Two Hilbert schemes in computer vision

We study multiview moduli problems that arise in computer vision. We show that these moduli spaces are always smooth and irreducible, in both the calibrated and uncalibrated cases, for any number of views. We also show that these moduli spaces always admit open immersions into Hilbert schemes for more than two views, extending and refining work of Aholt-Sturmfels-Thomas. We use these moduli spaces to study and extend the classical twisted pair covering of the essential variety.

[1]  W. V. Hodge,et al.  Methods of algebraic geometry , 1947 .

[2]  Joe Kileel,et al.  Minimal Problems for the Calibrated Trifocal Variety , 2016, SIAM J. Appl. Algebra Geom..

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

[4]  Joe Kileel,et al.  The Chow form of the essential variety in computer vision , 2016, J. Symb. Comput..

[5]  Rekha R. Thomas,et al.  On the Existence of Epipolar Matrices , 2015, International Journal of Computer Vision.

[6]  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..

[7]  Angelo Vistoli.  Notes on Grothendieck topologies, fibered categories and descent theory , 2004 .

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

[9]  A. Grothendieck,et al.  Éléments de géométrie algébrique , 1960 .

[10]  Thomas S. Huang,et al.  Theory of Reconstruction from Image Motion , 1992 .

[11]  Daniel Huybrechts,et al.  Fourier-Mukai transforms in algebraic geometry , 2006 .

[12]  M. Artin,et al.  Versal deformations and algebraic stacks , 1974 .

[13]  Rekha R. Thomas,et al.  A Hilbert Scheme in Computer Vision , 2011, Canadian Journal of Mathematics.

[14]  Edoardo Sernesi,et al.  Deformations of algebraic schemes , 2006 .

[15]  Martial Hebert,et al.  The Joint Image Handbook , 2015, 2015 IEEE International Conference on Computer Vision (ICCV).

[16]  L. Illusie.  Complexe cotangent et déformations II , 1971 .

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

[18]  Luke Oeding,et al.  The Quadrifocal Variety , 2015, ArXiv.

[19]  Richard Szeliski,et al.  Computer Vision - Algorithms and Applications , 2011, Texts in Computer Science.

[20]  S. Kovács,et al.  Reflexive pull-backs and base extension , 2004 .

[21]  Richard I. Hartley,et al.  Critical Configurations for Projective Reconstruction from Multiple Views , 2005, International Journal of Computer Vision.

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

[23]  Compact moduli of plane curves , 2003, math/0310354.

[24]  H. Opower.  Multiple view geometry in computer vision , 2002 .