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Advanced Techniques in Consistent Parameterizations for Mesh Representation and Surface Mapping

This document discusses methods in consistent parameterization, focusing on mapping surfaces from various domains like planes, spheres, and simplicial complexes. It outlines critical properties such as one-to-one mapping, minimizing distortion, and preserving geometric attributes like length, angles, and area. Key techniques include planar and spherical parameterizations, constrained and inter-surface mappings. The applications in digital geometry processing, morphing, and attribute transfer are emphasized, along with algorithms for achieving robust and efficient mappings essential for modern geometric computation.

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Advanced Techniques in Consistent Parameterizations for Mesh Representation and Surface Mapping

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  1. Consistent Parameterizations Arul Asirvatham Committee Members Emil Praun Hugues Hoppe Peter Shirley

  2. Parameterization • Mapping from a domain (plane, sphere, simplicial complex) to surface • Motivation: Texture mapping, surface reconstruction, remeshing …

  3. Desirable Properties • One-to-one • Minimize some measure of distortion • Length preserving • Angle preserving • Area preserving • Stretch minimizing

  4. Outline • Background • Commonly used Domains • Plane, Simplicial Complex, Sphere • Constrained Parameterizations • Consistent Parameterizations • Consistent Spherical Parameterizations • Inter-Surface Mapping • Summary and future work

  5. Planar Parameterizations • Convex combination maps • p = i pi , i=1,…,n i =1 • [Tutte 63] • [Floater 97] • [Floater et al 03] • Conformal Maps • [Sheffer et al 01] • [Levy et al 02] • [Desbrun et al 02] • Stretch preserving maps • [Sander et al 01] g G

  6. Simplicial Parameterizations • Planar parameterization techniques cut surface into disk like charts • Use domain of same topology • Work for arbitrary genus • Discontinuity along base domain edges [Eck et al 95, Lee et al 00, Guskov et al 00, Praun et al 01, Khodakovsky et al 03]

  7. Spherical Parameterization • No cuts  less distortion • Restricted to genus zero meshes • [Shapiro et al 98] • [Alexa et al 00] • [Sheffer et al 00] • [Haker et al 00] • [Gu et al 03] • [Gotsman et al 03] • [Praun et al 03]

  8. Constrained Parameterizations • Texture mapping [Levy et al 01, Eckstein et al 01, Kraevoy et al 03]

  9. Consistent Parameterizations Base Domain DGP Applications • Motivation • Digital geometry processing • Morphing • Attribute transfer • Principal component analysis [Alexa 00, Levy et al 99, Praun et al 01] Input Meshes with Features Semi-Regular Meshes

  10. Contributions • Consistent Spherical Parameterizations • Inter-surface maps

  11. Consistent Spherical Parameterizations

  12. Stretch Minimizing Spherical Parameterization [Praun & Hoppe 03] • Use multiresolution • Convert model to progressive mesh format • Map base tetrahedron to sphere • Add vertices one by one, maintaining valid embedding and minimizing stretch

  13. Stretch Metric [Sander et al. 2001] linear map singular values: γ , Γ g G 2D texture domain surface in 3D

  14. Conformal vs Stretch Conformal metric: can lead to undersampling Stretch Conformal Stretch metric encourages feature correspondence Conformal

  15. Approach • Find “good” spherical locations • Use spherical parameterization of one model • Assymetric • Obtain spherical locations using all models • Constrained spherical parameterization • Create base mesh containing only feature vertices • Refine coarse-to-fine • Fix spherical locations of features

  16. Finding spherical locations

  17. Algorithm UCSP + step 1 CSP step 4 UCSP CSP step 5 step 2 step 3 step 6 • Find initial spherical locations using 1 model • Parameterize all models using those locations • Use spherical parameterizations to obtain remeshes • Concatenate to single mesh • Find good feature locations using all models • Compute final parameterizations using these locations

  18. Constrained Spherical Parameterization

  19. Approach

  20. Consistent Partitioning • Compute shortest paths (possibly introducing Steiner vertices) • Add paths not violating legality conditions • Paths (and arcs) don’t intersect • Consistent neighbor ordering • Cycles don’t enclose unconnected vertices • First build spanning tree

  21. Swirls • Unnecessarily long paths

  22. Heuristics to avoid swirls • Insert paths in increasing order of length • Link extreme vertices first • Disallow spherical triangles with any angle < 10o • Sidedness test • Unswirl operator • Edge flips

  23. Sidedness test E C E D B D B C A A

  24. Morphing [Praun et al 03]

  25. Morphing

  26. Morphing

  27. Attribute Transfer + Color Geometry

  28. Attribute Transfer + Color Geometry

  29. Face Database avg =

  30. Timing • 2.4 GHz Pentinum 4 PC, 512 MB RAM

  31. Inter Surface Maps

  32. Introduction • No intermediate domain • Reduced distortion • Natural alignment of features

  33. Comparison to CSP • No intermediate domain • Arbitrary genus • Limited to 2 models • Applications • Morphing • Digital geometry processing • Transfer of surface attributes • Deformation transfer

  34. Contributions • Directly create inter-surface map • Symmetric coarse-to-fine optimization • Symmetric stretch metric  Automatic geometric feature alignment • Robust • Very little user input • Arbitrary genus • Hard constraints

  35. Algorithm Overview • Consistent mesh partitioning • Constrained Simplification • Trivial map between base meshes • Coarse-to-fine optimization

  36. Consistent Mesh Partitioning • Compute matching shortest paths (possibly introducing Steiner vertices) • Add paths not violating legality conditions

  37. Legality Conditions • Paths don’t intersect • Consistent neighbor ordering • Cycles don’t enclose unconnected vertices • First build maximal graph without sep cycles • genus 0: spanning tree • genus > 0: spanning tree + 2g non-sep cycles

  38. Separating/Non-separating cycles • Separating cycle breaks surface into 2 disjoint components Separating cycle Non separating cycle

  39. Non-separating cycle test • Grow 2 fronts starting on both sides of AB • Non-separating if fronts meet B A

  40. Tracing non separating cycle • Shortest path between AC is separating A C B

  41. Tracing non separating cycle • Grow contour around AC • Contour wraps around and meets itself at O O A C B

  42. Tracing non separating cycle • Trace paths from O to A and C O A C B

  43. Automatic Insertion Of Feature Points Add features if not enough to resolve genus

  44. Genus-0 example

  45. Genus-1 example

  46. Genus-2 example

  47. Contributions • Consistent Spherical Parameterizations for several genus-zero surfaces • Robust method for Constrained Spherical Parameterization • Robust partitioning of two meshes of arbitrary genus • Methods to avoid swirls and to correct them when they arise

  48. Future Work • Improve overall exectution time • Multiresolution path tracing algorithm • Linear stretch optimization • Construct maps between surfaces of different genus • Handle point cloud and volumetric data

  49. Publications Consistent Spherical Parameterizations, Arul Asirvatham, Emil Praun, Hugues Hoppe, Computer Graphics and Geometric Modelling, 2005. Inter-Surface Mapping, John Schreiner, Arul Asirvatham, Emil Praun, Hugues Hoppe, ACM SIGGRAPH 2004.

  50. Thank You

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