Growing Surface Structures

Hendrik Annuth and Christian-A. Bohn

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

Keywords:

Unsupervised Learning, Competitive Learning, Growing Cell Structures, Surface Reconstruction, Surface

Fitting.

Abstract:

Strictly iterative approaches derived from unsupervised artiﬁcial neural network (ANN) methods have been

surprisingly efﬁcient for the application of surface reconstruction from scattered 3D points. This comes from

the facts, that on the one hand, ANN are able to robustly cluster samples of arbitrary dimension, size, and

complexity, and on the second hand, ANN algorithms can easily be adjusted to speciﬁc applications by in-

venting simple local learning rules without loosing the robustness and convergence behavior of the basic ANN

approach.

In this work, we break up the idea of having just an “adjustment” of the basic unsupervised ANN algorithm

but intrude on the central learning scheme and explicitly use learned topology within the training process. We

demonstrate the performance of the novel concept in the area of surface reconstruction.

1 INTRODUCTION

Due to the rapid development in 3D scanning technol-

ogy, real world objects can be scanned faster, more

accurately and at a higher resolution. This allows

creating high quality virtual representations of these

objects that can be utilized for many different pur-

poses in digital data processing. In archaeology and

crime scene investigation a site can be analyzed in-

dependently from its location and time. Architecture

and plant manufacturing involve accurate construc-

tion planning and precise measuring, which can be ef-

ﬁciently done on a digital model of a construction site.

In ﬁlms reconstructed surfaces are often used as a sup-

port structure for manually created models. Computer

games use 3D scanning technology to incorporate real

world objects and environments into their otherwise

virtual worlds. Reverse Engineering, the most impor-

tant ﬁeld of reconstruction, is a process where exit-

ing products or physical prototypes are used as a tem-

plate to implement a virtual 3D model for production

and quality control purposes. Mobile robots can proﬁt

from surface models as maps for path planning, local-

ization, scene interpretation and grasping.

Laser scanning devices take samples of present

surfaces as three dimensional data points. These

points accumulate as an unorganized cloud of points.

Surface reconstruction is the process which creates a

virtual model of the surfaces from which these points

Figure 1: A photography of Michealangelo’s David (left), a

point cloud of the David statue (middle), and a surface ﬁtted

into that point cloud (right).

originate (see Fig. 1). Such a point set typically in-

cludes noise, outliers, non-uniform sample densities,

and holes. Since the combination of these problems

is often inherently ambiguous, a vast number of dif-

ferent reconstruction approaches, and pre- as well as

post-processing methods exist.

Due to the remaining challenges in the ﬁeld of re-

construction there is a strong tendency to focus on

ANN based solutions since they are strong with in-

complete and noisy data while being ﬂexibly adapt-

able. Thus, there is hope that by a more intuitive, “ad-

hoc” manner, ANN training can be modiﬁed to match

the problems under consideration without the need of

349

Annuth H. and Bohn C..

Growing Surface Structures.

DOI: 10.5220/0004529203490359

In Proceedings of the 5th International Joint Conference on Computational Intelligence (NCTA-2013), pages 349-359

ISBN: 978-989-8565-77-8

 2013 SCITEPRESS (Science and Technology Publications, Lda.)

a deterministic mathematical model.

2 PREVIOUS WORK

2.1 Classical Surface Reconstruction

Many surface reconstruction approaches have been

suggested over the last decades. Range image meth-

ods can achieve very high resolutions and accu-

racy, while necessitating a very controlled scanning

setup with a limited sensing area (Curless and Levoy,

1996). Region growing approaches (Gopi and Krish-

nan, 2002; Bernardini et al., 1999) extended an ini-

tial surface incrementally at its boundaries. Some

methods reduce a 3D Delaunay triangulation of the

samples to a ﬁnal surface (Edelsbrunner and M

ucke,

1992), some derive it from the Voronoi diagram of the

points (Mederos et al., 2005; Amenta et al., 1998).

