ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
2) The Decoding Scheme: every parameter of coefficients C, 
is decoded as: 
V =¥,+(C, ~128)x 2 
256 
  
I 
Where V is a parameter value to be acquired, V, is the initial 
guess of the parameter; V, is the range of the parameter, and 
C, is decoded on 8 bits which corresponds to the parameter in 
the chromosome and is located in the range of between 0 and 
255. In 2.2.3, we will address methods for computing each 
parameter's initial guess and its range. 
3) The Fitness evaluation function: a chromosome is decoded 
to acquire a set of parameter values of coefficients C, , and 
then these parameters are put into equation (1) and equation (3) 
in order to gain an energy value. The energy value is regarded 
as the fitness of the chromosome. 
4) The Procedure for GAs: 
i) Initialization: the initial population is generated at random. 
The population size is equal to nx10 , where n is the number 
of dimension of coefficients C, . 
ii) At each generation 
Compute each chromosome's fitness and find the best 
chromosome C,,, with the maximal fitness. 
Reproduce the next population. The selection probability 
is determined by the ranking of a chromosome. 
Mutation operates on each chromosome, and the mutation 
probability is about 0.07. 
One-point crossover operates on two chromosomes, and 
the crossover probability is 0.3. 
iii) Stop when a predefined limit MAXGAP on the number of 
iterations is reached. Decode the best chromosome C,,, and 
get the optimal coefficients C, . 
2.2.3 Determine C, initial guesses and their ranges 
GAs need each parameter's initial guess and its range in the 
C, . Through equation (1), we know that the second order 
coefficients in the C, must be located at the small range close 
to zero, thus, we may assume that these parameters will be 
smaller than a threshold 7, . Therefore these parameters’ initial 
guesses are equal to 0, and their ranges are 27, . 
Now, we discuss how to appoint the other parameters' initial 
guesses and their ranges. In the C, , there are six parameters, 
i.e., a,,a,,a,,b,,b,,b, . These parameters can represent a general 
affine transformation. When considering there is no need of 
high accuracy in the estimation of each parameter's initial 
guesses, we assume that the geometric transformation between 
two images is composed of the Cartesian operations of 
(10). 
scaling (s), translation (Ax, Ay) , and rotation (0) , that is, 
x cos@ sind | x, AX 
Yı —sinO  cosO | y, Ay 
Comparing equation (1) with equation (11), we have the 
relationship between parameters  a,,4,4,,b,b,b, and 
parameters s,0,Ax,Ay . Therefore, each parameter’s initial 
guess in the C, can acquired as follows: firstly two pairs of 
control points are manually selected, and then utilized to 
compute parameters s,0, Ax, Ay . The range of each parameter 
may be manually chosen to be large enough to cover the 
optimal values. 
3. EXPERIMENTAL RESULTS 
In this section, we give the experimental results divided into 
three parts. First, in order to evaluate the accuracy of the 
registration, we test our approach by using a pairs of synthetic 
images. Second, we apply our approach to register multi- 
temporal images. Lastly, we show some results of the image 
registration for a different sensor. Meanwhile, in order to test 
our approach, we compare the registration results of our 
approach with those of manual methods. In existing methods, 
the root mean square error (RMSE) between the match points 
provides a measure of registration. But RMSE is not suitable 
for evaluating our approach because we do not extract any 
match points. Here, we define a measure that is similar to but 
more precise than RMSE: 
RMSE = DX yy fv x 2) (12). 
y=0 x= 
Where W , H are width and height of the sensed image 
respectively, and the distant D(x,y) between the transformed 
point and the true point is defined as: 
D(x, y)= (F,, (5,9)-G. (6 »f +(F,, (6 3)-G, (x y)f — 13. 
Where F, (x, y) and P, (x, y) are computed by equation 
(1), and (G, [C y) G (x, y) is the true coordinate of point 
(x, y) in the reference image, we define the maximum distant 
D(x, y) as another measure for evaluating the accuracy of the 
registration, 
max.) = max D(x, y) (14). 
O<x<W,0<y<H 
To compute D(x, y) in equation (13), we must know 
(G, (x, y), G, (x, y)) of the point (x, y) in advance. It is 
feasible for synthetic images pairs, but very difficult for real 
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