Helmut Kager 
  
2.4 Gestalts with Form-Parameters 
Bearing in mind the inconvenience of interpreting the 274 order coefficients contained in a quadric, we desire a more 
direct formulation using the length of its axes resp. of the radii of circles as parameters. 
To achieve this, a further generalization of the basic concept allows the usage of up to four form-parameters € (q,r, s, t] 
for any GESTALT-surface. 
9 9 9 
9 9 89 9 
a DEE Ba ey (6) 
i=0 j=0 k=0 I=0 
=0 m=0 n=0 o=0 
  
  
  
  
  
  
  
Surface count form-parameters 
Sphere 1 T radius r 
Ellipsoid 3 r,s,t | length of three semi-axes a, b,c 
Rotational Ellipsoid 2 r,t | length of equator- and polar-semi-axis a — b,c 
Torus 2 r,t | inner and outer radius r, R 
  
Table 1: Surface description by form-parameters 
Examples for sets of coefficients c;jkimno for some surfaces from table 1 (literally given values for some coefficients 
indicate them being constant, whereas the others are to be determined by adjustment; for coding scheme see section 5.3): 
- Asphere 72 + 3% + 22 =r? in canonical form using form-parameter r as radius: 
{cijkimno } = {co000200 =-—1, €2000000 = 1, Co200000 = 1, Coo20000 = 1} 
A 3-axes ellipsoid 5? .1? 2? -- r2 12 - g? - r2 - 8? . 2? — r2 . s? .1? in canonical form using the length of the 
semi-axes (a,b,c) as form-parameters 7 = a,s=b,t=c: 
{Cijkimno } = {Coooo222 = —1, €2000022 0200202 C0020220 } 
The same ellipsoid but using the reciprocal semi-axes as form-parameters r = 1/a,s = 1/b,t = 1/c 
{Cijkimno } = (co000000 — —]1, C2000200; C0200020; C0020002 ) 
A 2-axes (rotational) ellipsoid r? - t? - z? 4- r? 12? + 14 - 2? = 14 -t? using form-parameters r =a = b,t = ¢ 
for length of the semi-axes: 
Íckimno] = {Coo00402 = —1, €2000202; C0200202; Co020400) 
2.5 Gestalts in General Design 
We prefer to describe GESTALT-surfaces with form-parameters in pure vector notation; later on, we will see that this will 
ease differentiation for deriving the linearized observation equations. 
A 3D-vector of surfaces (corresponding to (1)) may be set-up by the list of ADPAR-vectors Gijkimno :7 (a,b, c), 
ijklmno : 
9 9 
9 9 9 9 9 
= VE XXL ikimno : (£1) - (652) - (652)* - (61) - (65a) - (5a)" - (e1a)® 
i=0 j—0 kz0 [—0 mz0 n-0 o-0 
T(Z,a,q) (7) 
where a := { Gijkimno } is the set of relevant coefficient vectors, and q = (q,r, s, t)" is the vector of form-parameters. 
This definition allows for substitution of x, into (1) as function of the parameters z, a, and q. 
In return, it is possible to select one scalar surface-equation for the g^^ component from this vector of surfaces (with 
g € {1,2, 3} corresponding to a gestalt’s z-, y-, or z-equation (3)): 
T 
(2), — & % = (x(2,0,0))y = €, æ(E,a,q) = 
9 9 9 9 9 9 
Na >) 2, 2. €, Gijkimno) : (€) - (e5) - (e) -(€1a) -(e5a)^ -(&5a)^-(&,a)" — (8) 
j—0 kz0 1-0 m—0 n 
- 
il 
o 
Me 
Choosing g = 3, (8) comes out to be identical to (6) 
In the implementation, a GESTALT is such a vector of surfaces - see section 5.1. 
  
474 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.