Faber, Petko 
  
Note, the fitting of an ellipsoid will be supported only, not forced, if this constraint is selected. Following Bookstein 
(Bookstein, 1979), the constrained fitting problem can be written as: 
A= X a11%Ÿ + a22Y? + a332? + 2a12%;Y; + 2a13%;2; + 2a23Yy; 2; 
X; 
i€[1,n] (5) 
2 
+a14%; + A24Yıi + A342; + au — Minimum 
subject to the constraint z/ Cz = 1. The numerical solution of the resulting non-linear equation system will be obtained 
by a quadratically constrained least squares minimization as proposed in (Gander, 1981). First, by applying the Lagrange 
multipliers we get the conditions for the solution of z: Bz — ACz with z — [011 422 033 012 d13 023 014 024 034 044], 
the matrix B — U7U and the matrix 
2 à 9 
1 | ZI own £103 uu 8 Yı 4 1 
à 73-19 
TF 3 7% Pin 9272. Waza Es Yo Un I 
U = un 2 2 7 
Sí Jf nm ont wpU 4; 4; uy l 
2 2 2 
Ln Un Zn InYn nn Ynzn In Yn Zn 1 
Solving the generalized eigenvalue problem, we are looking for the eigenvector z; corresponding to the minimal positive 
eigenvalue Ag. Finally, after a proper scaling ensuring (z&)TC(2zx) = 1, we get a solution of the minimization problem 
(Eq. 5) which represents the best-fit 2-D center surface in 3-D for the given point set. 
3.3 Verification. 
Now, in the step of verification we decide wether the obtained eigenvector represents the desired solution, from which 
geometrically interpretable parameters of the ellipsoid can be derived. Therefore, we use the standard measures respecting 
to the DIN 33402/2. Concretely, we refer to the values bounding between the 5th percentile female to the 95th percentile 
male of persons aged between 16 and 65. Secondly, we check, if the actual hypothesis is supported by additionally 3-D 
points. If the hypothesis is confirmed, we accept the fitted ellipsoid. 
To derive geometrically interpretable parameters from the parameter vector z, = [011 422 à44] three different trans- 
formations are needed as given below: 
l. Firstly, the coordinate system is moved by a translation vector [£ 9 v]T. And it is required, that the linear terms 
vanish: 
O11 (Zm t £)? c azz(y + 7)? + a33(7 + v)?+ 
a12(2q -- £)(y - 9) -- a1s(zq +E)(z + V) t a»s(y - m)(z 4- v)4- 
a14(Tm + €) -- az4(y - 9) + a34(2 + V) + 044 = 0. 
Now we can estimate the coordinates of the center: 
iy 2 (222833014 — a33012024 — a22013034) + a23 (013034 + 01324 — 023014) 
m reru * 
2 (4011022033 + a12013023 — 411023 — a35a24 — 433025) 
sn 2 (2011033024 — 033012014 — 011023034) + 13 (012034 + 023014 — 013024) 
v 2 (4a11022433 4- 812013823 — 2 — 2 2) 
z 11022033 12013023 — 011054 — 422013 — 03301» 
> 2 (2011022034 — 022013014 — 011023024) + 012 (013024 + 023014 — 012034) 
m = 
2 (4011022033 + à12013023 — 411023 — a29024 — 43302») 
The resulting equation of the ellipsoid in the shifted € — 1 — v coordinate system is: 
€ au  G12/2 G13/2 € 
(Cnv) M} n |= (Env)| a1/? az ay)? 7) 
v a13/2 a23/2 ass V 
a11€? + azon? + aszv® + a126n + a13Év + axzny = h 
with 
h 
n 2 2 2 
= 011£,, -F 02294, t 03325, + A122mYm + 413ZmIm + 023Ym 2m + A14%m + A24Um + A342, + G44 . 
  
234 International Archives of Photogrammetry and Remote Sensing. Vol. XXXIII, Part B5. Amsterdam 2000. 
  
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