Combined concepts use region-growing approaches

for the triangulation and an additional global graph

like a 3D Delaunay triangulation as a guidance (Kuo

and Yau, 2005) or a medial scaffold (MS) (Chang

et al., 2009). Balloon models construct a volumet-

ric object surface by the “inﬂation” of a small sur-

face, as if it would be a balloon, inside a point cloud

(Sharf et al., 2006). Another huge class of reconstruc-

tion methods demands points that are augmented by

its normals to deﬁne an incomplete distance function.

This function is completed and the subspace in R

for

which it returns zero — the zero-level-set — is the

surface. The function can be composed of a multitude

of linear functions (Hoppe et al., 1992), quadratic

functions (Kazhdan et al., 2006; Ohtake et al., 2003)

or radial base functions (Carr et al., 2001). Model

based reconstruction approaches compose a surface

of a multitude of predeﬁned models or components,

which are recognized and ﬁtted into the point cloud

(Schnabel et al., 2009; Gal et al., 2007). Warping

algorithms approximate the surface by deforming an

initial surface to match the given points (Yu, 1999;

Baader and Hirzinger, 1993).

2.2 Artiﬁcial Neural Network based

Reconstruction

Many neural computation techniques have been ap-

plied to the problem of surface reconstruction and are

based on unsupervised learning concepts. Algorithms

such as the k-means clustering approach (MacQueen,

1967) use reference vectors to accomplish classiﬁ-

cation and clustering tasks on huge and challenging

data sets (“hard competitive learning”). Kohonen pre-

sented the Self-Organizing-Map (SOM) (Kohonen,

1982) — additionally reference vectors are connected

adding a topology (“soft competitive learning”) which

enables the construction of a surface over the sample

set. Kohonen’s approach has the disadvantage of a

ﬁxed resolution, which strongly relates the results to

the initial setting and size of the network. Fritzke pre-

sented the Growing-Cells-Structure (GCS) approach

(Fritzke, 1993) where the network grows over time

by dynamically adding reference vectors. The grow-

ing process can be determined by the approximation

error toward a likelihood distribution or a quantiza-

tion error, both of which are measured in relation to

the reference vectors. Based on this algorithm many

convincing surface reconstruction — sometimes re-

ferred to by the term “reﬁnement strategies” — meth-

ods have been presented (Annuth and Bohn, 2012;

Ivrissimtzis et al., 2003a; Ivrissimtzis et al., 2003b;

Vrady et al., 1999). The main disadvantage of the

methods above is the fact that 2D subspaces or sur-

faces are approximated by point distributions instead

of surface models. This becomes most apparent when

modeling ﬂat surface areas where the granularity of

the ANN surface depends on the distribution of sam-

ples and not on the complexity of the underlying sur-

face.

In this work we present an approach which lifts this

handicap. Basic ANN learning is changed to a sur-

face oriented learning saving the advantages of neu-

ral networks but concurrently implementing a reason-

able “surface learning”. The difference to former ap-

proaches which modify ANNs by adding additional

constraints to the learning rules our approach intro-

duces an actual novel learning scheme.

3 SURFACE RECONSTRUCTION

Surface reconstruction creates a 2D subspace S in a

3D space R

that represents a real world physical sur-

face S

phy

. The information given about S

phy

is a ﬁnite

collection of surface samples P = {p

... p

} ⊂ R

If closest neighbors in P always indicate a connec-

tion on S

phy

, surface neighborhood relations can be

investigated by accessing P in a 3D search pattern.

Real world scenarios however, involving noise, non-

uniform sample densities and incompletely sampled

areas. This makes a 3D search unreliable (see Fig.

2), which is the basic problem to overcome in a recon-

struction approach. Note that many of the following

illustrations are in 2D and are therefore curve recon-

structions, but to avoid confusions by swapping termi-

nology, we proceed in using the terms of 3D surface

reconstruction.

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350

Figure 2: The topology of a surface has to be derived from

3D samples. But searching in 3D for sample neighbors

might produce misleading results. First, a highly sampled

surface, where the two surface neighbors to a certain sam-

ple can be easily found (left). Then the same surface with a

lower sampling, the search space for the two closest neigh-

bors has grown (middle). And at last, a low sampling where

the two closest neighbors are not the correct topological (on

surface) neighbors (right).

Figure 3: Different successive surface stages in a GCS sur-

face reconstruction.

