ON THE IMPROVEMENT OF THE TOPOLOGICAL ACTIVE

VOLUMES MODEL

A Tetrahedral Approach

N. Barreira, M. G. Penedo, M. Ortega and J. Rouco

VARPA Group, Department of Computer Science, Universidade da Coru˜na, Spain

{nbarreira, mgpenedo, mortega, jrouco}@udc.es

Keywords:

3D Image Segmentation, Deformable models, Active models, Topological Active Volumes.

Abstract:

The Topological Active Volumes model is a 3D active model focused on segmentation and reconstruction

tasks. The segmentation process is based on the adjustment of a 3D mesh composed of polyhedra. This

adjustment is guided by the minimisation of several energy functions related to the mesh. Even though the

original cubic mesh achieves good segmentation results, it has difﬁculties in some cases due to its shape.

This paper proposes a new topology for the TAV mesh based on tetrahedra that overcomes the cubic mesh

difﬁculties. Also, the paper explains an improvement in the tetrahedral topology to increase the accuracy of

the results as well as the efﬁciency of the overall process.

1 INTRODUCTION

Deformable models are well-known tools for image

segmentation and reconstruction. They were intro-

duced in 2D by Kass et al. (Kass et al., 1988) and

generalised to 3D by Terzopoulos et al. (Terzopou-

los et al., 1988). The active nets model was ﬁrst

proposed by Tsumiyama and Yamamoto (Tsumiyama

and Yamamoto, 1989) as a variant of the deformable

models that integrates features of region–based and

boundary–based segmentation techniques. To this

end, this model has two different kind of nodes: in-

ternal nodes, for modelling the inner topology, and

external nodes, for surface adjustment. The former

is related to the region information whereas the latter

uses boundary information. The Topological Active

Volumes (TAV) model (Barreira and Penedo, 2005)

is a 3D extension of the active nets model. It has

an advantage over other models since it not only ﬁts

the surfaces but also models the whole volume. A

TAV consists of a set of nodes organised in a poly-

hedral mesh. Just like any other deformable model,

the mesh deformation is guided by energy functions

in such a way that the mesh energy has a minimum

when the model is over the objects of the scene. Also,

the TAV model is able to perform topological changes

in its structure in order to adjust to concavities, detect

holes, and ﬁnd separate objects in the scene.

The TAV model, originally developed as a cubic

mesh, achieves goodsegmentation results in both syn-

thetic and real images (Barreira and Penedo, 2004).

The cubic mesh is simple and able to adapt to a

wide range of surfaces. However, in objects with

pronounced curvatures, the cubic mesh has difﬁculty

in the adjustment due to the four-sided faces of the

cubes. In these cases, a tetrahedral mesh could

ﬁll the space better and, thus, improve the results.

Since tetrahedral meshes are widely used in modeli-

sation and reconstruction (Archip et al., 2006; Sitek

et al., 2006) as well as in segmentation tasks with de-

formable models (Sermesant et al., 2003; Pons and

Boissonnat, 2007), this paper proposes a new mesh

topology for the TAV model based on tetrahedra. The

development of a mesh topology is not straightfor-

ward and implies new ways of initialising and per-

forming topological changes in the structure. Also,

since the new node relationships affect the calculus

of the energies, they have inﬂuence on the computa-

tion times. For this reason, some improvements were

developed in order to increase the efﬁciency of the

model.

This paper is organised as follows. Section 2 ex-

plains the characteristics of the model and the seg-

mentation process. Section 3 introduces the tetrahe-

dral topology as well as the strategy developed to im-

prove the results. Section 4 shows some results of the

529

Barreira N., G. Penedo M., Ortega M. and Rouco J. (2008).

ON THE IMPROVEMENT OF THE TOPOLOGICAL ACTIVE VOLUMES MODEL - A Tetrahedral Approach.

In Proceedings of the Third International Conference on Computer Vision Theory and Applications, pages 529-534

DOI: 10.5220/0001081305290534

Copyright

c

SciTePress

new topology. Finally, section 5 explains the conclu-

sions and the future work.

2 MODEL

A Topological Active Volume (TAV) is a three-

dimensional structure composed of interrelated nodes

located at the vertices of a polyhedron (Barreira and

Penedo, 2005). This polyhedron is repeated through-

out the mesh and deﬁnes the neighbouring relation-

ships among nodes. There are two types of nodes:

internal, inside the mesh, and external, on the sur-

faces. Each type of node represents different object

features. The external nodes ﬁt the surface of the ob-

ject whereas the internal nodes model its inner topol-

ogy. Figure 1 depicts a TAV where the repeated poly-

hedron is a cube.

