SMART GROWING CELLS

Hendrik Annuth and Christian-A. Bohn

Computer Graphics & Virtual Reality, Wedel University of Applied Sciences, Feldstr. 143, Wedel, FR, Germany

Keywords:

Neural networks, Unsupervised learning, Self-organization, Growing cells structures, Surface reconstruction.

Abstract:

General unsupervised learning or self-organization places n-dimensional reference vectors in order to match

the distribution of samples in an n-dimensional vector space. Beside this abstract view on self-organization

there are many applications where training — focused on the sample distribution only — does not lead to a

satisfactory match between reference cells and samples.

Kohonen’s self-organizing map, for example, overcomes pure unsupervised learning by augmenting an addi-

tional 2D topology. And although pure unsupervised learning is restricted therewith, the result is valuable in

applications where an additional 2D structure hidden in the sample distribution should be recognized.

In this work, we generalize this idea of application-focused trimming of ideal, unsupervised learning and

reinforce it through the application of surface reconstruction from 3D point samples. Our approach is based

on Fritzke’s growing cells structures (GCS) (Fritzke, 1993) which we extend to the smart growing cells (SGC)

by grafting cells by a higher-level intelligence beyond the classical distribution matching capabilities.

Surface reconstruction with smart growing cells outperforms most neural network based approaches and it

achieves several advantages compared to classical reconstruction methods.

1 INTRODUCTION

The idea of developing the smart growing cells ap-

proach is driven by the need for an algorithm for

robust surface reconstruction from 3D point sample

clouds.

The demand for efﬁcient high quality reconstruc-

tion algorithms has grown signiﬁcantly in the last

decade, since the usage of 3D point scans has widely

been spread into new application areas. These in-

clude geometric modeling to supplement interactive

creation of virtual scenes, registering landscapes for

navigation devices, tracking of persons or objects in

virtual reality applications, medicine, or reverse engi-

neering.

3D points, retrieved by laser scanners or stereo

cameras, introduce two vital questions. First, how

can one recognize a topology of the originating 2D

surfaces just from independent 3D sample points and

without any other information from the sampled ob-

jects? Second, for further processing, how is it pos-

sible to project this topological information on a data

structure like a triangle mesh — meeting given con-

straints concerning mesh quality and size?

Although this issue has intensely been tackled

since the early eighties (Boissonnat, 1984) a general

concept that addresses all the problems of surface

reconstruction has not been determined up to now.

Noise contained in the sample data, anisotropic point

densities, holes and discontinuities like edges, and ﬁ-

nally, handling vast amounts of sampling data with

adequate computing resources are still a big chal-

lenge.

Previous Work. The issue of surface reconstruc-

tion is a major ﬁeld in computer graphics. There are

numerous approaches with different algorithmic con-

cepts. In (Hoppe et al., 1992) and (Hoppe, 2008) an

implicit surface is created from point clouds which

then is triangulated by the marching cubes approach.

(Edelsbrunner and Mcke, 1994) and (Kolluri et al.,

2004) reduce a delaunay tetrahedralization of a point

cloud until the model is carved out. Approaches like

(Storvik, 1996) or (Huang et al., 2007) utilize tech-

niques based on the Bayes’ theorem.

In the area of artiﬁcial neural networks a famous

work is (Kohonen, 1982). They propose the Self Or-

ganizing Map (SOM) which iteratively adapts its in-

ternal structure — a 2D mesh — to the distribution of

a set of samples and enables clustering or dimension-

ality reduction of the sample data. While a SOM has

a ﬁxed topology, the growing cells structures concept

227

Annuth H. and Bohn C..

SMART GROWING CELLS.

