research-article

A Splitting Scheme for Flip-Free Distortion Energies

Authors:

Justin SolomonAuthors Info & Claims

SIAM Journal on Imaging Sciences, Volume 15, Issue 2

Pages 925 - 959

https://doi.org/10.1137/21M1433058

Published: 01 January 2022 Publication History

Abstract

We introduce a robust optimization method for flip-free distortion energies used, for example, in parametrization, deformation, and volume correspondence. This method can minimize a variety of distortion energies, such as the symmetric Dirichlet energy and our new symmetric gradient energy. We identify and exploit the special structure of distortion energies to employ an operator splitting technique, leading us to propose a novel alternating direction method of multipliers (ADMM) algorithm to deal with the nonconvex, nonsmooth nature of distortion energies. The scheme results in an efficient method where the global step involves a single matrix multiplication and the local steps are closed-form per-triangle/per-tetrahedron expressions that are highly parallelizable. The resulting general-purpose optimization algorithm exhibits robustness to flipped triangles and tetrahedra in initial data as well as during the optimization. We establish the convergence of our proposed algorithm under certain conditions and demonstrate applications to parametrization, deformation, and volume correspondence.

References

[1]

4MULE8, The Helm of Glencairn Mesh, https://www.thingiverse.com/thing:1243621 (2016).

[2]

N. Aigerman and Y. Lipman, Injective and bounded distortion mappings in $3$d, ACM Trans. Graph., 32 (2013), 106.

[3]

N. Aigerman, R. Poranne, and Y. Lipman, Lifted bijections for low distortion surface mappings, ACM Trans. Graph., 33 (2014), 69.

[4]

AJade, Pegasus for 28mm Tabletop Roleplaying Mesh, https://www.thingiverse.com/thing:3955356 (2019).

[5]

K. Akeley, J. D. Foley, D. F. Sklar, M. McGuire, J. F. Hughes, A. can Dam, and S. K. Feiner, Computer Graphics -- Principles and Practice, Addison-Wesley, Upper Saddle River, NJ, 2013.

[6]

N. Al-Badri and J. N. Nelles, Nefertiti Mesh, https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/ (2020).

[7]

H. Attouch, J. Bolte, P. Redont, and A. Soubeyran, Proximal alternating minimization and projection methods for nonconvex problems: An approach based on the Kurdyka-łojasiewicz inequality, Math. Oper. Res., 35 (2010), pp. 438--457.

[8]

H. Attouch, J. Bolte, and B. F. Svaiter, Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward--backward splitting, and regularized Gauss--Seidel methods, Math. Program., 137 (2013), pp. 91--129.

[9]

I. Baran, Stuffedtoy Mesh, https://github.com/libigl/libigl-tutorial-data (2007).

[10]

billyd, Cat Stretch Mesh, https://www.thingiverse.com/thing:1565405 (2016).

[11]

Blender Foundation, Blender -- Free and Open Source 3D Creation Suite, https://www.blender.org (2020).

[12]

J. Bolte, A. Daniilidis, and A. Lewis, The Łojasiewicz inequality for nonsmooth subanalytic functions with applications to subgradient dynamical systems, SIAM J. Optim., 17 (2007), pp. 1205--1223.

[13]

D. Bommes, H. Zimmer, and L. Kobbelt, Mixed-integer quadrangulation, ACM Trans. Graph., 28 (2009), 77.

[14]

S. Bouaziz, M. Deuss, Y. Schwartzburg, T. Weise, and M. Pauly, Shape-up: Shaping discrete geometry with projections, Comput. Graph. Forum, 31 (2012), pp. 1657--1667.

[15]

S. Bouaziz, S. Martin, T. Liu, L. Kavan, and M. Pauly, Projective dynamics: Fusing constraint projections for fast simulation, ACM Trans. Graph., 33 (2014), 154.

[16]

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2011), pp. 1--122.

[17]

G. Brown and R. Narain, WRAPD: Weighted rotation-aware ADMM for parametrization and deformation, ACM Trans. Graph., 40 (2021), 82.

