1. Automatic Reconstruction of B-spline Surfaces of Arbitrary Topological Type Matthias Eck Hugues Hoppe Matthias Eck Hugues Hoppe University of Darmstadt * Microsoft Research SIGGRAPH 96 * now at ICEM Systems S

2. Surface reconstruction points P surface reconstruction surfaceS n Reverse engineering n Traditional design (wood,clay) n Virtual environments

3. smooth B-spline Previous work subdivision implicit meshes simple surface topological type [Schumaker93], … arbitrary [Sclaroff-Pentland91],... - [Schmitt-etal86],[Forsey-Bartels95],... [Hoppe-etal92], [Turk-Levoy94],... [Moore-Warren91],[Bajaj-etal95] [Hoppe-etal94] [Krishnamurthy-96],[Milroy-etal95],... B-spline[Krishnamurthy-96], [Milroy-etal95],... arbitrary

4. Problem statement points P reconstruction procedure B-spline surface S n automatic procedure n surface of arbitrary topological type S smooth  G 1 continuity S smooth  G 1 continuity error tolerance  error tolerance 

5. Main difficulties S: arbitrary topological type single B-spline patch N 3 Difficulties: 1) Obtaining N 2) Parametrizing P over N 3) Fitting with G 1 continuity  require network N of B-spline patches

6. Comparison with previous talk Krishnamurthy-LevoyEck-Hoppe patch network N parametrized P continuity curve painting automaticpartitioning hierarchicalremeshing harmonic maps stitching step G 1 construction

7. Overview of our procedure 5 steps: 5 steps: 1) Initial parametrization of P over mesh M 0 2) Reparam. over triangular complex K  3) Reparam. over quadrilateral complex K ÿ 4) B-spline fitting 5) Adaptive refinement

8. (1) Find an initial parametrization a) construct initial surface: dense mesh M 0 b) parametrize P over M 0 P [Hoppe-etal92] Using: [Hoppe-etal93] M0M0M0M0

9. (2) Reparametrize over domain K  Use parametrization scheme of [Eck-etal95]: Use parametrization scheme of [Eck-etal95]: M0M0M0M0 a) partition M 0 into triangular regions partitioned M 0 base mesh K  b) parametrize each region using a harmonic map

10. Harmonic map [Eck-etal95] planar triangle triangular region map minimizing metric distortion

11. Reparametrize points M0M0M0M0 KKKK using harmonic maps

12. (3) Reparametrize over K ÿ Merge faces of K  pairwise Merge faces of K  pairwise n cast as graph optimization problem KKKK KÿKÿKÿKÿ

13. Graph optimization two  regions planar square For each pair of adjacent  regions, let edge cost = harmonic map “distortion” For each pair of adjacent  regions, let edge cost = harmonic map “distortion” harmonic map n Solve MAX-MIN MATCHING graph problem

14. KÿKÿKÿKÿ reparametrize P using harmonic maps

15. (4) B-spline fitting Use surface spline scheme of [Peters94]: Use surface spline scheme of [Peters94]: n G 1 surface n tensor product B-spline patches n low degree l Other similar schemes [Loop94], [Peters96], … [Loop94], [Peters96], …

16. Overview of [Peters94] scheme MxMxMxMx 2 Doo-Sabin subdiv. affineconstruction McMcMcMc dfdfdfdfS B-spline bases KÿKÿKÿKÿ Fitting using [Peters94] scheme optimization MxMxMxMxaffine (Details: linear constraints, reprojection, fairing, …)

17. (5) Adaptively refine the surface Goal: make P and S differ by no more than  Goal: make P and S differ by no more than  Strategy: adaptively refine K ÿ Strategy: adaptively refine K ÿ 4 face refinement templates KÿKÿKÿKÿ K#K#K#K# S#S#S#S#  : 1.02%  0.74%

18. Reconstruction results P 20,000 points 29 patches  = 1.20% rms = 0.20% SÿSÿSÿSÿ

19. Reconstruction results 29 patches  = 1.20% rms = 0.20% 156 patches  = 0.27% rms = 0.03%

20. Approximation results S0S0S0S0 70,000 triangles P 30,000 points S#S#S#S# 153 patches  = 1.44% rms = 0.19%

21. Analysis Krishnamurthy-LevoyEck-Hoppe patch network N parametrized P continuity curve painting automaticpartitioning hierarchicalremeshing harmonic maps stitching step G 1 construction refinement noyes displac. map yesno

22. Future work l Semi-automated layout of patch network l Allowing creases and corners in B-spline construction Error bounds on surface approximation have: d(P,S)<  want: d(S 0,S)<  Error bounds on surface approximation have: d(P,S)<  want: d(S 0,S)< 