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Matching of IKONOS Stereo and Multitemporal GEO Images for DSM Generation
3. 3 Modified MPGC
MPGC is described in detail in Baltsavias (1991). It combines least squares matching and geometric constraints formulated either in image or object space. The constraints lead to a 1-D search space along an epipolar line, thus to an increase of success rate, accuracy and reliability, and permit a simultaneous determination of pixel and object coordinates. Any number of images (more than two) can be used simultaneously. The achieved accuracy is in the sub-pixel range. The algorithm also provides criteria for the detection of observation errors and blunders, and adaptation of the matching parameters to the image and scene content. Usually, the sensor model used for IKONOS is the rational function model, other models like polynomial mapping functions (Baltsavias and Stallmann, 1992) or DLT can be treated as a special case of the rational polynomial functions. For description of the RPCs see Grodecki (2000). Their form is given by: 
Where, (xn, Yn) and (Xn, Yn, Zn) are normalised image and object coordinates respectively. To compute the normalised coordinates, the following equations are used: 
Where, (x, y) and (X, Y, Z) are image and object coordinates; x0, y0 and xs, ys are offset and scale values for the image coordinates respectively, and similarly X0,Y0, Z0 and Xs, Ys, Zs are offset and scale values for the object coordinates. In the RPCs, the maximum power of each object coordinate and the total power of all object coordinates are limited to 3. In such a case and following the SI definition of coefficient sequence, each polynomial has the following form (for convenience, the subscripts are omitted), which leads to the 80 RPCs per IKONOS image: 
The geometric constraints were derived from equation (4) and have the following form: 
Equations (5) can be treated as weighted observation equations in MPGC, where (?x, ?y) are the unknown shifts in pixels, which are the common unknowns, appearing also in the affine parameters of the MPGC. Weighted geometric constraint force the matching to search for a conjugate point only along the epipolar curve. If the initial match of the point in the search image does not lie on this epipolar curve, at the first iteration of MPGC matching, it jumps onto this curve. The grid point matching results based on relaxation provide quite good approximations for the MPGC procedure and increase the convergence rate. In our implementation of the modified MPGC, the adjustment starts only with the two shift parameters and after the first convergence all affine transformation parameters are used. Also, at the first two iterations, the weight value for the geometric constraints takes a large value in order to speed up the convergence and then it decreases to consider remaining errors in the orientations/epipolar geometry and allow matching slightly off the epipolar curve. The modified MPGC matching procedure has the potential of obtaining sub-pixel matching results and exploiting more than 2 IKONOS images simultaneously. |