[18]

Catiav5ftw, Centrifugal Compressor Mesh, https://www.thingiverse.com/thing:1165042 (2015).

[19]

I. Chao, U. Pinkall, P. Sanan, and P. Schröder, A simple geometric model for elastic deformations, ACM Trans. Graph. 29 (2010), 38.

[20]

E. Chien, Z. Levi, and O. Weber, Bounded distortion parametrization in the space of metrics, ACM Trans. Graph., 35 (2016), 215.

[21]

G. P. T. Choi and C. H. Rycroft, Density-equalizing maps for simply connected open surfaces, SIAM J. Imaging Sci., 11 (2018), pp. 1134--1178.

[22]

S. Claici, M. Bessmeltsev, S. Schaefer, and J. Solomon, Isometry-aware preconditioning for mesh parameterization, Comput. Graph. Forum, 36 (2017), pp. 37--47.

[23]

colinfizgig, Maltese Falcon Mesh, https://www.thingiverse.com/thing:46631 (2013).

[24]

K. Crane, Blub mesh, https://www.cs.cmu.edu/~kmcrane/Projects/ModelRepository/ (2015).

[25]

T. Davis, SuiteSparse: A Sparse Matrix Algorithm Suite (Version 5.8.1), https://people.engr.tamu.edu/davis/welcome.html (2020).

[26]

M. Desbrun, M. Meyer, and P. Alliez, Intrinsic parameterizations of surface meshes, Comput. Graph. Forum, 21 (2002), pp. 209--218.

[27]

X. Du, N. Aigerman, Q. Zhou, S. Z. Kovalsky, Y. Yan, D. M. Kaufman, and T. Ju, Lifting simplices to find injectivity, ACM Trans. Graph., 39 (2020), 120.

[28]

M. Eck, T. DeRose, T. Duchamp, H. Hoppe, M. Lounsbery, and W. Stuetzle, Multiresolution analysis of arbitrary meshes, In Proceedings of the 22nd Annual Conference on Computer Graphics and Interactive Techniques, SIGGRAPH '95, ACM, New York, pp. 173--182, 1995.

[29]

D. Ezuz, J. Solomon, and M. Ben-Chen, Reversible harmonic maps between discrete surfaces, ACM Trans. Graph., 38 (2019), 15.

[30]

Y. Fang, M. Li, C. Jiang, and D. M. Kaufman, Guaranteed globally injective 3D deformation processing, ACM Trans. Graph., 40 (2021), 75.

[31]

L. P. Franca, An algorithm to compute the square root of a 3x3 positive definite matrix, Comput. Math. Appl., 18 (1989), pp. 459--466.

[32]

X.-M. Fu, Y. Liu, and B. Guo, Computing locally injective mappings by advanced MIPS, ACM Trans. Graph., 34 (2015), 71.

[33]

W. Gao, D. Goldfarb, and F. E. Curtis, ADMM for multiaffine constrained optimization, Optim. Methods Softw., 35 (2020), pp. 257--303.

[34]

V. Garanzha, I. Kaporin, L. Kudryavtseva, F. Protais, N. Ray, and D. Sokolov, Foldover-free maps in 50 lines of code, ACM Trans. Graph., 40 (2021), 102.

[35]

M. Gillespie, B. Springborn, and K. Crane, Discrete conformal equivalence of polyhedral surfaces, ACM Trans. Graph., 40 (2021), 103.

[36]

J. C. Gower and G. B. Dijksterhuis, Procrustes Problems, Oxford Statist. Sci. Ser. 30. Oxford University Press, Oxford, 2004.

[37]

X. Gu and S.-T. Yau, Global conformal surface parameterization, in Proceedings of the 2003 Eurographics symposium on Geometry Processing, ACM, New York, pp. 127--137, 2003.

[38]

B. C. Hall, Lie Groups, Lie Algebras, and Representations -- An Elementary Introduction. Springer, Cham, 2015.

[39]

E. F. Hefetz, E. Chien, and O. Weber, A subspace method for fast locally injective harmonic mapping, Comput. Graph. Forum, 38 (2019), pp. 105--119.