DOI: 10.5220/0003085202270237

In Proceedings of the International Conference on Fuzzy Computation and 2nd International Conference on Neural Computation (ICNC-2010), pages

227-237

ISBN: 978-989-8425-32-4

 2010 SCITEPRESS (Science and Technology Publications, Lda.)

(Fritzke, 1993; Fritzke, 1995) allows the network for

dynamically ﬁtting its size to the sample data com-

plexity. SOM and GCS are suitable for processing

and representing vector data like point samples on

surfaces. (Hoffmann and Vrady, 1998) uses a SOM

and (Vrady et al., 1999) and (Yu, 1999) a GCS for the

purpose of surface reconstruction. Further improve-

ments are made by (Ivrissimtzis et al., 2003b) where

constant Laplacian smoothing (Taubin, 1995) of sur-

faces is introduced, and in (Ivrissimtzis et al., 2003a)

the curvature described by the input sample distribu-

tion is taken to control mesh density. In (Ivrissimtzis

et al., 2004a) the GCS reconstruction process is fur-

ther enhanced in order to account for more complex

topologies. (Ivrissimtzis et al., 2004b) use several

meshes of the same model for a mesh optimization

process, and (Yoon et al., 2007) present a concept for

combining common deterministic approaches and the

advantages of the GCS approach.

Overview. In the following, we outline the basis of

our approach — the growing cells structures — and

then derive our idea of the smart growing cells, which

matches the speciﬁc requirements of reconstruction.

Afterwards, an analysis is compiled discussing mesh

quality and performance of our approach, and ﬁnally,

we close with a summary and a list of future options

of this work.

2 RECONSTRUCTION WITH

SMART GROWING CELLS

Classical growing cells approaches for reconstruction

tasks are based on using the internal structure of the

network as a triangulation of the object described by a

set of surface sample points. A 2D GCS with 3D cells

is trained by 3D points. Finally, the cells lie on the

object surface which the 3D points represent and the

network structure — a set of 2D simplices (triangles)

— is directly taken as triangulation of the underlying

3D object.

The reason for using a GCS scheme for recon-

struction tasks are its obvious advantages compared

to deterministic approaches.

• They can robustly handle arbitrary sample set

sizes and distributions which is important in case

of billions of unstructured points.

• They are capable of reducing noise and ply dis-

continuities in the input data.

• They are capable of adaption — it is not required

to regard all points of the sample set on the whole.

Further, incrementally retrieved samples can be

used to retrain the network without starting the tri-

angulation process from scratch.

• They guarantee to theoretically ﬁnd the best solu-

tion possible. Thus, approximation accuracy and

mesh quality are automatically maximized.

Nevertheless, these advantages partly clash with the

application of reconstruction. On the one hand, dis-

continuities are often desired (for example, in case of

edges or very small structures on object surfaces). On

the other hand, smoothing often destroys important

aspects of the model under consideration (for exam-

ple, if holes are patched, if separate parts of the un-

derlying objects melt into one object, or if the object

has a very complex, detailed structure). In such cases,

GCS tend to generalize which may be advantageous

from the physical point of view, but which mostly lets

vanish visually important features which the human is

quite sensitized to.

The presented smart growing cells approach ac-

counts for these application-focused issues and em-

phasizes that modiﬁcation of the general learning task

in the classical GCS is suitable for many novel appli-

cation ﬁelds.

2.1 Unsupervised Learning and

Growing Cells Structures

General unsupervised learning is very similar to k-

means clustering (MacQueen, 1967) which is ca-

pable of placing k n-dimensional reference vectors

in a set of n-dimensional input samples such that

they are means of those samples which lie in the n-

dimensional Voronoi volume of the reference vectors.

Adaption of reference vectors is accomplished by

randomly presenting single n-dimensional samples

from the input sample set to the set of n-dimensional

reference vectors and moving them in n-dimensional

space, described as follows.

Place k reference vectors c

∈ R

, i ∈ {0..k −

1} randomly in nD space of input samples.

repeat

Chose sample s

∈ R

randomly from the

input set.