Parametrically, a TAV is deﬁned as v(r, s, t) =

(x(r, s, t), y(r, s, t), z(r, s, t)), where (r, s, t) ∈ ([0, 1] ×

[0, 1] × [0, 1]). The state of the model is governed by

an energy function deﬁned as follows:

E(v) =

R

1

0

R

1

0

R

1

0

E

int

(v(r, s, t)) + E

ext

(v(r, s, t))drdsdt

(1)

where E

int

and E

ext

are the internal and the external

energy of the TAV, respectively. The former controls

the shape and the structure of the net. Its calculus

depends on ﬁrst and second order derivatives which

control contraction and bending, respectively. The in-

ternal energy term is deﬁned by:

E

int

(v(r, s, t)) =

α(|v

r

(r, s, t)|

2

+ |v

s

(r, s, t)|

2

+ |v

t

(r, s, t)|

2

) +

β(|v

rr

(r, s, t)|

2

+ |v

ss

(r, s, t)|

2

+ |v

tt

(r, s, t)|

2

) +

2γ(|v

rs

(r, s, t)|

2

+ |v

rt

(r, s, t)|

2

+ |v

st

(r, s, t)|

2

)

(2)

where subscripts represent partial derivatives and α,

β and γ are coefﬁcients that control the smoothness of

the net. In order to compute the energy, the parameter

domain [0, 1] × [0, 1] × [0, 1] is discretized as a regu-

lar grid deﬁned by the internode spacing (k, l, m) and

the ﬁrst and second derivativesare estimated using the

ﬁnite differences technique in 3D.

E

ext

represents the characteristics of the scene that

guide the adjustment process and is deﬁned as fol-

lows:

E

ext

(v(r, s, t)) = ωf[I(v(r, s, t))]

+

ρ

|ℵ(r,s,t)|

∑

p∈ℵ(r,s,t)

1

||v(r,s,t)−v(p)||

f[I(v(p))]

(3)

where ω and ρ are weights, I(v(r, s, t)) is the intensity

value of the original image in the position v(r, s, t),

f is a function related to the image intensity, and

ℵ(r, s, t) is the neighbourhood of the node (r, s, t).

This way, given that the repeated polyhedron in the

mesh deﬁnes the node neighbourhoods, the shape of

Figure 1: A 4 × 4 × 3 TAV mesh where the base polyhe-

dron is a cube. The dark nodes represent the external nodes

whereas the light ones are the internal nodes.

the polyhedron inﬂuences not only the ﬂexibility of

the mesh, but also the way the nodes are adjusted to

the objects.

Since the internal and external nodes model differ-

ent parts of the objects, f should be adapted for both

types of nodes. On one hand, if the objects to detect

are dark and the background is light, the energy of an

internal node will be minimum when it is on a point

with a low grey level. On the other hand, the energy

of an external node will be minimum when it is on a

discontinuity and on a light point outside the object.

In this situation, function f is deﬁned as:

f[I(v)] =

h[I(v)

n

] for internal nodes

h[I

max

− I(v)

n

+ ξ(G

max

− G(v))]

+GD(v) for external nodes

(4)

where ξ is a weighting term, I

max

and G

max

are the

maximum intensity values of image I and the gradient

image G, respectively, I(v) and G(v) are the intensity

values of the original image and the gradient image in

the node position v(r, s, t), I(v)

n

is the mean intensity

in a n× n × n cube, h is an appropriate scaling func-

tion, and GD(v) is the gradient distance, this is, the

distance from the node position v(r, s, t) to its nearest

edge.

Otherwise, if the objects are bright and the back-

ground is dark, the energy of an internal node will be

minimum when it is on a point with a high grey level

and the energy of an external node will be minimum

when it is on a discontinuity and on a dark point out-

side the object. In such a case, function f is deﬁned

as:

f[I(v)] =

h[I

max

− I(v)

n

] for internal nodes

h[I(v)

n

+ ξ(G

max

− G(v))]

+GD(v) for external nodes

(5)

where the symbols have the same meaning as in equa-

tion 4.

