1. 1 Multi-resolution Tetrahedral Meshes Leila De Floriani Department of Computer and Information Sciences University of Genova, Genova (Italy) http://www.disi.unige.it/person/DeflorianiL/

2. 2 Joint research activity with At DISI - University of Genova: Enrico Puppo, Paola Magillo and Emanuele Danovaro At Italian National Research Council, Pisa: Paolo Cignoni and Roberto Scopigno At CS Department - University of Maryland: Michael Lee and Hanan Samet

3. 3 Tetrahedral Meshes for Volume Data Analysis

4. 4 Outline Introduction and related work Multiresolution models based on tetrahedral meshes: regular Multi-Tessellation (Hierarchy of Tetrahedra) edge-based Multi-Tessellation Level-Of-Detail (LOD) queries Experiments and comparisons Current and future work

5. 5 Introduction The problem: modeling large sets of three-dimensional data describing scalar fields for analysis and rendering Applications: scientific data visualization,simulation, finite elements analysis, etc. Volumetric data set: set of points in three-dimensional Euclidean space with a scalar value associated with each point A volumetric data set is described by a mesh with vertices at the data points, usually a tetrahedral mesh

6. 6 Introduction Modeling volumetric data sets of large size: need for approximating meshes Accuracy of an approximating mesh in describing a scalar field is related to the mesh resolution (density of its cells) Accuracy may vary in different parts of the field domain, or in the proximity of interesting field values Two ways of producing approximating meshes: on-the-fly construction of an approximating mesh through a simplification process extract from a multiresolution representation

7. 7 Multiresolution Model

8. 8 Comprehensive structure built off-line which preprocesses and organizes a collection of alternative mesh representations of a spatial object can be efficiently queried according to parameters specified by an application task to extract adaptively refined meshes on-line

9. 9 Related Work Multiresolution triangle meshes: Hierarchical models in terrain modeling, finite element analysis, subdivision surfaces, wavelet analysis, etc. Discrete and continuous LOD models based on irregular triangle meshes. Simplification algorithms for tetrahedral meshes Hierarchical three-dimensional regular meshes Discrete multiresolution models based on irregular tetrahedral meshes: pyramidal models and progressive simplicial meshes The Multi-Tessellation (MT)

10. 10 How do we generate an approximating mesh? In a regular mesh: top-down refinement of a coarse mesh by recursive tetrahedron bisection In an irregular mesh bottom-up decimation of the mesh at full resolution by edge collapse top-down refinement of a coarse mesh by vertex insertion (on-going work)

11. 11 Recursive tetrahedron bisection Generates a nested tetrahedral mesh by recursively bisecting a tetrahedron along its longest edge Cubic domain initially splits into six tetrahedra:

12. 12 Recursive tetrahedron bisection Three basic tetrahedral shapes: 1/2 pyramid 1/4 pyramid 1/8 pyramid

13. 13 Splitting rule may generate non-conforming meshes Splitting rule A non-conforming mesh

14. 14 Why conforming meshes? Use of conforming meshes as decompositions of the domain of a scalar field Conforming meshes are a way of ensuring (at least C 0 ) continuity in the resulting approximation

15. 15 Conforming modifications in a regular mesh: tetrahedral clusters Tetrahedra around a bisected edge must be split at the same time to generate conforming meshes; such set of tetrahedra forms a cluster Three types of modifications corresponding to a cluster of 1/2 pyramids 1/8 pyramids a cluster of 1/4 pyramids

16. 16 Edge collapse in an irregular mesh Replace an edge e=(v’,v”) with a new vertex v (e.g., the middle point of e)

17. 17 Vertex insertion in an irregular mesh Insertion of a vertex P in a Delaunay tetrahedral mesh: Remove the sub-mesh subdividing the region of influence of P Re-triangulate the region of influence by joining P to the vertices of such region

18. 18 Multiresolution Meshes Multiresolution mesh: a coarse mesh plus a collection of modifications organized according to a partial order All subsets of modifications closed with respect to the partial order describe all possible tetrahedral meshes which can be extracted from a multiresolution mesh A conforming coarse mesh plus a set of conforming modifications produce conforming extracted meshes

19. 19 Multiresolution Meshes: dependency relation In a refinement or a decimation process: initial mesh undergoes a sequence of modifications Refinement sequence: 0, 1, 2, 3

20. 20 Multiresolution Meshes: dependency relation Dependency relation between pairs of modifications: Given two modifications m1 and m2, we say that m2 directly depends on m1 if m2 removes some tetrahedra inserted by m1 1 and 2 are independent. 3 depends on both 1 and 2.

21. 21 Closed sets and extracted meshes The closed sets of the partial order are in one-to-one correspondence with the conforming meshes which can be extracted from a multiresolution mesh

22. 22 Regular Multi-Tessellations (Hierarchies of Tetrahedra) Modification: splitting tetrahedral clusters Each modification replaces 4, 6 or 8 tetrahedra with 8,12 or 16 tetrahedra, respectively 1/2 pyramids 1/8 pyramids 1/4 pyramids

23. 23 Edge-based Multi-Tessellations Modification: vertex split into an edge (inverse of edge collapse) On average each modification replaces 27 tetrahedra with 32 tetrahedra

24. 24 Level-Of-Detail (LOD) Queries A set of basic queries for analysis and visualization of a volumetric data set at different levels of detail Instances of selective refinement: extract from a multiresolution model a mesh with the smallest possible number of tetrahedra satisfying some user-defined criterion based on LOD (for instance, uniform LOD, variable LOD based on a region of interest or a set of field values)