Table 1 Precision and accuracy measures for Kratky's SPOT model using different versions
RMS of control points [m] Number RMS of check points [m]
Version" t of
» X Y Z check X Y Z
points
L6 6.1 2,7 21 3.6 130 8.6 10.0 14.0
Q6 8.8 23 0.5 4.0 130 8.7 9.7 16.0
L10 M 5.4 23 3.0 37 126 8.8 9.1 11.5
Q10 S 51 2.5 21 33 126 8.8 8.9 11.8
L30 6.0 4.2 3.6 42 106 8.5 9.1 11.4
Q30 6.0 4.2 3.4 4.3 106 8.5 9.1 11.3
L6 5.6 1.4 2.9 1.0 130 8.9 10.9 12.0
Q6 3.8 1.0 0.2 1.4 130 8.9 10.3 14.1
L10 us 52 2.0 33 3.5 126 9.0 10.3 6.3
Q10 8 4.1 1.7 1.8 3.1 126 9.1 9.9 6.4
L30 6.0 4.0 4.2 4.7 106 9.3 9.3 7.0
Q30 5.7 3.8 39 4.3 106 9.5 9.3 7.0
MT an seeeeeesececténes linear and quadratic model respectively
Y Sg Eee a posteriori standard deviation of unit weight
FMA dias pixel coordinates in second image measured manually
M*mateh.........-—- pixel coordinates in second image measured by least squares matching
4. FAST POLYNOMIAL MAPPING
FUNCTIONS
After the strict SPOT model is estimated the PMFs are
derived by the following approach (Figure 2). A5 x 5
regular grid is defined in the left image. By using the
results of the rigorous solution and three heights (the
minimum and maximum of the scene, and their average),
map coordinates for 75 object points are computed.
These are projected in the right image again using the
rigorous solution. By using the known coordinates in all
three spaces, the coefficients of polynomial functions to
map from image to image, image to object, and object to
image space (in both directions, i.e. 6 polynomials
altogether) are computed by least squares adjustment.
Thereby, the object space is reduced to two dimensions
by extracting the elevation, i.e. Z is an independent
parameter connecting all three spaces. One polynomial is
computed for each coordinate to be determined, and for
the mappings involving the object space separate
polynomials are determined for left and right image. The
degree of the transformation, the number and the form of
needed terms were determined experimentally. The
degree of the polynomials is 3 - 4 with 11 - 16 terms.
Kratky provides for each mapping two sets of
polynomials, a basic and an extended. The extended has
two more terms involving mainly powers of y or Y. It
should be used if the quadratic model was used in the
rigorous solution. If the linear model was used, then the
basic polynomials suffice. A similar, although less
accurate, approach with five, instead of three, heights is
also used by the algorithm of the company TRIFID which
is integrated in the Intergraph Digital Photogrammetric
360
Station 6287 for SPOT modelling and digital orthophoto
generation.
In our tests the PMFs (basic model) were determined
after the previously mentioned rigorous solution with the
linear model, 10 control points and the matching
measurements. The pixel and object coordinates of the
136 points were determined by the PMFs and compared
to their known values. The differences did not exceed 1
m in object and 1 jum in image space, thus verifying
Kratky's results. The accuracy of PMFs was also tested
by another method. By using the image to image PMFs
and three out of the four pixel coordinates
(x', y', x", y") of each point, the heights can be
determined and compared to the known values. This was
done for the triplets (x',y', x"), (x",y",x),
(x, y, y"), (x",y", y). The last two cases gave
RMS errors of ca. 135 m, which is not surprising since
the image base is approximately in the x direction. The
first two cases gave the same RMS of 6.2 m which is
identical to the results of the rigorous solution.
Having established that the PMFs are fast and accurate
enough the next step was their integration in image
matching for DTM generation.
5. MODIFIED MPGC USING PMFs FOR
AUTOMATIC DTM GENERATION
Automatic DTM generation from SPOT images has been
extensively pursued and is particularly attractive for
poorly mapped countries. Many algorithms have been
developed but none exploits geometric information from
the SPOT sensor to guide and support the matching. A
