ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
following. 
2.2.1 Search the Optima by A SM 
SM is a local search technique that uses the evaluation of the 
current data set to determine the promising search direction. It 
is an iterative procedure that runs as follows: 
1) Initialization: construct a simplex with » +1 points in the n 
dimension search space (for function Energy(C,), n is equal to 
6, and for function Energy(C,), n is equal to 12) and determine 
initial guesses for n+1 points X, (1<i<n+1). We will estimate 
the initial guesses in the following 2.2.2. 
2) Form new simplexes by replacing the worst point in the 
simplex with a new point generated by reflecting, expanding or 
contracting: 
i) Initializing: at first, for each point in the simplex, compute its 
energy value according to equation (3). Secondly, find the 
minimum f„, the second minimum fy and the maximum 
f, among these energy values and their respective points X yh 
X, and X,. Thirdly, compute a centroid X of the remaining 
N points except the point X, as follows: 
X«L (X eX, Xe Xa Xu) (4. 
n 
ii) Reflecting: generate a new point X, by reflecting X, over 
the centroid X : 
Xxx.) (5). 
Then compute the energy value /, at the point X, . 
iii) Expanding: for f, » /;, a new point X, further along the 
reflection direction is generated using the equation: 
X, 2X e y(X - x,) (6), 
where y>1 is called as the expansion coefficients. Then 
compute the energy value f, atthe point X, . 
iv) Contracting: for f, « f, « f,, a new point X. close to the 
centroid X on the opposite side of X, is generated by: 
Xe - X « B(X - x,) (7), 
where £(0« <1) is called as the contraction coefficient; for 
f4 € fy , a new point X, close to the centroid X on the same 
side of À, is generated using the contraction coefficient 3 
A - 396 
X. YA) (8). 
Then compute the energy value f. atthe point X,. 
v) Replacing: for f,» f,, X, isreplaced by X,;for f,» fy, 
Xy is replaced by X,; for f.» f,, X, is replaced by X,; 
otherwise X, isreplaced by X,. 
3) Stop when 
  
nt A 
Sh rer, )- evr} | <E ©). 
i=l 
Where the & is a predetermined threshold. The X, is the 
global optimal coefficients C, . 
2.2.2 Determine the C, initial guesses by GAs 
GAs can be used to determine the initial guesses for the SM. 
Before discussing these, we will first consider how to apply 
GAs to optimize the energy function Energy(C, ) . 
GAs [7] are global search and optimization techniques modeled 
from natural genetics, exploring search space by incorporating 
a set of candidate solutions in parallel. A GA maintains a 
population of candidate solutions where each solution is 
usually coded as a binary string called a chromosome. A 
chromosome encodes a parameter set (i.e., a candidate solution) 
for a set of variables being optimized. A set of chromosomes 
forms a population, which is evaluated and ranked by a fitness 
evaluation function. The initial population is usually generated 
at random. The evolution from one generation to the next 
involves three steps. First, the current population is first 
evaluated using the fitness evaluation function, then ranked 
based on its fitness values. Second, GA's stochastically select 
"parents" from the current population with a bias that better 
chromosomes are more likely to be selected. This is 
accomplished using a selection probability that is determined 
by the fitness value or the ranking of a chromosome. Third, the 
GA reproduces “children” from the selected “parents” using 
two genetic operations: crossover and mutation. This cycle of 
evaluation, selection, and reproduction terminates when an 
acceptable solution is found, when a convergence criterion is 
met, or when a predetermined limit on the number of iterations 
is reached. 
In fact, only three components of GAs, such as decoding, and 
fitness evaluation of each chromosome, are related to a real 
optimization problem. In the following, we will first discuss the 
three components to the energy optimization, and then give out 
a detailed procedure for energy optimization. 
1) The Encoding Scheme: In this paper, a binary string is 
adopted to represent a chromosome, and 8 bits is used to 
encode every parameter of coefficients C, . Thus, the lengths 
of each chromosome for C, and C, are equal to 48 and 96 
respectively.