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It serves not only for Model Transformation but also as basics for Perspective Transformation as well as for Polar-point 
Transformation as well as for GESTALT S (i.e. surfaces) in ORIENT (ORIENT, 1995), (Kraus, 1996, B3.5). 
2.2 The Concept of Fictitious Observations and Simple Gestalts 
A fictitious observation is conceived as observation done in the mind - not using any physical measuring device - merely 
using experience and knowledge doing some geometric interpretation (Kager, 1976). 
Anyway, a fictitious observation is defined as "distance of zero" between some point and some surface or curve. That is 
considered to be the only possible "measurement" to be done without using some physical instrument. 
The first application of this idea is to use one (e.g. the third) row of (1) and "observe" this coordinate-component of x 
as zero-distance: z = 0 . Formally this is done, simply multiplying (1) from the left with e, . Since ez, = æ, we have 
found the one "inner" parameter of this plane: the off-set of the local zy-plane (its attitude is defined by R ) - from the 
"external reference point" X, . 
We can repeat this game with e! and e, as well; So we get two other planes: an yz-plane x — 0 anda zz-plane y — 0, 
any pair of them being orthogonal provided the same R-matrix has been used. 
The three types of Simple Gestalts are: 
z-Gestalt y-Gestalt z-Gestalt 
er =z ex =y er 
€ = EX = % e3% = % e 
dE =i e$ zj e$ i 
Another important feature is contained in that proposal: Parallelity of planes can be achieved by using one type of these 
planes with the same R-matrix but different “inner” parameters a, or different "external reference points" X, . 
2.3 Gestalts with Curvature 
It seemed a good idea to formulate (patches of) street-surfaces as bi-variate polynomials: 
9 ,9 
v= YY 0 FF 4) 
i=0 j=0 
the c;; being coefficients for the Z'j -terms, the upper summing index “9” stemming from the coding scheme (see 
section 5.3). It can easily be seen that the above definition of a fictitious observation €3x = z = 0 together with (4), 
interpreted as "inner" parameters €5a — 2», does not only the tilting of the planes but also the desired generalization 
from planes to surfaces with curvature. This being merely a question of the set of c;; used which also determines the 
degree of the actual polynomial. 
Sincethese c;; "add" the higher degrees to the "simple GESTALTS" we call them "additional parameters" ADPAR. This 
is also an analogy to the introduction of additional parameters describing distortion and film shrinkage for photographs 
(Kraus, 1996, B3.5.6). 
Pure formal generalization yields a GESTALT-surface as tri-variate polynomial: 
9 9 9 
m m DY ong A (5) 
Since this equation can also be written for z, and y replacing c;jx by aijk resp. bij , we have a common scheme 
for any of them. 
Some time after, it was discovered by chance that (5) contains also closed surfaces which was exploited to adjust circles 
and ellipses intersecting cylindric gestalts with planes in hybrid bundle adjustment (Kager, 1981). 
Examples: 
Set of coefficients c;;, for a 3-axes ellipsoid in canonical form: 
{Cijk} = (C000; C200; C020; Coo2 ) 204 
Set of coefficients c;;x for a 3-axes ellipsoid with attitude: 
(cix) = {€000, ¢200, C110; Co20; C101; Co11; Coo2] : 
Set of coefficients c;; for a 3-axes ellipsoid with attitude and shift: 
{Cijk} = Ícooo; C100; C010; C001 ; C200; C110; C020; C101; C011; C902) 
  
International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000. 473