Determine reference vector c

(best

matching or winning unit) closest to s

Move c

in the direction of s

according to

a certain strength ε

, like c

new

= c

old

(1 −

) + s

· ε

Decrease ε

until ε

≤ certain threshold ε

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228

Surface reconstruction with pure unsupervised

learning would place a set of reference vectors on

object surfaces, but does not determine information

about the underlying surface topology. This leads to

the Kohonen Self Organizing Map.

Kohonen Self Organizing Map. The SOM is

based on reference vectors which now are connected

as a regular 2D mesh. The learning rule is extended to

account for the direct neighborhood of a best match-

ing unit as follows.

for all c

∈ neigborhood of c

Move c

in the direction of s

accord-

ing to a certain strength ε

, like c

new

old

(1 − ε

) + s

· ε

Decrease ε

end for

Insertion of this neighborhood loop into the general

unsupervised learning algorithm (after moving of c

)

leads to the phenomenon that the reference vertices

now are moved by accounting for the regular 2D mesh

topology of the SOM. Training a plane-like sample

set leads to an adaption of the SOM grid to this im-

plicit plane — the sample topology is recognized and

ﬁnally represented by the SOM mesh.

Nevertheless, the mesh size of a SOM is ﬁxed

and cannot adjust to the sample structure complexity.

The growing cells structures approach overcomes this

drawback.

Growing Cells Structures. To a certain degree,

GCS may be seen as SOM which additionally are ca-

pable of growing and shrinking according to the prob-

lem under consideration which is deﬁned by the sam-

ple distribution. This mechanism is based on a so

called resource term contained in every reference vec-

tor and which — in the original approach — is a sim-

ple counter. It counts the reference vector being a best

matching unit. A high counter value signalizes the re-

quirement for insertion of new reference vectors.

With a GCS we could train a sample set lying

on a certain object surface and the network structure

would ﬁt the object surface at a certain approxima-

tion error. The problem is that in reconstruction tasks

sample distributions are often not uniform. The rep-

resented surfaces usually contain discontinuities like

sharp edges and holes, and the objects to be recon-

structed are not that simple like a plane or a tetrahe-

dron — which usually are chosen as initial network

and which can hardly adapt to complex topologies.

Only objects which are homeomorphic to the start ob-

ject can be represented satisfactorily.

Thus, general unsupervised learning should

evolve to a kind of constrained unsupervised learning

which detects and adapts to certain structures which

the sample set implicitly contains.

2.2 Smart Growing Cells

Smart growing cells are an application-focused, six-

way adaption of the general learning scheme of the

classical growing cells structures approach. The SGC

basic structure is identical to general GCS. There are

n-dimensional cells which we now term neural ver-

tices connected by links through an m-dimensional

topology.

We let n = 3 since neural vertices are directly

taken as vertices of the triangulation mesh and m = 2

since we aim at 2D surfaces to be reconstructed.

The main training loop is outlined in Fig. 1. Here

del

and k

ins

are simple counter parameters deﬁned

below (see section 2.3). Movements of vertices and

their neighbors slightly differ from the classical SOM.

Again, there are two parameters for the learning rates,

for the winner and ε

for its neighbors, but these

are not decreased during learning since vertex con-

repeat

for j = 1 to k

del

for i = 1 to k

ins

Choose sample s from point cloud ran-

domly, ﬁnd closest neural vertex and

move it together with neighbor ver-

tices towards s.

Increase signal counter at s (the re-

source term mentioned above) and de-

crease the signal counters of all other

vertices.

end for

Find best performing neural vertex (with

highest signal counter value) and add

new vertex at this position (see Fig. 2).

end for

Find worst performing neural vertices,

delete them and collapse regarding edges

(see Fig. 2).

until certain limit like approximation error, or

number of vertices is reached.

Figure 1: Classical growing cells structures algorithm.

