ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision*, Graz, 2002 
  
where A is the rotation matrix. By computing the vector 
quantities we derived translation invariant geometric 
primitives (vectors). 
Let 
ae Xo 7X, 
X X 2 1 
vi | 52 À and Vv, = : : 
Yj2 X ee 
V Ya 
the squared norm of equation (3) we will end up with the 
following equation: 
. By computing 
il - à npa @ 
Identity 
RTR=1, since R is an orthogonal matrix. By computing 
vector norms, one can derive rotation invariant geometric 
primitives. One can conclude from equation (4) that the scale 
factor can be determined by the following equation: 
[| 
= (5) 
[v2] 
The rotation angle (0) between the two images will be 
recovered as a difference between the directions of the 
normal to each vector. The directions of the normals can be 
obtained by the following equations: 
  
Vı= cos”! ^H (6) 
Ini] 
vis oot 2 (7) 
Ii] 
üzy)-yw, (8) 
where U/ , and V/ 2 are the directions of the normals to the 
two vectors. The direction of the normal to the vector is 
preferred over the direction of the vector to avoid the 
difficulties in vertical vectors. After solving for the scale 
factor and the rotation angle using equations (5) and (8) the 
translations along the x- and-y axes can be solved for by: 
Ty, 2 xj — (s(cos0)x; — s(sin 0)y;1 ) (9) 
T, =yj —(s(sin0)x; + s(cos0)y;1) (10) 
2.2 Parameter Space Clustering 
As was mentioned above, the basic idea behind this approach 
is a pairing process between two data sets. In statistical 
sense, the pairing process is nothing more than a 
determination of a parameter distribution function of the 
specified unknown parameter(s). Equations (5), (8), (9), and 
(10) were used to pair the extracted features from the first 
and the second image. For instance, equation (5) was used to 
pair the derived norms from the first and the second data sets 
to recover the parameter distribution function of the scale 
factor. By the same token, equations (8), (9), and (10) were 
used to recover the parameter distribution functions of the 
rotation and translations parameters respectively. The results 
A - 320 
of pairing can be encoded in the parameter space, and is 
implemented by an accumulator array. The correct pairs will 
generate a peak in the parameter space. This peak will be 
evaluated as a consistency measure between the two images 
to be registered. Incorrect pairing will give rise to non- 
peaked clusters in the parameter space. In this method and 
other ones, which adopt similar approaches to match data 
sets, the admissible range of the transformation parameters, 
encoded in the parameter space, defines a probability 
distribution function, as indicated previously. Then, the best 
transformation parameters are estimated by the mode, that is, 
by the maximum value (the peak). It is well known that the 
mode is a robust estimator (Rosseeuw and Leroy, 1987) since 
it is not biased to outliers. In automatic image registration, 
outliers correspond to transformation parameters originate by 
matching some image features to noise or to some features 
that do not exist in the other image. Hence, we can conclude 
that parameter space clustering should be capable of handling 
incorrect matches in a way that do not affect the expected 
solution. 
2.3 Least Squares Solution 
In order to propagate the accuracy of the extracted feature 
(points) into the registration parameters in an optimal way, a 
least squares solution was used. Equations (11) and (12) 
below describe the similarity transformation with the 
uncertainty associated with extracted points. 
à Xi 7 Épi 
Xj 7€y1 7 T, *(s(cosO) —s(sinO) (11) 
Ya T Évil 
. Xni Cal 
Jj 7 €yi 7 T, * (s(sin0) * s(cosQ)) (12) 
Ji 7 Épil 
i Y 0 
"equi! ) 
e: is the true error associated with each coordinate, —: stands 
for the normal distribution and E,, E, are the variance- 
covariance matrices associated with each data set. We 
assume that the two data sets are stochastically independent. 
The proper stochastic model of equations (11) and (12) is the 
condition equations with parameters (see Schaffrin, 1997), 
which can be stated as follows: 
bY = AE + be (13) 
b: is the partial derivatives with respect to the observation 
(extracted features), A: is the partial derivatives with respect 
to the registration parameters, = : is the correction values to 
the registration parameters, and e: is the true error. 
3. EXPERIMENTAL RESULTS 
This section presents a complete experiment of a typical 
example of workflow of automatic image registration using 
GIPSC. Two subimages of SPOT scenes of size 
1024x1024, were used in this experiment, see Fig. (1). 
These subimages shared a common overlap area and 
separated by a time difference of four years. The two images 
were corrected up level 1A. In order to remove the random