1. Fifth International Conference on Curves and Surfaces Incremental Selective Refinement in Hierarchical Tetrahedral Meshes Leila De Floriani University of Genova, Genova (Italy) Michael Lee University of Maryland, College Park, MD (USA)

2. Fifth International Conference on Curves and Surfaces Outline Introduction Related work Hierarchical tetrahedral meshes Selective refinement queries for volume data analysis A data structure for hierarchical tetrahedral meshes Selective refinement algorithms Neighbor finding in a hierarchical tetrahedral mesh Summary and future work

3. Fifth International Conference on Curves and Surfaces Introduction Objective: modeling large sets of 3D data describing scalar fields for analysis and rendering Applications: scientific data visualization, simulation, finite-element analysis, etc. Volume data set: finite set of points in the three-dimensional Euclidean space with a scalar value associated with each point. A volume data set is described by a mesh with vertices at the data points, usually a tetrahedral mesh.

4. Fifth International Conference on Curves and Surfaces Introduction Modeling volume data sets of large size requires computing (simplified) 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 Use of multiresolution (also called Level-Of-Detail (LOD)) models: compact way of encoding the steps performed by a refinement, or a decimation process a virtually continuous set of adaptive meshes at different LODs can be extracted

5. Fifth International Conference on Curves and Surfaces LOD Model

6. Fifth International Conference on Curves and Surfaces Related work Nested three-dimensional meshes: octree-based methods (Wilhelms and Van Gelder, 1994; Shekhar et al., 1996; Westermann et al., 1999) recursive tetrahedron bisection (Rivara and Levin, 1992; Hebert, 1994; Maubach, 1995; Zhou et al., 1997, Ohlberger and Rumpf, 1997; Gerstner et al., 1999-2000; Lee, et al., 2001) red/green tetrahedra refinement (Grosso et al., 1997; Greiner and Grosso, 2000) Multiresolution models based on irregular tetrahedral meshes: Pyramidal models (Cignoni et al., 1994; De Floriani et al., 1995) Progressive simplicial meshes (Gross and Staadt, 1998; Popovic and Hoppe, 1997) The 3D Multi-Tessellation: a continuous irregular multiresolution model (De Floriani et al., 2001)

7. Fifth International Conference on Curves and Surfaces Tetrahedral meshes Tetrahedral mesh: connected collection of tetrahedra such that their union covers the field domain any two distinct tetrahedra have disjoint interiors Regular mesh: mesh generated by a recursive subdivision process based on points on a regular grid

8. Fifth International Conference on Curves and Surfaces Conforming tetrahedral meshes The intersection of any two elements consists of a common lower- dimensional cell (face, edge, or vertex), or it is empty. In 2D: In 3D: Conforming Non-conforming

9. Fifth International Conference on Curves and Surfaces Why conforming meshes? Conforming meshes used as decompositions of the domain of a scalar field They are a way of ensuring a (at least C 0 ) continuity in the resulting approximation

10. Fifth International Conference on Curves and Surfaces Hierarchical tetrahedral meshes Nested tetrahedral mesh generated by recursively bisecting a tetrahedron along its longest edge Cubic domain initially splits into six tetrahedra

11. Fifth International Conference on Curves and Surfaces Hierarchical tetrahedral meshes The subdivision process generates three classes of congruent tetrahedral shapes 1/2 pyramid 1/4 pyramid 1/8 pyramid

12. Fifth International Conference on Curves and Surfaces Tetrahedron bisection may generate non-conforming meshes Tetrahedron bisection A non-conforming mesh

13. Fifth International Conference on Curves and Surfaces Extracting conforming meshes Tetrahedra around a bisected edge must split simultaneously to generate conforming meshes. Such tetrahedra form a cluster. Three types of clusters, corresponding to the three basic shapes Cluster of 1/2 pyramids Cluster of 1/4 pyramids Cluster of 1/8 pyramids

14. Fifth International Conference on Curves and Surfaces Extracting conforming meshes Each cluster should be replaced during mesh extraction with the corresponding split set Three types of split sets, corresponding to the three types of clusters 1/2 pyramids 1/8 pyramids 1/4 pyramids

15. Fifth International Conference on Curves and Surfaces Selective refinement queries A set of basic queries for analysis and visualization of a volume data set at different levels of detail General formulation for selective refinement: extract from a hierarchical mesh H a conforming mesh with the smallest possible number of tetrahedra covering the domain of H and satisfying some user-defined criterion based on Level-Of-Detail LOD criterion  : Boolean predicate defined on the tetrahedra of a hierarchical mesh:  (  ) =true if tetrahedron  is necessary in order to achieve the required resolution

16. Fifth International Conference on Curves and Surfaces LOD criteria LOD based on an approximation error associated with each tetrahedron of the hierarchical mesh LOD can be uniform on the whole domain, or variable at each point of the domain Uniform LOD: –  (  )= true if the error associated with tetrahedron  is less or equal to a constant threshold  Variable LOD: –  (  )= true if the error associated with tetrahedron  is less or equal to a threshold function  defined at each point of the domain (e.g., a view-dependent threshold function)

17. Fifth International Conference on Curves and Surfaces Uniform LOD meshes Error threshold: 0.5% of the absolute range of the field value Size of the extracted mesh: 44.7% of size of the full resolution mesh