4 GROWING CELL

STRUCTURES

A GCS network is composed of simplices of an ini-

tially chosen dimension. In case of surface recon-

struction, a 2D surface S built of triangles as sim-

plices. The initial surface is a very simple network

of triangles such as a tetrahedron. It is positioned

roughly at the center of P . Since the GCS algorithm is

inspired by growing organic tissue, the reference vec-

tors are termed cells. GCS use an iterative reﬁnement

process to ﬁt the current surface into the point data P

(see Fig. 3). The reﬁnement process randomly se-

lects a sample of P and deforms the current surface in

order to progressively minimize its distance. This ba-

sic step is repeated and in each iteration the local ap-

proximation error is measured. These errors are used

to determine surface areas which need to be reﬁned

by local subdivision processes. Subdivision and fur-

ther iterations lead to a better match of the surface

to the sample distribution and the process is stopped

when the chosen average approximation error reaches

a threshold or a certain number of reference vectors

is reached. In the following section we analyze the

algorithm in detail for the application of surface re-

construction and show different kinds of handling the

error such as likelihood distribution, error minimiza-

tion and topology preservation which each lead to dif-

ferent results. Since the algorithm represents S as a

triangular mesh, we will use the term vertex instead

of reference vector, since it is more common in this

context.

Algorithm 1 : An overview of the GCS algorithm.

Conditions 1 and 2 can be deﬁned as simple counters.

1: Given a point cloud P = {p

... p

} and an ini-

tial surface S in form of a tetrahedron repre-

sented as an interconnected network of vertices

S = {v

...v

2: repeat

3: repeat

4: repeat

5: Select random sample p

of P and search

the winning vertex v

with smallest Eu-

clidian distance to p

6: Move v

towards p

7: Move all direct neighbors of v

with a

lesser factor towards p

8: Adapt the approximation error of v

9: until condition 1 holds.

10: Add new vertex in the area of highest ap-

proximation error through a vertex split op-

eration.

11: until condition 2 holds.

12: Search the least winning vertex in the network

and delete it by an edge collapse operation.

13: until accuracy exceeds a certain threshold.

4.1 Likelihood Distribution

The approximation error (algorithm 1 line 8) can be

altered toward a likelihood distribution or a quantiza-

tion error (see section 4.2). The vertices create a like-

lihood distribution if for every given vertex v ∈ S the

likelihood to be the closest neighbor to a randomly

chosen sample p ∈ P is equal. If we see vertices

being closest neighbors as the result of a probabil-

ity experiment this approximation resembles entropy

maximization. This means that the information car-

ried by any given sample is of the same importance.

These representations are especially important in pat-

tern recognition and statistical analysis.

To achieve this every vertex carries a signal counter.

To approximate the likelihood distribution (algorithm

line 8) these counters are simply incremented when

a vertex is closest to an input sample. If a new ver-

tex is added (algorithm line 10) the highest error term

refers to the space where most samples share the same

vertex.

Since older signals tend to be less representative, all

signal counters are decreased at every iteration cycle

by a certain factor (algorithm line 8). By using a like-

lihood distribution signal counters can also be used

GrowingSurfaceStructures

351

to determine misplaced vertices in spaces that contain

few or even no samples, since their signal counters

are very low due to constant decreasing. This concept

has therefore been used in most implementations for

surface reconstruction.

These algorithms however determine the area for

which the likelihood of the vertices is highest and not

the surface. If a ﬂat surface is approximated, the al-

gorithm will create lots of vertices in relation to the

amount of samples, although the area could be accu-

rately approximated with a few triangles only.

4.2 Distance Minimization

When the approximation error (algorithm line 8) is

changed to account for a quantization error, vertices

are placed exposing the smallest Euclidian distance

to the samples in P . If the samples P are equally

distributed the goals of a likelihood distribution com-

pared to an error minimization are nearly the same.

