Numerical Geometry of Nonrigid Shapes. CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher

Transcription

1 Numerical Geometry of Nonrigid Shapes CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher

2 Intrinsically far Extrinsically close

3 Straightest Geodesics on Polyhedral Surfaces (Polthier and Schmies) Local vs. global optimality

4 Not OK OK Extrinsic may suffice for near vs. far

5 Locally OK Single source Multi-source All-pairs

6 Approximate geodesics as paths along edges Meshes are graphs

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10 ` = p 2 ` = 2

11 Asymmetric Anisotropic May not improve under refinement

12 Graph shortest-path does not always converge to geodesic distance.

13 Graph shortest-path does not always converge to geodesic distance.

14 Graph shortest path algorithms are well-understood.

15 Shortest path had to come from somewhere. All steps of a shortest path are optimal.

16 v 0 = Source vertex d i = Current distance to vertex i S = Vertices with known optimal distance Initialization: d 0 = 0 d i = 1 8i > 0 S = fg

17 v 0 = Source vertex d i = Current distance to vertex i S = Vertices with known optimal distance Iteration k: k = arg min vk 2V ns d k S Ã v k d` Ã minfd`; d k + d k`g 8 neighbors v` of v k

18 v 0 = Source vertex d i = Current distance to vertex i S = Vertices with known optimal distance Iteration k: k = arg min vk 2V ns d k S Ã v k d` Ã minfd`; d k + d k`g 8 neighbors v` of v k During each iteration, S remains optimal.

19 CS 468, 2009

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21 Dijkstra s algorithm, modified to approximate geodesic distances.

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25 d 1 = ~n > ~x 1 + p ~n ~x d 2 2 = ~n > ~x 2 + p ~x 1 Given: ~d = ~n > X + p1 2 1 Find: 0 d 3 = ~n > ~x 3 + p p

26 ~d = ~n > X + p1 2 1 # ~n = V > ( ~ d p1 2 1 ) 1 = ~n > ~n = ( d ~ p1 2 1 ) > X 1 X > ( d ~ p1 2 1 ) = p 2 1 > 2 1Q p 1 > 2 1Q d ~ + d ~ > Q d ~ Q (X > X) 1

27 1 = p 2 1 > 2 1 Q p 1 > 2 1 Q~ d + ~ d > Q ~ d Quadratic equation for p

28 Bronstein, Numerical Geometry of Nonrigid Shapes Two orientations for the normal

29 Bronstein, Numerical Geometry of Nonrigid Shapes Two orientations for the normal

30 Update should be from a different triangle! Bronstein, Numerical Geometry of Nonrigid Shapes Front from outside the triangle

31 QX > ~n < 0 Bronstein, Numerical Geometry of Nonrigid Shapes Front from outside the triangle

32 Bronstein, Numerical Geometry of Nonrigid Shapes Must reach x 3 after x 1 and x 2

33 Alternative edge-based update: d 3 Ã minfd 3 ; d 1 + kx 1 k; d 2 + kx 2 kg Add connections as needed [Kimmel and Sethian 1998]

34 Bronstein, Numerical Geometry of Nonrigid Shapes

35 Modified update step Update all triangles adjacent to a given vertex

36 krdk = 1 Greek: Image 1 = ~n > ~n = ( d ~ p1 2 1 ) > X 1 X > ( d ~ p1 2 1 ) = p 2 1 > 2 1Q p 1 > 2 1Q d ~ + d ~ > Q d ~ Q (X > X) 1 Solutions are geodesic distance

37 STILL AN APPROXIMATION

38 STILL AN APPROXIMATION

39 [Novotni and Klein 2002]: Circular wavefront

40 Raster scan and/or parallelize Bronstein, Numerical Geometry of Nonrigid Shapes Grids and parameterized surfaces

41 Crane, Weischedel, and Wardetzky. Geodesics in Heat. TOG, to appear.

42 Trace gradient of distance function

43 Equal left and right angles Polthier and Schmies. Shortest Geodesics on Polyhedral Surfaces. SIGGRAPH course notes Trace a single geodesic exactly

44 Equal left and right angles Polthier and Schmies. Shortest Geodesics on Polyhedral Surfaces. SIGGRAPH course notes Trace a single geodesic exactly

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46 Surazhsky et al. Fast Exact and Approximate Geodesics on Meshes. SIGGRAPH Dijkstra-style front with windows explaining source.

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48 Locally minimizing distance is not enough to be a shortest path!

49 Cut point: Point where geodesic ceases to be minimizing Set of cut points from a source p

50 G ¾ p;q (x) exp ( jd M(p; x) + d M (x; q) d M (p; q)j=¾) Function on surface expressing difference in triangle inequality p q Stable version of geodesic distance

51 G ¾ p;q (x) exp ( jd M(p; x) + d M (x; q) d M (p; q)j=¾) Function on surface expressing difference in triangle inequality p q Intersection by pointwise multiplication Stable version of geodesic distance

52 Morphological operators to fill holes rather than remeshing Campen and Kobbelt. Walking On Broken Mesh: Defect-Tolerant Geodesic Distances and Parameterizations. Eurographics 2011.

53 Sample points Geodesic field Triangulate (Delaunay) Fix edges Query (planar embedding) Xin, Ying, and He. Constant-time all-pairs geodesic distance query on triangle meshes. I3D 2012.

54 Numerical Geometry of Nonrigid Shapes CS 468, Spring 2013 Differential Geometry for Computer Science Justin Solomon and Adrian Butscher