[40]

H. Hencky, \itÜber die form des elastizitätsgesetzes bei ideal elastischen stoffen, Zeitschr. Tech. Phys., 9, 1928, pp. 215--220.

[41]

J. Holinaty, Brucewick Mesh, https://github.com/odedstein/meshes/tree/master/objects/brucewick (2020).

[42]

J. Holinaty, Mushroom Mesh, https://github.com/odedstein/meshes/tree/master/objects/mushroom (2020).

[43]

M. Hong, Z.-Q. Luo, and M. Razaviyayn, Convergence analysis of alternating direction method of multipliers for a family of nonconvex problems, SIAM J. Optim., 26 (2016), pp. 337--364.

[44]

Y. Hu, Q. Zhou, X. Gao, A. Jacobson, D. Zorin, and D. Panozzo, Tetrahedral meshing in the wild, ACM Trans. Graph., 37 (2018), 60.

[45]

hugoelec, Goat 5k Yuanmingyuan 12 Zodiac Animals Mesh, https://www.thingiverse.com/thing:42256 (2013).

[46]

hugoelec, Symmetrical Deer Head 10k Mesh, https://www.thingiverse.com/thing:149354 (2013).

[47]

A. Jacobson, Algorithms and Interfaces for Real-Time Deformation of 2D and 3D Shapes. PhD thesis, ETH Zürich, Zürich, 2013.

[48]

A. Jacobson, Geometry Processing - Parametrization (course), 2020.

[49]

A. Jacobson et al, libigl: A simple C++ geometry processing library, https://libigl.github.io/ (2018).

[50]

Z. Jiang, S. Schaefer, and D. Panozzo, Simplicial complex augmentation framework for bijective maps, ACM Trans. Graph., 36 (2017), 186.

[51]

F. Kälberer, M. Nieser, and K. Polthier, Quadcover - surface parameterization using branched coverings, Comput. Graph. Forum, 26 (2007), pp. 375--384.

[52]

S. Khashin, Solution of Cubic and Quartic Equations C++, manuscript.

[53]

S. Z. Kovalsky, N. Aigerman, R. Basri, and Y. Lipman, Large-scale bounded distortion mappings, ACM Trans. Graph., 34 (2015), 191.

[54]

B. Lévy, S. Petitjean, N. Ray, and J. Maillot, Least squares conformal maps for automatic texture atlas generation, ACM Trans. Graph., 21 (2002), pp. 362--371.

[55]

J. Li, A. M.-C. So, and W.-K. Ma, Understanding notions of stationarity in nonsmooth optimization: A guided tour of various constructions of subdifferential for nonsmooth functions, IEEE Signal Process. Mag., 37 (2020), pp. 18--31.

[56]

M. Li, D. M. Kaufman, V. G. Kim, J. Solomon, and A. Sheffer, OptCuts: Joint optimization of surface cuts and parameterization, ACM Trans. Graph., 37 (2018), 247.

[57]

Y. Lipman, Bounded distortion mapping spaces for triangular meshes, ACM Trans. Graph., 31 (2012), 108.

[58]

Y. Lipman, Bijective mappings of meshes with boundary and the degree in mesh processing, SIAM J. Imaging Sci., 7 (2014), pp. 1263--1283.

[59]

L. Liu, C. Ye, R. Ni, and X.-M. Fu, Progressive parameterizations, ACM Trans. Graph., 37 (2018), 41.

[60]

L. Liu, L. Zhang, Y. Xu, C. Gotsman, and S. J. Gortler, A local/global approach to mesh parameterization, in Proceedings of the Symposium on Geometry Processing, SGP '08, ACM, New York, 2008. pp. 1495--1504.

[61]

M3DM, Dead Tree - Tabletop Scater Terrain Mesh, https://www.thingiverse.com/thing:4105303 (2020).

[62]

H. Maron, M. Galun, N. Aigerman, M. Trope, N. Dym, E. Yumer, V. G. Kim, and Y. Lipman, Convolutional neural networks on surfaces via seamless toric covers, ACM Trans. Graph., 36 (2017), 71.

