Susumu Hattori
4.1 Case of Flat Terrain
(0)
A O A
j^
Ground / Ground
level
level
Pe
a) Flat terrain b) Undulating terrain
Figure 1: Image transformation from central perstective to affine projection
Let us first assume that the terrain is flat and approximate height is known. Fig.la depicts the case with unit
scale. The line scanner image intersects the ground at the principal point. Point p(v) in the central projection
image is transformed to p,(v,) in the affine projection via ground point Pg and according to Eq.9.
: (9)
Vg = —
“ 1-vtanw/c
va is easily proved to be virtually a linear function of v within a practical value of w. This means that the effect
of projection inconsistency is negligible for flat terrain.
4.2 Case of Undulating Terrain
If the terrain is highly undulating, errors due to projection inconsistency are eliminated by correcting the focal
length c at each image point. The error, Av, is determined by Eq.10:
Av = AZ(tan(w + a) — tanw) cosw (10)
where AZ is the difference from ground level, as shown in Fig.1b. The error is proportional to AZ and it can
be eliminated by the following iterative procedure;
1) Assuming the terrain to be flat, the height Z is measured by bundle adjustment of the stereo observations
based on Eqs.5. The resulting height will include a significant error at this stage.
2) The projection error is compensated by transforming pa(v,) to p’,(v’) using Eqs.11 with the estimated value
of AZ:
AZ vc v'
e! —6-p Ac, Ac — , v! — — vm TTL;
COS c 1 — v'tanw/c
(11)
3) The terrain height is again measured by bundle adjustment, this time using the updated affine image coor-
dinates.
4) The above process, steps 1 to 3, is repeated until convergence. Experience suggests that even in quite
mountainous terrain, only two or three iterations are needed.
4.3 Correction of Earth Curvature
If the 3D-GK coordinate system X,Y, Z and the local Cartesian coordinate system X 9» Yg, Z9 are both set with
their origins at the scene center, the difference in X, and X or Y, and Y will likely be negligible, but for Z and
Zg the difference will be appreciable as a result of Earth curvature. The height error at a ground point S km
away from the origin is given by the well-known expression:
AZ — Y?/2R km (12)
where R= 6367 km. This effect amounts to 67m in the margin of the SPOT scene used for the reported
experiments. An alternative to compensation for Earth curvature is utilisation of Level 2 SPOT images, which
are already free of projection errors. Otherwise the compensation can be imbedded into the process of image
transformation from central perspective to affine projection.
362 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
