4.1 Discrete Differential Geometry Hao Li http://cs599.hao-li.com - - PowerPoint PPT Presentation

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4.1 Discrete Differential Geometry Hao Li http://cs599.hao-li.com - - PowerPoint PPT Presentation

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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

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SLIDE 1

CSCI 599: Digital Geometry Processing

Hao Li

http://cs599.hao-li.com

1

Spring 2015

4.1 Discrete Differential Geometry

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SLIDE 2

Outline

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  • Discrete Differential Operators
  • Discrete Curvatures
  • Mesh Quality Measures
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SLIDE 3

Differential Operators on Polygons

Differential Properties

  • Surface is sufficiently differentiable
  • Curvatures → 2nd derivatives

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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

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Local Averaging

Local Neighborhood of a point

  • often coincides with mesh vertex
  • n-ring neighborhood or local geodesic ball

N(x) x vi Nn(vi)

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Local Averaging

Local Neighborhood of a point

  • often coincides with mesh vertex
  • n-ring neighborhood or local geodesic ball

N(x) x vi Nn(vi)

Neighborhood size

  • Large: smoothing is introduced, stable to noise
  • Small: fine scale variation, sensitive to noise

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Local Averaging: 1-Ring

N(x) x

Barycentric cell

(barycenters/edgemidpoints)

Voronoi cell

(circumcenters) tight error bound

Mixed Voronoi cell

(circumcenters/midpoint) better approximation

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Barycentric Cells

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Connect edge midpoints and triangle barycenters

  • Simple to compute
  • Area is 1/3 o triangle areas
  • Slightly wrong for obtuse triangles
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SLIDE 9

Mixed Cells

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Connect edge midpoints and

  • Circumcenters for non-obtuse triangles
  • Midpoint of opposite edge for obtuse triangles
  • Better approximation, more complex to compute…
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Normal Vectors

Continuous surface Normal vector Assume regular parameterization

x(u, v) =   x(u, v) y(u, v) z(u, v)   n = xu xv ⇥xu xv⇥ xu xv ⇥= 0

n p xu xv

normal exists

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Discrete Normal Vectors

xi xj xk n(T)

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xi xj xk n(T) n(v)

Discrete Normal Vectors

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xi xj xk n(T) n(v) θT

Discrete Normal Vectors

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tessellated cylinder

Discrete Normal Vectors

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Simple Curvature Discretization

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∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami mean curvature

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Simple Curvature Discretization

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∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami mean curvature How to discretize?

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Laplace Operator on Meshes

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Extend finite differences to meshes?

  • What weights per vertex/edge?

½ ½

  • 1

¼ ¼ ¼ ¼

  • 1

? ? ? ? ? ? ?

1D grid 2D grid 2D/3D grid

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SLIDE 18

Uniform Laplace

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Uniform discretization

  • What weights per vertex/edge?

Properties

  • depends only on connectivity
  • simple and efficient
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Uniform Laplace

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Uniform discretization Properties

  • depends only on connectivity
  • simple and efficient
  • bad approximation for irregular triangulations
  • can give non-zero for planar meshes
  • tangential drift for mesh smoothing

∆unixi := 1 |N1(vi)|

  • vj∈N1(vi)

(xj − xi) ≈ −2Hn H

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Gradients

Discrete Gradient of a Function

  • Defined on piecewise linear triangle
  • Important for parameterization and deformation

∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami gradient

  • perator

mean curvature

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Gradients

xi xj xk

triangle piecewise linear function

u = (u, v)

linear basis functions for barycentric interpolation on a triangle

fi = f(xi)

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Gradients

piecewise linear function

u = (u, v)

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Gradients

piecewise linear function

u = (u, v)

gradient of linear function

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Gradients

piecewise linear function

u = (u, v)

gradient of linear function partition of unity gradients of basis

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Gradients

piecewise linear function

u = (u, v)

gradient of linear function partition of unity gradients of basis gradient of linear function

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Gradients

gradient of linear function

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SLIDE 27

Gradients

gradient of linear function with appropriate normalization:

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SLIDE 28

Gradients

gradient of linear function with appropriate normalization: discrete gradient of a piecewiese linear function within T

fi = f(xi)

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∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami gradient

  • perator

mean curvature

Discrete Laplace-Beltrami

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∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami gradient

  • perator

mean curvature divergence theorem

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Discrete Laplace-Beltrami

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SLIDE 31

∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami gradient

  • perator

mean curvature divergence theorem

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vector-valued function local averaging domain boundary

Discrete Laplace-Beltrami

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∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami gradient

  • perator

mean curvature divergence theorem

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Discrete Laplace-Beltrami

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SLIDE 33

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Discrete Laplace-Beltrami

average Laplace-Beltrami

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gradient is constant and local Voronoi passes through a,b:

  • ver triangle

Discrete Laplace-Beltrami

average Laplace-Beltrami

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discrete gradient gradient is constant and local Voronoi passes through a,b:

  • ver triangle

Discrete Laplace-Beltrami

average Laplace-Beltrami

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average Laplace-Beltrami within a triangle

Discrete Laplace-Beltrami

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average Laplace-Beltrami within a triangle

Discrete Laplace-Beltrami

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Discrete Laplace-Beltrami

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average Laplace-Beltrami over averaging region

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Discrete Laplace-Beltrami

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average Laplace-Beltrami over averaging region discrete Laplace-Beltrami

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Discrete Laplace-Beltrami

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Cotangent discretization

∆Sf(vi) := 1 2A(vi)

  • vj∈N1(vi)

(cot αij + cot βij) (f(vj) − f(vi))

vi vj A(vi)

βij αij

for full derivation, check out: http://brickisland.net/cs177/

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SLIDE 41

Discrete Laplace-Beltrami

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Cotangent discretization

∆Sf(v) := 1 2A(v)

  • vi∈N1(v)

(cot αi + cot βi) (f(vi) − f(v))

Problems

  • weights can become negative
  • depends on triangulation

Still the most widely used discretization

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SLIDE 42

Outline

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  • Discrete Differential Operators
  • Discrete Curvatures
  • Mesh Quality Measures
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SLIDE 43

Discrete Curvatures

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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 average over local neighborhood

  • is a mesh vertex
  • within one-ring neighborhood

x A(x) A(x) x x A(x)

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Discrete Curvatures

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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 average over local neighborhood

x A(x) K(v) ≈ 1 A(v)

  • A

(v)

K(x) dA A(x) x

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SLIDE 45

Discrete Curvatures

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Which curvatures to discretize?

  • Discretize Laplace-Beltrami operator
  • Laplace-Beltrami gives us mean curvature
  • Discretize Gaussian curvature
  • From and we can compute and

H K K H κ1 κ2 ∆Sx = divS ∇Sx = −2Hn

Laplace-Beltrami mean curvature

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Discrete Gaussian Curvature

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Discrete Gauss Curvature

K = (2π −

  • j

θj)/A χ = 2 − 2g Z K = 2πχ

Gauss-Bonnet

V − E + F = 2(1 − g)

Verify via Euler-Poincaré

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SLIDE 47

Discrete Curvatures

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Mean curvature (absolute value) Gaussian curvature Principal curvatures

H = 1 2 ∆Sx K = (2π −

  • j

θj)/A κ1 = H +

  • H2 − K

κ2 = H −

  • H2 − K

A

θj

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SLIDE 48

Outline

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  • Discrete Differential Operators
  • Discrete Curvatures
  • Mesh Quality Measures
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Mesh Quality

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Visual inspection of “sensitive” attributes

  • Specular shading
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Mesh Quality

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Visual inspection of “sensitive” attributes

  • Specular shading
  • Reflection lines
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Mesh Quality

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Visual inspection of “sensitive” attributes

  • Specular shading
  • Reflection lines
  • differentiability one order lower than surface
  • can be efficiently computed using GPU

C0 C2 C1

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Mesh Quality

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Visual inspection of “sensitive” attributes

  • Specular shading
  • Reflection lines
  • Curvature
  • Mean curvature
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Mesh Quality

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Visual inspection of “sensitive” attributes

  • Specular shading
  • Reflection lines
  • Curvature
  • Gauss curvature
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Mesh Quality Criteria

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Smoothness

  • Low geometric noise
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Mesh Quality Criteria

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Smoothness

  • Low geometric noise

Fairness

  • Simplest shape
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Mesh Quality Criteria

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Smoothness

  • Low geometric noise

Fairness

  • Simplest shape

Adaptive tesselation

  • Low complexity
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Mesh Quality Criteria

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Smoothness

  • Low geometric noise

Fairness

  • Simplest shape

Adaptive tesselation

  • Low complexity

Triangle shape

  • Numerical Robustness
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Mesh Optimization

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Smoothness

  • Smoothing

Fairness

  • Fairing

Adaptive tesselation

  • Decimation

Triangle shape

  • Remeshing
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Summary

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Invariants as overarching theme

  • shape does not depend on Euclidean motions (no stretch)
  • metric & curvatures
  • smooth continuous notions to discrete notions
  • generally only as averages
  • different ways to derive same equations
  • DEC: discrete exterior calculus, FEM, abstract measure

theory.

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Literature

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  • Book: Chapter 3
  • Taubin: A signal processing approach to fair surface design,

SIGGRAPH 1996

  • Desbrun et al. : Implicit Fairing of Irregular Meshes using Diffusion

and Curvature Flow, SIGGRAPH 1999

  • Meyer et al.: Discrete Differential-Geometry Operators for

Triangulated 2-Manifolds, VisMath 2002

  • Wardetzky et al.: Discrete Laplace Operators: No free lunch, SGP

2007

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Next Time

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3D Scanning

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http://cs599.hao-li.com

Thanks!

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