skip to main content
research-article

Controlling singular values with semidefinite programming

Published: 27 July 2014 Publication History

Abstract

Controlling the singular values of n-dimensional matrices is often required in geometric algorithms in graphics and engineering. This paper introduces a convex framework for problems that involve singular values. Specifically, it enables the optimization of functionals and constraints expressed in terms of the extremal singular values of matrices.
Towards this end, we introduce a family of convex sets of matrices whose singular values are bounded. These sets are formulated using Linear Matrix Inequalities (LMI), allowing optimization with standard convex Semidefinite Programming (SDP) solvers. We further show that these sets are optimal, in the sense that there exist no larger convex sets that bound singular values.
A number of geometry processing problems are naturally described in terms of singular values. We employ the proposed framework to optimize and improve upon standard approaches. We experiment with this new framework in several applications: volumetric mesh deformations, extremal quasi-conformal mappings in three dimensions, non-rigid shape registration and averaging of rotations. We show that in all applications the proposed approach leads to algorithms that compare favorably to state-of-art algorithms.

Supplementary Material

MP4 File (a68-sidebyside.mp4)

References

[1]
Aigerman, N., and Lipman, Y. 2013. Injective and bounded distortion mappings in 3d. ACM Trans. Graph. 32, 4, 106--120.
[2]
Alexa, M., Cohen-Or, D., and Levin, D. 2000. As-rigid-as-possible shape interpolation. Proc. SIGGRAPH, 157--164.
[3]
Alexa, M. 2002. Linear combination of transformations. ACM Trans. Graph. 21, 3 (July), 380--387.
[4]
Allen, B., Curless, B., and Popović, Z. 2003. The space of human body shapes: Reconstruction and parameterization from range scans. ACM Trans. Graph. 22, 3 (July), 587--594.
[5]
Andersen, E. D., and Andersen, K. D. 1999. The MOSEK interior point optimization for linear programming: an implementation of the homogeneous algorithm. Kluwer Academic Publishers, 197--232.
[6]
Anguelov, D., Srinivasan, P., Koller, D., Thrun, S., Rodgers, J., and Davis, J. 2005. Scape: Shape completion and animation of people. ACM Trans. Graph. 24, 3 (July), 408--416.
[7]
Besl, P. J., and McKay, N. D. 1992. A method for registration of 3-d shapes. IEEE Trans. Pattern Anal. Mach. Intell. 14, 2 (Feb.), 239--256.
[8]
Bommes, D., Campen, M., Ebke, H.-C., Alliez, P., and Kobbelt, L. 2013. Integer-grid maps for reliable quad meshing. ACM Trans. Graph. 32, 4 (July), 98:1--98:12.
[9]
Boyd, S., and Vandenberghe, L. 2004. Convex Optimization. Cambridge University Press, New York, NY, USA.
[10]
Boyer, D. M., Lipman, Y., St. Clair, E., Puente, J., Patel, B. A., Funkhouser, T., Jernvall, J., and Daubechies, I. 2011. Algorithms to automatically quantify the geometric similarity of anatomical surfaces. Proceedings of the National Academy of Sciences 108, 45, 18221--18226.
[11]
Brown, B. J., and Rusinkiewicz, S. 2007. Global non-rigid alignment of 3-d scans. ACM Trans. Graph. 26, 3 (July).
[12]
Candès, E. J., and Recht, B. 2009. Exact matrix completion via convex optimization. Foundations of Computational mathematics 9, 6, 717--772.
[13]
Chao, I., Pinkall, U., Sanan, P., and Schröder, P. 2010. A simple geometric model for elastic deformations. ACM Trans. Graph. 29, 4, 38.
[14]
Ecker, A., Jepson, A. D., and Kutulakos, K. N. 2008. Semidefinite programming heuristics for surface reconstruction ambiguities. In ECCV 2008. Springer, 127--140.
[15]
Floater, M. S., and Hormann, K. 2005. Surface parameterization: a tutorial and survey. In Advances in Multiresolution for Geometric Modelling, Springer, 157--186.
[16]
Freitag, L. A., and Knupp, P. M. 2002. Tetrahedral mesh improvement via optimization of the element condition number. International Journal for Numerical Methods in Engineering 53, 6, 1377--1391.
[17]
Giorgi, D., Biasotti, S., and Paraboschi, L., 2007. Shape retrieval contest 2007: Watertight models track.
[18]
Goemans, M. X., and Williamson, D. P. 1995. Improved approximation algorithms for maximum cut and satisfiability problems using semidefinite programming. J. ACM 42, 6 (Nov.), 1115--1145.
[19]
Hernandez, F., Cirio, G., Perez, A. G., and Otaduy, M. A. 2013. Anisotropic strain limiting. In Proc. of Congreso Español de Informática Gráfica.
[20]
Hormann, K., and Greiner, G. 2000. MIPS: An efficient global parametrization method. In Curve and Surface Design: Saint-Malo 1999. Vanderbilt University Press, 153--162.
[21]
Hormann, K., Lévy, B., and Sheffer, A. 2007. Mesh parameterization: Theory and practice. In ACM SIGGRAPH 2007 Courses, ACM, New York, NY, USA, SIGGRAPH '07.
