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The International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B7. Beijing 2008
In addition, the clonal select algorithm can control the quantity
of the same antibodies by calculating the expected survival
value. It contains a memory cell set which saves the better
antibodies, so the immune algorithm can converge very fast.
Feature selection problem has been transformed to an
optimization problem under the constrain domain of S(m). Thus,
it can be proved that dimensional reduction problems are multi
objective optimization problems.
3. THE DIMENSIONAL REDUCTION MODEL IN
HYPERSPECTRAL IMAGE
3.1 Evaluation criteria during the process of feature
selection
In feature selection, many indexes are designed to evaluate the
selected bands. There are Joint entropy, determinant of
covariance matrix, Bhattacharyya distance, OIF (Optional Index
Factor), and so on (Swain P H, 1987; Sheffield C, 1985; Chavez
P S, 1982). In this paper, OIF is used as the evaluation criterion.
Is,
OIF = —^ (3)
n n
I IK I
¿=1 j=‘
In the above formula, Sj- is the standard deviation of the ith
band, and R tj is the correlation coefficient between the ith band
and the jth band.
The amount of information of the selected bands conforms to
direct ratio with OIF value of bands set.
3.2 The mathematical expression of feature selection
problem
Suppose that P is the number of bands in the image, and A is
the wavelength. All the wavelengths construct the
Set S- j/l], A 2 • • • A p }. The progress of feature select will
select N bands from S^ , and the result can be expressed as
/1 = ^, A 2 ... /l N } . The entire probable As constitute a
set S(m) . Then the object function F(A) could describe the
progress of calculating OIF value of the selected bands
i.e. \A i ,A 2 }■ Feature selection problem can be described
in math as formula (4).
min F(A) = OIF
s.t. A. g S(m)
Select A from S(m), and the A should minimize F(A). A is
the decision variables( A = {A ] , A 2 ... A-^}), constrain domain
A = {A | Aj e S(m),i = is said to the feasible
domain of decision variables. The formula of calculating OIF
has been shown on formula (3).
3.3 Hyperspectral dimensional reduction model
Inspired of immune system, this paper has built a Hyperspectral
dimensional reduction model on the base of clonal selection
algorithm. HDRM is displayed as below:
1) All the wavelengths construct the set , and N bands
constitute individual (feasible solution)/4,. In the space S(m)
which is constituted of all the probable A , Select M A from
S(m) to constitute original population S(/l) , which is a set of
antibodies. With no influence on the result, Aj (i.e. the elements
of A) are arranged from large to small.
2) Select n best individuals to constitute a set Sn from S(A)
through calculating the value of F{A) . On the progress of
calculating F(A) , the A is referenced to the wavelengths Aj,
and then to the corresponding images. So the OIF value which
is also the value of F{A) is computed. The whole calculating is
a progress of affinity estimation.
3) Clone these n best individuals, and generate temporary
population T(Ab). The clone size increases as the increasing of
its OIF value. The quantity of clonal bodies is described as
formula (5).
^clone
---- round
(5)
n c ione is the quantity of antibodies after an antibody is cloned.
P is the increment index, n is the quantity of all the antibodies
in T(Ab). round() is the rounding operation, i is the antibody's
index which indicates the order of the OIF value(decreasing).
4) A.fter cloning, the next step is mutation. The purpose is to
carry out the effective global search. A mutation operation is
used to temporary population T(Ab), and then a mature mature
population T*(Ab) is generated. The larger OIF value, the more
information the selected bands contain, and the less part the
individual should be changed. The rule is shown in formula (6).
Mutation rule: Suppose that A (A = {A\, A 2 ,... K N } )is selected
to mutate. The model defines a variable CD to control the
mutation, k elements( A { ) in A are chosen to change to the other
wavelengths randomly, k conforms to inverse ratio with F(A) .
k = round
coxn
(6)