A Dirac Operator for Extrinsic Shape Analysis Hsueh - Ti Derek Liu 1 - - PowerPoint PPT Presentation

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A Dirac Operator for Extrinsic Shape Analysis Hsueh - Ti Derek Liu 1 - - PowerPoint PPT Presentation

May 17, 2023 314 likes •809 views

relative Dirac operator Laplace-Beltrami A Dirac Operator for Extrinsic Shape Analysis Hsueh - Ti Derek Liu 1 , Alec Jacobson 2 , Keenan Crane 1 1 Carnegie Mellon University 2 University of Toronto 1 Outline Goal : extend spectral geometry

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A Dirac Operator for Extrinsic Shape Analysis

Hsueh-Ti Derek Liu1, Alec Jacobson2, Keenan Crane1

Laplace-Beltrami relative Dirac operator

1Carnegie Mellon University 2University of Toronto

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Outline

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Goal: extend spectral geometry processing

  • Traditionally: intrinsic only (point-to-point distance)
  • Today: extrinsic information (bending in space)

Basic idea: develop new differential operators

  • Instead of standard Laplacian, use relative Dirac operator

Applications:

  • Classification, segmentation, correspondence
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What is Spectral Geometry Processing?

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differential

  • perator

eigenvalues eigenvectors

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What is Spectral Geometry Processing?

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Analogy: “Fourier transform” for surfaces

= …

max min

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Why - Coordinate Invariant

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#

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eigen value

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#

5 10 15

eigen value

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eigenvalues eigenvalues

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#

5 10 15

eigen value

5 10 15 20 25 30

#

5 10 15

eigen value

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Why - (Almost) Invariant to Tessellation

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eigenvalues eigenvalues

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#

5 10 15

eigen value

5 10 15 20 25 30

#

5 10 15

eigen value

5 10 15 20 25 30

Why - (Almost) Invariant to Tessellation

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eigenvalues eigenvalues

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Why - Isometry Invariant

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Benefits from Laplace-Beltrami operator

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Isometry Invariance Is it a feature or a bug?

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Sensitivity to Metric Distortion

https://en.wikipedia.org/wiki/USP2

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(Discrete) Differential Operators

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Laplace-Beltrami Operator (intrinsic)

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  • Discrete cotangent Laplacian

Solomon, et al. 2014

∆pi = 1 2Ai X

j∈N (i)

(cot αij + cot βij)(pi − pj)

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Laplace-Beltrami Operator (intrinsic)

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  • Discrete cotangent Laplacian
  • Key idea: 


Laplace only depends on edge lengths!
 Edge length is a intrinsic quantity 


Solomon, et al. 2014

∆pi = 1 2Ai X

j∈N (i)

(cot αij + cot βij)(pi − pj)

i)

(cot αij + cot βij)(

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Not Purely Intrinsic Operators

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  • Mixture of intrinsic and

extrinsic information

  • Existing operators:

  • Anisotropic Laplace

  • Modified Dirichlet energy


5 10 15 10 20 30 40 50 60 70

anisotropic Laplace

Andreux et al. 2014, Hildebrandt et al. 2012

How sensitive?

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Quaternionic Dirac Operator

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  • Definition:

Crane, et al. 2011

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Quaternionic Dirac Operator

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  • Definition:

Crane, et al. 2011

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  • Discrete Dirac:
  • Key idea: 


depends on edge vectors 
 (rather then edge length)

Quaternionic Dirac Operator

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Crane, et al. 2011

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Square of Dirac Operator

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intrinsic extrinsic

Laplace Normal

Relative Dirac Operator

  • Crane. 2013
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Discretization

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  • Discrete relative Dirac
  • Matrix form
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Basic Properties of Relative Dirac

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Crane, et al. 2011

DNψ = −dN ∧ dψ |d f|2

(Dirac) (relative Dirac)

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Basic Properties of Relative Dirac

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Crane, et al. 2011

DNψ = −dN ∧ dψ |d f|2

(Dirac) (relative Dirac)

