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Multi-resolution Tetrahedral Meshes

Leila De Floriani

Department of Computer and Information SciencesUniversity of Genova, Genova (Italy)

http://www.disi.unige.it/person/DeflorianiL/

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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

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Tetrahedral Meshes for Volume Data Analysis

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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

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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

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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

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Multiresolution Model

MultiresolutionModel

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Multiresolution Model

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

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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)

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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)

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Recursive tetrahedron bisection

• Generates a nested tetrahedral mesh by recursively bisecting a tetrahedron along its longest edge

• Cubic domain initially splits into six tetrahedra:

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Recursive tetrahedron bisection

Three basic tetrahedral shapes:

1/2 pyramid

1/4 pyramid

1/8 pyramid

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Splitting rule may generate non-conforming meshes

Splitting rule

A non-conforming mesh

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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 C0) continuity in the resulting approximation

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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 a cluster of 1/2 pyramids 1/8 pyramids

a cluster of1/4 pyramids

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Edge collapse in an irregular mesh

• Replace an edge e=(v’,v”) with a new vertex v (e.g., the middle point of e)

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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

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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

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Multiresolution Meshes: dependency relation

• In a refinement or a decimation process: initial mesh undergoes a sequence of modifications

Refinement sequence: 0, 1, 2, 3

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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.

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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

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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

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Edge-based Multi-Tessellations

• Modification: vertex split into an edge (inverse of edge collapse)

• On average each modification replaces 27 tetrahedra with 32 tetrahedra

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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)