[63]

Max Planck Society e.V, SMPL Human Body Mesh, https://smpl.is.tue.mpg.de/en (2021).

[64]

A. McAdams, A. Selle, R. Tamstorf, J. Teran, and E. Sifakis, Computing the Singular Value Decomposition of 3x3 Matrices with Minimal Branching and Elementary Floating Point Operations, Technical report 1690, University of Wisconsin-Madison, Department of Computer Sciences, Madison, Wisconsin, 2011.

[65]

D. Minor, Making space for cloth simulations using energy minimization, in ACM SIGGRAPH 2018 Talks, ACM, New York, 2018, 41.

[66]

D. Minor and D. Corral, Smeat: ADMM based tools for character deformation, in SIGGRAPH Asia 2018 Technical Briefs, ACM, New York, 2018, 2.

[67]

P. Mullen, Y. Tong, P. Alliez, and M. Desbrun, Spectral conformal parameterization, Comput. Graph. Forum, 27 (2008), pp. 1487--1494.

[68]

MustangDave, Berry Bear Mesh, https://www.thingiverse.com/thing:1161576 (2015).

[69]

A. Myles, N. Pietroni, and D. Zorin, Robust field-aligned global parametrization, ACM Trans. Graph., 33 (2014), 135.

[70]

Alexander N., E. Saucan, and Y. Y. Zeevi, Geometry-based distortion measures for space deformation, Graph. Models, 100 (2018), pp. 12--25.

[71]

P. Neff, B. Eidel, and R. J. Martin, Geometry of logarithmic strain measures in solid mechanics, Arch. Ration. Mech. Anal., 222 (2016), pp. 507--572.

[72]

Yurii N., Lectures on Convex Optimization, Springer Optim. Appl. 137, Springer, Cham, Switzerland, 2018.

[73]

X. Nian and F. Chen, Planar domain parameterization for isogeometric analysis based on Teichmüller mapping, Comput. Methods Appl. Mech. Engi., 311 (2016), pp. 41--55.

[74]

W. Ouyang, Y. Peng, Y. Yao, J. Zhang, and B. Deng, Anderson acceleration for nonconvex ADMM based on Douglas-Rachford splitting, Comput. Graph. Forum, 39 (2020), pp. 221--239.

[75]

M. Overby, G. E. Brown, J. Li, and R. Narain, ADMM $\supseteq$ projective dynamics: Fast simulation of hyperelastic models with dynamic constraints, IEEE Trans. Vis. Comput. Graph., 23 (2017), pp. 2222--2234.

[76]

M. Overby, D. Kaufman, and N. Rahul, Globally injective geometry optimization with non-injective steps, Comput. Graph. Forum, 40 (2021), pp. 111--123.

[77]

paBlury, Small Cactus with Pot Mesh, https://www.thingiverse.com/thing:2720011 (2017).

[78]

K. Pal, C. Schüller, D. Panozzo, O. Sorkine-Hornung, and T. Weyrich, Content-aware surface parameterization for interactive restoration of historical documents, Comput. Graph. Forum, 33 (2014), pp. 401--409.

[79]

Z. Pan, H. Bao, and J. Huang, Subspace dynamic simulation using rotation-strain coordinates, ACM Trans. Graph., 34 (2015), 242.

[80]

M. Pauly, R. Keiser, L. P. Kobbelt, and M. Gross, Shape modeling with point-sampled geometry (3D octopus mesh), ACM Trans. Graph., 22 (2003), pp. 641--650.

[81]

U. Pinkall and K. Polthier, Computing discrete minimal surfaces and their conjugates, Exp. Math., 2 (1993), pp. 15--36.

[82]

Publicdomainvectors.org, Blue Robot Image, https://publicdomainvectors.org/en/free-clipart/Blue-robot/62916.html (2017).

[83]

Publicdomainvectors.org. Glossy Red Lips Image, https://publicdomainvectors.org/en/free-clipart/Glossy-red-lips/80649.html (2019).

[84]

M. Rabinovich, R. Poranne, D. Panozzo, and O. Sorkine-Hornung, Scalable locally injective mappings. ACM Trans. Graph., 36 (2017), 18.

