Firstly, the 
preprocessing step is employed to whiten the mixing data in 
order to remove the correlation between variables and reduce 
data dimension (if necessary).Secondly, a subjective function 
L(W) using the demixing matrix W as argument is defined to 
measure the independence of output variables y .Finally, a 
optimization algorithm is used to find a optimal estimation 
W ,which makes L(PF) has the extremum, while W is 
equal to W. The implementation of ICA can be seen as a 
combination of a objective function and a optimization 
algorithm[7]. The key point of algorithm is the definition of 
statistical impendence. In general, there are three kinds of 
objective functions: the maximum of non-gaussianity, the 
minimum of mutual information and maximum likelihood.lt has 
been improved that three criterions are equal to each other in 
the term of information’s[8],the difference between them 
is laying in the optimization algorithm, which means different 
calculating complexity. For non-gaussianity, kurtosis and 
negentropy are often used as performance criterion. For a 
random variable y with zero mean, its kurtosis can be defined 
as follows: 
kurt(y) = E{y 4 }- 3 (E{y 2 }) 2 ( 4) 
Meanwhile, its negentropy is presented in (6), which is the 
difference between the entropy of a gauss rand variable that is 
same as y in standard deviation and the entropy of y . 
H (*) = P ( X = a i ) l0 § P ( X = a i ) (5) 
i 
"e{y) = H(y gaas )-H(y) <6) 
A large collection of literature on optimization algorithms have 
been proposed in the last ten years. Among them, a fast and 
robust ICA (Fast-ICA) algorithm proposed by Hyvarinen [8] is 
widely used, which is a fixed-point algorithm based on an 
minimum of negentropy. Initially, this algorithm is introduced 
using (6) as a criterion function, and is subsequently extended 
into function (7) in order to reduce complexity of computation. 
ne(y) = [E{G(y)}-E{G(y)}f (7) 
y = w T x , <8) 
In which, V is a Gaussian variable of zero mean and unit 
variance and y is a variable of zero mean and unit variance. 
Function G(«) can be practically any non-quadratic functions. 
Two commonly used functions are listed below: 
G \ OO = — log cos(a, y) a x e [1,2] 
a 
G i OO = ~ exp(->> 2 / 2) 
In our experiment, G 2 (y) = — exp(—y 2 / 2) is used for 
iteration. Our target is finding an appropriate value for vector 
W in order that E{G(w T x)} has a maximum, which is 
equivalent that the deviation of Zs{G(w r x)} is equal to 0. In 
formula (7), g(*) is the deviation of G(*) 
E'{G{y)} = E'{Gxg(w T x))} = 0 
According to Newton Iteration algorithm , we can get iteration 
method simplifies to: 
+ E{xg{w T x)} 
W = W 
E{g\w T x)} 
Then we get following fixed-point algorithm for ICA: 
1. Take a random initial vector W of norm 1. 
2. Let W + = ~E{g'(\V T X)}\V +E{xg(\V T X)} 
3. Divide W + by its norm, then get a new W . 
4. If new W is close enough to W + , output the vector 
W .Otherwise, go back to step2. 
3. MULTIRESOLUTION DATA FUSION BSAED ON ICA 
The different bands in remote sensing imagery have strong 
correlated, which is caused by some interference factors such as 
weather, atmosphere condition, etc. On other hand, the 
established models for ground object imaging have no enough 
refutation accuracy under some uncertain disturbed conditions. 
So we use ICA to remove the interference, and get an 
independent representation of original bands. Then the high 
frequency information extracted by Atrous wavelet is added 
on three independent bands got by ICA. The diagram on the 
whole process can be found in Fig 1. 
Fig.l the fusion diagram 
In detail, three multispectral bands are firstly changed into three 
vectors R, G, B. Then the fast ICA algorithm mentioned above 
is employed to get three independent components, IC1, IC2, 
IC3. In formula 9, F means an ICA operation. 
[/Cl, IC2, /C3] = F(R, G, B) (9) 
Thirdly, the normalizing step is used to remove the mean and 
standard deviation of a panchromatic band P . AtroUS 
wavelet is applied on the result band P'. This procedure can 
be expressed as follow formula 10, 11.