Provably Good Moving Least Squares

Transcription

1 Provably Good Moving Least Squares Ravikrishna Kolluri Computer Science Division University of California at Berkeley 1

2 Problem Definition Given a set of samples on a closed surface build a representation of the surface. Surface Reconstruction Point-based Methods Refinement 2

3 Our Contribution First implicit surface reconstruction algorithm with theoretical guarantees. All previous work on provably good surface reconstruction was Delaunay-based. 3

4 Approach Samples with normals Cut Function I() 4

5 Approach Samples with normals Cut Function I() Ideally, the cut function should be the signed distance function of the sampled surface. Positive outside the surface. Negative inside the surface. 5

6 Zero Set Points at which the cut function is zero. Normals of the zero set are parallel to the gradient of the cut function. Cut Function I() p I n Zero Set p 6

7 Implicit Moving Least Squares(MLS) Shen, O'Brien, and Shewchuk, Interpolating and Approimating Surfaces from Polygon Soup, SIGGRAPH Input: Samples with normal vectors. Weighted average of point functions (Gaussian weight functions). Build point function for each sample. s n 7

8 Related Work Implicit Methods Implicit Methods Hoppe, Derose, Duchamp, McDonald, and Stuetzle (1992) Levoy and Curless (1996) O' Brien and Turk (1999) MLS-Projection Alea, Behr, Cohen-Or, Fleishman, Levin, and Silva (2003) Pauly, Keiser, Kobbelt, and Gross (2003) Amenta and Kil (2004) MLS-Implicit Ohtake, Belyaev, Alea, Turk, and Seidel (2003) Shen, O'Brien, and Shewchuk (2004) No guarantees on the reconstructed surface! 8

9 Related Work Delaunay Methods Amenta and Bern (1998) Amenta, Choi, Dey, and Leekha (2000) Amenta, Choi, and Kolluri (2001) Boissonnat and Cazals (2001) Dey and Goswami (2004) Provably Good 9

10 Results: Geometry Under certain sampling conditions we prove: The reconstructed surface is near the samples. The gradient of the implicit function approimates the sample normals. s _ n I Zero surface 10

11 Results: Topology X Under certain sampling conditions we prove: The reconstructed surface has the same topology as the sampled surface. Zeroset 11

12 Works in Practice The MLS algorithm builds smooth surfaces from noisy data. Noisy Data MLS Surface 12

13 Works in Practice 13

14 Algorithm Input: Sample points with normal vectors. 14

15 15 s n Point Functions Signed distance to tangent plane. P s s n Sample Point Any point in space P s

16 16 s n Implicit Function I s P s W s t W t Weighted average of point functions. Σ Σ = Point function Weight function

17 Medial Ais Points with at least two closest points on the surface. Curve Medial ais 17

18 Local Feature Size (Amenta and Bern) LFS(p): Distance to the medial ais from p. Assume that the smallest local feature size is 1. Point with smallest LFS. LFS(p) = d(,p) 18

19 Sampling Conditions: Sample Spacing Conditions that must be satisfied to guarantee the algorithm's correctness: Distance from a point on the surface to its closest sample is small. Closest sample d p, s Surface point

20 Sampling Conditions: Noise Samples and sample normals can be noisy. Sample d r, q n q n r 2 Closest point on surface 20

21 Varying Sampling Density Sampling density cannot change suddenly. Ball of radius 2ε. Ball of radius ε. 21

22 Weight Functions s 2 2 W s a s e s a s = 1 Number of samples in ball of radius at s. 22

23 Analysis Does the cut function have the correct sign? I s P s t W t W s 23

24 Analysis Does the cut function have the correct sign? I s P s t W t W s p Some point functions are correct. 24

25 Analysis Does the cut function have the correct sign? I s P s t W t W s q Some point functions are correct... but some are wrong. 25

26 Analysis I s P s t W t W s Good samples What are the good samples for a point? Bad samples 26

27 Analysis I s P s t W t W s Good samples What are the good samples for a point? Can the bad samples distort the value of the cut function? Bad samples 27

28 What are the Good Samples? Samples that are approimately nearest to will always be good samples. Shell of width 3ε s H 28

29 What is the effect of the bad samples? Eponential decay of Gaussian functions means that the weight of bad samples is much smaller than the weight of the good samples. Bad W s 0.05 Good W s s H 29

30 What is the effect of the bad samples? As the Gaussian function falls off eponentially, the weight of the bad samples is much smaller than the weight of the good samples. Bad W s 0.05 Good W s H s All samples outside H can be ignored when computing the cut function at! 30

31 Geometric Results Cut function is close to the signed distance function. I Signed distance function 2 I The Hausdorff distance between the reconstructed surface and the sampled surface is less than 2ε. 31

32 Gradient For a point in the zero surface, the gradient of the implicit function is close to the sample normals. I n 1 O Point on the zero surface n F s n B Gradient of the point functions 32

33 Topology The reconstructed surface is a manifold. Homeomorphic to the sampled surface. X y Negative The value of the cut function is monotonically decreasing along y. Positive 33

34 Topology The reconstructed surface is a manifold. Homeomorphic to the sampled surface. X y Negative y Positive 34

35 Future Work Adaptive sample spacing. Interpolating reconstruction. Different weight functions. Reconstruction with boundaries and sharp corners. Different point functions. 35

36 Acknowledgements Jonathan Shewchuk. James O'Brien. U.C. Berkeley François Labelle. Nina Amenta U.C. Davis 36