Efficient Solvers for Minimal Problems by Syzygy-Based Reduction

In this paper we study the problem of automatically generating polynomial solvers for minimal problems. The main contribution is a new method for finding small elimination templates by making use of the syzygies (i.e. the polynomial relations) that exist between the original equations. Using these syzygies we can essentially parameterize the set of possible elimination templates. We evaluate our method on a wide variety of problems from geometric computer vision and show improvement compared to both handcrafted and automatically generated solvers. Furthermore we apply our method on two previously unsolved relative orientation problems.

[1]  Yubin Kuang,et al.  Pose estimation from minimal dual-receiver configurations , 2012, Proceedings of the 21st International Conference on Pattern Recognition (ICPR2012).

[2]  Henrik Stewenius,et al.  Gröbner Basis Methods for Minimal Problems in Computer Vision , 2005 .

[3]  Yubin Kuang,et al.  Minimal Solvers for Relative Pose with a Single Unknown Radial Distortion , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[4]  Zuzana Kukelova,et al.  Fast and Stable Algebraic Solution to L2 Three-View Triangulation , 2013, 2013 International Conference on 3D Vision.

[5]  Jan-Michael Frahm,et al.  Minimal Solvers for 3D Geometry from Satellite Imagery , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[6]  Yubin Kuang,et al.  Exploiting p-fold symmetries for faster polynomial equation solving , 2012, Proceedings of the 21st International Conference on Pattern Recognition (ICPR2012).

[7]  Pascal Vasseur,et al.  A Homography Formulation to the 3pt Plus a Common Direction Relative Pose Problem , 2014, ACCV.

[8]  Roland Siegwart,et al.  Finding the Exact Rotation between Two Images Independently of the Translation , 2012, ECCV.

[9]  Richard I. Hartley,et al.  Lines and Points in Three Views and the Trifocal Tensor , 1997, International Journal of Computer Vision.

[10]  Stergios I. Roumeliotis,et al.  Two Efficient Solutions for Visual Odometry Using Directional Correspondence , 2012, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[11]  Marc Pollefeys,et al.  A Minimal Case Solution to the Calibrated Relative Pose Problem for the Case of Two Known Orientation Angles , 2010, ECCV.

[12]  David A. Cox,et al.  Using Algebraic Geometry , 1998 .

[13]  Bernhard P. Wrobel,et al.  Multiple View Geometry in Computer Vision , 2001 .

[14]  Zuzana Kukelova,et al.  R6P - Rolling Shutter Absolute Camera Pose , 2015, CVPR 2015.

[15]  David Nistér,et al.  An efficient solution to the five-point relative pose problem , 2003, 2003 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2003. Proceedings..

[16]  A. Heyden.  Geometry and algebra of multiple projective transformations , 1995 .

[17]  Marc Pollefeys,et al.  A minimal solution to the rolling shutter pose estimation problem , 2015, 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS).

[18]  Andrew Zisserman,et al.  Minimal projective reconstruction for combinations of points and lines in three views , 2004, Image Vis. Comput..

[19]  Viktor Larsson,et al.  Uncovering Symmetries in Polynomial Systems , 2016, ECCV.

[20]  Zuzana Kukelova,et al.  Singly-Bordered Block-Diagonal Form for Minimal Problem Solvers , 2014, ACCV.

[21]  Matthew A. Brown,et al.  Minimal Solutions for Panoramic Stitching with Radial Distortion , 2009, BMVC.

[22]  Kostas Daniilidis,et al.  Optimizing polynomial solvers for minimal geometry problems , 2011, 2011 International Conference on Computer Vision.

[23]  Kalle Åström,et al.  Absolute pose for cameras under flat refractive interfaces , 2015, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[24]  Matthew Turk,et al.  Solving for Relative Pose with a Partially Known Rotation is a Quadratic Eigenvalue Problem , 2014, 2014 2nd International Conference on 3D Vision.

[25]  Yubin Kuang,et al.  Pose Estimation with Unknown Focal Length Using Points, Directions and Lines , 2013, 2013 IEEE International Conference on Computer Vision.

