The relationship between image coordinates on the left 
(XL,yL) and (xR,yR) on the right is given by assuming an 
affine transformation exists between the windows 
(Foerstner, 1982). Furthermore, no information of the 
object is taken into account in the matching process. 
Matching is solely based on the intensity values of the 
pixels and the assumed affine transformation. However, if 
the collinearity conditions were to be used, the 
transformation could take into consideration the shape of 
the object. If some information about the object's surface 
is availabale, then a sensible surface model can be 
introduced and the model could be improved. This would 
then constrained the matching to the surface model. In 
doing so, image coordinates (XR,yR) are expressed in 
terms of (xL,yL) using coordinates (X,Y,Z) on the object 
surface. 
Consider a window of size n x n pixels (where n is odd) 
on the left image with its centre (i.e. the central point) 
having the coordinates of (x ,y| ). It should be noted that, 
‘central point‘ does not have to be the centre of the 
window. The term is chosen merely for the convenience 
of explanation. By using an appropriate surface model, 
the corresponding position on the right image (xp,YyR), 
where xR,yR are not neccessarily integers, can be in 
terms of the central point on the left (xL,yL) and the 
corresponding coordinates (X,Y,Z) on the surface. In 
addition to the central point, the relationships of 
neighbouring points, say, (xi -AxL , yL+AyL) on the left 
image and (XxR+AXR, YR+AYR) are also needed. Values 
AX| and Ay|_ are known while AxR and AyR are not known 
but are defined by the relevant transformation between 
the windows. 
Use of the collinearity equations, would introduce three 
additional parameters, (X,Y,Z) for the central point and 
also for each of the neighbouring points. Supposing that 
the six relative orientation parameters are known, then a 
relationship can be established that relates (xL,yL) to 
(X, Y,Z). By adopting a suitable surface model coordinates 
of neighbouring points on the surface can be related to 
the central point and the replacement of the affine 
transformation is achieved. 
2.1 Functional Model 
The coordinates (xL,,yL,) of the central point, O, of the 
window on the left image can be represented by :- 
XL 7 f (Xo. Yo,Zo) ; yL 7 fy, (Xo.Yo,Zo) ... (ii) 
where, xj and yj, are known; (Xo,Yo,Zo) are not known 
(but needed) and an initial estimation can be obtained; 
fx and fy, are the collinearity conditions. For a 
neighbouring point, P, with shifts (Ax|,Ay|) from the 
origin on the left image, then its coordinates can be 
represented by :- 
XL*AxL — fx, (Xp, Yp,Zp) N 
YL+AYL =fy, (XP,YP,ZP) … (iii) 
where Xp,Yp,Zp are the coordinates of the neighbouring 
points on the surface. If the corresponding shifts on the 
object are AX, AY and AZ, then eqn (iii) can be written 
as :- 
556 
International Archives of Photogrammetry and Remote Sensing. Vol. XXXI, Part B3. Vienna 1996 
XL+AXL m fq (Xo* AX, YoAY Zo AZ) 
YL+AYL = fyı (Xo+AX, YotAY,Zo+AZ) s (iV) 
Supposing that matching is to be done for a flat surface 
then a surface model can now be introduced across the 
window to represent the surface. This is given by 
(considering only the first order terms, i.e., a planar 
surface model) :- 
AZ = 2 None ny NAV) 
ox oY 
where (0Z/0X) and (0Z/dY) are the gradients of Z in the X 
and Y directions respectively. These gradients define the 
model surface and they are to be evaluated in the 
solution. In turn, the terms AX and AY can now be 
expressed in terms of AxL, AyL :- 
X 
AX 2 — Ax +—Ay … (vi)a 
Sx D yo 
oY oY 
AY = —A —A … (vi)b 
Ox. Carr YL (vi) 
where (oX/ox;), (0X/dyL), (aY/oxL), (dY/dyL ) are derived 
from the collinearity equations and Ax; & Ay] are 
known shifts on the left image. By substituting eqns (vi)a 
and (vi)b into eqn(v), AZ can be expressed as a function 
of the shifts Ax| , Ay| :- 
= 9Z X AX kdX ay 4 92 on ony 
oX|ox Lay 'L av|ax Loy Lb 
"I 
This implies that AX, AY and AZ can be written in terms 
of the known shifts Ax|_ and Ay, as well as the surface 
gradient (0Z/0X) and (0Z/dY). Since AX, AY,AZ, (0Z/0X) 
and (0Z/dY) represent the planar surface model of the 
object, these terms are therefore common to both the left 
and right windows. In other words, given a neighbouring 
point on the left image whose are (x +Ax;, yL*AyL), its 
corresponding coordinates on the right (xg+Axg, yR+A 
YR) can be estimated via the planar surface model. Thus 
eqn (i) can now be expanded to represent neighbouring 
points by :- 
ILXL+AXL,YL+AY| ) + N(X+AXp,y+Ay| ) = 
IR(XR+AXR,YR+AYR ) ... (Viii) 
Linearising eqn(viii) in xa, yp, Axq and AyR will yield :- 
IL(X_+AXL,YL+AYL) + N(X+AX ‚y+Ayı ) = 
IR° xR +axR%yR%+AYR®) 
dlp dlp dig dlp 
— Id —— d ——- dA — dA 
ES XR «| 2: YR «2n XR + Wn YR 
... (ix) 
      
    
    
   
   
    
     
    
   
   
   
   
   
    
   
  
    
    
      
   
    
    
    
    
  
     
   
    
    
   
    
   
     
    
    
      
   
    
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