SMART GROWING CELLS

229

Figure 2: Neural vertex split operation (from left to right) to

increase mesh granularity locally, and edge collapse (from

right to left) to shrink mesh locally.

nections automatically become smaller together with

the learning rates. For drawing the neighboring ver-

tices, a smoothing process like described in (Ivris-

simtzis et al., 2003b) and (Taubin, 1995) is applied,

which replaces the classical movement, and which

makes the adaption of the topology more robust.

As initial network, usually a tetrahedron or a plane

with random vertices is suitable.

Vertex Split. A neural vertex split operation adds

three edges, two faces, and a new neural vertex. The

longest edge at the neural vertex with the highest re-

source term is split and a new vertex is added in the

middle. The signal counter value is equally spread

between the two vertices (see Fig. 2).

Edge Collapse. All neural vertices with resource

terms below a certain threshold r

min

are removed to-

gether with three edges and two connected faces (see

Fig. 2). The determination of the edge to be removed

is driven by connectivity irregularities as proposed in

(Ivrissimtzis et al., 2003b).

It follows our adaption of the mentioned learning

cycle by six modiﬁcations driven by the application

needs of surface reconstruction.

2.2.1 Cell Weeding

Aggressively deleting neural vertices which are not

part of a sound underlying mesh structure is the most

important new training rule of the SGC approach.

It is essential for giving the network the chance of

adapting to any topology despite of its initial topol-

ogy (overcoming the homomorphic restriction). Be-

fore the edge collapse operation is applied at a vertex,

it will be tested if the vertex is contained in a degen-

erated mesh region (deﬁnition follows below). If so,

an aggressive cut out of the vertex and its surrounding

vertices is started.

It has been shown that degeneration of a part of

a mesh serves as perfect indicator for a mesh topol-

ogy which does not ﬁt the underlying sample struc-

ture correctly. For example, consider a region where

Figure 3: Statue’s bottom is not represented by samples. On

the right, the acute-angled triangles expose a degenerated

mesh region.

sample densities equal zero. Although vertices are

not directly drawn into it by training adjustment, their

neighbors may be moved there through their mesh

connections. Due to their resource terms, these ver-

tices will be deleted by edge collapse operations, but

their links remain and mistakenly represent the exis-

tence of some topology. In this case, the structure of

the links is degenerated, i.e., it usually shows a sur-

passing number of edges with acute-angled

vertices

(see Fig. 3).

The reason for terming this deletion ”aggressive”

are the triggering properties which are easy to match

— suspicious neural vertices will be cut out early.

Criterion for Degenerated Mesh Regions. In

(Ivrissimtzis et al., 2004a) a large area of a triangle

is taken as sign for a degenerated mesh structure, but

it has been shown that this criterion warns very late.

Also, anisotropic sample densities are mistakenly in-

terpreted as degenerated mesh regions. Our proposal

is a combination of vertex valence

, triangle quality,

and quality of neighboring vertices. If all of the fol-

lowing conditions hold, a deleting of the mesh struc-

ture at that vertex is started.

1. Vertex valence rises above a certain threshold

degvalence

2. The vertex is connected to at least n

degacute

acute-

angled triangles.

3. The vertex has more then n

degnb

neighboring ver-

tices for which condition (1) or (2) hold.

The latter condition says that deletion is only started

if at least one or two neighbors have the same incon-

sistencies in their local mesh structure. This is rea-

A triangle is termed acute-angled if the ratio of its area

and the area which is spanned by a second equilateral tri-

angle built from the longest edge of the ﬁrst lies below a

certain threshold ε

acute

Vertex valence is the number of connected vertices.

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230

Figure 4: Curing a boundary with a spike.

sonable since single degenerated vertices do not nec-

essarily expose a problem but may arise by accident.

Curing Boundaries after Weeding. It is obvious,

that after an aggressive extinction of a neural vertex

and its surrounding faces, a boundary will be left be-

hind which may consist of unfavorable mesh struc-

ture elements. Curing ﬁnds these structures along the

boundary and patches them discriminating between

four cases.

