Level-Of-Detail (LOD) Data Structures for Level-Of-Detail (LOD) Data Structures for Tetrahedral Meshes Tetrahedral Meshes Leila De Floriani Leila De Floriani University of Maryland, College Park (USA) and University of Genova, Genova (Italy) Overview Overview l Introduction l Background l Ingredients for a Level-Of-Detail (LOD) model l Selective refinement queries on volume data sets l Data structures for irregular meshes l Data structures for regular meshes l Selective refinement algorithms l Conclusions: results and research issues VIS 2003 Tutorial, October 20, 2003 1 1
Why Level-Of-Detail (LOD) Models? Why Level-Of-Detail (LOD) Models? Aim: modeling and visualizing large volume data sets large volume data sets Aim: l l LOD Model: LOD Model: l l – reduces the size size of the output data set – allows varying the accuracy varying the accuracy in different parts of the field domain – can be generated off-line generated off-line through an accurate simplification algorithm We focus on LOD models LOD models based on tetrahedral meshes tetrahedral meshes l VIS 2003 Tutorial, October 20, 2003 LOD Models LOD Models They are built off-line off-line through a simplification process (top-down refinement l or bottom-up decimation) They encode all atomic updates performed during simplification l They are queried on-line on-line to extract variable-resolution meshes l off-line on-line VIS 2003 Tutorial, October 20, 2003 2 2
Overview Overview l Introduction l Background Background l l Ingredients for a Level-Of-Detail (LOD) model l Selective refinement queries on volume data sets l Data structures for irregular meshes l Data structures for regular meshes l Selective refinement algorithms l Conclusions: results and research issues VIS 2003 Tutorial, October 20, 2003 Tetrahedral Meshes Tetrahedral Meshes Tetrahedral mesh: Tetrahedral mesh: connected set of tetrahedra such that – their union covers the field domain – any two distinct tetrahedra have disjoint interiors Regular (structured) mesh: Regular (structured) mesh : generated by a recursive subdivision process based on points on a regular grid VIS 2003 Tutorial, October 20, 2003 3 3
Conforming Tetrahedral Meshes Conforming Tetrahedral Meshes Conforming meshes: : any two intersecting cells meet at Conforming meshes a common lower-dimensional cell (face, edge, or vertex) In 3D In 3D In 2D In 2D Conforming Conforming Conforming Conforming mesh mesh mesh mesh t Non-conforming Non-conforming Non-conforming Non-conforming mesh mesh mesh mesh VIS 2003 Tutorial, October 20, 2003 Why Conforming Meshes? Why Conforming Meshes? Tetrahedral meshes are used as decompositions of the domain of a l scalar field Conforming meshes are a way of ensuring (at least C 0 ) continuity in l the resulting approximation Non-continuous Non-continuous C 0 surface C surface surface surface Conforming Conforming Non-conforming Non-conforming mesh mesh mesh mesh VIS 2003 Tutorial, October 20, 2003 4 4
Data structures for tetrahedral meshes Data structures for tetrahedral meshes Indexed data structure: Indexed data structure: – vertex coordinates – for each tetrahedron: links to its four vertices Indexed data structure with adjacencies: Indexed data structure with adjacencies: – vertex coordinates – for each tetrahedron: • links to its four vertices • links to its four face-adjacent tetrahedra VIS 2003 Tutorial, October 20, 2003 Overview Overview l Introduction l Background l Ingredients for a Level-Of-Detail (LOD) model Ingredients for a Level-Of-Detail (LOD) model l l Selective refinement queries on volume data sets l Data structures for irregular meshes l Data structures for regular meshes l Selective refinement algorithms l Conclusions: results and research issues VIS 2003 Tutorial, October 20, 2003 5 5
Ingredients for a LOD model Ingredients for a LOD model The coarsest coarsest mesh mesh approximation l The collection of all updates collection of all updates performed l on a mesh during simplification (refinement or decimation) A dependency relation dependency relation among updates l The dependency relation The dependency relation drives the l l extraction of meshes at intermediate resolutions VIS 2003 Tutorial, October 20, 2003 Ingredients for a LOD model: updates Ingredients for a LOD model: updates In simplification simplification - refinement refinement or decimation decimation - an initial mesh undergoes a sequence of updates. Refinement: Refinement: from coarse from coarse to fine to fine Update = replacement of a set of tetrahedra in a mesh with another Update set which covers the same portion of the domain VIS 2003 Tutorial, October 20, 2003 6 6