  • First order, self-adjoint and elliptic operator


countable eigenvalues and eigenvectors

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Basic Properties of Relative Dirac

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Crane, et al. 2011

  • First order, self-adjoint and elliptic operator


countable eigenvalues and eigenvectors

  • The operator is not coordinate invariant


eigenvalues are coordinate invariant 
 eigenvectors are determined up to a constant

DNψ = −dN ∧ dψ |d f|2

(Dirac) (relative Dirac)

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Basic Properties of Relative Dirac

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Crane, et al. 2011

  • First order, self-adjoint and elliptic operator


countable eigenvalues and eigenvectors

  • The operator is not coordinate invariant


eigenvalues are coordinate invariant 
 eigenvectors are determined up to a constant

DNψ = −dN ∧ dψ |d f|2

(Dirac) (relative Dirac)

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Basic Properties of Relative Dirac

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Crane, et al. 2011

  • First order, self-adjoint and elliptic operator


countable eigenvalues and eigenvectors

  • The operator is not coordinate invariant


eigenvalues are coordinate invariant 
 eigenvectors are determined up to a constant

DNψ = −dN ∧ dψ |d f|2

(Dirac) (relative Dirac)

N

  • t

i s

  • m

e t r y i n v a r i a n t ! !

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From Intrinsic to Extrinsic

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Laplace relative Dirac

…

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Applications

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Surface Texture Classification

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Patch Classification

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Laplace

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eigenvalues

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Infinite Potential Well

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  • Modified sigmoid function
  • Operator with potential well


Ex:

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eigenvalues

Laplacian with Infinite Potential Well

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Patch Classification

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Maaten, et al. 2008

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Segmentation

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Step 1: Adapt global point signature to the magnitude

  • f Dirac

Rustamov, et al. 2007

point on the surface eigenvalues eigenvectors

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Segmentation

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Step 2: Apply the consensus segmentation algorithm

Rodola, et al. 2014

random initialized k-means

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modified Dirichlet anisotropic Laplace Dirac (ours)

Segmentation

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Segmentation

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Modified Dirichlet Anisotropic Laplace Dirac (ours)

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Segmentation

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Modified Dirichlet Anisotropic Laplace Dirac (ours)

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  • Laplace cannot differentiate between bumped out/in

Correspondence

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?

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Heat Kernel Signature

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Sun, et al. 2009

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Correspondence

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  • Adapt heat kernel signature to the Dirac operator

heat kernel signature Dirac kernel signature

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Correspondence

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  • Adapt heat kernel signature to the Dirac operator

heat kernel signature Dirac kernel signature

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Correspondence

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heat kernel signature Dirac kernel signature

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Conclusion

  • Extrinsic properties play an important role in shape

analysis

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Conclusion

  • Extrinsic properties play an important role in shape

analysis

  • Our family of operators gives user the flexibility to

emphasize intrinsic/extrinsic properties

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Conclusion

  • Extrinsic properties play an important role in shape

analysis

  • Our family of operators gives user the flexibility to

emphasize intrinsic/extrinsic properties

  • Many other possible applications (machine learning?

quad mesh?)

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Conclusion

  • Extrinsic properties play an important role in shape

analysis

  • Our family of operators gives user the flexibility to

emphasize intrinsic/extrinsic properties

  • Many other possible applications (machine learning?

quad mesh?)

  • What are other operators can we use to capture

different geometric quantities?

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Conclusion

  • Extrinsic properties play an important role in shape

analysis

  • Our family of operators gives user the flexibility to

emphasize intrinsic/extrinsic properties

  • Many other possible applications (machine learning?

quad mesh?)

  • What are other operators can we use to capture

different geometric quantities?

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https://github.com/alecjacobson/gptoolbox.git

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Thank you!

Hsueh-Ti Derek Liu, hsuehtil@andrew.cmu.edu

Laplace-Beltrami relative Dirac operator