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.