If however some regions are represented by a denser

sampling than others, these regions will be repre-

sented by less vertices in the error minimization sce-

nario, since the error which is measured as the Euclid-

ian distance can be lowered more signiﬁcantly in re-

gions where samples lay farther apart, hence vertices

are more likely to be added there. This approximation

is typically used for vector quantization in data com-

pression. To implement this behavior every vertex

carries an error value which is increased (algorithm

line 8) by the distance or the squared distance between

the winning vertex and the given sample. The high-

est approximation error refers to the space where the

samples lay farthest away from a vertex, thus a new

vertex will be added there (algorithm line 10). In con-

trast to a likelihood value, removing a vertex with low

distance errors would make no sense, since these ver-

tices indicate that they are well placed. However in

case the created topology matters, as in surface re-

construction, it is reasonable to remove such vertices

for memory efﬁciency reasons, since they might be

redundant geometry wise.

The basic problem is the difference between the ap-

proximation of the right topology and achieving a

lowest distance error. We will discuss this problem in

more detail in section section 4.3. When this approxi-

mation error is used, the deletion process of misplaced

vertices need to be handled separately. Despite of this

disadvantage minimizing the distance error might be

more convenient for surface reconstruction. But this

is not the case if the approximation minimizes the dis-

tance to the vertices instead of the surface (see Fig.

4), since a surface approximation aims to ﬁt S as close

a possible to P .

Figure 4: Samples and an approximated surface (top), dis-

tance between vertices and samples (middle), distance be-

tween surface and samples (bottom). In case of surface ap-

proximation the distance to the surface is obviously more

worthy.

Many implementations have tackled this prob-

lem indirectly. (Annuth and Bohn, 2012) presents a

roughness adaptation where the average surface cur-

vature is compared to the one of a winning vertex and

curved areas lead to higher signals leading to more

subdivisions in such areas. In (Jeong et al., 2003)

vertices additionally have normals and the algorithm

counts how much these normals are moved to increase

subdivisions in such areas.

These changes lead to an implicit representation of

the approximation error within the algorithm, since

curved surface regions need more subdivisions to be

correctly approximated. But the surface approxima-

tion error itself is not explicitly represented.

4.3 Topology Preservation

The SOM (Kohonen, 1982) introduced an unsuper-

vised learning concept with an additional topology.

In the given network S vertices are not allowed to

move independently. When a vertex is moved to-

ward a sample its neighbors are also moved to de-

crease the created surface tension (algorithm line 7).

This principal adds elasticity to the network — the

behavior of a continuous surface is manifested implic-

itly by creating dependencies between the vertices. A

topology can increase the performance in placing ref-

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352

Figure 5: Two surfaces with the same vertices, the same

samples, and with the same approximation error, that ex-

pose an undesired (left) and a desired (right) solution. A tri-

angle placed in an empty space (top) and an incorrect dent

in the surface (bottom).

erence vectors since the dependencies between them

make their movements more stable and thereby make

smoother distributions more likely. But the created

topology itself can be used in many different ways as

well. Data of a high dimensional input space can be

mapped into a space of lower dimension and can then

be visualized or analyzed with less computational ef-

fort (dimensionality reduction). The topology can

also be used for regression analysis where P is known

to originate from an unknown continuous function to

be reconstructed from the data (function approxima-

tion). The SOM uses a static topology, which usually

resembles a square shaped grid. The standard GCS

algorithm also uses a static surface topology while

the connectivity of the network can change (note that

the network connectivity is often also referred to as

topology, perceiving the network as a graph). This

means a network area can be increased in resolution

and thereby gather new vertices and connections, but

the surface topology of a sphere, inherited from the

initial tetrahedron shape, cannot be changed.

So functions that have a different topology can not

be correctly resembled. GCS however has better sur-

face approximation capabilities than the SOM, since

it builds newly created surfaces by reﬁning a former

version of that surface. This adapts the vertex resolu-

tions in different surface areas toward the target func-

tion and gives a surface a certain inertia when being

modiﬁed, which avoids local failures if a surface is

ﬁtted into a challenging point constellation.

If the information of a sample placed in a 3D space

is processed by the algorithm, this information is al-

ways set considering a pre-existing current surface S .

This is the strategy of the GCS algorithm to over-

Figure 6: Samples that originate from a curved surface

(top,left), a ﬁtting approximation of this surface and a mag-

niﬁcation that shows the vertex-sample distance and the

surface-sample distance (top,right), the vertex Voronoi re-

gions which assign a sample at the wrong surface (bot-

tom,left), the surface Voronoi regions which assign the same

sample correctly (bottom,right).

come the 3D search problem (see section 3). But still

the GCS can get stuck in local minima and the initial

topology can mismatch the target surface.

