Variational Formulations and PDEs ononto Implicit Surfaces Good Bye Triangulated Surfaces

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Variational Formulations and PDEs on/onto
Implicit SurfacesGood Bye Triangulated Surfaces?

• Guillermo Sapiro
• University of Minnesota
• With M. Bertalmio, L. T. Cheng, S. Osher, and F.
  Memoli

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The problem

• Solve PDEs and variational problems for data
  defined from a generic surface onto a generic
  surface
• Surfaces from the data bases at Stanford and
  GATECH

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Variational problems and PDEs on surfaces?

• Mean curvature motion (Ilmanen, etc)
• Mathematical physics
• Computer graphics
• Image processing
• Regularization of inverse problems (e.g.,
  EEGMRI, e.g., Faugeras et al.)
• 3D surface mapping

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Examples

• Images from G. Turk , J. Dorsey, P. Thompson

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Classical approach

• Work with triangulated/meshed surfaces
• Discretization on non-uniform grids
• Projections onto triangulated surfaces
• Limit to functions (e.g., Kimmel)
• Very limited framework
• Work on surfaces mapped to the plane
• Loose the geometry, ads complexity
• No work on target surfaces reported

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Our approach

• Define the variational problems and PDEs
  following the theory of Harmonic Maps
• Well defined framework
• Represent the surfaces in implicit form
• Classical numerics on Cartesian grids
• No projections
• Motivated by Osher-Sethian level-sets and Osher
  variational levels-sets (though here the surface
  is not moving)

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Harmonic Maps Theory (Tang-Sapiro-Caselles)

• Find a map between manifolds (M,g) and (N,h)
  minimizing
• Gradient descent (p2)

See also Perona, Chan-Shen, Sochen et al., Hoppe
et al, Zorin et al., etc
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ExampleIsotropic direction smoothing
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ExampleColor Image Enhancement
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The embedding

• IM -gt N
• M is a generic surface (domain)
• With Bertalmio, Cheng, Osher
• N is a generic surface (target)
• With Memoli, Osher

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Embedding the domain surface

• Example IM-gtR
• A map from a generic domain surface onto the real
  line

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Embedding the domain surface (cont.)
Figure from G. Turk
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Embedding the domain surface (cont.)
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Embedding the domain surface (cont.)

• The gradient descent flow Heat flow on intrinsic
  surfaces
• All the computations are done in the Cartesian
  grid!

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Example
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Example
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Example
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Example
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L1 Denoising on Implicit Surfaces
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Example Curvature Smoothing
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Example
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Example L1 denoising with constraints
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Example Debluring
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Unit vector/color denoising on implicit surfaces

• I is a map from the 3D surface to the 3D unit
  sphere

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Example Chroma denoising
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Example General vector denoising
Original
L1
L2
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Pattern formation on implicit 3D surfaces

• Follows Turing, Kass-Witkin, Turk

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Examples
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Example
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Example
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Example
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Vector field visualization
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Vector field visualization
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Embedding the target manifold

• IM-gtN

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Embedding the target manifold (cont.)
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Concluding remarks

• No more need for triangulated surfaces for
  variational problems and PDEs
• Results locally independent of embedding function
• Extended to open domain and target surfaces
• Open problems More theory