Ingredients for a LOD model: updates Ingredients for a LOD model: updates Refinement Refinement Decimation: Decimation: from fine to from fine to coarse coarse VIS 2003 Tutorial, October 20, 2003 Ingredients for a LOD model: dependency Ingredients for a LOD model: dependency relation relation Updates 1 and 2 are independent ndependent Dependency relation between pairs of Dependency relation l l refinement updates: an update B directly depends directly depends on an update A if and only if if and only if some tetrahedron introduced by A is removed by B Update 3 depends depends on both both updates 1 and 2 The transitive closure transitive closure of the dependency l relation defines a partial order partial order VIS 2003 Tutorial, October 20, 2003 7 7
LOD Models: properties LOD Models: properties From a LOD model , we can extract any mesh obtained from the coarsest mesh by applying any sequence of refinement updates compatible with the dependency relation Mesh A: Mesh A: initial mesh initial mesh Mesh B: A + update 1 Mesh B: A + update 1 Mesh C: Mesh C: A + update 2 A + update 2 Mesh D: A + updates 1 and 2 Mesh D: A + updates 1 and 2 Mesh E: A + updates 1, 2, 3 (or 2,1,3) Mesh E: A + updates 1, 2, 3 (or 2,1,3) (or 2 and 1) (or 2 and 1) VIS 2003 Tutorial, October 20, 2003 Overview Overview l Introduction l Background l Ingredients for a Level-Of-Detail (LOD) model l Selective refinement queries on volume data sets Selective refinement queries on volume data sets l l Data structures for irregular meshes l Data structures for regular meshes l Selective refinement algorithms l Conclusions: results and research issues VIS 2003 Tutorial, October 20, 2003 8 8
Selective refinement queries [ Selective refinement queries [Cignoni Cignoni at el., 2003] at el., 2003] A set of basic queries for analysis analysis and l visualization of a volume data set at different visualization levels of detail All instances of selective refinement selective refinement: l extract from a LOD model a mesh extract mesh with the smallest possible number of tetrahedra smallest satisfying some user-defined criterion based criterion based on LOD on LOD LOD depends on approximation error approximation error l VIS 2003 Tutorial, October 20, 2003 Uniform LOD Uniform LOD Input Input parameters parameters: : an accuracy threshold E Error Error(t) (t) £ £ E E for every tetrahedron t t in the extracted mesh Buckyball E = 5% of the field range Buckyball E = 5% of the field range 12.5 million 12.5 million tetrahedra tetrahedra 274,460 274,460 tetrahedra tetrahedra VIS 2003 Tutorial, October 20, 2003 9 9
Variable Variable LOD LOD based on spatial location based on spatial location Input Input parameters parameters: : a Region Region of interest of interest ( ( ROI) R ROI) R and two two accuracy thresholds E E 1 and and E E 2 (E 1 <E 2 ): accuracy thresholds – Error(t) £ E 1 for each tetrahedron t interesecting R – Error(t) £ E 2 for any other tetrahedron ROI = box ROI = box ROI =cross plane ROI =cross plane E=0.01% in the ROI E=0.01% in the ROI E= 2% in the ROI E= 2% in the ROI size = 1/3 of reference mesh size = 1/3 of reference mesh size = 7% of reference mesh size = 7% of reference mesh BluntFin BluntFin: : 222,528 222,528 tetrahedra tetrahedra Turbine Blade 576,566 576,566 tetrahedra tetrahedra Turbine Blade VIS 2003 Tutorial, October 20, 2003 Variable LOD based on field values Variable LOD based on field values Input Input parameters parameters: : a collection collection of of field values field values FV FV and two accuracy thresholds E 1 and E 2 (E 1 <E 2 ): two accuracy thresholds – Error(t) £ E 1 for every tetrahedron t interesecting the isosurfaces of values in FV – Error(t) £ E 2 for any other tetrahedron E = 0.1% along the E = 0.1% along the blue isosurface blue isosurface 25% size of the mesh 25% size of the mesh at uniform LOD with at uniform LOD with error = 0.1% error = 0.1% Plasma: 1,500,282 Plasma: 1,500,282 tetrahedra tetrahedra VIS 2003 Tutorial, October 20, 2003 10 10
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