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.