[85]

R. Sanyal, Sk. M. Ahmed, M. Jaiswal, and K. N. Chaudhury, A scalable ADMM algorithm for rigid registration, IEEE Signal Process. Lett., 24 (2017), pp. 1453--1457.

[86]

R. Sawhney and K. Crane, Boundary first flattening, ACM Trans. Graph., 37 (2017), 5.

[87]

schlossbauer, Brain Ooze Mesh, https://www.thingiverse.com/thing:4169343 (2020).

[88]

schlossbauer, Slime Mold Mesh, https://www.thingiverse.com/thing:4756744 (2021).

[89]

P. Schmidt, J. Born, M. Campen, and L. Kobbelt, Distortion-minimizing injective maps between surfaces, ACM Trans. Graph., 38 (2019), 156.

[90]

P. Schmidt, M. Campen, J. Born, and L. Kobbelt, Inter-surface maps via constant-curvature metrics, ACM Trans. Graph., 39 (2020), 119.

[91]

J. Schreiner, A. Asirvatham, E. Praun, and H. Hoppe, Inter-surface mapping, ACM Trans. Graph., 23 (2004), pp. 870--877.

[92]

S. Sellán, N. Aigerman, and A. Jacobson, Developability of heightfields via rank minimization, ACM Trans. Graph., 39 (2020), 109.

[93]

N. Sharp and K. Crane, Variational surface cutting, ACM Trans. Graph., 37 (2018), 156.

[94]

A. Sheffer, B. Lévy, M. Mogilnitsky, and A. Bogomyakov, ABF++ : Fast and robust angle based flattening, ACM Trans. Graph., 24 (2004), pp. 311--330.

[95]

H. Shen, Z. Jiang, D. Zorin, and D. Panozzo, Progressive embedding, ACM Trans. Graph., 38 (2019), 32.

[96]

H. Si, Tetgen, a Delaunay-based quality tetrahedral mesh generator, ACM Trans. Math. Softw., 41 (2015), 11.

[97]

B. Smith, F. De Goes, and T. Kim, Analytic eigensystems for isotropic distortion energies, ACM Trans. Graph., 38 (2019), 3.

[98]

J. Smith and S. Schaefer, Bijective parameterization with free boundaries, ACM Trans. Graph., 34 (2015), 70.

[99]

Y. Soliman, D. Slepčev, and K. Crane, Optimal cone singularities for conformal flattening, ACM Trans. Graph., 37 (2018), 105.

[100]

J. Solomon, R. Rustamov, L. Guibas, and A. Butscher, Earth mover's distances on discrete surfaces, ACM Trans. Graph., 33 (2014), 67.

[101]

O. Sorkine and M. Alexa, As-rigid-as-possible surface modeling, in Proceedings of the Fifth Eurographics Symposium on Geometry Processing, SGP '07, A.K. Peters, Wellesleg, MA, 2007, pp. 109--116.

[102]

O. Sorkine and D. Cohen-Or, Camelhead Mesh, https://github.com/libigl/libigl-tutorial-data (2004).

[103]

B. Springborn, P. Schröder, and U. Pinkall, Conformal equivalence of triangle meshes, ACM Trans. Graph., 27 (2008), pp. 1--11.

[104]

J.-P. Su, X.-M. Fu, and L. Liu, Practical foldover-free volumetric mapping construction, Comput. Graph. Forum, 38 (2019), pp. 287--297.

[105]

J.-P. Su, C. Ye, L. Liu, and X.-M. Fu, Efficient bijective parameterizations, ACM Trans. Graph., 39 (2020), 111.

[106]

The Stanford 3D Scanning Repository, Bunny and Armadillo Meshes. http://graphics.stanford.edu/data/3Dscanrep/ (2020).

[107]

Toawi, Bread -- Robinson Crusoe Mesh, https://www.thingiverse.com/thing:4414133 (2020).

