699 
In: Wagner W., Szekely, B. (eds.): ISPRS TC VII Symposium - 100 Years ISPRS, Vienna, Austria, July 5-7, 2010, IAPRS, Vol. XXXVIII, Part 7B 
Z(i-/u m ) 2 PM £U~^) 2 P(%) .... 
.2 _ ,=0 _2 7=0 
I . — — rj — JL 
m s-1 ^ nj /-1 
I A®„) 
1=0 7=0 
¿(/-/J d ) 2 PK) X(j-PcjfP(%) . . 
.2 _j=s 2 _J=i ('5) 
Cl ¿-1 ’ °Cj L-1 
Z p K) Z P K) 
According to Equation 2, we can derive that 2D-Fisher criterion 
function is: 
Figure 4. distribution of changed and no changed pixels and 
noise area Difference image. 
The two-dimensional joint probability density of Pixel gray 
value and its neighbourhood average gray-scale is as follows: 
]l(P(vM, ^KK) 2 +(%> g -P(co ni k,) 2 (16) 
+p(%h 
m,j) 
N 
i=0 7=0 
When J(s,t) reaches the maximum the corresponding Split 
point should be the best change detection threshold, then the 
2D-Fisher threshold selection rule is: 
Given any two-dimensional threshold value, the corresponding 
changed class and no changed class are G) and (O n . Their 
gray and average neighborhood gray are: 
(s*,t*) = Arg max rj(s,/)l (17) 
0<5<I-1 L J 
0</<i-l 
A®J = Ep 9 ,7 = 0,1,-,¿-1 (6) 
/=0 
P(%) = f iPiJ ,i = 0,l,-,L-l (7) 
7=0 
A®J = Ep s ,7 = 0,l,-,i-l (8) 
i=s 
A%) = Ep 9 ,1 = 0,1,-,¿-1 (9) 
J=t 
The corresponding mean and variance vectors: 
Mc=(Mc„M cJ ) T A=(i“m>Ay) 
ri=(<7>*) r =(rt,crt,f 
And 
A, 
7=0 
" ’ Pnj ~ /-1 
¿=0 
¿-1 
7=0 
¿-1 
“ci L 
/=5 , . j- 1 
Pcj L-\ 
Z P K) 
(10) 
(11) 
(12) 
(13) 
7=' 
3.2 The improvement of 2D-Fisher criterion function 
2D-Fisher criterion function can make use of the gray 
information of a single pixel and related information of pixel’s 
neighborhood space. The considering scope is extended from a 
single point gray value into the point and its neighborhood 
gray-spatial information. Relative to the classical 
one-dimensional Fisher criterion function, anti-noise 
performance has been greatly improved in the process of image 
change detection. According to the basic principles and 
formulas of 2D-Fisher criterion function, it can be known that 
along with the increasing of solution space dimension, the entire 
solution space need to traverse when to find the optimal 
threshold is [0,L-l]x[0,Z,-l] .At this time, computing 
time is too long, and real-time is bad. To some extent, it limits 
the 2D-Fisher criterion function method in the practical 
application of remote sensing image change detection. 
Therefore, the following content will consider how to improve 
the timeliness of 2D-Fisher criterion function method. 
In order to improve the speed of remote sensing image 
change detection of 2D-Fisher criterion function, it can be 
considered from two aspects to improve the proposed 2D-Fisher 
criterion function: for one thing, narrow the solution space of 
2D-Fisher criterion function; for another, transform the 
two-dimensional threshold into two one-dimensional thresholds. 
The following will discuss these two mentioned-above aspects. 
1) Narrow the scope of the original solution space, so as to 
achieve the purpose of improving the speed of detection. 
Since the extraction of the change area is similar to the two 
types of clustering problem, to maximize the distance between 
classes, the initial cluster centers generally select the point 
where the rule function value is the largest. Because it is two 
types of clustering, the initial cluster centers always select the 
two points where the function gets two largest values. In 
specific clustering process, the optimal threshold value will fall