A Stable and Invariant Three-polar Surface Representation: Application to 3D Face Description

We intend in this paper to study a novel curved

surface representation, that we qualify by three-polar,

introduced in [Jri13]. It is an attempt to generalize what

is known by local coordinates around one reference

It is constructed from the superposition of

point.

the three geodesic potentials generated from three

reference points of the surface. The proposed repre-

sentation is obtained by sampling the sum of these

three geodesic potentials. The stability of this novel

representation under errors on the positions of the

reference points is established.

Its accuracy for 3D

human face description is performed in the mean of the

Hausdorff shape distance.

Thus,

this paper will be structured as follows:

We present in the second section the mathematical

formulation of the three-polar representation. The

used similarity metric to compare between shapes

is illustrated in the third section. We establish in

the fourth section a comparative study between the

three-polar representation and the unipolar one in the

sense of the stability under errors on the positions

of the reference points. We apply finally the novel

three-polar representation for human face description

in the fifth section. The cases of correct and wrong

positions of the reference points are considered.

2 CONSTRUCTION OF THE THREE-

POLAR REPRESENTATION

Let consider here a two dimensional differential mani-

fold S, and let denote by Ur the geodesic potential gen-

erated from a reference point r of S. Ur is the function

that computes for each point p of S the length of the

geodesic curve joining p to r. For a given real value

λ , the points of S with geodesic potential values from

r equal to λ form a curve that we denote by Cλ

r . It is

called the geodesic of level λ and it belongs to the sur-

face.

We describe in this section the construction process of

the three-polar representation. It is based on the super-

position of the three geodesic potentials generated from

three reference points of S. Thus, let consider {ri, i =

1..3} three points of the 2-differential manifold S. We

denote by {Uri, i = 1..3} their corresponding potentials

functions. U3 = ∑3

i=1 Uri is the geodesic potential con-

structed by the sum of the three geodesic potentials gen-

erated from the three reference points {ri, i = 1..3} .

Let p∗ be a point of S. Thus, there exist three real val-

ues {λ ∗

i , i = 1..3} such that p∗ belongs to the three level

curves {Cλ ∗

3 = min{U3}. The points

of the surface with the same geodesic sum are invari-

ant under the rotations’ group SO(3). By selecting a

levels set of this sum, we construct a system of invari-

ant points under the same transformations group . The

representation that we propose is obtained by varying

these levels from 0 to the integer K. This integer rep-

resents the maximum value determined by the interest

region extend on the surface. Therefore, the three-polar

representation can be formulated as following:

, i = 1..3}. Let U ∗

r

i

Mk

3(S) = {p∗ ∈ S;U3(p∗) = U ∗

3 +

k

K

(αK −U ∗

3 ), k = 0..K}

(1)

Where αK is the maximum of the geodesic sum.

3 SIMLARITY METRIC

We present here the used similarity metric to compare

between different shapes. We choose the well known

Hausdorff shape distance introduced by Ghorbel in

[Gho98, Gho12]. By following the same process,

we denote by G the group representing all possible

normalized parametrisations of surfaces. It can be the

real plane R2 or the unit sphere S2. we consider the

space of all surface pieces as the set of all 3D objects

assumed diffeomorphic to G. It can be assimilated to

a subspace of L2

R3(G) formed by all square integrated

maps from G to R3. The direct product of the Euler

rotations group SO(3) by the group G , acts on such

space in the following sense:

SO(3) × G × L2

R3(G) → L2

{A, (u0, v0), S(u, v)} → AS(u + u0, v + v0)

R3(G)

(2)

The 3D Hausdorff shape distance ∆ can be written for

every S1 and S2 belonging to L2

R3(G) and g1 and g2 to

SO(3) as follows:

∆(S1, S2) = max(ρ(S1, S2), ρ(S2, S1))

(3)

Where:

ρ(S1, S2) = sup

g1∈SO(3)

inf

g2∈SO(3)

(cid:107) g1S1 − g2S2 (cid:107)L2

(4)

WSCG 2014 Conference on Computer Graphics, Visualization and Computer VisionFull Papers Proceedings186ISBN 978-80-86943-70-1(cid:107) S (cid:107)L2 denotes the norm of the functional banach space

L2

R3(G).

Since the euclidean rotations preserve this norm, it is

easy to show that this distance is reduced to the follow-

ing quantity:

∆(S1, S2) = inf

h∈SO(3)

(cid:107) S1 − hS2 (cid:107)L2

(5)

We consider after that, a normalized version of ∆ so that

its variations are confined to the interval [0,1]. In prac-

tice, this distance is obtained by using the well known

Itertive Closest Point (ICP) algorithm [Bes92].

