Faber, Petko 
human’s head: the head length range between 192 and 257 [mm], the head height between 163 and 205 [mm] and the 
head breadth between 137 and 168 [mm], respectively to the DIN 33402/2 "Human body dimensions; values". 
The available data are defined by the set of all corresponding points. That means, all points are potential candidates 
belonging to an ellipsoid describing the human's head. But, our main problem is, that the points are irregularly distributed 
in the 3-D space and, moreover, additionally no information about the relations between the points is available. The 
resulting problem is a defined selection of a subset of 3-D points, which are representative of the searched ellipsoid. 
The detection is performed using a RANSAC procedure (Fischler and Bolles, 1981). Here, however, we need to randomly 
select nine points and check their validity. The best solution is then refined using a direct least squares method with a 
condition, which supports, but nor enforces the approximation of an ellipsoid. The least squares method is similar to the 
approach of (Fitzgibbon et al., 1996). 
3.2 Ellipsoid specific fitting algorithm 
A general second order surface can be present in 3-D implicitly F(z, x) 2 x - A- xT +x-a+a4 = 0 witha = 
[414 424 das. x = [x y z] and the real symmetrical matrices 
411 diz 013 14 
431  d22 423 0424 
431 432 033 034 
41 42 043 44 
aj; 412 13 
A = | 421 an 423 , A= 
431 432 433 
Explicitly the general second order surface can be read as: 
— „2 a 2 2 €) PR € iia €). > A = ; - _ 
F(z, x) — a112? + Agoy” 4- 332^ + 201909 + 213% 7 + 202397 + 414% + A24Y + A347 + 444 = 0. (1) 
F(z, x;) is called the algebraic distance of a point x; = [z; yi z;] to the surface F(z, x) — 0. The fitting of a general 
surface may be approached (Haralick and Shapiro, 1993) by minimizing the sum of the squared algebraic distances: 
n 
Ax > F (z, xi)? — Minimum . (2) 
iz] 
In order to avoid the trivial solution for the parameter, and recognizing that any multiple of a solution represents the same 
surface, the formulation of a constraint is necessary. The selection of a suitable constraint is important for the fitting. 
Note, if a constraint is selected which has insufficiently discriminant characteristics, a set of 3-D points can be fitted by 
an ellipsoid, a hyperboloid, a double cone, or any surface, which does not present a center surface. On the other hand, the 
equation to describe an ellipsoid can be written in a normalized form: 
x? y? 22 
amp =1 with A>0,8B>0,C>0. 
In order to fit ellipsoids specifically we would like to constrain the parameter vector z. A constraint which forces the 
fitting of an ellipsoid always can be formulated as 
det (A) = aq1az2a33 + a12a23a31 + a13a32a21 — 011023032 — 022013031 — 033012021 — l . (3) 
However, on the basis of the obvious structural contradiction between the actual formulation and the demanded structure to 
solve the generalized eigenvalue problem Bz — ACz with zT Cz — land: = [a11 422 433 412 ... a44] this formulation 
is not suitable. Concluding, a "good" approximation of the constraint has to been selected to exclude at least the surfaces, 
which do not present a center surface. Therefore it is necessary, that det (A) # 0. Assuming, that every projection of the 
3-D point set into the three 2-D plane X — Y, X — Z, and Y — Z are approximated optimal by an ellipse the following 
and finally used constraint is motivated: 
4 (a11422 -- 411433 + 422033) — (412021 + 413031 + 423032) = | . (4) 
This is a quadratic constraint which may be expressed in the matrix form 2 Cz = | as 
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International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 233