[108]

W. T. Tutte, How to draw a graph, Proc. London Math. Soc.(3), 13 (1963), pp. 743--767.

[109]

S. Vaswani, A. Mishkin, I. Laradji, M. Schmidt, G. Gidel, and S. Lacoste-Julien, Painless stochastic gradient: Interpolation, line-search, and convergence rates, in Advances in Neural Information Processing Systems, H. Wallach, H. Larochelle, A. Beygelzimer, F. d'Alché Buc, E. Fox, and R. Garnett, ed., Curran Associates, Red Hook, NY, 2019, pp. 3727--3740.

[110]

J. Wang and L. Zhao, Nonconvex generalization of alternating direction method of multipliers for nonlinear equality constrained problems, Results Control Optim., 2 (2021), 100009.

[111]

Y. Wang, Y. Wotai, and J. Zheng, Global convergence of ADMM in nonconvex nonsmooth optimization, J. Sci. Comput., 78 (2019), pp. 29--63.

[112]

O. Weber and D. Zorin, Locally injective parametrization with arbitrary fixed boundaries, ACM Trans. Graph., 33 (2014), 75.

[113]

Z. Xu, M. A. T. Figueiredo, X. Yuan, C. Studer, and T. Goldstein, Adaptive relaxed ADMM: Convergence theory and practical implementation, in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), IEEE Computer Society, Los Alamitos, CA, 2017, pp. 7389--7398.

[114]

YahooJAPAN, Monkey Mesh, https://www.thingiverse.com/thing:182232 (2013).

[115]

L. Yang, T. K. Pong, and X. Chen, Alternating direction method of multipliers for a class of nonconvex and nonsmooth problems with applications to background/foreground extraction, SIAM J. Imaging Sci., 10 (2017), pp. 74--110.

[116]

M.-H. Yueh, T. Li, W.-W. Lin, and S.-T. Yau, A novel algorithm for volume-preserving parameterizations of 3-manifolds, SIAM J. Imaging Sci., 12 (2019), pp. 1071--1098.

[117]

J. Zhang, Y. Duan, Y. Lu, M. K. Ng, and H. Chang, Bilinear constraint based ADMM for mixed Poisson-Gaussian noise removal, Inverse Probl. Imaging, 15 (2021), pp. 339--366.

[118]

J. Zhang, S. Ma, and S. Zhang, Primal-dual optimization algorithms over Riemannian manifolds: An iteration complexity analysis, Math. Program., 184 (2020), pp. 445--490.

[119]

J. Zhang, Y. Peng, W. Ouyang, and B. Deng, Accelerating ADMM for efficient simulation and optimization, preprint, arXiv:1909.00470, 2019.

[120]

T. Zhang and Z. Shen, A fundamental proof of convergence of alternating direction method of multipliers for weakly convex optimization, J. Inequal. Appl., 2019 (2019), pp. 1--21.

[121]

Y. Zhu, R. Bridson, and D. M. Kaufman, Blended cured quasi-Newton for distortion optimization, ACM Trans. Graph., 37 (2018), 40.

Cited By

Wang YGuo MSolomon J(2023)Variational quasi-harmonic maps for computing diffeomorphismsACM Transactions on Graphics10.1145/359210542:4(1-26)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592105
Edelstein MGuillen NSolomon JBen-Chen M(2023)A Convex Optimization Framework for Regularized Geodesic DistancesACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591523(1-11)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591523

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cover image SIAM Journal on Imaging Sciences

SIAM Journal on Imaging Sciences Volume 15, Issue 2

EISSN:1936-4954

DOI:10.1137/sjisbi.15.2

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© 2022, Society for Industrial and Applied Mathematics.

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Published: 01 January 2022

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Cited By

Wang YGuo MSolomon J(2023)Variational quasi-harmonic maps for computing diffeomorphismsACM Transactions on Graphics10.1145/359210542:4(1-26)Online publication date: 26-Jul-2023
https://dl.acm.org/doi/10.1145/3592105
Edelstein MGuillen NSolomon JBen-Chen M(2023)A Convex Optimization Framework for Regularized Geodesic DistancesACM SIGGRAPH 2023 Conference Proceedings10.1145/3588432.3591523(1-11)Online publication date: 23-Jul-2023
https://dl.acm.org/doi/10.1145/3588432.3591523

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