SMI 2002 Multiresolution Tetrahedral Meshes: an Analysis and a Comparison Emanuele Danovaro, Leila De Floriani University of Genova, Genova (Italy) Michael Lee, Hanan Samet University of Maryland, College Park, MD (USA)

SMI 2002 Outline Introduction Related Work Updates in a Multiresolution Mesh Multiresolution Tetrahedral Meshes Hierarchy of Tetrahedra Edge-based Multi-Tessellation Level-Of-Detail (LOD) Queries for Volume Data Analysis Experimental Results and Comparisons Summary and Future Work

SMI 2002 Introduction A volume data set a set of points in the 3D Euclidean space with a scalar field value associated with each of them often modeled as a tetrahedral mesh, which can be regular or irregular depending on the vertex distribution Analysis and rendering of volumetric data sets of large size through multiresolution models: compact way of encoding the steps performed by a simplification process a virtually continuous set of adaptive meshes at different Levels Of Details (LODs) can be extracted

SMI 2002 Contribution Analysis and comparison of multiresolution models based on tetrahedral meshes: Hierarchies of Tetrahedra: regular nested meshes generated through recursive tetrahedron bisection Edge-based Multi-Tessellations: multiresolution irregular meshes built through edge collapse Definition of the two models as instances of a general multiresolution model for simplicial meshes Experimental comparison of the two models on a basic set of queries for analyzing and rendering a volume data set at a variable resolution.

SMI 2002 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, Zhou et al., 1997, Ohlberger and Rumpf, 1997; Gerstner et al., ; Lee, et al., 2001) red/green tetrahedra refinement (Grosso et al., 1997; Greiner and Grosso, 2000) Simplification algorithms for tetrahedral meshes: Vertex insertion: Renze and Oliver, 1996; DeFloriani et al., 1994; Hamann and Chen, Edge collapse: Gross and Staadt, 1998; Trotts et al., 1999; Cignoni et al., 2000.

SMI 2002 Related Work Discrete multiresolution models based on irregular tetrahedral meshes: Pyramidal models (Cignoni et al., 1994; De Floriani et al., 1995) Progressive simplicial meshes (Gross and Stadt, 1998; Popovic and Hoppe, 1997) The Multi-Tessellation (MT) (De Floriani et al., ): A continuous dimension-independent multiresolution framework based on regular simplicial complexes with a manifold domain:

SMI 2002 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

SMI 2002 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

SMI 2002 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

SMI 2002 Multiresolution tetrahedral meshes Basic idea: collect the updates performed on a mesh during simplification (refinement or decimation) and organize them by defining suitable dependency relations. Dependency relations drive the extraction of meshes at intermediate resolutions Updates must satisfy consistency rules to allow extracting conforming meshes

SMI 2002 Updates in a multiresolution mesh An update of a mesh  : pair of meshes u=(  1,  2 )  1 is a sub-mesh of   2 replaces  1 in  by filling the hole left by  1.

SMI 2002 Updates in a multiresolution mesh An update u=(  1,  2 ) is conforming when  1 and  2 are conforming meshes the combinatorial boundary of  1 consists of the same set of cells as that of  2.

SMI 2002 Tetrahedron bisection It consists of bisecting a tetrahedron along its longest edge It generates three classes of congruent tetrahedral shapes 1/2 pyramid 1/4 pyramid 1/8 pyramid

SMI 2002 Conforming updates defined by tetrahedron bisection Tetrahedra around a bisected edge must split simultaneously to generate conforming meshes. Such tetrahedra form a cluster. Three types of clusters (and, thus, of updates) a cluster of 1/2 pyramids 1/8 pyramids a cluster of 1/4 pyramids

SMI 2002 Edge collapse/vertex split Contract an edge e=(v’,v”) into a new vertex v (full-edge collapse), or into an existing one (half-edge collapse) Inverse of collapse: vertex split Edge collapse and vertex split are conforming updates

SMI 2002 Multiresolution Tetrahedral Meshes A set of conforming updates A partial order defined by the following dependency relation: update B directly depends on update A if B replaces some tetrahedra introduced by A a sequence of updates corresponding multiresolution mesh

SMI 2002 Closed sets and extracted meshes There exists a one-to-one correspondence between the closed sets of the partial order and the meshes which can be extracted from a multiresolution mesh All extracted meshes are conforming

SMI 2002 Hierarchy of Tetrahedra (Regular Multi-Tessellation) Update: splitting clusters of tetrahedra at the mid-point of their common edge Each update replaces 4, 6 or 8 tetrahedra with 8,12 or 16 tetrahedra, respectively 1/2 pyramids 1/8 pyramids 1/4 pyramids

SMI 2002 Edge-based Multi-Tessellation (MT) Update: vertex split (inverse of a full-edge collapse) On average, each update replaces 27 with 33 tetrahedra

SMI 2002 Encoding a Hierarchy of Tetrahedra Data structure describing the nested subdivision of the cubic domain. It consists of: a table of field values six almost full binary trees (without the mesh at full resolution): each tree node stores the error associated with the corresponding tetrahedron Storage cost: 14n bytes (assuming 2 bytes for the error and for the field value), where n is the number of vertices in the mesh at full resolution

