TY - JOUR
T1 - Controlling singular values with semidefinite programming
AU - Kovalsky, Shahar Z.
AU - Aigerman, Noam
AU - Basri, Ronen
AU - Lipman, Yaron
N1 - European Research Council (ERC Starting Grant SurfComp) [307754]; U.S.-Israel Binational Science Foundation [331/10]; Israel Science Foundation [1284/12, 764/10]; I-CORE program of the Israel PBC and ISF [4/11]; Israeli Ministry of Science; Citigroup FoundationThis research was supported in part by the European Research Council (ERC Starting Grant SurfComp, Grant No. 307754), U.S.-Israel Binational Science Foundation, Grant No. 331/10, by the Israel Science Foundation Grants No. 1284/12 and 764/10, I-CORE program of the Israel PBC and ISF (Grant No. 4/11), the Israeli Ministry of Science, and by the Citigroup Foundation. The authors would like thank Gilles Tran for the airplane and ladybug models, Johan Lofberg for providing and supporting Yalmip, Ethan Fetaya for helpful discussions, and the anonymous reviewers for their useful comments and suggestions.
PY - 2014
Y1 - 2014
N2 - 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.
AB - 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.
UR - http://www.scopus.com/inward/record.url?scp=84905734219&partnerID=8YFLogxK
U2 - 10.1145/2601097.2601142
DO - 10.1145/2601097.2601142
M3 - مقالة من مؤنمر
SN - 0734-2071
VL - 33
JO - ACM Transactions on Graphics
JF - ACM Transactions on Graphics
IS - 4
M1 - 68
T2 - 41st International Conference and Exhibition on Computer Graphics and Interactive Techniques, ACM SIGGRAPH 2014
Y2 - 10 August 2014 through 14 August 2014
ER -