[26]  Richard I. Hartley,et al.  Projective reconstruction from line correspondences , 1994, 1994 Proceedings of IEEE Conference on Computer Vision and Pattern Recognition.

[27]  Karl Johan Åström,et al.  Solutions to Minimal Generalized Relative Pose Problems , 2005 .

[28]  H. M. Möller,et al.  Multivariate polynomial equations with multiple zeros solved by matrix eigenproblems , 1995 .

[29]  Dieter Schmalstieg,et al.  A Minimal Solution to the Generalized Pose-and-Scale Problem , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[30]  Yubin Kuang,et al.  A Minimal Solution to Relative Pose with Unknown Focal Length and Radial Distortion , 2014, ACCV.

[31]  David Nister,et al.  Recent developments on direct relative orientation , 2006 .

[32]  Zuzana Kukelova,et al.  Automatic Generator of Minimal Problem Solvers , 2008, ECCV.

[33]  Donal O'Shea,et al.  Ideals, varieties, and algorithms - an introduction to computational algebraic geometry and commutative algebra (2. ed.) , 1997, Undergraduate texts in mathematics.

[34]  Fredrik Kahl,et al.  City-Scale Localization for Cameras with Known Vertical Direction , 2017, IEEE Transactions on Pattern Analysis and Machine Intelligence.

[35]  Long Quan,et al.  Invariants of Six Points and Projective Reconstruction From Three Uncalibrated Images , 1995, IEEE Trans. Pattern Anal. Mach. Intell..

[36]  Gaku Nakano,et al.  Globally Optimal DLS Method for PnP Problem with Cayley parameterization , 2015, BMVC.

[37]  Changchang Wu,et al.  P3.5P: Pose estimation with unknown focal length , 2015, 2015 IEEE Conference on Computer Vision and Pattern Recognition (CVPR).

[38]  Zuzana Kukelova,et al.  New Efficient Solution to the Absolute Pose Problem for Camera with Unknown Focal Length and Radial Distortion , 2010, ACCV.

[39]  Yubin Kuang,et al.  Revisiting Trifocal Tensor Estimation Using Lines , 2014, 2014 22nd International Conference on Pattern Recognition.

[40]  Yubin Kuang,et al.  Partial Symmetry in Polynomial Systems and Its Applications in Computer Vision , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[41]  Luke Oeding,et al.  The ideal of the trifocal variety , 2012, Math. Comput..

[42]  Marc Pollefeys,et al.  Relative Pose Estimation for a Multi-camera System with Known Vertical Direction , 2014, 2014 IEEE Conference on Computer Vision and Pattern Recognition.

[43]  Zuzana Kukelova,et al.  3D reconstruction from image collections with a single known focal length , 2009, 2009 IEEE 12th International Conference on Computer Vision.

[44]  Yubin Kuang,et al.  Stratified sensor network self-calibration from TDOA measurements , 2013, 21st European Signal Processing Conference (EUSIPCO 2013).

[45]  Zuzana Kukelova,et al.  Making minimal solvers fast , 2012, 2012 IEEE Conference on Computer Vision and Pattern Recognition.

[46]  Yubin Kuang,et al.  Revisiting the PnP Problem: A Fast, General and Optimal Solution , 2013, 2013 IEEE International Conference on Computer Vision.

[47]  Joel A. Hesch,et al.  A Direct Least-Squares (DLS) method for PnP , 2011, 2011 International Conference on Computer Vision.

[48]  Martin Byröd,et al.  Fast and Stable Polynomial Equation Solving and Its Application to Computer Vision , 2009, International Journal of Computer Vision.

[49]  Marc Pollefeys,et al.  A 4-point algorithm for relative pose estimation of a calibrated camera with a known relative rotation angle , 2013, 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems.

[50]  Yubin Kuang,et al.  Numerically Stable Optimization of Polynomial Solvers for Minimal Problems , 2012, ECCV.

[51]  Robert C. Bolles,et al.  Random sample consensus: a paradigm for model fitting with applications to image analysis and automated cartography , 1981, CACM.