Helmut Kager
025
0 Qijkimno
9 9 9 9
9
= YY PTY asim (68)-(62) (2) (34)
9
1=0 j=0 k=0 [=0 m=0 n=0 0=0
(€12)" - (52) - (€2)* - (€1)' - (65a) - (e5a)^ - (e4a)^ 1 (33)
9
Il
[+ (ea) * - (e5a)”" - (e5a)” - (e19)® - €; + m - (et a)! - (ea) "7! - (eL.9)” - (eha)” - € +
*n- (£a) - (e5a)" - (e5a)”"* - (&4a)* - & & o- (ea)! - (5) - (5a)^ - (€,a)*7! - €; ]
with N resp. N containing as columns the normals of the three potential surfaces in the local resp. global system, and:
9.9 9 9 9 9 9
E ECSSSSS SS age Ea) (Ca) oa) (da) (35)
i=0 j=0 k=0 I=0 m=0 n=0 o—0
[i (613) - (653 - (58) à j- (12) (05) - (5D) - & + k- (3) (5 - (58) e]
Notice well the dyadic products @;jkimno - €; with d € (1,2,3, 4} in above derivatives!
4.3 Derivatives for Normalized Gestalts
To make things easier, we handle only implicit GESTALTs now: A — 0. So, (28) becomes in accordance with (21):
vog meg cm aq. V on (36)
Ill] lImell Ine(Z, a, q)]|
Since the denominator is also a function, the derivatives become more complicated. Additionally to section 4.2, the
quotient-rule has to be taken into. account to get the various ohn, Due to the space-limitation of this paper, the reader is
invited to try it himself. He should then explore further the important role of the normal vectors contained in N resp.
Nand he might come finally to the derivatives of these normals: the Hessian Matrix being the 2™¢ derivative of the
surface.
x,
5 PROGRAMMING ASPECTS
5.1 Implementation
A GESTALT is a vector of surfaces (7) but not all of its components are validated (i.e. turned on). If one component is
validated, this GESTALT represents a surface; if two components are validated, this GESTALT represents a curve. The
ADPAR representing ea = { e, Gijkimno } build a list of "1D-points", the identifier of every one contains g and the
actual index for a;;4;,,5, (8) (see section 5.3). A "status" controls (besides others) whether that coefficient is treated as
fixed (—constant) parameter or as free unknown to be determinable in adjustment.
5.2 Numerics
Since polynomials are known as numerically critical under certain circumstances, two measures were taken to ensure
numerical stability for the polynomial expressions in e.g. (7). Numerical mathematics suggests the domain of polynomials
being the interval [—1,--1] to grant stability. The definition of the point-argument Z (2) used for all polynomial
expressions does as well the centering around zero as the normalization to that unit-interval.
The centering, done by the choice of the "external reference point" X, is responsible mainly for avoidance of numeric
extinction, whereas the "normalization radius" o keeps the values of the coefficients in a moderate range avoiding exponent
overflows. It has to be stressed that the definition of higher degree/order GESTALTs requires some skillness of the
operator: to define a suitable reference point in the center of the area of interest and to estimate the radius of that area.
GESTALT: defined in canonical form may be considered stable, usually.
5.3 Coding Scheme
Every required individual ADPAR coefficient ea; kimno has an identifier; it is coded as "point number":
identifier i — type g of equation and exponents : - -- 0
e.g. = 2 =: + Co210200 - §2Zr? + - - - in scalar notation like (6)
base x.y
form par. + q
form par. >+r
form par. — 8S
form par. 3 t
478 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.
