anbul 2004 
(9) 
nd I is the 
yes over to 
| À is very 
Normally, 
; parameter 
(10) 
G 
‘hing and 
NS, 
y) 1) 
he moment. 
ructed such 
hange, the 
(12) 
(13) 
master and 
i value for 
apply cross 
white noise 
; -- f'n. (14) 
s follows: 
International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B3. Istanbul 2004 
S ron 
FA =" (15) 
EA(r'n)(rn)'| [Ed 
The maximum SN ratio means that optimal matching is 
OR ; te ; 
accomplished. From zo , optimal matching is achieved 
Ö 
{ef f 
f (s UD (16) 
f'g f'g 
If the slave image g 1s equivalent to the master image f, the SN 
ratio becomes a maximum. The cross correlation is then 
optimal in the sense of the SN ratio, and is robust against noise. 
when 
3.3 Expansion of Regression Model 
So far, since it has been. assumed that the errors exist only in the 
slave image (g-axis), a minimization of the sum of squared 
differences from a regression line in the g-axis has been 
performed (Figure 1). 
S ^ 
  
> 
> 
7 
Figure 1. Errors only in slave image 
However, it is plausible that the errors exist in both the master 
and slave image. In such a case, we have to minimize the sum 
of squared distances from observed points to the regression line 
(Figure 2). 
  
» 
» 
f 
Figure 2. Errors in both master and slave images 
This kind of regression model is known as principal component 
regression (Greene, 2000). The principal component regression 
is applied to the intensity values of both master and slave 
images: 
X= fx ‚NM ) f(x. » V2 ) uo wf (xo > VMN ) (17) 
2 {x N ) g(x yh ) rt Ces iw) 
z=PX (18) 
; S s W W, : 
where, z is the principal component vector, P | 3 il is 
an orthogonal matrix consisting of the eigen vectors of XX". 
Since only the z;-axis is related to minimization of Sw ‚the 
objective function of the principal component regression is 
formulated as follows: 
M 
1 
E, Ya FY 
P jul i 
N 
ist ja 
Mf (x.v,)^ wag(x'.v',)) — min. (19) 
1 
To perform the minimization, the Levenberg-Marquardt method 
is again applied: 
(A+2AI)6p=b, (20) 
a», 
Ala] AM »-[]-|-Y« a. (21) 
i k 1 i 
where, 
  
et W,, (ER. og e : (22) 
Op, 
Ox'Op,  Oy'Op, 
The optimization can be carried out in the same way as 
described in Section 2.2. 
3.4 Matching Process 
The results obtained using both methods, least-squares 
matching and cross correlation matching, will very much 
depend on the initial values of parameters. In order to assign 
initial values, a matching process is followed: 
I. roughly specify conjugate points at the four corners of 
the images by manual means; 
2. transform the master image to slave image by using 
affine transformation; 
3. set the image patch at the feature points in the master 
image; 
4. apply cross correlation matching, including only 
translation to search for the initial location; 
5. apply least squares matching or cross correlation 
matching (including deformation); and 
6. continue the iterative procedure of the Levenberg- 
Marquardt method until the solution converges or a 
given number of iterations is reached. 
4. EXPERIMENT 
4.1 Comparison of least-squares and cross correlation 
matching 
Least-squares matching and cross correlation matching have 
been evaluated using an IKONOS Geo panchromatic stereo 
image pair covering an area of 7 x 7 km'over central Melbourne 
(Figure 3). Figure 4 shows an enlarged portion of the stereo 
image with feature points, these having been selected to as 
clearly image identifiable points.