The segmentation process consists of several

stages. First, a mesh with an homogeneous distribu-

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

530

tion of nodes is created and located over the whole im-

age. Then, the mesh energy is minimised iteratively

using a greedy algorithm. The energy functions reach

a minimum when the mesh is located around the ob-

jects. After that, the number of nodes in each axis

is recomputed to adapt the mesh size to the object

size; for example, if the object is longer than wider,

the number of nodes in the x-axis will be increased

whereas the number of nodes in the y-axis will be

decreased. The mesh is also centred over the de-

tected objects and its energy is minimised again. Fi-

nally, topological changes are performed to increase

the ﬂexibility of the model and, after a local energy

minimisation step (Barreira et al., 2006), to adjust the

mesh to concave surfaces, holes, or separate objects.

The topological changes involve the breaking of

links between external nodes wrongly located, this

is, external nodes far away from the objects. These

nodes are identiﬁed and sorted by their distance to the

objects in order to break the links between the worst

located nodes (Barreira and Penedo, 2005). However,

some link breakings are not allowed because of the

fact that the mesh should keep the polyhedral struc-

ture since isolated nodes or planes does not provide

volumetric information.

3 TETRAHEDRAL MESHES

The TAV model has been developed using a cube

as the base polyhedron of the mesh. Even though

the segmentation results using this conﬁguration were

good (Barreira and Penedo, 2004), the topology based

on cubes has limitations in the adjustment to surfaces.

Speciﬁcally, the four-sided faces of the cubes have

difﬁculty in the adjustment to pronounced curvatures.

Although this problem can be partly solved by in-

creasing the density of nodes, a new mesh topology

that could improve the segmentation results is nec-

essary. Figure 2 shows the reconstruction of an ob-

ject with several small holes that the cubic topology

is not able to detect with a small mesh size and de-

tects roughly with a larger mesh size.

The triangular meshes have the advantage of be-

ing able to model very complex geometries so that

they can improve the results of a quadrilateral mesh.

Therefore, most of the surface reconstruction tech-

niques use triangular meshes to represent surfaces.

Since the TAV model works in 3D with polyhedra,

the new mesh topology will consist of triangle-based

polyhedra, this is, tetrahedra.

Given that there are ﬁve tetrahedra in a cube (see

ﬁgure 3), the tetrahedral mesh is built from a cubic

mesh in such a way that each cube is the specular im-

Figure 2: Adjustments of meshes based on cubes and tetra-

hedra. First row: original object. Second row: results using

a cubic mesh. Third row: results with tetrahedral meshes.

The meshes have 10× 10× 10 nodes in the ﬁrst column and

20 × 20 × 20 in the second one. The tetrahedral mesh is

able to detect the small rounded holes in the surface of the

object even with the 10× 10× 10 mesh. On the contrary,

although the results are improved when the mesh size is in-

creased, the cubic mesh only detects roughly the holes with

the 20× 20× 20 mesh.

age of its neighbours. Figure 4 shows an example of

a tetrahedral mesh with 3× 3× 3 nodes.

The mesh topology change does not affect the seg-

mentation process whereas the accuracy of the results

is improved as ﬁgure 2 shows. Nevertheless, the new

neighbourhoodof the nodes will affect the calculus of

the external energy function (see eq. 3) in the minimi-

sation stage. Since the number of neighboursgrowsin

the tetrahedral meshes and the calculus of the external

energy depends on the neighbourhood,the tetrahedral

meshes will be slower than the cubic meshes.

In order to speed up the adjustment process and

given that the cubic meshes often produce good re-

sults, a mixedapproach has been developed in this pa-

per. In this approach, the segmentation process starts

with a cubic mesh. Its energy is minimised and, after

that, its size is recomputed. When the minimisation

ON THE IMPROVEMENT OF THE TOPOLOGICAL ACTIVE VOLUMES MODEL - A Tetrahedral Approach

531

Figure 3: Decomposition of a cube in ﬁve tetrahedra. The

tetrahedral mesh is built from this decomposition and its

specular image.

Figure 4: A tetrahedral mesh with 3×3×3 nodes. The dark

nodes are the external nodes whereas the light central one is

the internal one.

process ﬁnishes for the second time, the cubic mesh is

able to detect the objects but maybe an improvement

in the adjustment is needed. At this point, a tetrahe-

dral mesh is built from the adjusted cubic mesh and

its energy is minimised again. This way, the cubic

mesh obtains a coarse but fast segmentation whereas

the tetrahedral mesh achieves a ﬁne adjustment. Fig-

ure 5 shows how the mixed approach is also able to

detect the small holes in the object in ﬁgure 2 as ac-

curately as the the tetrahedral approach.

