SMI 2002

Multiresolution Tetrahedral Meshes:an Analysis and a Comparison

Emanuele Danovaro, Leila De FlorianiUniversity of Genova, Genova (Italy)

Michael Lee, Hanan SametUniversity of Maryland, College Park, MD (USA)

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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

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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

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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.

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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., 1999-2000; 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, 1994.

• Edge collapse: Gross and Staadt, 1998; Trotts et al., 1999; Cignoni et al., 2000.

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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., 1997-1999):A continuous dimension-independent multiresolution framework based on regular simplicial complexes with a manifold domain:

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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

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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

Conforming Non-conforming

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Why conforming meshes?

• Conforming meshes used as decompositions of the domain of a scalar field

• They are a way of ensuring a (at least C0) continuity in the resulting approximation

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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

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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.

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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.

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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

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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 a cluster of 1/2 pyramids 1/8 pyramids

a cluster of1/4 pyramids

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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

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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

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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

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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

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Edge-based Multi-Tessellation (MT)

• Update: vertex split (inverse of a full-edge collapse)

• On average, each update replaces 27 with 33 tetrahedra

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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

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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)

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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

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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.

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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

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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

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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.

Smallbucky Plasma

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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%

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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

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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%

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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)

PlasmaSmallbucky

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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

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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

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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

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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

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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

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Meshes at a uniform LOD from PlasmaError 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

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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