SMI 2002 Encoding a Hierarchy of Tetrahedra Use of location codes to uniquely identify the tetrahedra in the forest Location code for a tetrahedron  : level of  in the tree path from the root of the tree to  Location codes used to index the field table Dependency relation implicitly encoded by the forest Clusters defining the updates computed when extracting a mesh by using a worst-case constant time neighbor finding algorithm (Lee et al., SMI 2001)

SMI 2002 Encoding an Edge-based MT Dependencies implicitly encoded through an extension of a technique proposed by El Sana and Varshney (1999) for triangle meshes Compact data structure for encoding full-edge collapses in (De Floriani et al., IEEE TVCG (to appear)) Storage cost: (a) 30n bytes when error is associated with updates (b) 82n bytes when error is associated with tetrahedra Storage cost: (a) between 22% and 44% of the cost of storing the mesh at full resolution (b) between 60% and 120% of the cost of storing the mesh at full resolution

SMI 2002 Level-Of-Detail (LOD) Queries A set of basic queries for analysis and visualization of a volume 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 LOD based on approximation error LOD can be uniform on the whole domain, or variable at each point of the domain.

SMI 2002 Experiments on two regular volume data sets Smallbucky: portion of the Bucky-ball data set (courtesy of AVS): 32,768 vertices and 196,608 tetrahedra Plasma: synthetic data set (courtesy of Visual Comp. Group, CNR, Italy): 262,144 vertices and 1,572,864 tetrahedra

SMI 2002 Uniform LOD Error threshold: 0.5% of the absolute range of the field values FR mesh : mesh at full resolution Full resolution mesh From an edge-based MT: 24.2% of size of FR mesh From a HT: 44.7% of size of FR mesh

SMI 2002 Results: uniform LOD Edge-based MT performs better than a HT for queries at a uniform resolution. Size of the mesh extracted from a HT, from an MT with error on updates, and from an MT with error on tetrahedra, over different error thresholds. SmallbuckyPlasma

SMI 2002 Variable LOD in a Region Of Interest Error threshold inside the ROI: 0.1% of the range of the field values Size of the mesh (extracted from an HT): 6.2% of the size of the mesh at uniform LOD with error equal to 0.1%

SMI 2002 Results: variable LOD in a ROI HT shows higher selectivity than MT ROI: axis-aligned box. Error threshold: specified inside the box, any error allowed outside the box Smallbucky Plasma

SMI 2002 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 (from a HT): 26.3% of the size of the mesh at uniform LOD with error equal to 0.1%

SMI 2002 Results: variable LOD based on the field Both HT and MT with error on tetrahedra perform better than an MT with error on updates Size of the mesh extracted from a HT, from an MT with error on updates, and from an MT with error on tetrahedra, over different error thresholds (averaged on several field values) Plasma Smallbucky

SMI 2002 Results: variable LOD based on the field HT at error threshold 0.5%: 65,856 tetrahedra (10.61% wrt uniform LOD); 20,429 faces MT at error threshold 0.5%: 98,162 tetrahedra (38.24% wrt uniform LOD); 12,060 faces

SMI 2002 Results: variable LOD based on the field HT at error threshold 0.5%: 65,856 tetrahedra (10.61% wrt uniform LOD); 20,429 faces MT at error threshold 0.23%: 172,359 tetrahedra (27.58% wrt uniform LOD); 20,250 faces

SMI 2002 Summary Expressive power: Edge-based MT suitable for both irregular and regular data sets HT specific for regular data sets Approximation quality: HT produces Delaunay tetrahedral meshes Circumradius-to-shortest-edge ratio: HT: ~ 0.9 Edge-based MT: ~1.3

SMI 2002 Summary Storage cost: HT more economical than edge-based MT since the topology is implicit Data structures for edge-based MT more economical than data structures for encoding the mesh at full resolution Selectivity: Edge-based MT has less tetrahedra for queries at a uniform LOD; the opposite for spatial selection queries Extracted meshes: Meshes with connectivity and adjacencies are extracted

SMI 2002 On-going and future work A compact data structure for a edge-based MT built through half-edge collapse (Danovaro and De Floriani, 3DVPT, 2002): higher selectivity (since updates are smaller) lower storage cost when errors are associated with tetrahedra Client/server applications: progressive transmission and selective refinement of tetrahedral meshes in a client/server environment Out-of-core algorithmic issues: data structures for HTs and MTs construction algorithms algorithms for selective refinement

SMI 2002 Meshes at a uniform LOD from Plasma Error threshold: 1.0% Full resolution From a MT: 7.7% of size of FR mesh From an HT: 21.5% of size of FR mesh

SMI 2002 Results: variable LOD based on the field Error threshold: specified along the isosurfaces, any error otherwise Experiments showing the size of the mesh Smallbucky Plasma

SMI 2002 Results: variable LOD based on the field Error threshold: specified along the isosurfaces, any error otherwise Experiments showing the size of the extracted isosurfaces Smallbucky Plasma