Generalized Convexity in Multiple View Geometry

Recent work on geometric vision problems has exploited convexity properties in order to obtain globally optimal solutions. In this paper we give an overview of these developments and show the tight connections between different types of convexity and optimality conditions for a large class of multiview geometry problems. We also show how the convexity properties are closely linked to different types of optimization algorithms for computing the solutions. Moreover, it is also demonstrated how convexity can be used for detection and removal of outliers. The theoretical findings are accompanied with illustrative examples and experimental results on real data.

[1]  Carl Olsson,et al.  A polynomial-time bound for matching and registration with outliers , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[2]  F. Kahl.  Multiple View Geometry and the L-infinity Norm , 2005, ICCV 2005.

[3]  Robert J. Vanderbei,et al.  An Interior-Point Algorithm for Nonconvex Nonlinear Programming , 1999, Comput. Optim. Appl..

[4]  Richard I. Hartley,et al.  Recovering Camera Motion Using L\infty Minimization , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[5]  Anders P. Eriksson,et al.  Efficient optimization for L∞-problems using pseudoconvexity , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[6]  Steven M. Seitz,et al.  Fast algorithms for L∞ problems in multiview geometry , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[7]  Fredrik Kahl,et al.  Multiple view geometry and the L/sub /spl infin//-norm , 2005, Tenth IEEE International Conference on Computer Vision (ICCV'05) Volume 1.

[8]  Richard I. Hartley,et al.  An L∞ approach to structure and motion problems in ID-vision , 2007, 2007 IEEE 11th International Conference on Computer Vision.

[9]  Jirí Matousek.  On geometric optimization with few violated constraints , 1994, SCG '94.

[10]  Richard I. Hartley,et al.  Projective least-squares: Global solutions with local optimization , 2009, 2009 IEEE Conference on Computer Vision and Pattern Recognition.

[11]  Mokhtar S. Bazaraa,et al.  Nonlinear Programming: Theory and Algorithms , 1993 .

[12]  Takeo Kanade,et al.  Uncertainty Models in Quasiconvex Optimization for Geometric Reconstruction , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[13]  Stephen P. Boyd,et al.  Convex Optimization , 2004, Algorithms and Theory of Computation Handbook.

[14]  Takeo Kanade,et al.  Quasiconvex Optimization for Robust Geometric Reconstruction , 2005, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[15]  Richard I. Hartley,et al.  Multiple-View Geometry Under the {$L_\infty$}-Norm , 2008, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[16]  R. Hartley,et al.  L/sub /spl infin// minimization in geometric reconstruction problems , 2004, Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004. CVPR 2004..

[17]  Steven M. Seitz,et al.  Fast algorithms for L INFINITY problems in multiview geometry. , 2008, CVPR 2008.

[18]  Richard I. Hartley,et al.  Global Optimization through Rotation Space Search , 2009, International Journal of Computer Vision.

[19]  Matthijs C. Dorst.  Distinctive Image Features from Scale-Invariant Keypoints , 2011 .

[20]  Yongduek Seo,et al.  Verifying global minima for L2 minimization problems , 2008, 2008 IEEE Conference on Computer Vision and Pattern Recognition.

[21]  R. Tyrrell Rockafellar,et al.  Convex Analysis , 1970, Princeton Landmarks in Mathematics and Physics.

[22]  Andrew Zisserman,et al.  Multiple View Geometry in Computer Vision (2nd ed) , 2003 .

[23]  Jos F. Sturm,et al.  A Matlab toolbox for optimization over symmetric cones , 1999 .

[24]  J. Ponstein,et al.  Seven kinds of convexity , 1967 .

[25]  Kristy Sim,et al.  Removing outliers using the L∞ Norm , 2006, CVPR 2006.

[26]  Micha Sharir,et al.  Efficient algorithms for geometric optimization , 1998, CSUR.

[27]  Hongdong Li,et al.  A practical algorithm for L triangulation with outliers , 2007, 2007 IEEE Conference on Computer Vision and Pattern Recognition.

[28]  Agostino Dovier,et al.  Reconstruction with Interval Constraints Propagation , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).

[29]  R. Hartley,et al.  Multiple-View Geometry under the L 1-Norm , 2007 .

[30]  Richard I. Hartley,et al.  Removing Outliers Using The L\infty Norm , 2006, 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition (CVPR'06).