Spring 2015 CSCI 599: Digital Geometry Processing 4.1 Discrete Differential Geometry Hao Li http://cs599.hao-li.com 1
Outline • Discrete Differential Operators • Discrete Curvatures • Mesh Quality Measures 2
Differential Operators on Polygons Differential Properties � • Surface is sufficiently differentiable • Curvatures → 2nd derivatives 3
Differential Operators on Polygons Differential Properties � • Surface is sufficiently differentiable • Curvatures → 2nd derivatives Polygonal Meshes � • Piecewise linear approximations of smooth surface • Focus on Discrete Laplace Beltrami Operator • Discrete differential properties defined over N ( x ) 4
Local Averaging N ( x ) Local Neighborhood of a point � x • often coincides with mesh vertex v i N n ( v i ) • n-ring neighborhood or local geodesic ball 5
Local Averaging N ( x ) Local Neighborhood of a point � x • often coincides with mesh vertex v i N n ( v i ) • n-ring neighborhood or local geodesic ball Neighborhood size � • Large: smoothing is introduced, stable to noise • Small: fine scale variation, sensitive to noise 6
Local Averaging: 1-Ring N ( x ) x Barycentric cell Voronoi cell Mixed Voronoi cell (barycenters/edgemidpoints) (circumcenters) (circumcenters/midpoint) tight error bound better approximation 7
Barycentric Cells Connect edge midpoints and triangle barycenters � • Simple to compute • Area is 1/3 o triangle areas • Slightly wrong for obtuse triangles 8
Mixed Cells Connect edge midpoints and � • Circumcenters for non-obtuse triangles • Midpoint of opposite edge for obtuse triangles • Better approximation, more complex to compute… 9
Normal Vectors Continuous surface x ( u, v ) y ( u, v ) x ( u, v ) = z ( u, v ) n Normal vector x u x v p x u � x v n = ⇥ x u � x v ⇥ Assume regular parameterization x u � x v ⇥ = 0 normal exists 10
Discrete Normal Vectors n ( T ) x k x i x j 11
Discrete Normal Vectors n ( v ) n ( T ) x k x i x j 12
Discrete Normal Vectors n ( v ) n ( T ) x k x i θ T x j 13
Discrete Normal Vectors tessellated cylinder 14
Simple Curvature Discretization mean curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami 15
Simple Curvature Discretization mean curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami How to discretize? 16
Laplace Operator on Meshes Extend finite differences to meshes? � • What weights per vertex/edge? ¼ ? ? ? ? ¼ ? ¼ -1 ½ -1 ½ ? ? ¼ 1D grid 2D grid 2D/3D grid 17
Uniform Laplace Uniform discretization � • What weights per vertex/edge? Properties � • depends only on connectivity • simple and efficient 18
Uniform Laplace Uniform discretization � 1 � ∆ uni x i := ( x j − x i ) ≈ − 2 H n |N 1 ( v i ) | v j ∈ N 1 ( v i ) Properties � • depends only on connectivity • simple and efficient • bad approximation for irregular triangulations • can give non-zero for planar meshes H • tangential drift for mesh smoothing 19
Gradients mean curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami gradient operator Discrete Gradient of a Function � • Defined on piecewise linear triangle • Important for parameterization and deformation 20
Gradients x k x i triangle x j u = ( u, v ) piecewise linear function f i = f ( x i ) linear basis functions for barycentric interpolation on a triangle 21
Gradients u = ( u, v ) piecewise linear function 22
Gradients u = ( u, v ) piecewise linear function gradient of linear function 23
Gradients u = ( u, v ) piecewise linear function gradient of linear function partition of unity gradients of basis 24
Gradients u = ( u, v ) piecewise linear function gradient of linear function partition of unity gradients of basis gradient of linear function 25
Gradients gradient of linear function
Gradients gradient of linear function with appropriate normalization: 27
Gradients gradient of linear function with appropriate normalization: f i = f ( x i ) discrete gradient of a piecewiese linear function within T 28
Discrete Laplace-Beltrami gradient mean operator curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami
Discrete Laplace-Beltrami gradient mean operator curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami divergence theorem 30