Regarding the topological changes, only break-

ings that preserve the tetrahedral structure are al-

lowed. To this end, the mesh integrity is checked

Figure 5: Adjustments of the mixed approach. Left image

was segmented with a 10 × 10 × 10 mixed mesh and the

right one, with a 20 × 20 × 20 mesh. The results of both

tetrahedral and mixed approaches are equivalent.

(a) (b) (c) (d)

Figure 6: Some slices of the 3D images used in the exam-

ples. Slices (a), (b), (c), and (d) correspond to the images in

ﬁgures 7, 8, 9, and 10, respectively.

Table 1: Parameters used in the examples.

Figures α β γ ρ ω ξ

7 1.5 10

−5

10

−5

3.0 3.0 5.0

8 3.5 10

−5

10

−5

4.0 3.0 6.0

9 2.0 10

−4

10

−4

4.0 4.0 6.0

10 3.0 10

−5

10

−5

3.0 3.0 5.0

before each breaking. This way, a link between two

nodes is broken only if all their neighboring nodes

belong to, at least, another tetrahedron after the link

breaking.

4 RESULTS

This section shows the results of applying the tetrahe-

dral meshes to several synthetic images. A 3D image

is built as a set of 2D stacked images. Figure 6 shows

some slices of the images used in the segmentation

examples. The input image was used in the calculus

of the external energy for both internal and external

nodes. The gradient images were computed using a

3D Canny detector. The model parameters, empiri-

cally chosen, are summarised in table 1. Since the

mixed strategy is faster than the tetrahedral approach

and produces similar results, all the tetrahedral results

in this section were obtained using the mixed strategy.

The efﬁciency and the adjustment of the topolo-

gies developed were analysed. With this aim, sev-

eral segmentation processes with different mesh sizes

were performed and the computation times of the cu-

bic, tetrahedral, and mixed approaches were com-

pared. The processes were run in an Intel Core 2

Duo at 2.40 GHz. The graphs in ﬁgures 7, 8, and

9 show the computation times of the adjustment pro-

cesses, prior to the stage of topological changes, for

three example images. The graphs show that not only

the computation times of the mixed segmentations

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

532

1000 2000 3000 4000

Nodes

0

500

1000

1500

2000

2500

3000

Computation times (s)

Mixed

Tetrahedron

Cube

Figure 7: Segmentation results in a synthetic image. Top:

results of the cubic (left) and the mixed (right) approaches.

In both cases, the initial meshes had 10 × 10 × 10 nodes.

Bottom: evolution of the computation times of the mesh

topologies with respect to the average mesh size of the ini-

tialisation and readjustment stages. The tetrahedral mesh

produces a more accurate adjustment to the hole surfaces.

1000 1500 2000 2500 3000 3500

Nodes

0

200

400

600

800

1000

1200

Computation times (s)

Mixed

Tetrahedron

Cube

Figure 8: Segmentation of a synthetic object. Top: results

with an initial 9 × 9 × 9 cubic mesh (left) and an initial

12× 12× 12 mixed mesh (right). Bottom: evolution of the

computation times of the mesh topologies with respect to

the average mesh size of the initialisation and readjustment

stages. The adjustment of the tetrahedral mesh is more ac-

curate in the concave area.

3000 4000 5000 6000

Nodes

0

1000

2000

3000

4000

5000

Computation times (s)

Mixed

Tetrahedron

Cube

Figure 9: Segmentation of a vertebra. Top: results of the cu-

bic (left) and mixed (right) approaches. The initial meshes

had 13× 13× 13 nodes. Bottom: evolution of the computa-

tion times of the mesh topologies with respect to the average

mesh size of the initialisation and readjustment stages. The

tetrahedral mesh produces better results since it detects both

lateral prominent areas of the vertebra.

are lower and similar to the cubic ones, but also the

time increase as the mesh size grows is greater in the

case of the tetrahedral meshes. Also, the tetrahedral

meshes improve the adjustment to surfaces.

Figure 10 shows a special case where the object

covers almost the whole image. In these cases, the

mixed strategy produces higher computation times

than the cubic approach. The mixed approach only

uses the fast cubic approach to detect the objects. For

this reason, when the objects cover the whole im-

age, the cubic step is very short and the adjustment

is achieved mainly by means of the tetrahedral mesh.

