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 p i, i=1,…,n  i =1 Stretch preserving maps Conformal Maps   [Tutte 63] [Floater 97] [Floater et al 03] [Sheffer et al 01] [Levy et al 02] [Desbrun et al 02] [Sander et al 01]

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

10 Contributions Consistent Spherical Parameterizations Inter-surface maps

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] 2D texture domain surface in 3D linear map singular values: γ, Γ

14 Conformal vs Stretch Conformal metric: can lead to undersampling Stretch metric encourages feature correspondence Conformal Stretch 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 1.Find initial spherical locations using 1 model 2.Parameterize all models using those locations 3.Use spherical parameterizations to obtain remeshes 4.Concatenate to single mesh 5.Find good feature locations using all models 6.Compute final parameterizations using these locations step 1 step 2step 3step 6 Algorithm + step 4 step 5 UCSP CSP

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 < 10 o Sidedness test Unswirl operator Edge flips

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

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 # models #tris1256Total (mins) 271k- 200k k- 200k k- 363k GHz Pentinum 4 PC, 512 MB RAM

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 1.Consistent mesh partitioning 2.Constrained Simplification 3.Trivial map between base meshes 4.Coarse-to-fine optimization Algorithm Overview

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 cycleNon separating cycle

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

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 A C O B

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

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, Inter-Surface Mapping, John Schreiner, Arul Asirvatham, Emil Praun, Hugues Hoppe, ACM SIGGRAPH 2004.

50 Thank You