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1) computing initial estimation as disparity measure
between original and reflected models and 2) correc-
tion of symmetry plane of a model.
3.1 Computing Disparity Measure
between Two Models
Main steps of algorithm for computing disparity mea-
sure between two models are:
1. Delaunay triangulation construction of each dis-
crete set;
2. location of each discrete set in triangulation of the
other set;
3. linear interpolation of each function on the other
set using barometrical coordinates;
4. constructing of general triangulation of two dis-
crete sets on basis of merger algorithm;
5. function comparison on particular cells of the gen-
eral triangulation. Positional relationships of the
spatial triangles given by functions are analyzed
during this comparison .
3.2 Searching Symmetry Plane of
Model
As we assume that model’s normalization in coordi-
nate system is approximate, we try to transform co-
ordinates using small shifts and rotations by small an-
gles about the coordinateaxes. One process of surface
comparison may be implemented very efficiently so it
is possible to organize a guided search of such dis-
crete set’s transformation that provides the maximum
matching. The aim of this correction is to find such
position of the Oyz symmetry plane that the value of
quantitative asymmetry estimation is minimum.
It can be assumed that we minimize estimation not
by all six parameters of elementary manipulations but
only by three of them because it is obvious that shifts
along the y and z axes doesn’t have an influence on
asymmetry estimation and we also don’t consider ro-
tation about the x axis as we have full face photogra-
phy.
We make small transformation of the coordinate
system: shift along the x axis, then rotation by the
angle ϕ about the z axis and, finally, rotation by the
angle ψ about the y axis. In such a way G
1
will trans-
form to G
1
(x, ϕ, ψ) and G
2
— to G
2
(x, ϕ, ψ). f and
g will also be transformed. Denote by Φ(x, ϕ, ψ) =
∆( f(x, ϕ, ψ), g(x, ϕ, ψ)) disparity measure between
transformed surfaces.
The problem of searching the optimal symmetry
plane reduces to minimization of Φ(x, ϕ, ψ). For this
purpose we use alternating-variable descent method
combining with algorithm of golden section.
Notice that function Φ is ravine, i.e. change δ of
variables ϕ or ψ causes a greater change of function
value than the same change δ of variable x. We are
taking into account this property of function during
minimization procedure.
In table 1 there are values of initial estimation of
facial asymmetry (after stage 1) and estimation after
symmetry plane correction for 4 different face models
of one person. To understand significance of these
values let us remark that volume of fluid in tablespoon
is approximately equal to 15000 cmm.
Table 1: Initial and corrected estimations of facial asymme-
try.
Model’s Initial asymmetry Corrected asym-
number estimation (cmm) metry estimation
(cmm)
1 47 466,361 24 072,518
2 49 192,110 25 205,272
3 43 473,767 24 421,316
4 46 280,040 22 263,813
An optimal shift along the x axis is about 2, 4−2, 6
mm for models from the database, values of optimal
angles about the y and z axes are about 0, 015 rad.
4 COMPUTING EXPERIMENTS
The described method for comparison of models was
implemented, and there also has been made multiple
computing experiments for all stages of algorithm.
As experimental estimations have shown, each of
stages, except stage of triangulation constructions,
is implemented in linear for number of mesh nodes
time. Delaunay triangulation is implemented in time
O(N logN). Triangulation construction defines com-
putational complexity of the proposed approach.
Running time for different stages of algorithm
during comparison of human’s face surfaces are ad-
duced in table (2). The three-dimensional portraits
consisting approximately from 3000 points were used
here. Computing experiments were conducted using
AMD Athlon 2600+ processor and 512 Mb operative
memory.
Results of computing experiments on the database
demonstrate that the proposed estimation is stable for
different models of one and the same person.
The experiments indicate that the initial estima-
tion varies strongly for several models od the same
person. Neverthelessthe stage of symmetry plane cor-
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