Consistent Shape Maps via Semidefinite Programming

Recent advances in shape matching have shown that jointly optimizing the maps among the shapes in a collection can lead to significant improvements when compared to estimating maps between pairs of shapes in isolation. These methods typically invoke a cycle‐consistency criterion — the fact that compositions of maps along a cycle of shapes should approximate the identity map. This condition regularizes the network and allows for the correction of errors and imperfections in individual maps. In particular, it encourages the estimation of maps between dissimilar shapes by compositions of maps along a path of more similar shapes.

[1]  Yi Ma,et al.  Robust principal component analysis? , 2009, JACM.

[2]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2009, Found. Comput. Math..

[3]  David K. Smith.  Theory of Linear and Integer Programming , 1987 .

[4]  Amit Singer,et al.  Exact and Stable Recovery of Rotations for Robust Synchronization , 2012, ArXiv.

[5]  Marc Pollefeys,et al.  Disambiguating visual relations using loop constraints , 2010, 2010 IEEE Computer Society Conference on Computer Vision and Pattern Recognition.

[6]  Bernard Chazelle,et al.  Shape distributions , 2002, TOGS.

[7]  Stephen DiVerdi,et al.  Exploring collections of 3D models using fuzzy correspondences , 2012, ACM Trans. Graph..

[8]  Stephen P. Boyd,et al.  Distributed Optimization and Statistical Learning via the Alternating Direction Method of Multipliers , 2011, Found. Trends Mach. Learn..

[9]  Vladimir G. Kim,et al.  Blended intrinsic maps , 2011, SIGGRAPH 2011.

[10]  Wotao Yin,et al.  Alternating direction augmented Lagrangian methods for semidefinite programming , 2010, Math. Program. Comput..

[11]  M. Fiedler.  Algebraic connectivity of graphs , 1973 .

[12]  Tiefeng Jiang,et al.  SPECTRAL DISTRIBUTIONS OF ADJACENCY AND LAPLACIAN MATRICES OF RANDOM GRAPHS , 2010, 1011.2608.

[13]  H. Pottmann,et al.  Reassembling fractured objects by geometric matching , 2006, SIGGRAPH 2006.

[14]  M. Hebert,et al.  Automatic three-dimensional modeling from reality , 2002 .

[15]  Leonidas J. Guibas,et al.  An optimization approach for extracting and encoding consistent maps in a shape collection , 2012, ACM Trans. Graph..

[16]  Leonidas J. Guibas,et al.  An Optimization Approach to Improving Collections of Shape Maps , 2011, Comput. Graph. Forum.

[17]  Emmanuel J. Candès,et al.  Decoding by linear programming , 2005, IEEE Transactions on Information Theory.

[18]  Emmanuel J. Candès,et al.  Exact Matrix Completion via Convex Optimization , 2008, Found. Comput. Math..

[19]  Sebastian Thrun,et al.  SCAPE: shape completion and animation of people , 2005, SIGGRAPH '05.

[20]  Thomas A. Funkhouser,et al.  Möbius voting for surface correspondence , 2009, ACM Trans. Graph..

[21]  Alexander M. Bronstein,et al.  Numerical Geometry of Non-Rigid Shapes , 2009, Monographs in Computer Science.

[22]  Maks Ovsjanikov,et al.  Functional maps , 2012, ACM Trans. Graph..

[23]  Leonidas J. Guibas,et al.  A concise and provably informative multi-scale signature based on heat diffusion , 2009 .