Xiunguang Zhou 
  
6.2 Geometry Constraint Cross Correlation 
The traditional cross-correlation is to determine the similarity of two points by measuring the similarities pixel by pixel 
in two rectangular windows that centered at the two points. It is given in Equation 10. 
Y Y (eG 5-2Ye 65-2) 
RU i=l j=1 de 
S X O-z nr] 
i=l jzl i=l j=1 
  
where m and n give the rectangular window size, g'(i, j) and g’ (i, j) are gray values in the left window and right 
window, respectively, 2/ and g" are the mean gray values of the two windows. 
Equation 10 presumes that the geometry in the two windows is identical. Obviously, if the two points have different 
geometry properties (different orientation and/or different scale), we can not measure the similarity by two identical 
rectangular windows. In order to get a correct measuring result, we introduced a geometry constraint in the traditional 
cross correlation. That is 
Y Y (r6 -2:Xe 6. 5-2) 
i=l j=l 
YY (sr aij SV Y (DEV 
i=l j=1 i=1 j=l 
  
RY (11) 
  
where TC, j) - €, li. ill is the new coordinates of (i, j) after an affine transform, g; is the mean gray value in 
the transformed window. 
Obviously, equation 11 takes geometry differences into account to measure the disparity of the two points. Equation 11 
can be equivalently explained in the way that a geometric transform is applied to the left window before measuring the 
disparity. This will guarantee that the disparity measure is performed under the same geometry. 
7 MATCHING AT LOWER LEVELS OF THE PYRAMIDS 
The sub-algorithms given in above sections were only carried out at the highest level of the image pyramid. At levels 
other than the highest level, two matching procedures were performed. They are 1) Iteration of geometry constraint 
cross-correlation at lower levels; 2) Least Square matching at base level. 
7.1 Iteration Of Geometry Constraint Cross-Correlation At Lower Levels 
As the image size at the highest level is quite small, it is not a big performance issue if a time consuming algorithm is 
carried out on such a small image. When the matching goes through different levels of the image pyramids, the image 
size gets larger as the level of the pyramid gets lower. Therefore, the computational costs become a critical issue. 
Fortunately, we already had the matching results at the higher level. Alternatively say, we had the pre-knowledge about 
the matching at lower levels. The pre-knowledge can be used as the matching guide at the present level. This greatly 
simplifies the matching process. Two items of the pre-knowledge can be derived from the higher level. One is the 
searching range for conjugate point pairs at the present level. The other is the local geometry correspondence at the 
present level. 
According to the WT algorithm (X. Zhou and E. Dorrer, 1994), a point at the higher level of the pyramid corresponds to 
four points at the present level. Let (x, y), be a point at level j of the left image pyramid, the four corresponding points 
at level j+1 are (2x, ay). (2x+1,2y) @x, 2y+1) and (2x41, 2y+1) u. In the same way we have the 
conjugate point (x,y);, and its four corresponding points (2x, 2y), (x+1,2y)., (2x,2y+1),. and 
(2x+1, 2y+1), . The matching task at level j+1 is to determine the correspondences of the four points in the left and 
right images. Though the coordinates between two levels have the above relationship, the correspondence of the left and 
right points at level j+1 can not be simply obtained on this analogy. This means that (2x, 2y), , and (2x, 2y);,.. may be 
Ay) 
not a conjugate pair even if (x, y), and (x, y), are a conjugate pair. In pixel accuracy, one point in the left image may 
  
1060 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 
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