1. Geometric Modeling with  -Complexes SPEAKER: Bart H.M. Gerritsen (TNO) Klaas van der Werff (DUT) Remco Veltkamp (UU)

2. Alpha complex 

3. Overview Modeling natural objects Modeling with alpha complexes Weighting Case studies Conclusions & further research

4. 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

5. Natural object example: the sun Vague boundary separation Heterogeneous an-isotropic material

6. What is exactly the problem? What is a natural object anyway? How to described a natural object mathematically? How to model natural objects, geometrically, topologically and numerically? With the current tools? From scratch? Should natural objects be alive in a virtual environment?

7. Current problems, tomorrows needs Tomorrow, we need: – more natural object models – object models live on, in a virtual world – object models need to adapt to and evolve in environment Current tools target primarily on engineering objects: – assume geometric and topological description – limited support for holes and separations – numerical models limit geometric freedom – no dynamic topology

8. Short term objectives & constraints Objectives – find geometric concept for natural objects using pre- shaped template objects (primitives) – numerical computations on geometric model – prepare for variational geometry – prepare for knowledge-based modeling Constraints – general dimension – dynamic modeling Evaluate  -complexes in this context

9. Holes Term: cavity pocket void handle Genus: 0 0 0 0 >1 Interior: 1 1 1 1 >1 Exterior: 1 1 1 >1 1

10. Modeling with alpha complexes Preprocess data set Design weight set Visualize and inspect Triangulate Compute  -family export

11. Modeling: data organization weighting sampling  -complex 

12. Hyper-spatial modeling: principle Small, medium and large spheres in 2D Same spheres cluster naturally in 3D hyper-space

13. Modeling: distances triangulationnearest- neighbor furthest neighbor Unequal weights Equal weights

14. Modeling: weights Euclidean distance: d(x 1,x 2 ) = | x 1 - x 2 | Laguerre distance: L(x 1,x 2 ) = d 2 (x 1,x 2 ) - ( w 1 +w 2 ) x2x2 x1x1 x1x1 r1r1 r =  w

15. - +  -value distance weight Modeling: effects

16.  Wd(x,y) leanerricher

17. Weighting strategy Comet West Property space Sample space Model space

18. Case studies Engineering Natural

19. Clinical objects: a hip

20. Scapula: rear view

21. Scapula: oblique view

22. Scapula: view from the thorax

23. Scapula: data analysis Anatomic landmarks Point processes Local distances Position in body Curvatures Mathematical landmarks Pseudo-landmarks Geometric/topological constraints

24. Scapula: nearest-neighbor analysis nearest-neighbor graphfurthest-neighbor graph

25. Scapula: singular face analysis Singular triangles

26. Scapula: finding the best alpha Fitting physical constraints, e.g., volume, curvature Absence / presence of holes A-priori knowledge and expertise Subjective matters

27. Evaluation: criteria Alpha complexes as a solid model description: – evaluate: solidity, continuity, finiteness, homogenous dimension, etc. Alpha complexes as a representation scheme: – compare to: faceted BRep, CSG, Cell Decomposition – evaluate: domain, validity, completeness, uniqueness

28. Evaluation: comparison Alpha complexes and faceted BRep: boundary description from (d-1)-simplexes (facets) of the alpha complex Alpha complexes and volumetric description: volumetric description, with cellular complex from (d)- simples of the alpha complex (may require regularization) Alpha complexes and part-whole description: part-whole description from (0..d)-simples of the alpha complex

29. Evaluation: findings As solid model description: – unconnected interiors (separations) – non-manifold topology – takes regularization to drop singular faces A representation scheme: – wide domain – allows unevaluated description – validation computationally expensive – homogenous properties per simplex or per point – not unique

30. 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

31. Modeling: level of abstraction Knowledge-based modeling Parametric modeling Variational geometry Current geometric modeling Level of abstraction 0%100%Relative effort Progress abstract modeling geometric modeling Alpha complex

32. 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

33. Outline longer term approach Scenario’s Templates Intra-object constraints Environment Object Inter-object constraints Object  -complex