In the standard algorithm a destructive method is pre-

sented which uses the average edge length as an in-

dicator to cut out triangles (Fritzke, 1993). In (Ivris-

simtzis et al., 2003a) triangles which are larger than

the average size are cut out, boundaries that fall be-

low a certain Hausdorff distance are joint, and in (An-

nuth and Bohn, 2012) high valences are used as an

indicator to cut the surface and low distances in com-

parison to edge length to join boundaries. With these

changes complex topologies can be created. The main

problem of all presented GCS based algorithms con-

cerning topology issues is the missing representation

of the actual surface within the adaptation process,

which makes many insufﬁcient approximation states

simply not measurable (see Fig. 5).

Even so the given 3D information of samples is set in

relation to the existing 2D surface, the surface is still

represented as a collection of Voronoi regions of the

vertices, since vertex-sample and not surface-sample

distances are considered. This concept implicitly in-

cludes the assumption that the Voronoi regions of two

connected vertices will not be interrupted within their

attached surface. But for close or complex shaped sur-

faces, this is not the case (see Fig. 6). Here the actual

representation of S becomes apparent being a perme-

able space of independent Voronoi volumes.

Topology preservation and error minimization are two

different things. Error minimization tries to create the

smallest possible distance to the samples in P . Topol-

ogy preservation however is concerned with reaching

GrowingSurfaceStructures

353

Figure 7: A surface and samples (left), an approximation of

that surface and the distance error between the surface and

the samples (middle), another approximation of that surface

with a topological error (right).

a topology with S that lies as close as possible to the

topology of S

phy

(see Fig. 7).

In order to be topologically correct every point on one

surface need to have a unique equivalent on the other

surface and vice versa, while neighbor relations are

preserved, meaning the shortest on-surface path be-

tween any two given points projected onto S should

always correspond to the one on S

phy

5 APPROACH

The GCS algorithm has proven to be a high quality

surface reconstruction tool. However, in our anal-

ysis of the algorithm we saw that topology is only

created implicitly and only accounted for through the

additional adjustment of neighboring vertices. In the

following section we will present our changes to the

general approximation concept of the basic algorithm

and then the improvements that can be made based on

these changes.

5.1 Topology focused Approximation

The basic algorithm concept focuses on placing ver-

tices in positions likely to decrease the chosen ap-

proximation error. To put the actually created surface

topology into focus, the approximation error needs to

be set in relation to the surface-sample distance. The

most important change is to search for the closest sur-

face element (algorithm line 5) instead of the clos-

est vertex. The adaptation process (algorithm line 6

and 7) can now also be set in relation to a sample be-

ing closest to a triangle or being closest to an edge,

which gives rise to more different local surface mod-

iﬁcations (see section 5.2.2). In the basic GCS im-

plementation the signal counter or error value is car-

ried by the vertices. The surface structure element

that most distance errors are measured towards and

that is also the building block of the discretization of

phy

is the triangle and is therefore the structure that

carries the local approximation error values in our im-

plementation. The most sensible place for the error

value of any topology focused function approxima-

tion is always the simplex of highest dimension in the

GCS algorithm. The distance between a sample and

its closest structure represents the actual distance er-

ror to the approximated surface and gives this approx-

imation error way more validity (algorithm line 8).

This allows for better judgments about a current lo-

cal approximation state and the choice of location for

subdivision (algorithm line 10). The new approxima-

tion error also allows and demands to distinguish be-

tween topology changing deletions to correct topolog-

ically misplaced surface structures and non-topology

changing deletions that remove geometrically redun-

dant structures from the surface (algorithm line 12).

Topology changing deletions are realized by adding

an “age” to every triangle and cutting them out when

they reach a certain age.

5.2 Adaptations of the Algorithm

Behavior

With the changes described in section 5.1 additional

and more accurate information about the current ap-

proximation state is available within the GCS process.

This information can be used to create a better ap-

proximation result in accuracy and topology. In the

following we will present our implementation details.