Discrete Laplace-Beltrami gradient mean operator curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami divergence theorem vector-valued function local averaging domain boundary 31
Discrete Laplace-Beltrami gradient mean operator curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami divergence theorem 32
Discrete Laplace-Beltrami average Laplace-Beltrami 33
Discrete Laplace-Beltrami average Laplace-Beltrami gradient is constant and local Voronoi passes through a,b: over triangle 34
Discrete Laplace-Beltrami average Laplace-Beltrami gradient is constant and local Voronoi passes through a,b: over triangle discrete gradient 35
Discrete Laplace-Beltrami average Laplace-Beltrami within a triangle 36
Discrete Laplace-Beltrami average Laplace-Beltrami within a triangle 37
Discrete Laplace-Beltrami average Laplace-Beltrami over averaging region 38
Discrete Laplace-Beltrami average Laplace-Beltrami over averaging region discrete Laplace-Beltrami 39
Discrete Laplace-Beltrami Cotangent discretization � 1 � ∆ S f ( v i ) := (cot α ij + cot β ij ) ( f ( v j ) − f ( v i )) 2 A ( v i ) v j ∈ N 1 ( v i ) A ( v i ) v i β ij α ij for full derivation, check out: http://brickisland.net/cs177/ v j 40
Discrete Laplace-Beltrami Cotangent discretization 1 � ∆ S f ( v ) := (cot α i + cot β i ) ( f ( v i ) − f ( v )) 2 A ( v ) v i ∈ N 1 ( v ) Problems � • weights can become negative • depends on triangulation Still the most widely used discretization 41
Outline • Discrete Differential Operators • Discrete Curvatures • Mesh Quality Measures 42
Discrete Curvatures How to discretize curvature on a mesh? � • Zero curvature within triangles • Infinite curvature at edges / vertices • Point-wise definition doesn’t make sense Approximate differential properties at point as x average over local neighborhood � A ( x ) • is a mesh vertex A ( x ) x • within one-ring neighborhood A ( x ) x 43
Discrete Curvatures How to discretize curvature on a mesh? � • Zero curvature within triangles • Infinite curvature at edges / vertices • Point-wise definition doesn’t make sense Approximate differential properties at point as x average over local neighborhood A ( x ) A ( x ) 1 � K ( v ) ≈ K ( x ) d A A ( v ) ( v ) A x 44
Discrete Curvatures Which curvatures to discretize? � • Discretize Laplace-Beltrami operator • Laplace-Beltrami gives us mean curvature H • Discretize Gaussian curvature K • From and we can compute and H K κ 2 κ 1 mean curvature ∆ S x = div S ∇ S x = − 2 H n Laplace-Beltrami 45
Discrete Gaussian Curvature Gauss-Bonnet Z K = 2 πχ χ = 2 − 2 g Discrete Gauss Curvature � K = (2 π − θ j ) /A j Verify via Euler-Poincaré V − E + F = 2(1 − g ) 46
Discrete Curvatures Mean curvature (absolute value) H = 1 2 � ∆ S x � A Gaussian curvature � K = (2 π − θ j ) /A θ j j Principal curvatures � � H 2 − K H 2 − K κ 1 = H + κ 2 = H − 47
Outline • Discrete Differential Operators • Discrete Curvatures • Mesh Quality Measures 48
Mesh Quality Visual inspection of “sensitive” attributes � • Specular shading 49
Mesh Quality Visual inspection of “sensitive” attributes � • Specular shading • Reflection lines 50 50
Mesh Quality Visual inspection of “sensitive” attributes � • Specular shading • Reflection lines • differentiability one order lower than surface • can be efficiently computed using GPU C 0 C 1 C 2 51 51 51
Mesh Quality Visual inspection of “sensitive” attributes � • Specular shading • Reflection lines • Curvature • Mean curvature 52
Mesh Quality Visual inspection of “sensitive” attributes � • Specular shading • Reflection lines • Curvature • Gauss curvature 53
Mesh Quality Criteria Smoothness � • Low geometric noise 54
Mesh Quality Criteria Smoothness � • Low geometric noise Fairness � • Simplest shape 55
Mesh Quality Criteria Smoothness � • Low geometric noise Fairness � • Simplest shape Adaptive tesselation � • Low complexity 56
Mesh Quality Criteria Smoothness � • Low geometric noise Fairness � • Simplest shape Adaptive tesselation � • Low complexity Triangle shape � • Numerical Robustness 57
Mesh Optimization Smoothness � • Smoothing Fairness � • Fairing Adaptive tesselation � • Decimation Triangle shape � • Remeshing 58
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