Spike. A boundary vertex with a valence of 2 (see

Fig. 4) is termed spike. Such a vertex is very unlikely

to support a correct reconstruction process since it

will be adjusted to an acute-angled triangle after few

iteration steps. A spike must be deleted completely.

Nasty Vertex. A nasty vertex is a neural vertex with

at least n

nastyacute

acute-angled triangles and/or trian-

gles with a valence greater than n

nastyval

(see Fig. 5).

These vertices are suspected to be part of a degener-

ated mesh region and are deleted.

Needle Eye. A needle eye is a neural vertex that is

connected to at least two boundaries (see Fig. 6). At

these locations the mesh does not have a valid mesh

structure. To delete a needle eye, all groups of con-

nected faces are determined. From these, the group

with the most faces is kept and all others are deleted.

Bridge. A bridge is very likely to be part of a de-

generated mesh region. If a mesh has a hole that con-

sist of three vertices, then it would soon be closed

by a coalescing process (see section 2.2.2). This is

not allowed if exactly one of the edges of this hole

would additionally be connected to a face (which we

term “bridge”, see Fig. 7) since an invalid edge with

Figure 5: Cut out process of a nasty vertex.

Figure 6: Cut out process of a needle eye.

three faces would arise. The entire bridge structure is

deleted and the hole will be closed with a new face.

Multiple Boundary Search Through. After the

deletion of a neural vertex by the cell weeding pro-

cess the curing mechanism will search for unfavorable

structures along the boundary. There is more than one

boundary to be considered, if the deletion destroys a

coherent set of faces and multiple separate groups of

faces arise.

Four cases may appear. First, the usual case with

no additional boundaries. Second, when a needle

eye is destroyed, the boundaries of all groups of con-

nected faces need to be tested. Third, when surround-

ing faces of a vertex are interrupted by boundaries.

And fourth, when a needle eye is connected to the

surrounding faces of a vertex (see Fig. 8). In other

words, these cases happen since the faces that are

deleted may not necessarily be connected to a further

face due to another deletion process.

2.2.2 Coalescing Cells

Like the mesh can be split through deletion of ver-

tices, it must also be possible to merge two mesh

boundaries during training. For that, a coalescing test

is accomplished each time a vertex at a mesh bound-

ary is moved.

Coalescing Test. It determines if two boundaries

are likely to be connected to one coherent area. For

that, a sphere is created with the following parame-

ters. Given the neigboring boundary vertices v

and

of c

, then we deﬁne c = 1/2(v

). A boundary

normal n

is calculated as the average of all vectors

originating at c and ending at neighbors of c

, where

and v

are not taken into account. The boundary

Figure 7: Curing a bridge.

SMART GROWING CELLS

231

Figure 8: Cut out of a needle eye with a row of faces. Here,

each face is not necessarily connected to another face. In

contrast, if a needle eye has several groups of connected

faces then there are some omissions of faces around it.

normal can be seen as a direction pointing to the op-

posite side of the boundary. We deﬁne a sphere with

the center at c+n

r with radius r as the average length

of the edges at c

The coalescing condition at two boundaries hold,

i.e., merging of the boundaries containing c

and q on

the opposite side happens, if

• q is contained in the deﬁned sphere, and

• scalar product of the boundary normals at c

and

q is negative.

Coalescing Process. After detecting the neural ver-

tex q to be connected with c

, the according faces

must be created starting with one edge from c

to q.

There are two cases which have to be considered.

Corner. A corner of the same boundary arises when

an q have one neighboring vertex in common (see

Fig. 9). A triangle of the three participating vertices

is created.

