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Figure 2: The Lambert reflectance model Figure 3: The Lommel-Seeliger reflectance model
at an arbitrary point is interpolated from the neighbouring grid heights, e. g. by bilinear interpolation. At each point of
the object surface the surface normal vector » thus becomes a function of the neighbouring Z4.
In the radiometric model a uniform albedo A is assigned to the object surface. Object surface clements of constant size
are defined within cach grid mesh. The size is chosen approximately equal to the pixcl size multiplied by the average
image scale factor. Assuming known parameters for the exterior orientation and thus known components of the vector
v model grey values denoted by G, and G; s, respectively, can be assigned to each object surface element. Gy and Gig em
be equated with the corresponding BDR values rand 1,5, and can thus be related analytically to the surface albedo A, the
surface normal n , and to thc DTM heights Zi:
Gc =r.= A cosin(Zk, n, s) (3)
cos (n( Zx, 1), s)
Gus =rus = 2 À = = = z (4)
cos (n(Zx, 1), S) * cos(n(Zk, 1), v)
If also the direction of illumination s is assumed to be known, the only unknown quantities in the equations (3) and (4)
are the parameters of the object surface models, the DTM heights Z,; and the surface albedo A. For each image j with
given interior oricntation parameters the considered object surface element can be projected into image space using the
well known collinearity equations, and image grey values g can be resampled from the original grey values at this posi
tion. The g are considered as observations in a least squares adjustment for the estimation of the unknowns. The corre:
sponding observation equations read (omitting the indices L and LS for the model grey values to arrive at a common
notation for both reflectance models):
vi = Gi(d, 2.1) — g(Zk.1) (5)
yj stands for the residuals of the adjustment. Since equation (5) is non-linear with respect to thc Z4, an iterative compu
tational scheme is needed, and initial values for the object space parameters must be available. In equation (6) the
structure of the linearised observation equations is given.
y = (0G/0Zk,1 — de / àZk, 1) AZk,1 + 9G /04 AA — (29 — Gr)e (6)
AZi.1 and A4 arc the changes of the unknowns from iteration to iteration, and (g — G;)- is the difference between the image
and the model grey value computed from the initial values for thc unknowns.
If two or morc images taken from different viewpoints are given, stereoscopic correspondence between DTM meshes
projected into the images is implicitly exploited, and therefore absolute heights can be computed. The model can also be
used if only one single image is available. In this case the classical indeterminability of SFS is overcome by the intro-
duction of the geometric surface model which stabilises the solution. However, only height differences rather than ab-
solute heights can be derived, because from a single image an image scale factor cannot be determined. In order to ob
tain absolute heights, it is sufficient to consider one of the DTM heights as constant.
While MI-SFS generalises classical SFS by allowing for a perspective transformation between image and object spa
and for the simultaneous processing of multiple images, it shares some of the assumptions often found in SFS ap
proaches. Interreflections and self shadowing are not modelled, and the object surface must have constant albedo and
must be piecewise smooth without breaklines. Also, occlusions are not accounted for in the model.
726 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
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