Laplacian Operator and Smoothing

Transcription

1 Laplacian Operator and Smoothing Xifeng Gao Acknowledgements for the slides: Olga Sorkine-Hornung, Mario Botsch, and Daniele Panozzo

2 Applications in Geometry Processing Smoothing Parameterization Remeshing

3 Laplacian Operator Laplacian operator gradient operator function in Euclidean space divergence operator

4 Laplacian Operator Laplacian operator gradient operator 2nd partial derivatives function in Euclidean space divergence operator

5 Laplacian Operator Laplacian operator gradient operator 2nd partial derivatives function in Euclidean space divergence operator Cartesian coordinates

6 Laplacian Operator Laplacian operator gradient operator 2nd partial derivatives Intuitive Explanation The Laplacian Δf(p) of a function f at a point p, is the rate at which the average value of f over spheres centered at p deviates from f(p) as the radius of the sphere grows. function in Euclidean space divergence operator Cartesian coordinates

7 Laplace-Beltrami Operator Extension of Laplace to functions on manifolds Laplace- Beltrami gradient operator function on surface M divergence operator

8 Laplace-Beltrami Operator For coordinate functions: Laplace- Beltrami gradient operator mean curvature P at function f divergence operator unit surface normal

9 Discrete Curvatures Mean curvature (sign defined according to normal)

10 Surfaces, Parametric Form Continuous surface n p u pv p(u,v) Tangent plane v u

11 Surfaces, Normal Surface normal: n pu pv p(u,v) v u

12 Surfaces, Curvature n t p u p pv Unit-length direction tin the tangent plane (if puand pvare orthogonal): j t Tangent plane

13 Surfaces, Curvature n The curve g is the intersection of the surface with the plane p u pv through n and t. t p Normal curvature: g kn(j)n = k(g(p))n = g (p) t j Tangent plane

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sha1_base64="kiisw+1xy+ht6ezsxtxlm4d9+h0=">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</latexit> <latexit sha1_base64="kiisw+1xy+ht6ezsxtxlm4d9+h0=">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</latexit> Surfaces, Curvature Mean curvature H = 1 2π Z 2π 0 k n (ϕ)dϕ M p= 2Hn k <latexit sha1_base64="l/xv12jk+wj/bcysedupnvwprky=">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</latexit> n (ϕ)n = γ 00 Z 2π M p= 1 π γ 00 dϕ 0

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sha1_base64="kiisw+1xy+ht6ezsxtxlm4d9+h0=">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</latexit> <latexit sha1_base64="kiisw+1xy+ht6ezsxtxlm4d9+h0=">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</latexit> Discrete Laplace-Beltrami Intuition for uniform discretization H = 1 2π Z 2π 0 k n (ϕ)dϕ M p= 2Hn k <latexit sha1_base64="l/xv12jk+wj/bcysedupnvwprky=">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</latexit> n (ϕ)n = γ 00 Z 2π M p= 1 π γ 00 dϕ 0

16 <latexit sha1_base64="gvfm7mfjarvj8nptzczam1+3bkc=">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</latexit> <latexit sha1_base64="gvfm7mfjarvj8nptzczam1+3bkc=">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</latexit> <latexit sha1_base64="gvfm7mfjarvj8nptzczam1+3bkc=">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</latexit> <latexit sha1_base64="gvfm7mfjarvj8nptzczam1+3bkc=">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</latexit> Discrete Laplace-Beltrami Intuition for uniform discretization vi-1 vi vi+1 g Z 2π M p= 1 π γ 00 dϕ 0

17 <latexit sha1_base64="kqu7ol1rkfabt7pd0xh3qkoujic=">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</latexit> <latexit sha1_base64="kqu7ol1rkfabt7pd0xh3qkoujic=">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</latexit> <latexit sha1_base64="kqu7ol1rkfabt7pd0xh3qkoujic=">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</latexit> <latexit sha1_base64="kqu7ol1rkfabt7pd0xh3qkoujic=">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</latexit> <latexit sha1_base64="l/xv12jk+wj/bcysedupnvwprky=">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</latexit> <latexit sha1_base64="l/xv12jk+wj/bcysedupnvwprky=">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</latexit> <latexit sha1_base64="l/xv12jk+wj/bcysedupnvwprky=">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</latexit> <latexit sha1_base64="l/xv12jk+wj/bcysedupnvwprky=">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</latexit> Discrete Laplace-Beltrami vj1 vj2 vj6 vi vj3 h =1 M p= 2Hn vj5 vj4

18 <latexit sha1_base64="wltcj+gmrgeva9ydtmieum6+cjk=">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</latexit> <latexit 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sha1_base64="wltcj+gmrgeva9ydtmieum6+cjk=">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</latexit> <latexit 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sha1_base64="0dzjqamajguuchrqfzi9qxk+plw=">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</latexit> Discrete Laplace-Beltrami Uniform discretization: L u (v i )=( 1 N(i) X j N(i) v j ) v i v i Depends only on connectivity = simple and efficient Bad approximation for irregular triangulations

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ij +cotβ ij ) vj vj

20 Surface Smoothing Motivation

21 Curvature and Smoothness

22 Curvature and Smoothness mean curvature plot

23 Curvature and Smoothness mean curvature plot

24 Curvature and Smoothness mean curvature plot

25 Curvature and Smoothness Smoothing = reducing curvature? Smoothing = make curvature vary less?

26 Example smoothing curves Laplace in 1D = second derivative:

27 Example smoothing curves Laplace in 1D = second derivative: In matrix-vector form for the whole curve

28 Example smoothing curves Laplace in 1D = second derivative: In matrix-vector form for the whole curve

29 Example smoothing curves Flow to reduce curvature: Scale factor 0 < l < 1 Matrix-vector form: Drawbacks?

30 Example smoothing curves Flow to reduce curvature: Scale factor 0 < l < 1 Matrix-vector form: May shrink the shape; can be slow

31 Filtering Curves Original curve

32 Filtering Curves 1st iteration; l=0.5

33 Filtering Curves 2nd iteration; l=0.5

34 Filtering Curves 8th iteration; l=0.5

35 Filtering Curves 27th iteration; l=0.5

36 Filtering Curves 50th iteration; l=0.5

37 Filtering Curves 500th iteration; l=0.5

38 Filtering Curves 1000th iteration; l=0.5

39 Filtering Curves 5000th iteration; l=0.5

40 Filtering Curves 10000th iteration; l=0.5

41 Filtering Curves 50000th iteration; l=0.5

42 Filtering Surfaces (Demo)

43 Thank you!