1 / 13

Grimm and Hughes

Grimm and Hughes. Input: arbitrary mesh Subdivide once (Catmull-Clark) and take dual Mesh with vertices of valence 4 Charts One for each vertex, edge, face Overlaps Adjacent elements Eg., vertex with 4 faces, 4 edges Transition functions

abram
Télécharger la présentation

Grimm and Hughes

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. Grimm and Hughes • Input: arbitrary mesh • Subdivide once (Catmull-Clark) and take dual • Mesh with vertices of valence 4 • Charts • One for each vertex, edge, face • Overlaps • Adjacent elements • Eg., vertex with 4 faces, 4 edges • Transition functions • Affine (rotate, translate) or projective where possible • Blend where not Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  2. Motivation • Maximize overlap • Three chart blend better than two • Co-cycle condition made > 3 hard • Affine transformations • (we got close) • Generalize spline construction process • Blend functions, not points Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  3. Charts • Vertex: Square • Always valence 4 • Edge: Diamond • Diamond shape determined by number of sides of adjacent faces • Face: N-sided unit polygon • Shrunk slightly Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  4. Overlaps • Vertex-face: corners • Vertex-edge: wedges • Edge-face: triangle • Edge-vertex: wedges • Face-vertex: corner quad • Face-edge: triangle Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  5. Transition functions • Edge-face: Affine • Translate, rotate, translate • Face-vertex: Projective • Square->quadrilateral • Edge-vertex: Composition Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  6. F1 Transition functions Vertex • Edge-vertex: F2 ? Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  7. Transition functions • C¥ continuous everywhere except blend area • Ck in blend area (determined by blend function) • At most three charts overlap anywhere • Reflexive: Use identity function • Symmetric: E-F, V-F both invertible • Co-cycle condition satisfied by blend function Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  8. f v e Abstract manifold Barycentric coords • “Glue” points that are related through transition functions • Each point appears in (at most) one each of a vertex, edge, and face chart f e v Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  9. Adding geometry • Blend and embed functions per chart • Fit to subdivision surface • 1-1 correspondence between manifold and dual mesh Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  10. Blend functions • Vertex: Spline surface basis function • Edge/face: Spline basis function “spun” in a circle • Inscribe • Partition of unity formed by dividing by sum of non-zero basis functions • Charts overlap sufficiently to guarantee sum non-zero Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  11. Embed functions • Spline patches • Fit to subdivision surface • One-to-one correspondence between dual mesh and charts • Implies correspondence between dual mesh and points in the chart Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  12. Plusses • Embed functions simple, well-behaved • Three-chart overlap • Transition functions (mostly) simple • Locality Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

  13. Minuses • Blending composition function is ugly • Difficult to analyze • Large number of charts Siggraph 2005, 8/1/2005 www.cs.wustl.edu/~cmg/Siggraph2005/siggraph.html

More Related