[22]
Huang, Q., and Guibas, L. 2013. Consistent shape maps via semidefinite programming. Proc. Eurographics Symposium on Geometry Processing 32, 5, 177--186.
[23]
Huang, Q.-X., Adams, B., Wicke, M., and Guibas, L. J. 2008. Non-rigid registration under isometric deformations. In Proc. Eurographics Symposium on Geometry Processing, 1449--1457.
[24]
Igarashi, T., Moscovich, T., and Hughes, J. F. 2005. As-rigid-as-possible shape manipulation. ACM Trans. Graph. 24, 3 (July), 1134--1141.
[25]
Jeuris, B., Vandebril, R., and Vandereycken, B. 2012. A survey and comparison of contemporary algorithms for computing the matrix geometric mean. Electronic Transactions on Numerical Analysis 39, 379--402.
[26]
Karcher, H. 1977. Riemannian center of mass and mollifier smoothing. Comm. pure and applied mathematics 30, 5, 509--541.
[27]
Kiwiel, K. 1986. A linearization algorithm for optimizing control systems subject to singular value inequalities. IEEE Transactions on Automatic Control 31, 7, 595--603.
[28]
Lévy, B., Petitjean, S., Ray, N., and Maillot, J. 2002. Least squares conformal maps for automatic texture atlas generation. ACM Trans. Graph. 21, 3 (July), 362--371.
[29]
Li, H., Sumner, R. W., and Pauly, M. 2008. Global correspondence optimization for non-rigid registration of depth scans. Proc. Eurographics Symposium on Geometry Processing 27, 5.
[30]
Lipman, Y. 2012. Bounded distortion mapping spaces for triangular meshes. ACM Trans. Graph. 31, 4, 108.
[31]
Lipman, Y. 2014. Bijective mappings of meshes with boundary and the degree in mesh processing. SIAM J. Imaging Sci., to appear.
[32]
Liu, L., Zhang, L., Xu, Y., Gotsman, C., and Gortler, S. J. 2008. A local/global approach to mesh parameterization. Proc. Eurographics Symposium on Geometry Processing 27, 5, 1495--1504.
[33]
Löfberg, J. 2004. Yalmip: A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference.
[34]
Lu, Z., and Pong, T. K. 2011. Minimizing condition number via convex programming. SIAM J. Matrix Analysis Applications 32, 4, 1193--1211.
[35]
Maréchal, P., and Ye, J. J. 2009. Optimizing condition numbers. SIAM Journal on Optimization 20, 2, 935--947.
[36]
Paillé, G.-P., and Poulin, P. 2012. As-conformal-as-possible discrete volumetric mapping. Computers & Graphics 36, 5, 427--433.
[37]
Polak, E., and Wardi, Y. 1982. Nondifferentiable optimization algorithm for designing control systems having singular value inequalities. Automatica 18, 3, 267--283.
[38]
Rentmeesters, Q., and Absil, P.-A. 2011. Algorithm comparison for karcher mean computation of rotation matrices and diffusion tensors. In Proc. European Signal Processing Conference, EURASIP, 2229--2233.
[39]
Rossignac, J., and Vinacua, A. 2011. Steady affine motions and morphs. ACM Trans. Graph. 30, 5 (Oct.), 116:1--116:16.
[40]
Rusinkiewicz, S., and Levoy, M. 2001. Efficient variants of the ICP algorithm. In Int. Conf. 3D Digital Imaging and Modeling.
[41]
Sander, P. V., Snyder, J., Gortler, S. J., and Hoppe, H. 2001. Texture mapping progressive meshes. Proc. SIGGRAPH, 409--416.
[42]
Schüller, C., Kavan, L., Panozzo, D., and Sorkine-Hornung, O. 2013. Locally injective mappings. Proc. Eurographics Symposium on Geometry Processing 32, 5, 125--135.
[43]
Shoemake, K. 1985. Animating rotation with quaternion curves. SIGGRAPH Comput. Graph. 19, 3 (July), 245--254.
[44]
Singer, A. 2011. Angular synchronization by eigenvectors and semidefinite programming. Applied and Computational Harmonic Analysis 30, 1, 20--36.
[45]
Sorkine, O., and Alexa, M. 2007. As-rigid-as-possible surface modeling. In Proc. Eurographics Symposium on Geometry Processing, 109--116.
[46]
Sorkine, O., Cohen-Or, D., Goldenthal, R., and Lischinski, D. 2002. Bounded-distortion piecewise mesh parameterization. In Proc. Conference on Visualization '02, VIS '02, 355--362.
[47]
Sumner, R. W., Schmid, J., and Pauly, M. 2007. Embedded deformation for shape manipulation. ACM Trans. Graph. 26, 3.
[48]
Sun, J., Ovsjanikov, M., and Guibas, L. 2009. A concise and provably informative multi-scale signature based on heat diffusion. In Proc. Eurographics Symposium on Geometry Processing, 1383--1392.
[49]
Vandenberghe, L., and Boyd, S. 1994. Semidefinite programming. SIAM Review 38, 49--95.
[50]
Wang, H., O'Brien, J., and Ramamoorthi, R. 2010. Multi-resolution isotropic strain limiting. In ACM SIGGRAPH Asia 2010, 156:1--156:10.
[51]
Weber, O., Myles, A., and Zorin, D. 2012. Computing extremal quasiconformal maps. Computer Graphics Forum 31, 5, 1679--1689.
[52]
Weinberger, K. Q., and Saul, L. K. 2009. Distance metric learning for large margin nearest neighbor classification. J. Mach. Learn. Res. 10 (June), 207--244.

