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where x, y, are the image coordinates and X,Y,Z the object point coordinates. As can be seen from Eqs. 4, a number of 
existing restitution algorithms for line scanner imagery are based on special formulations of the ratio-of-polynomials model 
(e.g. the DLT and polynomial expressions in which the denominator is reduced to unity). 
Rational functions have previously found application in photogrammetry as a restitution model for the real-time loop in 
analytical stereoplotters, and they have well established advantages and disadvantages. Among the principal advantages are 
sensor independence (a combination of different sensors is possible) and speed. They are fast enough for real-time 
implementation, can accommodate any object space coordinate system, and can be tuned (e.g. selection of a subset of 
parameters) for particular sensors. Among the disadvantages are the possible need to tune the functions for particular 
sensors, the subjectivity associated with parameter selection (there can be 80 parameters for a stereo pair), and the fact that 
they are prone to numerical instability and require more object space control. 
Hence, although developers of digital photogrammetric workstations can provide necessary restitution software based on 
rational functions, application of this model can require careful attention. For the experienced practitioner, the costs of 
polynomial-based restitution approaches are well known in terms of the need for extra ground control and tie points in a 
multi-image scene. These inconveniences may be avoided, however, if the rational functions are obtained along with the 
imagery from the satellite image provider, and not derived by the customer. The metric impact of this empirical modelling 
approach is yet to be fully quantified for 1m satellite imagery, though it is unlikely that the majority of end users will be too 
concerned about questions of minor accuracy differences. The remote sensing and photogrammetric communities have, after 
all, long been achieving accuracies with polynomial models for satellite line scanner imagery that can reach the one pixel 
level in favourable situations. Nevertheless optimal 3D feature extraction accuracy is, by design, unlikely to be achieved via 
vendor-supplied rational function coefficients. 
It is not inconceivable that shortly after high-resolution satellite imagery becomes available, photogrammetic research 
groups will achieve a self-calibration of the imaging sensors and publish their own camera models. In the meantime, 
however, all parties in the mapping industry, the image providers, the photogrammetric system developers and companies 
performing stereo-based value adding, will potentially be able to proceed with product generation by utilising vendor 
supplied rational function coefficients. 
4.3 A Direct Linear Transformation Approach 
Restitution models based on linear projective equations have already proved practical in close-range photogrammetry, in 
spite of some well-known shortcomings in comparison to the collinearity equation model. The most widely used of these 
models is the Direct Linear Transformation (DLT), a variation of which has been proposed by Wang (1999) for triangulation 
of satellite line scanner imagery. As was alluded to earlier, the DLT also represents a special case of the rational function 
model, with the form proposed by Wang (1999) being as follows: 
  
  
L1X + L2Y + L3Z + LA 
X. zc 
LoX + LioY - LuZ 41 
(5) 
Ls5sX + L6Y +L7Z + L8 
Ye = t Luo xt yt 
LoX +L10Y + L11Z +1 
This model is effectively the ‘standard’ DLT for frame imagery supplemented with an extra image coordinate correction 
parameter, L;,. Multi-image triangulation based on Eqs. 5 does not require knowledge of sensor interior orientation, and nor 
does it require preliminary estimates of sensor EO. 
Implicit in the formulation of Eqs. 5 is the assumption that the dynamic behaviour of the sensor trajectory and attitude can be 
adequately modelled with variation functions of first-order. This would imply that the DLT approach is most suited to 
scenes of limited geographical extent. Experiments reported by Wang (1999) indicated that, for SPOT imagery covering a 
60 x 60 km area, the DLT approach yielded the same level of triangulation accuracy as the more rigorous collinearity 
equation model with multiple projection centres (Eq. 3). Moreover, the model yielded 1 pixel triangulation accuracy in the 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B7. Amsterdam 2000. 457