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Curve cuspless reconstruction via sub-Riemannian geometry∗∗∗

Published online by Cambridge University Press:  27 May 2014

Ugo Boscain ,
Remco Duits ,
Francesco Rossi  and
Yuri Sachkov
Ugo Boscain
Affiliation:
Centre National de Recherche Scientifique (CNRS), CMAP, Ecole Polytechnique, Route de Saclay, 91128 Palaiseau Cedex, France, and Team GECO, INRIA-Centre de Recherche Saclay. ugo.boscain@polytecnique.edu
Remco Duits
Affiliation:
Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands. Department of Mathematics and Computer Science; R.Duits@tue.nl
Francesco Rossi
Affiliation:
Aix-Marseille Univ, LSIS, 13013, Marseille, France; francesco.rossi@lsis.org
Yuri Sachkov
Affiliation:
Program Systems Institute Pereslavl-Zalessky, Russia
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Abstract

We consider the problem of minimizing \hbox{$\int_{0}^\ell \sqrt{\xi^2 +K^2(s)}\, {\rm d}s $}∫0ℓξ2+K2(s) ds for a planar curve having fixed initial and final positions and directions. The total length ℓ is free. Here s is the arclength parameter, K(s) is the curvature of the curve and ξ > 0 is a fixed constant. This problem comes from a model of geometry of vision due to Petitot, Citti and Sarti. We study existence of local and global minimizers for this problem. We prove that if for a certain choice of boundary conditions there is no global minimizer, then there is neither a local minimizer nor a geodesic. We finally give properties of the set of boundary conditions for which there exists a solution to the problem.

Keywords

Curve reconstructiongeneralized pontryagin maximum principle
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 20 , Issue 3 , July 2014 , pp. 748 - 770
Copyright
© EDP Sciences, SMAI, 2014

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References

Agrachev, A., Compactness for sub-Riemannian length-minimizers and subanalyticity. Rend. Sem. Mat. Univ. Politec. Torino 56 (2001) 1–12. Google Scholar
Agrachev, A., Exponential mappings for contact sub-Riemannian structures. J. Dynam. Control Syst. 2 (1996) 321–358. Google Scholar
A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and Sub-Riemannian geometry, available at http://www.math.jussieu.fr/˜barilari/Notes.php
A.A. Agrachev, Yu. L. Sachkov,Control Theory from the Geometric Viewpoint. Encyclopedia of Math. Sci., vol. 87. Springer (2004).
A. Bellaiche, The tangent space in sub-Riemannian geometry. Sub-Riemannian Geometry, Progr. Math., vol. 144. Edited by A. Bellaiche and J.-J. Risler. Birkhäuser, Basel (1996) 1–78.
Boscain, U., Charlot, G. and Rossi, F., Existence planar curves minimizing length and curvature. Proc. Steklov Institute Math. 270 (2010) 43–56. Google Scholar
U. Boscain, R. Chertovskih, J.-P. Gauthier and A. Remizov, Hypoelliptic diffusion and human vision: a semi-discrete new twist on the Petitot theory. To appear in SIAM J. Imaging Sci.
Boscain, U., Duplaix, J., Gauthier, J.P. and Rossi, F., Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion. SIAM J. Control Opt. 50 1309–1336.
Boscain, U. and Rossi, F., Projective Reeds-Shepp car on S 2 with quadratic cost. ESAIM: COCV 16 (2010) 275–297. Google Scholar
Citti, G. and Sarti, A., A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vision 24 (2006) 307–326. Google Scholar
R. Duits, U. Boscain, F. Rossi and Y. Sachkov, Association fields via cuspless sub-Riemannian geodesics in SE(2). To appear in J. Math. Imaging Vision.
Duits, R. and Franken, E.M., Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, Part I: Linear Left-Invariant Diffusion Equations on SE(2). Quart. Appl. Math. 68 (2010) 293–331. Google Scholar
Duits, R. and Franken, E.M., Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores, Part II: nonlinear left-invariant diffusions on invertible orientation scores. Quart. Appl. Math. 68 (2010) 255–292. Google Scholar
M. Gromov, Carnot–Caratheodory spaces seen from within, in Sub-Riemannian Geometry, in vol. 144 Progr. Math., edited by A. Bellaiche and J.-J. Risler (1996) 79–323.
Hladky, R.K. and Pauls, S.D., Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model. J Math Imaging Vis 36 (2010) 1–27. Google Scholar
Hoffman, W.C., The visual cortex is a contact bundle. Appl. Math. Comput. 32 (1989) 137–167. Google Scholar
Hörmander, L., Hypoelliptic Second Order Differential Equations. Acta Math. 119 (1967) 147–171. Google Scholar
Hubel, D.H. and Wiesel, T.N., Receptive fields, binocular interaction and functional architecture in the cat’s visual cortex. The J. Phys. 160 (1962) 106. Google Scholar
Moiseev, I. and Sachkov, Yu. L., Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380–399. Google Scholar
R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications. Vol. 91 of Math. Surveys and Monogr. AMS (2002).
M. Nitzberg and D. Mumford, The 2.1-D sketch. ICCV (1990) 138–144.
Petitot, J., Vers une Neuro-géomètrie. Fibrations corticales, structures de contact et contours subjectifs modaux. Math. Inform. Sci. Humaines 145 (1999) 5–101. Google Scholar
J. Petitot, Neurogéomètrie de la vision – Modèles mathématiques et physiques des architectures fonctionnelles. Les Éditions de l’École Polytechnique (2008).
Petitot, J., The neurogeometry of pinwheels as a sub-Riemannian contact structure. J. Phys. – Paris 97 (2003) 265–309. Google Scholar
Sachkov, Y., Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018–1039. Google Scholar
Sachkov, Y.L., Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 17 (2011) 293–321. Google Scholar
Sachkov, Y.L., Discrete symmetries in the generalized Dido problem. Sb. Math. 197 (2006) 235–257. Google Scholar
G. Sanguinetti, G. Citti and A. Sarti, Image completion using a diffusion driven mean curvature flow in a sub-riemannian space, in Int. Conf. Comput. Vision Theory and Appl. (VISAPP’08), Funchal (2008) 22–25.
Sarychev, A.V., First and Second-Order Integral Functionals of the Calculus of Variations Which Exhibit the Lavrentiev Phenomenon. J. Dyn. Control Syst. 3 (1997) 565–588. Google Scholar
R. Vinter, Optimal Control. Birkhauser (2010).
E.T. Whittaker and G.N. Watson, A Course of Modern Analysis. An introduction to the general theory of infinite processes and of analytic functions; with an account of principal transcendental functions. Cambridge University Press, Cambridge (1996).