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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)