Long Side. Here, two boundaries appear to be

separated. After determining the new edge, there are

four possibilities for insertion of a new face contain-

ing the edge (see second picture in Fig. 10). The tri-

angle with edge lengths which vary fewest is taken in

our approach (see third picture in Fig. 10) since it is

the triangle with the best features concerning triangle

quality. Finally, to avoid a needle eye, a further tri-

angle must be added — again, we take the face with

Figure 9: Coalescing process at a mesh corner. On the left,

the search process of a coalescing candidate. In the middle,

one edge is created, on the right, the only face capable of

being added is the corner face.

Figure 10: Coalescing of two separate boundaries. In the

second picture, the edge is determined, in the third, the tri-

angle with smallest variance of edge lengths is added, in the

fourth, another triangle must be added to avoid a needle eye.

Figure 11: Roughness adaption correlates surface curvature

with mesh density, details of the model are exposed.

the greatest edge similarity (see fourth picture in Fig.

10).

2.2.3 Roughness Adaption

Up to now, the SGC are able to approximate an arbi-

trary sample set by a 2D mesh. What remains is an

efﬁcient local adaption of the mesh density in a way

that areas with a strong curvature are modeled by a

ﬁner mesh resolution (see Fig. 11). This also relieves

the inﬂuence of the sample density on the mesh gran-

ularity making the SGC less vulnerable to sampling

artefacts like holes or regions which are not sampled

with a uniform distribution.

Each time a vertex is adapted by a new sample we

calculate the estimated normal n

at a neural vertex

by the average of the normals at the surrounding

ICFC 2010 - International Conference on Fuzzy Computation

232

faces. The curvature c

∈ R at a vertex is determined

= 1 −



∑

∀n∈N

· n (1)

with the set N

containing the normals of the neigh-

boring neural vertices of v

. Each time a neural vertex

is selected as winner, its curvature value is calculated

and a global curvature value c is adjusted. Finally, the

curvature dependent resource term r

at v

is adapted

through r

new

= r

old

+ ∆r

, and

∆r



1, if (c

< c + σ

)



/(c + σ

)



(1 − r

min

) + r

min

else,

(2)

with the deviation σ

of the resource term r

, and a

constant resource r

min

that guarantees that the mesh

does not completely vanish at plane regions with a

very small curvature.

2.2.4 Curvature Cells

Each time after a vertex v

has been moved we apply

a smoothing mechanism like mentioned at the begin-

ning of section 2.2.

Roughness adaption (see section 2.2.3) leads to

the fact that in regions of high curvature the density

of neural vertices will increase. These vertices then

will get fewer sample hits, since they have a smaller

Voronoi region, and thus, Laplacian smoothing is ap-

plied fewer times.

We found out, that this signiﬁcantly reduces mesh

quality in areas of high curvature. To avoid this, neu-

ral vertices in regions with high curvature are marked

as such and smoothing these is strengthened by re-

peating it n

times where

= b(c

− c)/σ

c − 1 (3)

with c

and c like deﬁned in section 2.2.3 and σ

the

deviation of the curvature at vertex v

. The value is

limited to a maximum of N

to intercept looping at

extraordinary curvature values.

2.2.5 Discontinuity Cells

A sampled model that exposes discontinuities like

edges is difﬁcult to be approximated by the neural net-

work mesh. Discontinuities are smoothed out since

the network tries to create a surface over them. This

might be acceptable in many application areas since

the approximation error is fairly small, but this effect

is unfavorable in computer graphics since it is clearly

visible. And even worse: edges are quite common in

real world scenarios.

Therefore, we propose discontinuity neural ver-

tices which, ﬁrst, are only capable of moving in the

Figure 12: A dent (left picture) on a sharp edge is solved

(right picture) by an edge swap operation. Finally, connec-

tions of discontinuity vertices model object edges.

direction of an object edge to represent them more

properly, and second, the smoothing process is not ap-

plied to them.

Recognizing those vertices is accomplished as fol-

lows. We determine the curvature values of those

neighbors which have a distance of two connections

from the vertex (the “second ring” of neighbors).