Cited By

View all

Recommendations

Comments

Information & Contributors

Information

Published In

cover image ACM Transactions on Graphics
ACM Transactions on Graphics  Volume 33, Issue 4
July 2014
1366 pages
ISSN:0730-0301
EISSN:1557-7368
DOI:10.1145/2601097
Issue’s Table of Contents
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than ACM must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected]

Publisher

Association for Computing Machinery

New York, NY, United States

Publication History

Published: 27 July 2014
Published in TOG Volume 33, Issue 4

Permissions

Request permissions for this article.

Check for updates

Author Tags

  1. optimization
  2. semidefinite programming
  3. simplicial meshes
  4. singular values

Qualifiers

  • Research-article

Funding Sources

Contributors

Other Metrics

Bibliometrics & Citations

Bibliometrics

Article Metrics

  • Downloads (Last 12 months)4
  • Downloads (Last 6 weeks)0
Reflects downloads up to 17 Jan 2025

Other Metrics

Citations

Cited By

View all
  • (2024)Bijective Volumetric Mapping via Star DecompositionACM Transactions on Graphics10.1145/368795043:6(1-11)Online publication date: 19-Dec-2024
  • (2024)Generative Escher MeshesACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657452(1-11)Online publication date: 13-Jul-2024
  • (2024)In the Quest for Scale-optimal MappingsACM Transactions on Graphics10.1145/362710243:1(1-16)Online publication date: 3-Jan-2024
  • (2024)TutteNet: Injective 3D Deformations by Composition of 2D Mesh Deformations2024 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR)10.1109/CVPR52733.2024.02020(21378-21389)Online publication date: 16-Jun-2024
  • (2023)Orientable Dense Cyclic Infill for Anisotropic Appearance FabricationACM Transactions on Graphics10.1145/359241242:4(1-13)Online publication date: 26-Jul-2023
  • (2023)PCBend: Light Up Your 3D Shapes With Foldable Circuit BoardsACM Transactions on Graphics10.1145/359241142:4(1-16)Online publication date: 26-Jul-2023
  • (2023)Galaxy Maps: Localized Foliations for Bijective Volumetric MappingACM Transactions on Graphics10.1145/359241042:4(1-16)Online publication date: 26-Jul-2023
  • (2023)Meso-Facets for Goniochromatic 3D PrintingACM Transactions on Graphics10.1145/359213742:4(1-12)Online publication date: 26-Jul-2023
  • (2023)Dictionary Fields: Learning a Neural Basis DecompositionACM Transactions on Graphics10.1145/359213542:4(1-12)Online publication date: 26-Jul-2023
  • (2023)NeRO: Neural Geometry and BRDF Reconstruction of Reflective Objects from Multiview ImagesACM Transactions on Graphics10.1145/359213442:4(1-22)Online publication date: 26-Jul-2023
  • Show More Cited By

View Options

Login options

Full Access

View options

PDF

View or Download as a PDF file.

PDF

eReader

View online with eReader.

eReader

Media

Figures

Other

Tables

Share

Share

Share this Publication link

Share on social media