4 STABILITY OF THE THREE-POLAR

REPRESENTATION

The stability is one of the most important properties of

a given 3D surface representation. This property is of a

paramount importance since we wish that a small local

deformation on the surface don’t lead to a big change

of the corresponding representation.

In our work, we are interested in studying a special sta-

bility under errors of the reference points positions. We

propose to make here a comparison between the three-

polar representation and the unipolar one in the sense

of the proposed stability. We used the object "Stanford

Bunny" [Tur94] which is known as a standard model for

testing graphical algorithms. The figure 1(a) illustrates

this object and the figure 1(b) shows the used three ref-

erence points chosen randomly on this object.

Figure 1: (a) The "Stanford Bunny" object.

used three reference points.

(b) The

We denote by P1 the common reference point between

the three-polar representation and the unipolar one. P2

and P3 are the two other points of the three-polar repre-

sentation.

In the case of extraction errors of a reference point,

we assume that it belongs to a geodesic disc of ra-

dius equal to RGmax around its correct position. In our

work, we qualify by a reference representation the one

constructed from reference points that are all extracted

without errors. In order to establish this comparative

study, the experimentations are performed on two parts.

Each part corresponds to a variation of reference points

positions. In the first part, only the positions of the point

P1 are extracted with errors for the two representations.

In the second part, the positions of the three reference

Figure 2: The variation of the reference points. (a) the

first part of the experimentation. (b) the second part of

the experimentation.

points are incorrect for the three-polar representation.

We note that the unipolar representation and the three-

polar one undergo the same variation of the point P1 in

the two parts of experimentation. The figure 2 illus-

trates the variation areas of the reference points.

For each part of this study, the Hausdorff shape distance

is computed between:

• The three-polar representations with errors on the

positions of the reference points and the correspond-

ing reference one.

• The unipolar representation with errors on the po-

sitions of the reference point and the corresponding

reference one.

All the variations of the Hausdorff shape distance are

computed according to the position of the common

point P1 between the three-polar representation and the

unipolar one. Here, sixteen positions of the point P1 are

chosen randomly on the disc of radius value equal to

RGmax around its correct position. The figure 3 shows

the variation of this distance for the first part of study.

We can note that the three-polar representation is more

stable than the unipolar one in the case of one reference

point with errors of extraction. In fact, the distance val-

ues are smaller in the case of the three-polar represen-

tation than the unipolar one.

In the second part of study, for each wrong position

of the point P1, many incorrect positions of the points

P2 and P3 exist. For each wrong position of the point

P1, we compute the average of the Hausdorff shape dis-

tances obtained for incorrect positions of the points P2

and P3. The figure 4 illustrates the variation of the

Hausdorff shape distance for the second part of study

according to the wrong positions of the point P1. From

this figure, we can observe also that the three-polar rep-

resentation has ensured a more stability than the unipo-

lar one.

WSCG 2014 Conference on Computer Graphics, Visualization and Computer VisionFull Papers Proceedings187ISBN 978-80-86943-70-1Figure 3: First part of study: Hausdorff shape distance according to the positions of the point P1 chosen randomly

on the geodesic disc of radius value equal to RGmax around the correct position of the point P1.

Figure 4: Second part of study: Hausdorff shape distance according to the positions of the point P1 chosen ran-

domly on the geodesic disc of radius value equal to RGmax around the correct position of the point P1.

5 HUMAN FACE DESCRIPTION WITH

THE THREE-POLAR REPRESENTA-

TION

The 3D face description has received a great deal of at-

tention over the last few years because of its various

application domains like biometrics which are one of

the most important. We test here the performance of

the three-polar representation on the 3D meshes of the

database Bosphorus [Sav08] in the mean of the Haus-

dorff shape distance. We use a total of ten faces that can

be grouped into two classes. A first class contains five

faces of the same person with different expressions and

a second one contains five faces of different persons.

5.1 The reference points: choice and au-

tomatic extraction

The choice of the reference points consists the first

step of the three-polar representation construction. We

choose to use the two outer corners of the eyes as

two reference points for the proposed three-polar rep-

resentation. Indeed, there is a general agreement that

eyes are the most important facial features [Cam07].

In fact, they are a crucial source of information about

the state of the human being and their appearance is

less variant to certain face changes. Since the nose

tip is commonly used for the unipolar representation

[Sam06, Sri08, Gad12], it will be also chosen as a third

reference point in the three-polar representation. We

WSCG 2014 Conference on Computer Graphics, Visualization and Computer VisionFull Papers Proceedings188ISBN 978-80-86943-70-1[Sze09] for the

refer to the work of Szeptycki et al.

automatic extraction of these points. This method is

based on a curvatures analysis with the use of a generic

face model generated from a set of faces of the database

Bosphorus.

