SMI 2012 As-Conformal-As-Possible Discrete Volumetric Mapping - PowerPoint PPT Presentation

SMI 2012 As-Conformal-As-Possible Discrete Volumetric Mapping Gilles-Philippe Paillé Pierre Poulin University of Montreal, Canada May 23, 2012

Context Parameterization 2D 3D 2

Context Parameterization Some popular codomains Cube Sphere Polycube 3

Context Parameterization ● Energy – Distortion measure ● Measured transformation – Rotation – Uniform scale – Non-uniform scale – Shear 4

Context Parameterization ● Energy – Distortion measure ● Measured transformation – Rotation – Uniform scale – Non-uniform scale – Shear 5

Context Parameterization ● Energy – Distortion measure ● Desired transformation – Rotation – Uniform scale – Non-uniform scale – Shear 6

Context Parameterization ● Conformal maps – Local transformation = Uniform scale * Rotation 7

Context Conformal maps – Jacobian matrix 8

Context Conformal maps – Jacobian matrix 9

Context Conformal maps – Jacobian matrix 10

Context Conformal maps – Jacobian matrix 11

Context Conformal maps – 3D? ● Problem – 2D conformality is possible – 3D conformality is not possible ● Our solution – Be as-conformal-as-possible (ACAP) – Linear method based on Cauchy-Riemann 12

Context Conformal maps – 3D? ● Problem – 2D conformality is possible – 3D conformality is not possible ● Our solution – Be as-conformal-as-possible (ACAP) – Linear method based on Cauchy-Riemann 13

Context Conformal maps – 3D? Harmonic way 2D 3D First order Cauchy-Riemann ? Second order Harmonic Harmonic 14

Context Conformal maps – 3D? ACAP way 2D 3D First order Cauchy-Riemann Generalized CR Second order Harmonic ACAP 15

Method Idea Optimize jacobian matrix on 3 canonical planes 16

Method Formulation Optimize jacobian sub-matrices 17

Method Formulation A compact form 18

Method Formulation A more compact form 19

Method Rotation invariance ● Problem – Operator D supposes there's no rotation ● Solution – Estimate rotation contained in – Cancel rotation part with 20

Method Algorithm ● Goal – Minimize the energy functional ● Steps 1) Surface conformal map 2) Volume harmonic map 3) Local rotation estimation 4) Volume ACAP map 21

Results Tetrahedra Timing Harmonic : 48 secs 5569k ACAP : 154 secs 22

Results Remarks ● Approximately 3x longer to compute ● Results are always better than harmonic 23

Results Variations ● Balance uniform scale and orthogonality – Use a parameter – : Orthogonality only – : Uniform scale only 24

Conclusion Limitations ● Element inversion – Cannot be garanteed with linear energies – Especially in concave parts ● Fixed boundaries – Limit the movement of the interior – Conformal surface maps are not optimal 25

Conclusion Future work ● Volume-aware surface parameterization 26

Conclusions Future work ● Volume-aware surface parameterization 27

Conclusions Future work ● Smart singularity placement 28

Conclusions ● Generalized Cauchy-Riemann ● Fast ● Easy to implement ● Generalize to N dimensions ● To do – Free boundaries – Smart placement of singularities – Optimal codomain geometry 29

Conclusions Thank you! 30