RNG SE RR 
  
     
    
    
  
   
  
  
  
  
  
  
  
   
  
  
    
   
  
  
  
   
  
2. THE ESTIMATION MODEL 
The statistical estimation model used is a combined least squares adjustment. The observation 
equations consist of two parts, gray level matching and imposed geometric constraints, whereby 
the two parts are related through the shift parameters of the image patches. The matching 
observation equations include, besides the equations of the gray levels, equations for the 
shaping parameters, which are treated as observed parameters. By assigning them weights, 
their alterations can be controlled and constrained. This is justified by the fact that their 
alterations lie within certain limits and has the advantage of preventing their excessive increase - 
when they are not determinable - which leads to a wrong patch transformation. The geometric 
constraints used in this case make use of the collinearity equations. 
Although the model can be applied to a multiphoto arrangement, in the sequel, a two image 
configuration will be described for reasons of simplicity. 
Since the height should be determined at a given X,Y ground position, both the left and right 
image patches have to be transformed. In this case the shifts can be determined because of the 
collinearity constraints. The shaping parameters are not constrained by the collinearity equations, 
so they are necessary and determinable only for one image patch. 
The radiometric offset between the two image patches is not included in the model as a 
parameter, but it is accounted for prior to the least squares iterations by equalizing the average 
gray level of the two windows. This speeds up the computations and relieves the model from 
one additional parameter without leading to a deterioration of the results. 
The gray level matching observation equations are formulated as: 
-e(xy) 2g(xy)-f(xy) (1) 
where e(Xy )....true error vector 
f ( Xy ).......gray level function of left image patch 
g ( x,y )......gray level function of right image patch 
Assuming two shifts for the left patch and an affine transformation for the right patch, yields: 
XL =TxL +XLo 
YL=TyL+YLo 
Xp 7 Tx n* Sx Xno*Py Yno (2) 
Yn*lvn *Hy XRo* Sy YRo 
where X45 Lo * Ro Ÿ Ro" initial pixel coordinates of the left and right patch 
XLYLXR YR ooo. transformed pixel coordinates of the left and right patch 
T«pTvp Tx Ty p---Xy shift of the left and right patch 
Sos y mme x,y scale of the right patch 
R,.R ym x,y sheering of the right patch 
The X i5. Y Lo; X Ro: Y Ro refer to the centered pixel coordinates of the left and right patch before 
the iterations start. Since the dimensions of the patches, and thus the pixel coordinates, are the 
same for both patches, the indeces L,R are going to be left out in the sequel.