Locally Injective Mappings

Transcription

1 8/6/ Locally Injective Mappings Christian Schüller 1 Ladislav Kavan 2 Daniele Panozzo 1 Olga Sorkine-Hornung 1 1 ETH Zurich 2 University of Pennsylvania Locally Injective Mappings Popular tasks in geometric processing 2D / 3D deformations Parametrization Christian Schüller # 2 1

2 8/6/ Locally Injective Mappings Solve variational problems by minimize an energy Deformation energies Dirichlet (harmonic) Laplacian (bi-harmonic) ARAP (rotation invariant) [Sorkine and Alexa 2007] Christian Schüller # 3 Locally Injective Mappings In the discrete setting such deformation energies can introduce undesired inverted elements: Christian Schüller # 4 2

3 8/6/ Our goals 1. Deform a 2D/3D mesh minimizing an arbitrary deformation energy 2. Subject to positional constraints 3. Interactive response 4. Prevent inverted elements Christian Schüller # 5 Related work Penalize inverted elements [Irving et al. 2004], [Müller et al. 2004], [Adams et al. 2008], [Stomakhin et al. 2012],... Prevent inverted elements [Hormann et al. 2000], [Sander et al. 2001], [Degener et al. 2003], [Freitag-Diachin et al. 2004], [Angelidis et al. 2004], [Sheffer et al. 2005], [W. von Funck et al. 2006] Extremal Quasi Conformal Maps [Weber et al. 2012] Bounded Distortion Mapping [Lipman 2012] Extension to 3D at SIGGRAPH Christian Schüller # 6 3

4 8/6/ Problem Statement Minimize a deformation energy Subject to positional constraints non-flip constraints Christian Schüller # 7 As Hard-constraints As Soft-constraints Positional constraints? : linear combination of vertex positions : target positions : positional constraint weight Christian Schüller # 8 4

5 8/6/ Non-flip constraint We define a non-linear constraint based on the signed area/volume of a single element : signed area function of element j : minimal area/volume Christian Schüller # 9 Interior point method Solving the non-linear problem with a state of the art Interior Point method 15 min Christian Schüller # 10 5

6 8/6/ Our approach We apply the non-flip constraints as a sum of barrier terms to the current energy : barrier function of element j : barrier weight constant Christian Schüller # 11 Barrier term : rest pose area of element j Note that should be continuous Christian Schüller # 12 6

7 8/6/ Barrier term We propose to use an inverse barrier function We chose to be a simple cubic function Christian Schüller # 13 Barrier term result in final barrier: Christian Schüller # 14 7

8 8/6/ Modified deformation energy original deformation energy positional soft-constraints barrier term preventing inverted elements Christian Schüller # 15 Optimization To minimize we use a variant of the Levenberg-Marquardt method Basically the Newton method with adaptive line search and a regularized Hessian: chosen as small as possible to make the Hessian invertible using sparse Cholesky decomposition Christian Schüller # 16 8

9 8/6/ Problem Even for simple cases this optimization can fail Initial constraints Fail case What we expect Reason: For large the step size approaches zero causing numerical rounding problems Christian Schüller #17 Substepping A novel substepping strategy based on the condition number of the current Hessian Well-conditioned: Rely on search direction Ill-conditioned: Relax positional constraints constrained vertex target constraints current constraint Christian Schüller #18 9

10 8/6/ Stress Test Christian Schüller # 19 Results 10

11 8/6/ 2D Deformation ARAP our Christian Schüller # 21 Image Warping Christian Schüller # 22 11

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14 8/6/ Comparison to Bounded Distortion 2 iterations, 39s 154 iterations, 6.5s [Lipman 2012] Christian Schüller # 28 14

15 8/6/ Live Demo Limitations Inversion free starting point needed Barriers change global and local minima of original deformation energy No guarantees on fulfilling positional constraints or the convergence of the method Christian Schüller # 30 15

16 8/6/ Conclusion Method which creates locally injective mappings Handles arbitary deformation energies Allows user interaction Works for 2D and 3D meshes Christian Schüller # 31 Future Work Prevent global overlaps Extend to hard positional constraints Bound the conformal disortion Incorporate remeshing techniques Christian Schüller # 32 16

17 8/6/ Thank You Locally Injective Mappings Christian Schüller Ladislav Kavan Daniele Panozzo Olga Sorkine-Hornung Demo: We thank Yaron Lipman for sharing the MATLAB implementation of his Bounded Distortion method [Lip12].This work was supported in part by the ERC grant imodel (StG ). 17