The International Archives of the Programmetry, Remote Sensing and Spatial Information Sciences. Vol. XXXVII. Part B3b. Beijing 2008
Set a and b as the left and right threshold of transition region,
then
a = Ex c -3cr- He c — Ex c - 3En c - He c
b = Ex c +3(7 + He c = Ex c + 3En c + He c
(12)
Transitional region is defined as the two-dimensional pixels sets
that covered by edge cloud. That is
TR = {(Uj) e I \a< f(ij) < b |} = {(/,/) e I \Ex c - 3 En c - He c < f{i,j) < Ex c + 3 En c + He c |}
(13)
This algorithm has some advantage compared to traditional
algorithm. First, the algorithm obtains the digital characteristics
of edge cloud and two threshold of transition region by Boolean
calculation between intersectant clouds, so, the situation of low
extremum is bigger than high extremum can be prevented.
Second, the algorithm doesn’t refer complex algebraic operation,
it is simple, fast and operating cost is small. The last, algorithm
according to the image gray level forms object cloud, replace
microcosmic pixel to macroscopical cloud object, In the process
of cloud building , the operation which is similar to smooth
algorithm can weaken the influence of noise to extracted result.
4. THE EDGE DETECTION IN TRANSITIONAL
REGION.
4.1 Stochastic fuzzy feature plane and its characteristics
A two-dimensional image can be seen as a fuzzy matrix. Every
element of matrix has has the membership function JLIwhich
relative to a given gray level. The plane which formed with All
fij (z = 1,2,...,M; j = l,2,...,tV) is named image fuzzy
feature plane 114] , it is the base of fuzzy edge detection algorithm.
In this feature plane, each pixel is corresponding to an element
in matrix, the randomicity of image is not considered. This
method cannot solve the problem of uncertainty of spatial object
by representing adjacent degree of a fuzzy object to another
object by an exact membership.
For the membership of every element to cloud core of edge
cloud in transitional region changed from one to multi under the
influence of super entropy, so, the corresponding element in
fuzzy matrix to any pixel in transitional region is not a value but
a sets of membership, the stochastic fuzzy feature plane is
proposed. The three digital characters (Ex c ,En c ,Eie c ) of
edge cloud can be obtained by calculation and the membership
of each pixel can be retro-inferred by the algorithm of normal
cloud generator 1121 . Assume f (x') is pixel gray level in
transitional region, according to the mode
Efl k — G (En c , He c ) to build the normal stochastic data
En k which expected value is En c and standard deviation
is En k . The membership of each pixel can be calculated by
formula 14.
Mk = exp
(f(x)-Ex c f
2 En k 2
(14)
Where Ex c ,En c ,He c denote three digital characteristics
of edge cloud.
By the above calculation, we can get ji- (X) = {jU k . ] , so,
fuzzy feature plane can be expressed with following forms:
1M,
{a} 2 ,
{a)|2 •
M 22 •
•• KL~
}2N
'/ = 1,2 V;
(15)
Ia}«,
{Mi }
teL.
J = \,2,...,M\k = \,2,... l
Every element in stochastic fuzzy feature plane is a aggregate
of membership, it indicates that under the influence of
uncertainty of spatial object, the membership of each pixel
belong to another object is not a exact value but a probability
distribution.
4.2 Edge detection based on maximal fuzzy entropy
The gradient image G can be obtained by the gradient
operating in transition region image I that with L gray level,
its histogram is h r ,Y = \,2,...,L — 1 . Suppose G is
divided to strong edge region Re and transitional region Rs ,
set {jbi k },k = \,2,...,n is the probability of each pixel of
G to divided to Re ,\ —{/^jis the probability of each
pixel to divided to Rs . Build fuzzy partition aggregate
Qi -{g(Uj) = r},r = 0,1,...,1-1, g(i,j) is the gray
value in (z, 7) of gradient image G . Apparently,
Q = {Qo,Q 2 ,-,Ql j is a fuzzy division of G .
Though the condition entropy of fuzzy division, the condition
entropy of natural fuzzy division Q under Re is
//(e|/№)=-£
p{Q r Re)
=0 p(Re)
log
p{Q,Re)
p[Re)
hp{Re) g p(Re)
(16)
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