Then the average δ

ring

of the squared differences of

consecutive curvature values on the ring is calculated.

If a curvature value clearly deviates from the av-

erage curvature value, then we assume that it is a

discontinuity vertex if the average of the neighbors’

(second ring) curvature gradient differs to a certain

amount. Thus, we deﬁne a vertex v

being a disconti-

nuity vertex if

> 2σ

) ∧ (∀c ∈ C

: δ

ring

> 4σ

) (4)

with C

the set of curvature values of the second ring

of neighbors.

For approximating the edge normal we take the

average of the normals of two of the neighboring ver-

tices of v

, either those with the highest curvature

value, or those which are already marked as discon-

tinuity vertex. Finally, the normal is mirrored if the

edge angle lies above 180

◦

, which is indicated by the

average of the surrounding vertex normals; in the ﬁrst

case it points in the direction of v

Edge swap. If two connected discontinuity vertices

grow into an edge, they nicely represent this edge by

a triangle edge. But if the line is interrupted by a non-

discontinuity vertex, a dent arises since this vertex is

not placed on the edge. Thus, we propose an edge

swap process which minimizes this effect.

Each time a discontinuity vertex is moved towards

a sample, the need for an edge swap operation will be

determined by collecting the three consecutive faces

with the most differing face normals. In case of a

dent, the face in the middle is assumed to be the one

which is misplaced and an edge swap operation is ap-

plied (see Fig. 12). Then, if the difference of the nor-

mals is now lower than before, edge swap is accepted,

if not, the former structure is held.

Edge swap results in models where ﬁnally edges

are represented by mesh boundaries (see Fig. 13).

SMART GROWING CELLS

233

2.2.6 Boundary Cells

Similar to discontinuity vertices which are capable of

moving to object edges, boundary vertices are able to

move to the outer border of a surface (see Fig. 14).

They are recognized by being part of a triangle edge

which is connected to one face only.

Then, these vertices are moved only in the direc-

tion of the boundary normal like described in section

2.2.2 in order to avoid vertices just lying in the aver-

age of the surrounding samples but directly match the

surface boundaries at their locations.

2.3 Results

For the full algorithm of this approach see the pseu-

docode in Fig. 16. To keep it comprehensive, the

outermost loop of the algorithm is neglected, and ver-

tex split and edge collapse operations are triggered by

counters.

Parameters which have been proven to be reliable

for almost all sample sets we took for reconstruc-

tion are ε

= 0.1, ε

= 0.08, r

min

= 0.3, ε

acute

0.5, n

degacute

= 4, k

ins

= 100, k

del

= 5, n

degnb

= 1,

nastyacute

= 4, n

nastyval

= 3.

The following results have been produced on a

Dell



Precision M6400 Notebook with Intel



Core 2

Extreme Quad Core QX9300 (2.53GHz, 1066MHz,

12MB) processor with 8MB 1066 MHz DDR3 Dual

Channel RAM. The algorithm is not parallelized.

Some visual results are exposed in Fig. 15. All

pictures are drawn from an SGC mesh. Most models

stem from the Stanford 3D Scanning Repository.

Besides visual results, reconstruction with SGC

comes up with impressive numbers compared to clas-

sical approaches, listed in table 1.

It can be seen that mesh quality, i.e. the percent-

age of perfect triangles in the mesh lies at 96% at

average. This is an outstanding result, nevertheless

this is usually expected when using an approach from

the ﬁeld of unsupervised learning, since this guaran-

tees an ideal representation of the underlying training

Figure 13: Discontinuity vertices focus on edges. Edge

swap operations let mesh edges map to object edges.

Figure 14: Mesh boundary due to the missing bottom of the

statue is represented exactly by boundary cells.

Table 1: Results with sample sets from the Stanford

3D Scanning Repository. “Quality” means percent-

age of triangles which hold the Delaunay criterion.

