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Herbert Jahn 
Now some words concerning the calculation of the derivatives of gr and gg are necessary. The (continuous) image in the 
focal plane of an diffraction limited optical system is band-limited and hence an analytical function which can be 
differentiated (Jahn, Reulke, 1995). If the sampling frequency is greater than the highest spatial frequency transmitted 
by the optical system then Shannon’s sampling theorem is fulfilled and the intensity samples (gray values) represent 
the whole intensity function g(x,y). Then computing the derivatives of g(x,y) is possible using the sampling theorem. Of 
course, this ideal case is not fulfilled. Often the sampling condition is violated meaning that the sampling theorem is not 
fulfilled exactly. Furthermore, the gray values are digitized numbers and noise is present which prevents exact 
differentiation of g(x,y). Therefore, only approximations of the derivatives of g(x,y) are possible. Here, the 
approximation 
dg (x, j) 
——— 
ox 
  
et ue A 
= 7 l8 L j)- eG -1.j)] (7) 
-is used. 
Of course, if one applies (5) with (7) to the original images then the recursion (4) often will be trapped in false (local) 
minima, especially when the initial disparities (s, se) are too far from the real ones. Therefore, the recursion (4) is 
applied in a Gaussian pyramid (Jolion, Rosenfeld, 1994) starting with the coarse image 
2,#G,98- (8) 
Here & means convolution, and the kernel G, is the normal distribution function N(0,0). The operation (8) has the 
advantage that it transfers disparity information (which is available at edges, corners etc.) into homogeneous regions. 
Therefore, if &' is big enough also inside large homogeneous regions a disparity can be measured and determined with 
the algorithm (4). Of course, the convolution (8) also blurs both images and fine detail is destroyed. In those regions 
wrong disparities are generated. To overcome this, the values of & used in the pyramid must become smaller with 
increasing pyramid layer. Here, a pyramid with four layers / = 0, 1, 2, 3 was used and the c - values were chosen 
according to 
o, =" (9) 
Then the algorithm works as follows: 
(0) (0) 0) _ 
x Sy = 
1. Initial disparities (s ) are chosen. Here the initial condition (s: 03 = 0) can be used. In case of 
pushbroom images the epipolar condition is fulfilled approximately. Then, using an epipolar matching algorithm 
(0) 
X 
(e.g. that of Gimel’farb, 1999), one can determine §, . This reduces the number of pyramid layers (/ = 0 may be 
sufficient), the number of iterations, and hence the computing time. 
Layer / 2 0: The smoothed L- and R-images are computed according to (8) with o; — 8. Then the iteration (4) is 
; as ; as 0) (0) 1=0) _(I=0) 
started with the initial disparities (st Ns 4 8 
ba 
). The result are new disparities (s ) which are used as 
initial values in the next layer / = 1, and so on. 
3. At the end of the iteration in layer / = 3 the result of the algorithm are the final disparity estimates 
^ (423) ^ (123) 
(5, = ‚S, =S; 
x y 
) in each point (77) of the left (nadir) image. 
That algorithm has the drawback that it is slow on a sequential computer. To speed up the algorithm the smoothed 
images 9 o, Can be sub-sampled (for / = 3 one has N, - N, gray values, whereas in layer / there are only NU. N,/2*" 
sampling values). Then, the output of layer / are N,/2”" - Ny/2*" disparity values, but in layer /+1 N,/2”" - Ny2^ initial 
values are needed. Therefore, interpolation is necessary. This works but it is not considered here in detail. 
In order to transfer disparity information into the interior of large homogeneous regions the maximal value of & and 
hence the highest pyramid layer must be chosen big enough. Adaptivity seems to be necessary, but this was not studied 
up to now. 
Another method to enhance the performance of the algorithm is the use of 
g,70,8t0,: 8, (10) 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 439