Discrete Differential-Geometry Operators for Triangulated 2-Manifolds

1
Discrete Differential-Geometry Operators for
Triangulated 2-Manifolds

• Mark Meyer, Mathieu Desbrun, Peter Schroder, and
  Alan H. Barr
• Presented by Shadi Ashnai

2
Overview

• Motivation
• Definitions
• Derive differential geometry operators
• Accuracy analysis
• Generalization

3
Applications

• Mesh quality
• Mesh simplification
• Mesh modeling
• Denoising
• (a) mean curvature plot
• (b) principal directions
• (c-d) feature preserving denoising

4
Previous Works

• n(p) weighted average of the adjacent face
  normals
• Some introduced weights are
• Incident adjacent angles independent of the
  shape and length of the adjacent faces Thurmer
  and Wuthrich 98
• Computed by assuming that surface can locally be
  approximated by a sphere Max 99
• Taubin 95 introduced a complete derivation of
  surface properties approximating curvature
  tensors for polyhedral surface
• Hamann 93 simple way to determine principal
  curvature and direction using least-squared
  paraboloid fitting
• No easy way to selecting an appropriate tangent
  plane

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Definitions

• n Normal vector
• ?N(?) normal curvature for unit direction ?
• ?1 , ?2 principal curvatures
• e1 , e2 principal directions
• ?H mean curvature
• ?G gaussian curvature

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Definitions (cntd.)

• ?H (mean curvature)
• Mean curvature normal (Laplace-Beltrami)
  operator
• ?G (gaussian curvature)

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Discrete Geometry Attributes

• Mesh as linear approximation of an artibrary
  surface
• Define geometric quantities
• as spatial averages around each vertex
• Converge to the pointwise definitions knowing
  that
• The mesh is a good approximation of the initial
  surface
• The averages are made consistently
• The triangle mesh is non-degenerate
• etc.
• Example (Gaussian curvature)

8
Local Regions

• Select a linear finite element on each triangle
• Circumcenter - AVoronoi
• Barycenter - ABarycenter
• Generally we do the calculation with AM

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Discrete Mean Curvature
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Discrete Mean Curvature

• We compute mean curvature normal, know as
  Laplace-Beltrami operator.
• Once we compute this operator
• ?H is half the magnitude of K
• n is the normalized K

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Integral of Mean Curvature Normal

• Using conromal space parameters u,v
• Using Gausss theorm

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Minimizing Error Bounds

• Compare local spatial average with actual value
• Minimizing E
• Use voronoi region that minimizes x-xi (by
  definition)
• Sample with respect to curvature such that Cis
  vary slowly from patch to patch (assume)

13
Voronoi Region Area

• For non-obtuse triangle
• In presence of obtuse angles
• Use mixed regions

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Mixed Regions
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Mean curvature normal

• ?H half the magnitude of ?(xi)
• n
• normalized ?(xi), if ?Hltgt0
• Average of the 1-ring face normals, if ?H0

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Discrete Gaussian Curvature
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Integral of the Gaussian curvature

• Gauss-Bonnet theorm
• Discrete model
• For voronoi region

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Gaussian curvature

• We can use mixed regions the same as used for ?H
  and the region still satisfies the Gauss-Bonnet
  theorm

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Discrete Principal Curvatures
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Discrete Principal Curvatures

• ?G?1?2
• ?H(?1?2)/2

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Principal Directions

• Mean curvature as a quadrature of normal
  curvature samples

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Principal Directions

• Mean curvature interpretation
• Use normal curvature samples to fully determine
  curvature tensor
• Principal directions are eigenvectors of the
  curvature tensor

23
Least square fitting

• Symmetric curvature tensor
• For finding curvature in direction xixj
• Least square approximation

24
Results and Applications
25
Geometric Quality of Meshes

• (a) Loop surface from an 8-neighbor ring
• (b) Horse mesh
• (c-d) noisy scanned and smoothed head

26
Denoising

• (a) 3 noise added along the normal
• (b) Isotropic smoothing
• (c) Anisotropic smoothing

27
Generalization
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Generalization

• 2D-manifolds in nD
• Mean curvature normal
• generalize area and cot expressions for nD
• Gaussian curvature operator
• still holds (independent of embedding)
• 3D-manifolds in nD
• Compute gradient of the 1-ring volume for
  Beltrami operator

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Denoising of Arbitrary Fields

• (a) Original noisy vector field
• (b) Smoothed using Beltrami flow
• (c) Smoothed using anisotropic weighted flow