5.2.1 Search for Closest Element

Due to run time efﬁciency reasons we did not use an

actual triangle based spatial subdivision data struc-

ture, but still a vertex based octree. By searching for a

number num

of vertices and checking their surround-

ing triangles, we heuristically ﬁnd the closest struc-

ture to a given sample. We used num

= 3 which fails

when the degree of curved and ﬂat surface areas di-

verge too much and are close to each other, but we

consider this case to be rare (see Fig. 8). The new

search process has three possible outcomes: a vertex,

an edge, or a triangle (algorithm line 5).

5.2.2 Surface Movement

Instead of having only a vertex as a closest element,

we now can access additionally an edge and a trian-

gle in our implementation. We modeled three differ-

ent main movements (algorithm line 6). Since we

now know the distance of a given sample p to the

surface we can compare this distance d

to the av-

erage sample-to-surface distance d

. When d

is only

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354

Figure 8: The presented approach searches for the clos-

est surface structure, such as a vertex, an edge or a trian-

gle. However this process is emulated on a search inves-

tigating the surrounding elements of a number of vertices

num

, instead of having a search tree actually comprising

edges and triangles. The ﬁgure shows a sampled curved

and ﬂat surface close by (top, left), next to it its approxi-

mated surface (top,right). The search heuristic works cor-

rectly for num

= 3, where three vertices are investigated

(bottom,left). The search heuristic fails for num

= 2 (bot-

tom,right), where the vertex connected to the closest trian-

gle is not investigated.

a fraction lim

skip

of d

we entirely discard the adap-

tation since the sample already lies more or less close

to the surface. When d

is about the degree lim

single

lower than d

we only move the closest vertex, since

we consider the surface to be generally correct, but

the “joints” of the divisions can be optimized. We

use lim

skip

= 0.9 and lim

single

= 1.2. Note that dis-

tance error distribution within the process is not Gaus-

sian, thus we cannot describe the process in terms of

standard deviations. For higher distances we move

all vertices of the given structure towards the sam-

ple. Due to this, surface widening is less likely to

cause spikes and the surface is moved more uniﬁed,

which produces a smoother surface. The neighbor-

hood movement (algorithm line 7) is unchanged and

accomplished with the Laplace smoothing mecha-

nism (Taubin, 1995) for all ﬁrst neighbors of the given

structure.

5.2.3 Distance Error

A sample can be closest to a vertex, an edge, or a

triangle. When it is closest to a triangle the age of

this triangle is set to zero and the distance error is

changed. In case of an edge this is done for both tri-

angles connected to that edge. In case of a vertex it

is done for all triangles connected to this vertex. If

Figure 9: A triangle split operation (left) and a triangle col-

lapse operation (right).

we would set the error value directly to the given dis-

tance, all previous distance errors would be lost, if

we, on the other hand accumulate all distance errors,

old distance errors would totally determine the reﬁne-

ment process, since their distances where huge and it

would take many subdivisions to decrease them. We

could constantly decrease all error values as described

in section 4.1 but this would strongly decrease the

accuracy of the local distance errors since it implies

that the surface constantly improves everywhere with

a constant rate. We wanted the half-life λ of a distance

error to be 9 winning operation (see equation 1).

(

k−1

)

= 0.5

k =

−1

√

0.5−1

err

new

+err

old

(k−1)

(1)

5.2.4 Reﬁnement

Instead of the vertex we search the triangle with the

highest approximation error. Subdivision is done by

splitting the triangle’s surface from one of its three

vertices to the opposite edge and then also split the

other triangle in the mesh with this edge. The edge

with the additional triangle with the largest error term

is taken. Four new triangles are added, four new

edges, and one new vertex (see Fig. 9). The error

value of a new triangle is the half of the error value of

its predecessor.

