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

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 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 2

What is Spectral Geometry Processing? differential eigenvalues operator eigenvectors 3

What is Spectral Geometry Processing? max … = min Analogy: “Fourier transform” for surfaces 4

Why - Coordinate Invariant eigenvalues eigenvalues 15 15 10 10 eigen value eigen value 5 5 0 0 5 10 15 0 0 5 10 15 # # 5

Why - (Almost) Invariant to Tessellation eigenvalues eigenvalues 30 30 25 25 20 20 eigen value eigen value 15 15 10 10 5 5 0 0 0 5 10 15 0 5 10 15 # # 6

Why - (Almost) Invariant to Tessellation eigenvalues eigenvalues 30 30 25 25 20 20 eigen value eigen value 15 15 10 10 5 5 0 0 0 5 10 15 0 5 10 15 # # 7

Why - Isometry Invariant Benefits from Laplace - Beltrami operator 70 70 70 60 60 60 50 50 50 40 40 40 30 30 30 20 20 20 10 10 10 0 0 0 0 5 10 15 0 5 10 15 0 5 10 15 8

Isometry Invariance Is it a feature or a bug? 9

Sensitivity to Metric Distortion https://en.wikipedia.org/wiki/USP2 10

(Discrete) Differential Operators 11

Laplace - Beltrami Operator (intrinsic) • Discrete cotangent Laplacian 1 X ∆ p i = (cot α ij + cot β ij )( p i − p j ) 2 A i j ∈ N ( i ) Solomon, et al. 2014 12

Laplace - Beltrami Operator (intrinsic) • Discrete cotangent Laplacian 1 X ∆ p i = (cot α ij + cot β ij )( p i − p j ) (cot α ij + cot β ij )( 2 A i j ∈ N ( i ) i ) • Key idea: 
 Laplace only depends on edge lengths ! 
 Edge length is a intrinsic quantity 
 Solomon, et al. 2014 13

Not Purely Intrinsic Operators • Mixture of intrinsic and extrinsic information 70 • Existing operators: 
 60 50 - Anisotropic Laplace 
 40 - Modified Dirichlet energy 
 30 20 anisotropic How sensitive? 10 Laplace 0 0 5 10 15 Andreux et al. 2014, Hildebrandt et al. 2012 14

Quaternionic Dirac Operator • Definition: Crane, et al. 2011 15

Quaternionic Dirac Operator • Definition: Crane, et al. 2011 16

Quaternionic Dirac Operator • Discrete Dirac: • Key idea: 
 depends on edge vectors 
 (rather then edge length) Crane, et al. 2011 17

Square of Dirac Operator Normal Laplace Relative Dirac Operator intrinsic extrinsic Crane. 2013 18

Discretization • Discrete relative Dirac • Matrix form 19

Basic Properties of Relative Dirac (Dirac) (relative Dirac) D N ψ = − dN ∧ d ψ | d f | 2 Crane, et al. 2011 20

Basic Properties of Relative Dirac (Dirac) (relative Dirac) D N ψ = − dN ∧ d ψ | d f | 2 • First order, self - adjoint and elliptic operator 
 countable eigenvalues and eigenvectors Crane, et al. 2011 21

Basic Properties of Relative Dirac (Dirac) (relative Dirac) D N ψ = − dN ∧ d ψ | d f | 2 • 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 Crane, et al. 2011 22

Basic Properties of Relative Dirac (Dirac) (relative Dirac) D N ψ = − dN ∧ d ψ | d f | 2 • 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 Crane, et al. 2011 23

Basic Properties of Relative Dirac (Dirac) (relative Dirac) D N ψ = − dN ∧ d ψ | d f | 2 • First order, self - adjoint and elliptic operator 
 countable eigenvalues and eigenvectors ! ! t n a • The operator is not coordinate invariant 
 i r a v n i y r eigenvalues are coordinate invariant 
 t e m o s i t eigenvectors are determined up to a constant o N Crane, et al. 2011 24

From Intrinsic to Extrinsic … Laplace relative Dirac 25

Applications 26

Surface Texture Classification 27

Patch Classification 28

Laplace eigenvalues 29

Infinite Potential Well • Modified sigmoid function • Operator with potential well 
 Ex: 30

Laplacian with Infinite Potential Well eigenvalues 31

Patch Classification Maaten, et al. 2008 32

Segmentation Step 1: Adapt global point signature to the magnitude of Dirac eigenvectors point on the surface eigenvalues Rustamov, et al. 2007 33

Segmentation Step 2: Apply the consensus segmentation algorithm random initialized k - means Rodola, et al. 2014 34

anisotropic modified Dirac (ours) Laplace Dirichlet Segmentation 35

Segmentation Modified Dirichlet Anisotropic Laplace Dirac (ours) 36

Segmentation Modified Dirichlet Anisotropic Laplace Dirac (ours) 37

Correspondence • Laplace cannot differentiate between bumped out/in ? 38

Heat Kernel Signature Sun, et al. 2009 39

Correspondence • Adapt heat kernel signature to the Dirac operator Dirac kernel signature heat kernel signature 40

Correspondence • Adapt heat kernel signature to the Dirac operator Dirac kernel signature heat kernel signature 41

Correspondence Dirac kernel signature heat kernel signature 42

Conclusion • Extrinsic properties play an important role in shape analysis 43

Conclusion • Extrinsic properties play an important role in shape analysis • Our family of operators gives user the flexibility to emphasize intrinsic/extrinsic properties 44

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?) 45

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? 46

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? https://github.com/alecjacobson/gptoolbox.git 47

relative Dirac operator Laplace-Beltrami Thank you! Hsueh - Ti Derek Liu, hsuehtil@andrew.cmu.edu