RMS/Size is the root of the squared distances between

original point samples and the triangle mesh, divided

by the diameter of the sample set.

Samples Vertices

Time

[m:s]

Quality

RMS/

Size

36K 30K 0:39 95.6% 4.7e-5

438K 100K 2:47 95.5% 3.3e-5

544K 260K 9:15 93.1% 1.7e-5

14,028K 320K 12:17 98.5% 1.3e-5

5,000K 500K 21:5 95.9% 2.7e-5

511K 10K 0:11 99.8% 6.6e-5

38K 5K 0:6 99.0% 15e-5

346K 5K 0:6 98.3% 0.7e-5

sample distribution.

Further, the distance (RMS/object size) between

samples and mesh surface is negligible low — far be-

low 1% of the object size at average. This is even

more pleasant, since usually, the problem at edges

generate big error terms. Also the computing times

needed are very short, few minutes in each case.

All those measurements seem to be far better than

those from classical approaches, as long as we could

extract them from the regarding papers. Our algo-

rithm works very robustly. There are nearly no out-

liers visible in the mesh.

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234

Figure 15: Upper lines: mesh training stages with number of vertices, lower lines, assorted pictures of reconstructed models.

SMART GROWING CELLS

235

Adjust samples regarding roughness.

Calculate average curvature and deviations.

Recognize and sign discontinuity cells.

Recognize and sign curvature cells.

for all Boundary cells do

if ∃ coalescing candidate then

Melt boundary.

for all Weeding candidates do

Weeding process.

end for

end if

end for

if Edge collapse operation triggered then

Collapse edge.

for all Weeding candidates do

Weeding process

end for

end if

if Vertex split operation triggered then

Split vertex.

end if

Figure 16: Outline of the complete SGC algorithm.

3 CONCLUSIONS

We presented a new neural network approach, the

smart growing cells, which is a modiﬁcation of the

classical growing cells structures approach.

The modiﬁcation type is new in a way that

it changes the pure, general unsupervised learning

scheme ad hoc to match training requirements of spe-

ciﬁc applications.

Thus, drawbacks of using unsupervised learning

approaches can be avoided while advantages be re-

tained, and nevertheless, SGC training keeps its roots

at general unsupervised learning.

We encourage this idea by one speciﬁc application

case — surface reconstruction from 3D point sam-

ples. Here, we add six topics to the classical unsuper-

vised learning scheme, and ﬁnally the approach out-

performs classical approaches concerning quality, ef-

ﬁciency, and robustness. Surface reconstruction with

SGC is able to handle arbitrary topologies and mil-

lions of samples. It recognizes and solves discontinu-

ities in the sample data and it is capable of adapting to

varying sample distributions. Finally, the network is

able to reorganize its topology to match arbitrary sur-

face structures. Altogether these advantages can not

be found in any of the classical approaches of surface

reconstruction.

The essential issue which transforms GCS to SGC

is the mechanism of weeding cells as a network clean-

ing mechanism for ill-formed structures. Further, face

normals are regarded and included in the neural net-

work training loop to adapt to mesh roughness and

to make the reconstruction process independent from

the sample distribution. Additionally, we propose co-

alescing cells which can connect to others, curvature

cells which recognize very small structures, and dis-

continuity cells which account for certain discontinu-

ous structures like sharp edges.

The proof of concept of our approach is enriched

by the achieved quality and performance measures.

For the tested geometries which each hold speciﬁc

challenges of reconstruction, we got approximation

errors for comparable mesh resolutions that lie far be-

low 1% at average. Mesh quality, measured by the

percentage of triangles which comply the Delaunay

criterion, lies at 96% at average. And the time needed

to compute meshes of several hundreds of thousands

of polygons were just few minutes.

Future Work. This work shows that application-

focused unsupervised learning is able to solve prac-

tical problems efﬁciently. Computation times are that

small that we think of a real-time reconstruction ap-

proach through multithreaded sample adjustment.

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