This way, the computation times for both mixed and

tetrahedral approaches are similar. Nevertheless, the

improvement in the surface adjustment justiﬁes the

use of a tetrahedral mesh.

5 CONCLUSIONS

This paper proposes a new topology for the Topolog-

ical Active Volumes model. The former TAV topol-

ogy based on cubes has limitations to achieve a ﬁne

adjustment to curved surfaces. For this reason, a new

topologybased on tetrahedra was developed. The new

topology behaves like the cubic one, this is, the tetra-

hedral topology is able to detect several objects in the

scene and to adjust to object concavities and holes.

ON THE IMPROVEMENT OF THE TOPOLOGICAL ACTIVE VOLUMES MODEL - A Tetrahedral Approach

533

600 800 1000 1200 1400 1600 1800 2000 2200

Nodes

0

100

200

300

400

500

600

Computation times (s)

Mixed

Tetrahedron

Cube

Figure 10: Segmentation of a cube with holes. Top: seg-

mentation results with a cubic mesh (left) and a mixed

(right) mesh of 11× 11× 11 nodes. Bottom: evolution of

the computation times of the mesh topologies with respect

to the average mesh size of the initialisation and readjust-

ment stages. The mixed and tetrahedral approaches have

similar computation times because the object covers the

whole image.

However, the new relationships between nodes in the

tetrahedral meshes imply a higher complexity in the

segmentation process as well as an increase in the

computation time. In order to overcome these draw-

backs, a new strategy was proposed. A mixed topol-

ogy that combines a cubic mesh for a fast and rough

segmentation and a tetrahedral mesh for a ﬁne adjust-

ment. This strategy not only reduces the computation

times, but also improves the results of the segmenta-

tion process.

Future work includes the development of tech-

niques to change the structure of the tetrahedral mesh

by means of inserting or removing nodes in order to

achieve a ﬁne adjustment to complex areas.

ACKNOWLEDGEMENTS

This paper has been partly funded by the

Xunta de Galicia through the grant contracts

PGIDIT05SIN001E and PGIDIT06TIC10502PR.

REFERENCES

Archip, N., Rohling, R., Dessenne, V., Erard, P.-J., and

Nolte, L.-P. (2006). Anatomical structure modeling

from medical images. Computer Methods and Pro-

grams in Biomedicine, 82(3):203–215.

Barreira, N. and Penedo, M. G. (2004). Topological Active

Volumes for Segmentation and Shape Reconstruction

of Medical Images. Image Analysis and Recognition:

Lecture Notes in Computer Science, 3212:43–50.

Barreira, N. and Penedo, M. G. (2005). Topological Ac-

tive Volumes. EURASIP Journal on Applied Signal

Processing, 13(1):1937–1947.

Barreira, N., Penedo, M. G., and Penas, M. (2006). Local

energy minimisations: An optimisation for the topo-

logical active volumes model. In First International

Conference on Computer Vision Theory and Applica-

tions, volume 1, pages 468–473.

Kass, M., Witkin, A., and Terzopoulos, D. (1988). Active

contour models. International Journal of Computer

Vision, 1(2):321–323.

Pons, J.-P. and Boissonnat, J.-D. (2007). Delaunay de-

formable models: Topology-adaptive meshes based

on the restricted delaunay triangulation. In IEEE Con-

ference on Computer Vision and Pattern Recognition,

Minneapolis, USA.

Sermesant, M., Forest, C., Pennec, X., Delingette, H., and

Ayache, N. (2003). Deformable biomechanical mod-

els: Application to 4D cardiac image analysis. Medi-

cal Image Analysis, 7(4):475–488. PMID: 14561552.

Sitek, A., Huesman, R., and Gullberg, G. (2006). Tomo-

graphic reconstruction using an adaptive tetrahedral

mesh deﬁned by a point cloud. IEEE Transactions

on Medical Imaging, 25(9):1172 – 1179.

Terzopoulos, D., Witkin, A., and Kass, M. (1988). Con-

straints on deformable models: Recovering 3D shape

and nonrigid motion. Artiﬁcial Intelligence, 36(1):91–

123.

Tsumiyama, K. and Yamamoto, K. (1989). Active net: Ac-

tive net model for region extraction. IPSJ SIG notes,

89(96):1–8.

VISAPP 2008 - International Conference on Computer Vision Theory and Applications

534