GEOMETRICAL

DEGENERACY REMOVAL BY VIRTUAL

DISTURBANCES

An Application to Surface Reconstruction from Point Slice Samples

Oscar Ruiz

CAD CAM CAE Laboratory, EAFIT University, Medellin, Colombia

Eliana Vasquez

, Sebastian Pe

, Miguel Granados

Erasmus Universitaet, Nederlands

Fraunhofer Inst. Comp. Graphics, Germany

Max Planck Inst. Informatiks, Germany

Keywords:

Geometric degeneracy, Voronoi diagram, Delaunay triangulation, surface reconstruction, slice point sample.

Abstract:

In surface reconstruction from slice samples (typical in medical imaging, coordinate measurement machines,

stereolithography, etc.) the available methods attack the geometrical and topological properties of the surface.

Topological methods classify the transitions occurred in the 2-manifold between two consecutive slices i and

i + 1. Geometrical methods synthesize the surface based on local proximity of the contours in consecutive

slices. Superimposed 2D Voronoi Diagrams V D

and VD

i+1

for slices i and i + 1, respectively, present topo-

logical problems if, for example, a site of V D

lies on an site or an edge of V D

i+1

. The usual treatment of

this problem in literature is to apply a geometrical disturbance to either V D

or V D

i+1

, thus eliminating the

degeneracy. In contrast, this article presents the implementation of a method which identiﬁes the degener-

ate situation, constructs un-instantiated topological constructs, choses a geometrical instantiation based on a

virtual disturbance introduced to the actual conﬁguration. The algorithm was successfully applied to remove

non-manifold topologies produced by well known algorithms in surface reconstruction.

1 INTRODUCTION

Degenerate conditions in geometric algorithms have

been dealt with different ways: (i) by stating the

same problem in different spaces with better condi-

tioning, (ii) by increasing the real computation pre-

cision, (iii) by relying on rational numbers, with no

rounding errors, and (iv) by disturbing the input for

the geometrical algorithms, while at the same time

estimating the probability of respecting the original

problem topology. Strategies (i) and (ii) have been

extensively applied in Numerical Analysis, for ex-

ample, by generating equivalent linear systems with

better manipulation properties. Alternative (iii) has

been investigated, for example in Computational Ge-

ometry Algorithm Library (Burnikel et al., 1999),

with exact computation paradigms. Strategy (iv) has

given probability bounds for alteration of Voronoi-

Delaunay topology upon numerical disturbance of de-

generate events (Funke et al., 2005). Virtual Perturba-

tions have been used in other contexts (Edelsbrunner

and M

ucke, 1990). It should be noticed that none of

the mentioned strategies solves the degeneracy prob-

lem. Each is suited for a particular domain of prob-

lems.

The strategy presented here assumes the possibil-

ity of detecting the degenerate condition, and to cre-

ate a ﬁnite number of topological conﬁgurations for

the solution. The geometrical and topological objects

created are not numerically instantiated, until the very

end of the proposed scheme. The strategy presented

here is clearly convenient when there is a ﬁnite num-

ber of topological conﬁgurations, which can be enu-

merated and distinguished.

The particular context in which this strategy is

presented is the general problem of surface recon-

struction, from planar samples. Particular steps of

the Boissonat & Geiger algorithm (Boissonnat, 1988;

Geiger, 1993) have been changed in order to make

them more robust (Ruiz et al., 2002; Ruiz et al., 2005).

Section 2 gives the application context of the present

work and reviews related literature. Section 3 de-

113

Ruiz O., Vasquez E., Peña S. and Granados M. (2008).

GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface Reconstruction from Point Slice Samples.

In Proceedings of the Third International Conference on Computer Graphics Theory and Applications, pages 113-118

DOI: 10.5220/0001098101130118

 SciTePress

scribes the methodology applied and the procedures

followed. Section 4 gives an account of the results,

and section 5 concludes the article.