5.2 3D face description: Good extraction

of the reference points

We test in this section the performance of the three-

polar representation for human face description in

the case of a good extraction of the reference points.

The figure 5 shows the three-polar representation with

different resolutions which are linked to the number of

levels in the representation construction.

Figure 6: Matrix of pairwise normalized Hausdorff dis-

tances between the ten facial surfaces. The first five

faces correspond to the same person while others be-

long to different individuals

Figure 5: Row 1: A neutral face. Row 2: A face with a

surprise expression. (a) The three-polar representation

with 10 levels. (b) The three-polar representation with

(c) The three-polar representation with 30

20 levels.

levels.

In order to illustrate the effectiveness of the three-polar

the matrix representing the pairwise

representation,

normalized Hausdorff shape distance is computed

between the ten faces. The first five faces correspond

to the first class. The rest belongs to the second class.

The figure 6 illustrates this matrix.

This matrix shows that

the distances between the

faces of the same person are smaller compared with the

ones computed between faces of different individuals.

5.3 3D face description: Extraction errors

of the reference points

A comparative study between the three-polar represen-

tation and the unipolar one is established in the sense

of the 3D face description when extraction errors of the

reference points positions exist. Two study cases are

considered. For the first case, only the common point

between these two representations is extracted with er-

rors. It corresponds to the nose tip which is chosen ran-

domly in this case in a geodesic disc of radius value

equal to RGmax around its correct position. The figure 7

illustrates the variations area of the reference points in

Figure 7: Variation area of the reference points for the

first case of study: (a) the unipolar representation. (b)

the three-polar representation

the first case of study for the two representations.

In the second case of study, the three reference points

are chosen randomly in the three geodesic discs of ra-

dius values equals to RGmax around their correct posi-

tions for the three-polar representation. We note that

the nose tip has the same variations for the both repre-

sentations. The figure 8 illustrates the variation area of

the reference points in the second study case.

Figure 8: Variation area of the reference points for the

second case of study: (a) the unipolar representation.

(b) the three-polar representation

For each study case, the matrices of pairwise normal-

ized Hausdorff shape distance are computed for the two

representations. The results of the first study case are il-

lustrated in the figure 9.

From the observation of these two matrices (figure 9),

we can note that the three-polar representation better

WSCG 2014 Conference on Computer Graphics, Visualization and Computer VisionFull Papers Proceedings189ISBN 978-80-86943-70-1Figure 9: Matrices of pairwise normalized Hausdorff distances between the ten facial surfaces with errors on the

positions of the nose tip. (a): The unipolar representation. (b): The three-polar representation.

Figure 10: Matrices of pairwise normalized Hausdorff distances between the ten facial surfaces with errors on

the positions of the nose tip for the unipolar representation and on the positions of all the reference points for the

three-polar representation. (a): The unipolar representation. (b): The three-polar representation.

caracterizes 3D faces than the unipolar one in the case

of wrong positions of only one reference point.

In

fact, the distances between the faces of the same per-

son are smaller for the three-polar representation than

the unipolar one.

The figure 10 shows the two matrices corresponding to

each representation for the second study case. We can

note that the three-polar representation has shown also

better performances for the description of 3D faces in

the second case of study than the unipolar one.

6 CONCLUSION

In this paper, we have studied a novel 3D invariant

curved surface representation. It is qualified by three-

polar since it is constructed from the superposition of

the three geodesic potentials generated from three ref-

erence points of the surface. The goal was to generalize

the representation constructed from one reference point

and to ensure a more stability in the case of errors on

the reference point positions. A comparison study be-

tween the three-polar representation and the unipolar

one is established in the sense of the stability under er-

rors on the reference points positions. We applied the

novel 3D representation for 3D human face description

in the mean of the Hausdorff shape distance.

The perspectives of future works involve the applica-

tion of the three-polar representation on a larger num-

ber of 3D face surfaces. We intend also to find the op-

timal number of the levels of the novel representation.

Finally, we propose to generalize the three-polar repre-

sentation to n reference points. The value of n must be

optimal in the sense that it will not be necessary to add

other reference points.

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WSCG 2014 Conference on Computer Graphics, Visualization and Computer VisionFull Papers Proceedings191ISBN 978-80-86943-70-1WSCG 2014 Conference on Computer Graphics, Visualization and Computer VisionFull Papers Proceedings192ISBN 978-80-86943-70-1