5.2.5 Deletion

The deletion process is one of the most important

changes in the algorithm. When using a sample-to-

surface distance error, the error values can only be

used to determine triangles that might be redundant

and do not improve the approximation. In order to

have a model representation that is as memory efﬁ-

cient as possible, these triangles can be deleted by an

edge collapse operation of one of its three edges. The

GrowingSurfaceStructures

355

best candidate for the operation is the edge which is

surrounded by triangles with those normals that ex-

pose the least differences |∇n| to one another, since a

collapse of this edge will change the surface gradient

the least. This edge can be determined as the one with

the highest dot product of the normals of its two ver-

tices, since these normals are calculated based on their

surrounding triangles. It is reasonable to set a thresh-

old max

|∇n|

for |∇n|to avoid changes that decrease the

surface approximation quality. We choose max

|∇n|

be cos 10

◦

which allows for a maximum angles of 10

◦

between those normals. In addition to the surface er-

ror we need a triangle age a that indicates if a triangle

reached a maximum age max

. This should happen

when the triangle has been missed for a certain num-

ber of times η. Those triangles are considered to be

misplaced and topological wrong. A misplaced trian-

gle is detached from the rest of the network and then

deleted. Note that the deletion of a single triangle can

leave the surface in an undesired state, where for ex-

ample triangles are connected by single vertices only,

which need to be cleared. Since triangles are likely to

have different sizes and small triangles are less likely

to be winners, the real age a

real

of a triangle need to

be set in relation to its size, or in other words, small

triangles age slower. This can easily be achieved by

an additional age factor t

size

calculated as the size of

a triangle divided by the average triangle size. For

small triangles this value is very low, so their age is

strongly reduced. For every iteration the age of all tri-

angles is increased by a tiny factor µ that has a relation

to the overall number of triangles |T | (see equation

2). This is basically the reversed tumble tree (Annuth

and Bohn, 2010) principal. We use 1 for max

and

10 for η. This means, i.e., if one of four triangles of

the same size is not hit one time for forty iterations

it is considered to be misplaced. The deletion pro-

cess now explicitly distinguishes distance driven and

topologically driven deletion.

(|T |·η)

= max

µ =

(|T |·η)

√

max

new

= a

old

·µ

real

= t

size

·a

(2)

5.2.6 Finalization

One of the assets of the GCS algorithm is the fact that

S is an approximation of S

phy

at any time during the

running loop — the algorithm can be stopped and re-

sumed at any given time. With the novel surface dis-

tance approximation error a potential stopping point

for the algorithm can be chosen more sensibly.

Figure 10: A progression series of the dragon model from

left to right with 2500, 5000 and 10000 triangles with the

standard algorithm (top) and with the new algorithm (bot-

tom). With the new algorithm the surface diverges faster

toward the ﬁnal topology.

6 RESULTS

We accomplished different tests with the Stanford

Dragon model as a good example for a point cloud

that is relatively challenging by its shape and the sam-

ple distribution, the hand model exposes sharp fea-

tures, the Asian Dragon and the Thai Statue expose

a lot of curved areas, the Heating Pipes model in-

cludes some extremely noisy areas, non-uniform sam-

ple densities and open surface areas, the Happy Bud-

dha has regions of surfaces lying close together. From

the basic algorithm we used (Annuth and Bohn, 2012)

but deactivated roughness adaptation and sharp fea-

ture detection.

Theoretically, any surface would be correctly re-

constructed with the GCS algorithm, if inﬁnite sam-

ples, sufﬁcient memory and time were available. A

reasonable parameter to judge efﬁciency is the time to

reach a certain accuracy. In our experiments with the

new approach, for instance the topology of the dragon

model was approximated much faster, shown in Fig.

10.

Close surfaces can be handled correctly with our

presented method and a vast number of additional it-

erations to avoid permeating Voronoi regions is not

required any longer as we show in Fig. 11.

Although the standard algorithm is already quite ro-

bust when dealing with noise, we could show that

spikes and rough surface gradients could be greatly

reduced with our presented method. The moving of

entire substructures seems to have a smoothing effect

on the surface (see Fig. 12).

We compared the new and the old algorithm (see table

1). Generally the old algorithm creates a lower aver-

age point P to surface S distances, since it evenly dis-

tributes its subdivisions over S , whereas the presented

algorithm focuses its subdivisions on areas of high ap-

IJCCI2013-InternationalJointConferenceonComputationalIntelligence

356

Figure 11: Some thin areas of the Happy Buddha model,

reconstructed with 200K triangles with the old (top) and the

new (bottom) algorithm. The new algorithm is able to build

a correct topology in thin areas in an earlier algorithm stage.