2 CONTEXT AND LITERATURE

REVIEW

The scenario to apply the proposed algorithm for de-

generacy control is the following: Consider a solid

object B, whose boundary M = ∂B is a smooth 2-

manifold or shell in R

. Consider a set of parallel

planes Π

, i = 0, 1, 2, ... sectioning M, and therefore

producing traces of M which are Jordan curves Γ

j,i

or contours drawn on the planes Π

. Recovering M

from the contour set boils down to recovering M

i,i+1

the portion of M contained between the planes Π

and

i+1

The algorithm proposed and implemented by

Boissonat & Geiger in (Boissonnat, 1988; Geiger,

1993) (called here B+G ) builds tetrahedra ﬁlling the

space between two consecutive sampling planes i and

i + 1. B+G is a fairly fast and robust algorithm, origi-

nally presenting weaknesses that have been corrected

by complementary works ((Ruiz et al., 2005)). The

boolean union of such tetrahedra produces the por-

tion of the solid contained between the planes i and

i + 1. B+G basically uses local geometrical proximity

between contours to construct the surface M

i,i+1

. As

a consequence, over-stretched surfaces may be gener-

ated joining contours on the planes i with those on

plane i + 1 which have little to do with each other

in the global sense. This effect may be diminished

by applying a 2D shape similarity (2DSS) algorithm

(Ruiz et al., 2002; Ruiz et al., 2005). In B+G, the

tetrahedra are built by projecting the Voronoi Dia-

gram (VD) of the point set in level i, VD

, onto V D

i+1

or vice versa. A degeneracy condition for B+G occurs

when a Voronoi site of V D

exactly lies on either a

Voronoi site or a Voronoi edge of V D

i+1

. Such a con-

dition produces a non-manifold and self - intersection

condition in the surface built by B+G. The work pre-

sented in this article corresponds to the application

of late numerical instantiation of topological and ge-

ometrical objects to solve such a degeneracy.

2.1 Brief Review of the B+G Method

The B+G method divides the interior of the contours

in triangles by creating the Delaunay Triangulation of

the contour vertices (ﬁgures 2 and 3). After some pro-

cessing, the Voronoi Diagrams belonging to the lev-

els are used to create a planar graph named the Joint

Voronoi Diagram (ﬁgure 4). This graph states how

1 2

i+1

Figure 1: Original contours.

123

Figure 2: Delaunay Triangulation and Voronoi Diagram of

the contours.

ABC

Figure 3: Delaunay Triangulation and Voronoi Diagram of

the contours.

Figure 4: Delaunay Triangulation and Voronoi Diagram of

the contours.

Figure 5: Related tetrahedrons.

the triangles in the levels are linked, by translating

then to tetrahedrons (ﬁgure 5). Finally the triangles

of the tetrahedrons facing the exterior are taken as the

reconstructed surface.

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114

3 IMPROVEMENTS ON THE B+G

METHOD

The Polyhedral Surface Method, presented here an

in (Ruiz et al., 2005) is based on the B+G method.

In the following discussion it will be shown that the

B+G method may reconstruct incomplete surfaces

and present non manifold situations. With our method

the non manifold situations are minimized by preemp-

tively considering special conﬁgurations of the Joint

Voronoi Diagram (section 3.1).

3.1 Special Cases in the Creation of the

Joint Voronoi Diagram

The Joint Voronoi Diagram of two consecutive levels

results from intersecting the orthogonal projections

of their Voronoi Diagrams on a common plane. The

Joint Voronoi Diagram is formed by three kinds of

nodes: T

, T

and T

. The T

and T

nodes correspond

to the Voronoi vertices belonging to the Voronoi Dia-

gram on levels i or i + 1. The T

nodes correspond to

the intersection of Voronoi edges, coming from levels

i or i + 1.

Every node in the graph corresponds to a tetrahe-

dron, and the union of all these tetrahedrons forms the

3D Delaunay Diagram of the contour points P on both

levels i and i + 1. Because the tetrahedrons that are

translated from the graph are Delaunay tetrahedrons,

they satisfy the “empty-sphere” condition, namely is,

the sphere that circumscribes the tetrahedron does not

contain any other point in P except its vertices.

Each tetrahedron is created with four Delaunay

vertices. See ﬁgure 5 where a T

, T

and two T

tetrahedrons are translated from the Joint Voronoi Di-

agram in ﬁgure 4.

3.1.1 Case 1: Voronoi Vertex Vs. Voronoi Edge

When ﬁve Delaunay vertices are co-spherical (Figure

6), a Voronoi vertex belonging to level i is exactly

projected on a Voronoi edge belonging to level i + 1,

or vice versa (Figure 7).

The solution for such a degenerate condition is to

pretend that VV

123

is either above or below Voronoi

edge VV

ABD

←→ VV

BCD

. For the ﬁrst alternative, the

virtual conﬁguration appears in Figure 8, and its spa-

tial consequence is displayed in Figure 9. For the sec-

ond alternative, the virtual conﬁguration is shown in

Figure 10, while its spatial consequence appears in

Figure 11.

Figure 6: Voronoi vertex vs. Voronoi edge case. Vertices

, DV

and DV

are co-spherical.

Figure 7: Voronoi vertex vs. Voronoi edge case. Join

Voronoi Diagram of levels i and i + 1.

123

ABD

BCD

Figure 8: Voronoi vertex vs. Voronoi edge case. Delaunay

vertex DV

elected as apex.

Figure 9: Voronoi vertex vs. Voronoi edge case. Spatial

Conﬁguration.

