1 / 31

A Multi-resolution Topological Representation for Non-manifold Meshes

A Multi-resolution Topological Representation for Non-manifold Meshes. Leila De Floriani, Paola Magillo, Enrico Puppo, Davide Sobrero University of Genova Genova (Italy). Non-Manifold Multiresolution Modeling: Why Non-Manifold?.

rico
Download Presentation

A Multi-resolution Topological Representation for Non-manifold Meshes

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. A Multi-resolution Topological Representation for Non-manifold Meshes Leila De Floriani, Paola Magillo, Enrico Puppo, Davide Sobrero University of Genova Genova (Italy) Solid Modeling 2002

  2. Non-Manifold Multiresolution Modeling: Why Non-Manifold? • Need to represent and manipulate objects which combine wire-frames, surfaces, and solid parts in CAD/CAM applications • Complex spatial objects are described by meshes witha non-manifoldandnon-regular domain • Non-manifold and non-regular meshes are generated by topology-modifying simplification algorithms Solid Modeling 2002

  3. Non-Manifold Multiresolution Modeling: Why Multiresolution? • Availability of CAD models of large size • Need for a multiresolution representation to be able to extract selectively refined meshes • Our aim: multiresolution modeling for extracting adaptivemesheswith a complete topological description (to support efficient mesh navigation through adjacencies) Solid Modeling 2002

  4. Our contribution • Non-Manifold Multi-Tessellation (NMT) • A compact data structure for a 2D instance of the NMT • A new scalable topological data structure for 2D simplicial complexes • Algorithms for selective refinement on the previous data structures Solid Modeling 2002

  5. Related work • Topology-modifying simplification algorithms:e.g., Rossignac and Borrel, 1990; Garland and Heckbert, 1997; Popovic and Hoppe, 1997; El-Sana and Varshney, 1998. • Non-manifold data structures: Weiler, 1988; Gursoz et al., 1990; Yamaguchi and Kimura, 1995; Rossignac and O’Connor, 1990; Campagna et al., 1999; Lee and Lee, 2001. • Multiresolution models for unstructured triangle meshes:Hoppe, 1997; De Floriani et al., 1997; Xia et al., 1997; Gueziec et al., 1998; Luebke and Erikson, 1997; Klein and Gumhold, 1998; El-Sana and Varshney, 1999. Solid Modeling 2002

  6. Background notions Two-dimensional simplicial complexes in 3D (triangle-segment meshes) Wire-edge: no triangle incident into it Triangle-edge: at least one triangle incident into it Solid Modeling 2002

  7. Non-Manifold Multi-Tessellation (NMT) • Extension of the Multi-Tessellation(De Floriani, Magillo, Puppo, 1997) to the non-manifold domain • Basic ingredients: mesh modification + dependency relation • Modification of a mesh: replace a set of elements E of  with a new set E’ so that the result ’ is still a mesh Solid Modeling 2002

  8. Non-Manifold Multi-Tessellation (NMT) A modification: S S’ Solid Modeling 2002

  9. Non-Manifold Multi-Tessellation (NMT) Modification as a pair M=(M-,M+) where • M+: refinement modification defined by EE’ • M-: coarsening modification defined by E’E Solid Modeling 2002

  10. Non-Manifold Multi-Tessellation (NMT) • NMT: partially ordered set of nodes {M0, ….Mq}: each node Mi: refinement modification Mi+ and its inverse coarsening one Mi- • Partial order induced by dependency relation: by Mjdepends onMi iff some element introduced by Mi+ is deleted Mj+ • Minimum element in the order: creation of the coarsest mesh Solid Modeling 2002

  11. Non-Manifold Multi-Tessellation (NMT) Solid Modeling 2002

  12. Non-Manifold Multi-Tessellation (NMT) Solid Modeling 2002

  13. Selective refinement on a NMT • Selective refinement: extract a mesh from a NMT satisfying some application-dependent requirements (LOD criterion + maximal size) • Extracted meshes correspond to set S of modifications closed with respect to the partial order Solid Modeling 2002

  14. Selective refinement: an incremental approach • Modify a previously extracted mesh • by adding necessary nodes • by removing superfluous nodes • Primitives on the current mesh: • mesh refinement (apply M+) • mesh coarsening (apply M-) Solid Modeling 2002

  15. A data structure for a 2D instance of a NMT • NMT built through vertex-pair contraction: (v’,v”)  v • Each NMT node M: vertex-pair contraction (M-) and vertex expansion (M+) • NMT nodes : encoding technique by Popovic and Hoppe (1997) • Partial order : implicit encoding technique by El-Sana and Varshney (1999) Solid Modeling 2002

