Implicit Fairing of Arbitrary Meshes Using Diffusion and Curvature Flow

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Implicit Fairing of Arbitrary Meshes Using
Diffusion and Curvature Flow

• Mathieu Desbrun Mark Meyer Peter Schröder
  Alan Barr
• Caltech

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Smoothing Arbitrary Meshes

• Context
• Meshes often come from 3D scans
• Noisy
• Non uniform triangles
• No parameterization, just positions in space
• Smoothing removal of rough detail features
• while preserving the global shape

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Current Smoothing Results

• OK on semi-uniform meshes, Problems
  with non uniform
• yet slow on large meshes
  meshes distortion!

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Contributions (I)

• Implicit fairing
• Implicit integration scheme for efficiency
• Exact volume preservation
• Constraints during smoothing

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Contributions (II)

• Two new operators
• Laplace operator that respects length scale
• Curvature normal operator, for better shape
  denoising

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Previous Approaches (I)

• Early work (CAGD)
• Minimization of complex functionals
• Ex
• Taubin 95 (and follow-ons)
• Signal processing approach using L(M)

(low-pass filter)
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Previous Approaches (II)

• Iterative energy minimization
• Membrane energy
• Follows the Laplacian
• Iterations of an explicit Euler scheme
• 
  (Diffusion)

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Implicit Integration (I)

• Diffusion
• Strict stability requirements
• Small time steps for explicit integration
• Implicit schemes are strongly suggested
• BW98

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Implicit Integration (II)
Explicit Euler scheme
Implicit Euler scheme
vs.
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Introducing Implicit Fairing

• Explicit integration

• Small time steps for large meshes

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Introducing Implicit Fairing

• Implicit integration for larger time steps

• (PB) Conjugate Gradient for efficiency
• Equivalent to

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Volume Preservation

• Diffusion induces shrinkage!
• Enforce exact volume preservation
• Compute volume
• Rescale after integration step(s)
• Incorporate other invariants
• Volume, surface area, ...
• Build directly into PDE

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Constraint Enforcement

• Fixed points/Fixed regions
• Set the Laplacian to zero
• Soft constraints
• Locally adjust smoothing amount (l)
• Spray-paint the object directly
• Complex constraints BW98

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Where are we now? (I)

• Implicit fairing
• Implicit integration scheme for efficiency
• Exact volume preservation
• Constraints during smoothing

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Where are we now? (II)

• Two new operators
• Laplace operator that respects length scale
• Curvature normal operator, for better shape
  denoising

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Regular Diffusion

• Problem with the umbrella operator
• 
  smoothed as much as
• It assumes a regular parameterization!

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Regular Diffusion Effects
Initial mesh
Regular fairing
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Accurate Diffusion

• Length-scale weighted operator
• (Fujiwara operator)
• Creates scale-dependent operator
• But stability issues
• Explicit time step too small dt lt lmin2 / l
• Implicit integration a must!

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Regular/Accurate Diffusion
Scale-dependent fairing
Initial mesh
Regular fairing
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More Results With Diffusion
Initial mesh
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More Results With Diffusion
After upsampling and one fairing
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Shape Denoising

• Why diffuse the noise?
• Normal component
• Tangent component, distorting the shape!
• Alternatively curvature flow equation
  (Laplace-Beltrami)
• Problem how can we find k n reliably?

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Discrete Curvature

• Differential geometry definition
• After derivation
• Dependent on angles and lengths
• Zero curvature ensured when flat

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Curvature operator
Curvature visualization using false colors
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Curvature Flow Evaluation
Initial Mesh
Regular Diffusion
Curvature Flow
Improved Diffusion
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Results on 3D Scanned Data
Initial mesh
After one fairing
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Results on 3D Scanned Data
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Video
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Conclusion

• New tools for irregular meshes
• Implicit fairing
• Adds efficiency/stability
• Improved Laplace operator
• Limits distortion on arbitrary meshes
• New curvature normal operator
• Shape denoising with curvature flow

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Future Work

• More smoothing
• Other kinds of geometrical flow
• Edge preservation in computer vision
• Modeling
• Prescribed curvature normals
• Animation
• Applications of curvature operator
• Info? http//www.multires.caltech.edu/mathieu