Interpolating and Approximating Implicit Surfaces from Polygon Soup

Transcription

1 Interpolating and Approimating Implicit Surfaces from Polygon Soup Chen Shen, James F. O Brien, Jonathan R. Shewchuk University of California, Berkeley Geometric Algorithms Seminar CS 468 Fall 2005

2 Overview Talk Overview: Motivation Implicit Surface Fitting for Polygon Soup Computational Aspects Etensions Results 2

3 Motivation 3

4 What does the paper do? Goal: Convert polygon mesh into implicit surface and back again Applications: Mesh cleanup create consistent meshes from polygon soup Topological simplification Creating bounding volumes 4

5 Consistent Meshes Polygon models often show consistency issues: Holes and gaps no closed surface } T-junctions in meshing fied Non-manifold structure, self intersections Inconsistent normals Sometimes: Internal structure should be omitted } partially fied } preprocessing 5

6 Topological Simplification Create simplified bounding volume, allowing topological changes: Useful for... Mesh simplification Spatial queries e.g. collision detection [Shen et al. 04] 6

7 Related Work Reconstruction via Implicit Surfaces: Standard technique: see e.g. [Hoppe et al. 92], [Turk et al. 99], [Carr et al. 01], [Turk et al. 02],... General reconstruction procedure: Estimate normals Appro. signed distance function Apply marching cubes Possibly: Simplify result This paper 7

8 Reconstruction via Implicit Surfaces Initial data Estimate normals Signed distance func. Marching cubes Final mesh 8

9 Reconstruction via Implicit Surfaces Initial data Estimate normals Signed distance func. Marching cubes Final mesh 9

10 Reconstruction via Implicit Surfaces Initial data Estimate normals Signed distance func. Marching cubes Final mesh 10

11 Reconstruction via Implicit Surfaces Initial data Estimate normals Signed distance func. Marching cubes Final mesh This paper: technique for polygon meshes 11

12 Reconstruction via Implicit Surfaces Initial data Estimate normals Signed distance func. Marching cubes Final mesh 12

13 Reconstruction via Implicit Surfaces Initial data Estimate normals Signed distance func. Marching cubes Final mesh 13

14 What is this paper about? In this paper: Defining the implicit function this step But: Consider polygon models input: polygons 14

15 Utility Hole Filling: Create well-defined closed surface 15

16 Utility Also: Remeshing Marching Cubes T-verte, small hole fied: remeshed 16

17 Utility Missing Normals: Reconstruct a few missing normals??? 17

18 Implicit Surface Fitting for Polygon Soup 18

19 Least-Squares Least Squares Approimation: p i = i, φ i B 1 B 2 B 3 target values basis functions w weighting functions least squares fit 19

20 20 Least-Squares Least-Squares Least Squares Approimation: ~ 1 B c i n i i = = φ Best Fit: = n i i i i i w c 1 2 ~ argmin φ φ

21 21 Least-Squares Least-Squares φ 2 2 W B c B W B T T = Normal Equations: φ W B B W B c T T = Solution: φ φ 2 T 1 2 T T, ~ W B B W B b c b >= =< Evaluation: [ ] B n B b,..., : 1 = = : 1 n b b B M = φ n φ φ M 1 : = c n c c M 1 : : 1 n w w W = O Notation:

22 Moving Least-Squares Moving Least Squares Approimation: target values move basis and weighting function, ~ recompute approimation φ 22

23 Moving Least-Squares Moving Least Squares Approimation: target values approimation 23

24 Application... Approach: MLS-Approimation of implicit function in space: f: 3 Set polygons surface to zero Normal constraints: implicit function should grow in normal direction Coordinate frame: 3 no non-linear coord.-sys. estimation Weighting function: wr = 1/r 2 + ε 2 24

25 Weighting Function Weighting Function Used in This Paper: Vary ε to adjust tightness of fit. 25

26 Polygonal Constraints Problem: Need polygonal constraints not point constraints [Shen et al. 04] 26

27 27 Polygonal Constraints Polygonal Constraints Idea: Use infinite number of points i.e. integrate φ 2 2 W B c B W B T T = Normal Equations: i N i i i N i i T i i b w c b b w φ = = = ,, Can be rewritten: Sum of point constraints Just some algebra

28 28 Polygonal Constraints Polygonal Constraints Point Constraints: i N i i i N i i T i i b w c b b w φ = = = ,, dp p b p w c dp p b p b p w k N k Poly N k Poly T k k φ = = = ,, Polygonal Constraints: just integrate over all polygon points

29 Application... First Try: φ k = 0 Missing: Normal constraints Result: f 0 29

30 Normal Constraints Traditional Technique: Here: Add some +1, -1 points in space Does not work very well oscillation artifacts Directly approimate weighted signed distance function in space [Shen et al. 04] 30

31 Normal Constraints Basic Idea: w, r Add additional equations, f should match signed distance function: p i n i weighted by wpolygon, details not given 31

32 Application... With Normal Constraints: 32

33 Computational Aspects 33

34 Computing Integrals Problem: Evaluation of Integrals Normal Equations: = N k= 1 Poly N w k k= 1 Poly w k 2, 2, p b p b T p b p φ dp p dp c k Weighting function w, p = 1/p 2 + ε 2 : small ε almost singular 34

35 Computing Integrals Solution: One dimensional line integrals can be computed analytically Two dimensional area integrals are approimated numerically using an importance sampling approach Sum of line integrals Split triangle at eval.-point Eponentially increasing spacing 35

36 Compleity Problem Computation Time: ON for one evaluation of f Millions of function evaluations This won t work... Solution: Piecewise constant integral approimation Fewer evaluations at larger distance 36

37 Hierarchical Approimation Data Structure: Precomputed spatial hierarchy Precomputed distance independent integrals in each node with w 1 Eval.: Subdivide until node diameter = λ distance Multiply w with precomp. constant appro. error 37

38 Etensions 38

39 Tight Bounding Volumes Level-of-Detail Control: Vary ε in weighting function wr = 1/r 2 + ε 2 to smooth geometry Problem: Bounding volumes grow too much [Shen et al. 04] 39

40 Tight Bounding Volumes Create Tight Bounding Volumes: Adjust iso-value to best fit on average Iteratively: Adjust constraint values φ i of polygons to guarantee inclusion [Shen et al. 04] 40

41 Preprocessing Preprocessing Steps: no details given in paper Ensure consistent normals Standard technique: Region growing on surface For approimate bounding volumes: Remove interior, invisible polygons 41

42 Results 42

43 Eample original restauration [Shen et al. 04] 43

44 Sharp Features small ε larger ε [Shen et al. 04] 44

45 Bounding Volumes [Shen et al. 04] 45

46 Continuous Simplification [Shen et al. 04] 46

47 Timings Computationally Epensive: Heavy Loader : Min. 37K polygons in, 30K 2M vertices out Bunny : Machine Part : Min. 69K polygons in, 28K 1.5M vertices out Min. 800 polygons in, 20K 1M vertices out The larger ε less output polygons, the longer the running time presumably due to hierarchy traversal costs 47

48 Conclusions 48

49 Conclusions Conclusions: Polygonal models can be fied via implicit function fitting Direct approimation of signed distance: straightforward and effective idea normals required, might also be useful for other appro.-applications Epensive technique Not perfect at sharp features Implementation involved quadrature, hierarchical approimation 49

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