123

ABD

BCD

Figure 10: Voronoi vertex vs. Voronoi edge case. Delaunay

vertex DV

elected as apex.

GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface

Reconstruction from Point Slice Samples

115

Figure 11: Voronoi vertex vs. Voronoi edge case. Delaunay

vertex DV

elected as apex.

Figure 12: Voronoi Vertex vs. Voronoi Vertex case. Sub-

case 1a2b3c.

ABC

Figure 13: Voronoi Vertex vs. Voronoi Vertex case. Sub-

case 1a2b3c. Projection of Voronoi Diagrams of levels i

and i + 1.

3.1.2 Case 2: Voronoi Vertex Vs. Voronoi Vertex

This case occurs when six Delaunay vertices are co-

spherical. As a result, a Voronoi vertex of level i is

projected onto a Voronoi vertex of level i + 1.

The sub-cases are determined by the distribution

of the edges on the “intersecting star” created when all

the edges are projected on the same plane (see ﬁgure

12 to 17 ). There are only two possible distributions.

(a) the edges are intercalated or (b) they are not. When

two consecutive edges belong to the same level, the

sub-case is identiﬁed as the 1ab23c sub-case. If there

are no two consecutive levels belonging to the same

level, the sub-case is identiﬁed as the 1a2b3c one.

The sub-case 1a2b3c appears in Figure 12. The

Joint Voronoi Diagram for the levels i and i + 1 ap-

pears in Figure 13. The virtual disturbance to solve

this degeneracy appears in Figure 14, while its spatial

effects do so in Figure 15.

123

ABC

Figure 14: A virtual disturbance for the 1a2b3c. sub-case.

Figure 15: Spatial consequence for the virtual disturbance

of Figure 14.

For the same situation, the sub-case 1ab23c ap-

pears in Figure 16. For this case, the projection of the

Voronoi Diagrams onto each other is seen in Figure

17. The virtual disturbances to solve this problem are

shown in Figures 18, 19 and 20. A spatial conﬁgura-

tion for one of such virtual scenarios appears in 21.

Figure 16: Voronoi Vertex vs. Voronoi Vertex case. 1ab23c

sub-case.

4 RESULTS

As said before, the special cases dealt with here are

generated when more than four Delaunay vertices

lie on the surface of an empty sphere. In the B+G

method this situation leads to non-manifold surfaces,

like the one shown in ﬁgure 22. The virtual perturba-

tions proposed here avoid the uncertainty of the B+G

method (which uses random numerical disturbances).

Such virtual perturbations produce an improved re-

sult, shown in ﬁgure 23. Results for the Skull data set

(ﬁgure 24) are shown in ﬁgure 25. The results for the

data set Brain are shown in Figures 26 and 27. Other

examples will be published in more extended reports

of this investigation.

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116

123

ABC

Figure 17: Voronoi Vertex vs. Voronoi Vertex case. 1ab23c

sub-case.

123

ABC

Figure 18: Virtual disturbance for the 1ab23c sub-case. So-

lution with three T

123

ABC

Figure 19: Virtual disturbance for the 1ab23c sub-case. So-

lution with two T

123

ABC

Figure 20: Virtual disturbance for the 1ab23c sub-case. So-

lution with just one T

Figure 21: A spatial conﬁguration for the 1ab23c. sub-case.

Figure 22: Surface reconstructed using the B+G method di-

rectly.

Figure 23: Surface reconstructed by the virtual disturbance

method.

Figure 24: Detail of levels i and i + 1 of the set of contours

“skull”.

Figure 25: Reconstructed surface in a transparent material.

GEOMETRICAL DEGENERACY REMOVAL BY VIRTUAL DISTURBANCES - An Application to Surface

Reconstruction from Point Slice Samples

117

Figure 26: Set of contours.

Figure 27: Reconstructed surface in a transparent material.

5 CONCLUSIONS

A method has been designed and implemented, to

circumvent geometrical degeneracies arising from si-

multaneous processing of 2D superimposed Voronoi

Diagrams, in the context of Surface Reconstruction

from Slice Samples. In this particular problem, for

each degenerate condition an enumerable ﬁnite set of

non-degenerate counterparts is programmed, and in-

stantiated as the geometry of the degeneracy dictates.

In absence of the algorithm, self-intersecting and

therefore non - manifold constructions are produced.

With the algorithm, degenerate cases are mapped to

their non - degenerate counterparts. This allows the

normal downstream execution of the host algorithm

(B+G, by Boissonnat & Geiger, 1988, 1993). The

method presented classiﬁes actions to be taken, based

on the level of the degeneracy. The results show that

the method is successful in removing the degeneracy,

without further iterations and in a deterministic way.

This method can be applied when the number of cases

of degeneracy is known.

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