Figure 12: Very noise section of the Heating Pipes model

(left); Pointy vertices or spikes on the surface of the stan-

dard algorithm (middle) and a smoother surface with our

approach (right).

proximation error rates. This is visible through a 25%

decrease of the mean squared error. Especially for

curved models such as the Asian Dragon and the Thai

Statue this effect is very salient.

Although the search process is more complex, the

extra time costs are nearly leveled by the discarded

operations which are for the Dragon model 43.3% of

discarded adaptations and a rate of 26.3% inside the

surface movements of vertices only with our setting

for lim

skip

and lim

single

When we used the square distance as the approxima-

tion error, the results for both the average distance er-

ror as well as the square distance error where worse,

than the results of the standard algorithm. This is rea-

sonable since most triangles are used up to model tiny

but steep curvature (see Fig. 13). In addition, tri-

angles tended to clump even for low error half-lifes

λ. For the Happy Buddha model this led too many

clump-like artefacts. We consider this setting gener-

ally impractical.

Test Hardware A Dell



Precision M6400 with

Intel



QX9300 (2.53GHz) processor with 8GB 1066

MHz DDR3 Dual Channel RAM.

Parameter Settings num

= 3; max

= 1;

η = 10; max

|∇n|

= cos 10

◦

; λ = 9;

lim

skip

= 0.9; lim

single

= 1.2;

(See (Annuth and Bohn, 2012) for the parameter set-

tings of the basic algorithm).

Figure 13: Two magniﬁcations of the dragon model recon-

structed with100K triangles with the square of the point to

surface distance as the approximation error d

(top) and

with just the distance error d

(bottom). In the latter case,

the triangle resolutions at the back and between tail and

body are very high and are even for curved areas overrepre-

sented.

7 CONCLUSIONS AND FUTURE

WORK

In this paper we focused on the behavior of the grow-

ing cell structures approach as a function approxima-

tion algorithm. We analyzed the GCS with the clas-

sical adaption algorithm for matching requirements

of surface reconstruction. Derived from these obser-

vations we presented our new GCS learning model

and proved theoretically and by examples that it out-

performs the classical GCS approach. The basic

idea of the presented approach is to incorporate the

constructed topology into the GCS learning scheme.

GCS creates ideal distribution matching, clustering,

or dimensionality reduction by an implicit represen-

tation of a topology. In our work, we introduced an

explicit topology to model the approximation behav-

ior according to it, while saving the valuable ANN ca-

pabilities mentioned above. As result, we got a new

ANN learning strategy, which showed several advan-

tages compared to classical models. We see this paper

as proof of our conceptual change of the GCS algo-

rithm, giving rise to many improvements for the algo-

rithm in future work.

ACKNOWLEDGEMENTS

The authors would like to thank the Stanford Univer-

sity Computer Graphics Laboratory and the Hamburg

Hafen City University for making their laser scanned

data available to us. We also would like to thank Kai

Burjack for his assistance in rendering.

GrowingSurfaceStructures

357

Table 1: Our results for different models. We expose the time for the reconstruction process (time), the average distance to P

(dist) times 10

and the square distance to P (dist

) times 10

. We also tested different values for num

. Note that models are

normalized by setting the cubic diagonal of their bounding box to one.

GCS

GCS error=d

Model (# triangles)

time[s]

dist

time[s]

dist

time dist

dist

Hand (20K)

3.19

4.38

4.29

4.11

6 4.70 5.00

Dragon (100K)

2.29

3.42 65 2.55

2.97

66 3.05 4.34

Asian Dragon

(100K)

55 2.36 3.62 61

2.71

2.53 61 2.83 2.78

Thai Statue (200K)

146 3.21 15.4

147

3.00 4.02 148 4.05

7.17

Happy Buddha

(200K)

150 1.48

14.1

158

1.89

11.7

160 4.02 63.0

num

= 1

num

= 5

num

= 10

Model (# triangles)

time[s]

dist

time[s]

dist

time dist

dist

Dragon (100K)

62 2.35 3.01

2.55 3.05

121

2.57

2.95

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