Curve cuspless reconstruction via sub-riemannian geometry

Boscain, Ugo; Duits, Remco; Rossi, Francesco; Sachkov, Yuri

ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 748-770.

We consider the problem of minimizing ${\int }_0^{\ell }\sqrt{{\xi }^2+K^2\left(s\right)}\mathrm{d}s$ ∫ 0 ℓ ξ 2 + K 2 ( s )   d s 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.

[1] A. Agrachev, Compactness for sub-Riemannian length-minimizers and subanalyticity. Rend. Sem. Mat. Univ. Politec. Torino 56 (2001) 1-12. | MR | Zbl

[2] A. Agrachev, Exponential mappings for contact sub-Riemannian structures. J. Dynam. Control Syst. 2 (1996) 321-358. | MR | Zbl

[3] A. Agrachev, D. Barilari and U. Boscain, Introduction to Riemannian and Sub-Riemannian geometry, available at http://www.math.jussieu.fr/˜barilari/Notes.php

[4] A.A. Agrachev, Yu. L. Sachkov,Control Theory from the Geometric Viewpoint. Encyclopedia of Math. Sci., vol. 87. Springer (2004). | MR | Zbl

[5] 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. | MR | Zbl

[6] U. Boscain, G. Charlot and F. Rossi, Existence planar curves minimizing length and curvature. Proc. Steklov Institute Math. 270 (2010) 43-56. | MR | Zbl

[7] 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. | MR

[8] U. Boscain, J. Duplaix, J.P. Gauthier and F. Rossi, Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion. SIAM J. Control Opt. 50 1309-1336. | MR | Zbl

[9] U. Boscain and F. Rossi, Projective Reeds-Shepp car on S2 with quadratic cost. ESAIM: COCV 16 (2010) 275-297. | Numdam | MR | Zbl

[10] G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vision 24 (2006) 307-326. | MR | Zbl

[11] 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. | MR | Zbl

[12] R. Duits and E.M. Franken, 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. | MR | Zbl

[13] R. Duits and E.M. Franken, 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. | MR | Zbl

[14] 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. | MR | Zbl

[15] R.K. Hladky and S.D. Pauls, Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model. J Math Imaging Vis 36 (2010) 1-27. | MR

[16] W.C. Hoffman, The visual cortex is a contact bundle. Appl. Math. Comput. 32 (1989) 137-167. | MR | Zbl

[17] L. Hörmander, Hypoelliptic Second Order Differential Equations. Acta Math. 119 (1967) 147-171. | MR | Zbl

[18] D.H. Hubel and T.N. Wiesel, Receptive fields, binocular interaction and functional architecture in the cat's visual cortex. The J. Phys. 160 (1962) 106.

[19] I. Moiseev and Yu. L. Sachkov, Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 380-399. | Numdam | MR | Zbl

[20] R. Montgomery, A Tour of Subriemannian Geometries, Their Geodesics and Applications. Vol. 91 of Math. Surveys and Monogr. AMS (2002). | MR | Zbl

[21] M. Nitzberg and D. Mumford, The 2.1-D sketch. ICCV (1990) 138-144.

[22] J. Petitot, Vers une Neuro-géomètrie. Fibrations corticales, structures de contact et contours subjectifs modaux. Math. Inform. Sci. Humaines 145 (1999) 5-101.

[23] J. Petitot, Neurogéomètrie de la vision - Modèles mathématiques et physiques des architectures fonctionnelles. Les Éditions de l'École Polytechnique (2008). | MR

[24] J. Petitot, The neurogeometry of pinwheels as a sub-Riemannian contact structure. J. Phys. - Paris 97 (2003) 265-309.

[25] Y. Sachkov, Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 16 (2010) 1018-1039. | Numdam | MR | Zbl

[26] Y.L. Sachkov, Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV 17 (2011) 293-321. | Numdam | MR | Zbl

[27] Y.L. Sachkov, Discrete symmetries in the generalized Dido problem. Sb. Math. 197 (2006) 235-257. | MR | Zbl

[28] 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.

[29] A.V. Sarychev, First and Second-Order Integral Functionals of the Calculus of Variations Which Exhibit the Lavrentiev Phenomenon. J. Dyn. Control Syst. 3 (1997) 565-588. | MR | Zbl

[30] R. Vinter, Optimal Control. Birkhauser (2010). | MR

[31] 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). | MR | Zbl