18. Fifth International Conference on Curves and Surfaces Variable LOD based on a region of interest A mesh with a small approximation error inside a given region of interest (0.1% of the field value within the box) Size of the extracted mesh: 6 % of the size of the mesh at uniform LOD with error equal to 0.1%

19. Fifth International Conference on Curves and Surfaces Variable LOD based on the field value Error threshold on the tetrahedra intersected by isosurface of value 1.27: 0.1% of the range of the field values Size of the extracted mesh: 26% of the size of the mesh at uniform LOD with error equal to 0.1%

20. Fifth International Conference on Curves and Surfaces Data structure for a hierarchical mesh Table of field values Forest of six almost full bin-trees (without the full resolution mesh) describing the nested subdivision of the cubic domain 1/8, 1/2 and 1/4 pyramids alternate in the tree each tree node stores the approximation error associated with the corresponding tetrahedron each tree encoded in an array Storage cost: 14n bytes Field table: 2n bytes, assuming 2 bytes for the field value Forest: 12n bytes, assuming 2 bytes for the error n = # of vertices in the mesh at full resolution 6n = # of tetrahedra in the mesh at full resolution

21. Fifth International Conference on Curves and Surfaces Location codes in a hierarchical mesh Location codes used to uniquely identify the tetrahedra in the forest Location code for a tetrahedron  : level of  in the tree (also length of location code) path from the root of the tree to 

22. Fifth International Conference on Curves and Surfaces Location codes in a hierarchical mesh The parent and children of a given tetrahedron can be retrieved in constant time directly from the array indices Location codes are not explicitly stored, but they are computed at run-time Location codes are used when extracting a mesh to index the field table index forest containing error values retrieve neighbors

23. Fifth International Conference on Curves and Surfaces Selective refinement: incremental algorithm A conforming mesh satisfying the given LOD criterion is extracted by starting from a previously extracted mesh (the initial subdivision of the domain at first application and either splitting each tetrahedron, which does not satisfy the LOD criterion, along its longest edge e together with those tetrahedra sharing edge e

24. Fifth International Conference on Curves and Surfaces Selective refinement: incremental algorithm or, merging all tetrahedra in a split set such that the tetrahedra in the corresponding cluster satisfy the LOD criterion Basic ingredient: finding tetrahedral clusters split setcorresponding cluster

25. Fifth International Conference on Curves and Surfaces Algorithm behavior for a moving ROI

26. Fifth International Conference on Curves and Surfaces Finding tetrahedral clusters Given a tetrahedron  and an edge e of , find all the tetrahedra sharing edge e with , i.e., belonging the the cluster of  with respect to edge e. It requires traversing the cluster of  by navigating through face-adjacent neighbors Problem: finding the face-adjacent neighbors of a given tetrahedron We apply the traditional approach to neighbor finding in a quadtree combined with the use of locational codes

27. Fifth International Conference on Curves and Surfaces Finding face-neighbors in a hierarchical mesh Traditional approach to neighbor finding in a region quadtree: Find nearest common ancestor of input tetrahedron and its neighbor Find child of nearest common ancestor that contains the neighbor Reflect the path to the input node to obtain the path to the neighbor For pointer-less representation, we only need to change the location code of the input tetrahedron to obtain that of its face- adjacent neighbor

28. Fifth International Conference on Curves and Surfaces Finding face-neighbors in a hierarchical mesh Find the nearest common ancestor 1010101011 101010101 10101010 1010101 101010 10101 1010 101

29. Fifth International Conference on Curves and Surfaces Finding face-neighbors in a hierarchical mesh Reflect the path to find the neighbor 101 1011 10111 101110 1011101 10111010 101110101 1011101011

30. Fifth International Conference on Curves and Surfaces Finding neighbors in constant time Neighbor finding is performed in constant time The algorithm works on the location code of the tetrahedra No coordinate values are needed, or computed The algorithm yields the same results as traditional neighbor finding algorithm No need to perform tree traversals Use of bit manipulations and binary arithmetic (Lee,De Floriani and Samet et al., Shape Modeling 2001)

31. Fifth International Conference on Curves and Surfaces Summary and future work Efficient selective refinement based on extraction of clusters of tetrahedra through constant-time neighbor finding Fast navigation in a hierarchical tetrahedral mesh and efficient isosurface extraction On-going work: comparison with multiresolution structures based on irregular tetrahedral meshes Future work: external memory data structure for a hierarchical tetrahedral mesh out-of-core algorithms for selective refinement and iso- surface extraction

32. Fifth International Conference on Curves and Surfaces Selective refinement: depth-first algorithm A conforming mesh satisfying the given LOD criterion is extracted by starting from the initial subdivision of the cubic domain and recursively splitting each tetrahedron which does not satisfy the LOD criterion

33. Fifth International Conference on Curves and Surfaces Selective refinement: depth-first algorithm when a tetrahedron  is split along an edge e, all tetrahedra having edge e in common with , i.e. in the same cluster, must split simultaneously Basic ingredient: finding tetrahedral clusters

34. Fifth International Conference on Curves and Surfaces Depth-first extraction of a conforming mesh