  16. A data structure for a 2D instance of a NMT • Total cost:32n bytes (where n is the number of vertices in the mesh at full resolution) • Around 90% of the cost of storing the mesh at full resolution (by encoding only connectivity information) • Less than half than the cost of storing the mesh at full resolution with connectivity and face adjacencies Solid Modeling 2002

  17. A data structure for 2D simplicial complexes • For each vertex v: • all vertices to which v is connected through a wire-edge (Vertex-Vertex relation) • One triangle incident in v for each connected component (“fan”) of the link of v (partial Vertex-Triangle relation) Solid Modeling 2002

  18. A data structure for 2D simplicial complexes • For each triangle t: • link to its three vertices (Triangle-Vertex relation) • for each edge e of t: two triangles if e is non-manifold,one triangle otherwise(partial Triangle-Triangle relation) Solid Modeling 2002

  19. A data structure for 2D simplicial complexes • It can be traversed through edge adjacencies and around a vertex in optimal time (linear in the output size) • Scalability to manifold meshes: overhead of one byte per vertex wrt indexed data structure with adjacencies when applied to manifold meshes • More compact than specialization of existing non-manifold data structures to 2D simplicial complexes Solid Modeling 2002

  20. Implementation of mesh refinement • Input: • : current mesh encoded in the previous data structure • M+: feasible vertex expansion v  (v’,v”) as specified in the corresponding NMT node • Output:mesh ’ obtained from  by expanding v Solid Modeling 2002

  21. Basic operations in mesh refinement • Vertex into an edge (if v’ and v” were connected by an edge in the mesh at full resolution): v  e=(v’,v”) • On wire-edges: • move extreme vertex: (w,v)  (w,v’) • duplicate edge: (w,v) (w,v’) + (w,v”) • edge into triangle: (v,w)  (v’,v”,w) Solid Modeling 2002

  22. Basic operations in mesh refinement • On triangle-edges: • duplicate edge: (w,v) (w,v’) + (w,v”) • edge into triangle: (v,w)  (v’,v”,w) Solid Modeling 2002

  23. Basic operations in mesh refinement • On triangles: • move extreme vertex: (w1,w2,v)  (w1,w2,v”) • duplicate triangle: (w1,w2,v)  (w1,w2,v’) + (w1,w2,v”) Solid Modeling 2002

  24. Some experimental results • NMT built through a “simple” simplification strategy based on collapsing the shortest edge • Choice of the simplification strategy affects the shape of the NMT • Results: • Number of wire-edges incident at a vertex < 4 • Number of “fans” of triangles incident at a vertex < 4 • Average number of “fans” of triangles incident at a vertex < 1.2 • Only 2% of the vertices are extremes of wire-edges Solid Modeling 2002

  25. Selective refinement at work 21,648 triangles 0 wire edges 1,909 triangles 9 wire edges 3,509 triangles 7 wire edges 3,311 triangles 6 wire edges Solid Modeling 2002

  26. Summary • NMT: a model for multi-resolution simplicial complexes in arbitrary dimension • A compact data structure for a 2D instance of a NMT • A compact and scalable topological data structure for 2D simplicial complexes • Algorithms for performing vertex-pair contraction and vertex expansion (basic ingredients for selective refinement) on the previous data structure Solid Modeling 2002

  27. Current and future work • Our aim: extension of the work presented here to arbitrary dimensions • A mathematical framework for describing non-manifold simplicial complexes in arbitrary dimensions as assembly of simpler quasi-manifold components (De Floriani et al., DGCI, 2002). • An algorithm for decomposing a d-complex into a natural assembly of quasi-manifolds of dimension less or equal tod (on-going work). Solid Modeling 2002

  28. Solid Modeling 2002

  29. Background notions • Manifold edge: exactly one or two triangles incident into it manifold edges non-manifold edge Solid Modeling 2002

  30. Background notions • Manifold vertex: • No triangle incident in it and • one or two edges incident into it • OR • no wire-edges incident into it and • its incident triangles form a single fan manifold vertices non-manifold vertices Solid Modeling 2002

  31. Selective refinement: basic primitives • On the partial order: • node insertion test • node removal test • dependencies retrieval • On the current mesh: • mesh refinement (apply M+) • mesh coarsening (apply M-) Solid Modeling 2002

More Related