Helmut Kager 
  
ADJUSTMENT OF ALGEBRAIC SURFACES BY LEAST SQUARED DISTANCES 
Helmut Kager 
Institute of Photogrammetry and Remote Sensing 
Vienna University of Technology, Austria 
h.kager@tuwien.ac.at 
Working Group III/2 
KEY WORDS: Mathematical models, Orientation, Photogrammetry, Surface reconstruction, Software 
ABSTRACT 
GESTALT S are part of ORIENT, the hybrid adjustment program for photogrammetric, geodetic, and other observations. 
GESTALT surfaces are formalized as tri-variate polynomials. Further "form-parameters" allow convenient formulation of 
algebraic surfaces allowing the use of canonical forms (for e.g. ellipsoids or tori); nevertheless, these surfaces may have 
general position and attitude in space. 
GESTALT contain "fictitious" observations, such that points "lay on some surface” + some accuracy. 
The LLSQ-minimization principle for (implicit) functions is discussed: minimization of "algebraic error" versus mini- 
mization of "normal distance". 
Derivatives for doing iterative LLSQ adjustment are presented supplemented by implementation aspects. 
1 INTRODUCTION 
The original requirements for designing ORIENT (Kager, 1976) as hybrid adjustment program for photogrammetric, 
geodetic, and other observations contained also the handling of points in planes (horizontal, vertical, general), and on 
straight lines (horizontal, vertical, general). 
Moreover, the handling of street-surfaces for court expertises of traffic-accident scenes should be feasible. Curves in 
space seemed to be a further expansion of the concept allowing the orientation of (e.g.) photographs using line-features 
(tie-curves) additionally to (or instead of) tie-points. Investigating the algorithmics of the system, the question arose 
whether all these relations had to be implemented as separate features or if they could be handled in the frame of one 
common concept. The answer was the concept of GESTALTS introducing "fictitious" observations to photogrammetry. 
The applicability of the system was demonstrated in (Kager, 1980). 
After the discovery that the algebraic formalism for GESTALTSs also contained closed (-implicit) surfaces f(x, y, z) — 0, 
two problems appeared: first, the question of what to minimize when using an implicit equation, since it is homogeneus 
causing its residuals non-metric; second, the minor oddities that it is easy to adjust a general ellipsoid as a general quadric, 
but difficult to handle rotational ellipsoids in this way. The answers to these two problems shall be given in this paper: 
first, minimizing the squared distances from the surface; second, introducing "form-parameters" into the algorithm. 
2 TRANSFORMATIONS AND THE FORMALISM OF GESTALTS 
2. Spatial Similarity Transformation 
The Spatial Similarity Transformation turns out as the foundations of all: 
(x - x) = À HR -:{(X - X,)=\:0 8 (1) 
where: 2 = (ay, 2) ... some observable point (e.g. in a photo, model, on a surface, etc.) 
Io = (5, oio). ... the "internal reference point" (e.g. inner orientation of a photo) 
… some scale factor (e.g. model scale) 
R … the rotation matrix transposed, R = R(r) 
T … the vector of rotation parameters (three angles or axis components) 
X = (X,Y,Z) … the object point 
X,-—(Xo,Yo,Zo) … the ”external reference point” (e.g. projection center) 
0 . some "normalization radius" 
$z(6,95,.7) ... some local version of the normalized-rotated-reduced object point; 
a short-hand for: 
à : M -(X -X) = S(X,X.r) (2) 
  
472 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B3. Amsterdam 2000.