Geometric Modeling with -Complexes

SPEAKER:

Bart H.M. Gerritsen (TNO)

Klaas van der Werff (DUT)

Remco Veltkamp (UU)

Alpha complex

Overview

• Modeling natural objects

• Modeling with alpha complexes

• Weighting

• Case studies

• Conclusions & further research

Natural object features

Natural object

• Complex geometry and topology

• Scale dependent geometry, topology and material

• Fuzzy boundaries

• Embedded in “background”

• Holes, separations

• Heterogeneous, an-isotropic material

• Ruled by natural evolutionary processes

Engineering object

• Moderately complex geometry and low complexity topology

• Virtually scale independent geometry and topology

• Well-established boundaries

• assemblies of monolithic parts

• no separations

• homogenous, isotropic or ortho-tropic material

• Demands-driven features and functions

Modeling with alpha complexes

Preprocess data set

Design weight set

Visualize and inspect

Triangulate

Compute -family

export

Modeling: data organization

wei

ghtin

gsa

mpl

ing

-c

ompl

ex

Modeling: distances

triangulation nearest-neighbor

furthest neighbor

Un

eq

ual

weig

hts

Eq

ual

weig

hts

Modeling: weights

• Euclidean distance: d(x1,x2 ) = | x1 - x2 |

• Laguerre distance: L(x1,x2 ) = d2 (x1,x2 ) - ( w1+w2 )

x2

x1x1

r1

r = w

Modeling: effects

Wd(x,y)

leaner richer

Weighting strategy

Co

met

Wes

t

Property space

Sample space

Model space

Case studies

Engineering

Natural

Scapula: view from the thorax

Scapula: data analysis

• Anatomic landmarks

• Point processes

• Local distances

• Position in body

• Curvatures

• Mathematical landmarks

• Pseudo-landmarks

• Geometric/topological

constraints

Scapula: nearest-neighbor analysis

nearest-neighbor graph

furthest-neighbor graph

Scapula: singular face analysis

Singular triangles

Scapula: finding the best alpha

• Fitting physical constraints,

e.g., volume, curvature

• Absence / presence of holes

• A-priori knowledge and

expertise

• Subjective matters

Conclusions

+ Working by example + observed ‘landmarks’

+ Intuitive

+ Can cope with roughness and vagueness of natural objects

+ hooks up well with knowledge-based and variational geometry

- Weighting can be (overly) complicated

- Limited control

- Large models require heavy computing

- Serious lack of data on natural objects

Modeling: level of abstraction

Knowledge-based modeling

Parametric modeling Variational geometry

Current geometric modeling

Lev

el o

f ab

stra

ctio

n

0% 100%Relative effort

Progress

abstract modeling geometric modeling

Alpha complex

Further research

• Knowledge-frameworks for natural objects

• Hyper-spatial modeling

• Further weighting strategies

• An-isotropic weighting

• Interactive tools for weighting

• Improved numerical modeling with alpha complexes

• Data collection on natural objects

Outline longer term approach

Scenario’s TemplatesIntra-object constraints

Environment

Object

Inter-object constraints

Object

Object

-complex