Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-11T12:08:14.959Z Has data issue: false hasContentIssue false
  • English
  • Français

Parallel Algorithm and Software for Image Inpainting via Sub-Riemannian Minimizers on the Group of Rototranslations

Published online by Cambridge University Press:  28 May 2015

Alexey P. Mashtakov ,
Andrei A. Ardentov  and
Yuri L. Sachkov
Alexey P. Mashtakov*
Affiliation:
Program Systems Institute, Pereslavl-Zalessky, Russia
Andrei A. Ardentov*
Affiliation:
Program Systems Institute, Pereslavl-Zalessky, Russia
Yuri L. Sachkov*
Affiliation:
Program Systems Institute, Pereslavl-Zalessky, Russia
*
Corresponding author.Email address:alexey.mashtakov@gmail.com
Corresponding author.Email address:aaa@pereslavl.ru
Corresponding author.Email address:sachkov@sys.botik.ru
Get access

Abstract

The paper is devoted to an approach for image inpainting developed on the basis of neurogeometry of vision and sub-Riemannian geometry. Inpainting is realized by completing damaged isophotes (level lines of brightness) by optimal curves for the left-invariant sub-Riemannian problem on the group of rototranslations (motions) of a plane SE(2). The approach is considered as anthropomorphic inpainting since these curves satisfy the variational principle discovered by neurogeometry of vision. A parallel algorithm and software to restore monochrome binary or halftone images represented as series of isophotes were developed. The approach and the algorithm for computation of completing arcs are presented in detail.

Keywords

65M1078A48Image inpaintingsub-Riemannian geometry neurogeometry of visiongroup of rototranslations of a planeparallel software
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 6 , Issue 1 , February 2013 , pp. 95 - 115
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Chan, T.F., Kang, S.H. and Shen, J., Euler’s elastica and curvature based inpainting, SIAM J. Appl. Math., 63 (2002), 564–592.Google Scholar
[2] Citti, G. and Sarti, A., A cortical based model of perceptual completion in the roto-translation space, J. Math. Imaging Vis., 24 (2006), 307–326.Google Scholar
[3] Esedoglu, S. and Shen, J., Digital image inpainting by the Mumford-Shah-Euler image model, Europ. J. Appl. Math., 13 (2002), 353–370.CrossRefGoogle Scholar
[4] Kimia, B. B., Frankel, I. and Popescu, A.-M., Euler spiral for shape completion, Int. J. Comp. Vision, 54 (2003), 159–182.Google Scholar
[5] Petitot, J., The neurogeometry of pinwheels as a sub-Riemannian contact structure, J. Physiology - Paris, 97 (2003), 265–309.Google Scholar
[6] Petitot, J., Neurogeometrie de la vision — Modeles mathematiques et physiques des architectures fonctionnelles, Editions de l’Ecole Polytechnique, 2008.Google Scholar
[7] Moiseev, I. and Sachkov, Yu. L., Maxwell strata in sub-Riemannian problem on the group of motions of a plane, ESAIM: COCV, ESAIM: COCV, 16 (2010), 380–399, available at arXiv:0807.4731v1, 29 July 2008.Google Scholar
[8] Sachkov, Yu. L., 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
[9] Sachkov, Yu. 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
[10] Agrachev, A. A. and Sachkov, Yu. L., Control Theory from the Geometric Viewpoint, Springer-Verlag, Berlin, 2004.CrossRefGoogle Scholar
[11] Montgomery, R., A Tour of Subriemannian Geometries, Their Geodesics and Applications, American Mathematical Society, 2002.Google Scholar
[12] Laumond, J. P., Nonholonomic motion planning for mobile robots, Lecture notes in Control and Information Sciences, 229, Springer, 1998.Google Scholar
[13] Duits, R. and van Almsick, M., The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group, Quarterly of Applied Mathematics, 66(1) (2008), 27–67.Google Scholar
[14] 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, Quarterly of Applied Mathematics, 68(2) (2008), 293-331.Google Scholar
[15] 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), Quarterly of Applied Mathematics, 68(2) (2008), 255-292.Google Scholar
[16] Agrachev, A. A., Boscain, U., Gauthier, J. P. and Rossi, F., The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, Journal of Functional Analysis, 256(8) (2009), 2621-2655.CrossRefGoogle Scholar
[17] Pontryagin, L. S., Boltyanskii, V G., Gamkrelidze, R. V and Mishchenko, E. F., The Mathematical Theory of Optimal Processes, Wiley Interscience, 1962.Google Scholar
[18] Whittaker, E. T. and Watson, G. N., 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.Google Scholar
[19] Sachkov, Yu. L., Maxwell strata in Euler’s elastic problem, Journal of Dynamical and Control Systems, 14(2) (2008), 169–234.Google Scholar
[20] Sachkov, Yu. L., Conjugate points in Euler’s elastic problem, Journal of Dynamical and Control Systems, 14(3) (2008), 409-439.Google Scholar
[21] Ardentov, A. and Sachkov, Yu. L., Solution of Euler’s elastic problem (in Russian), Avtomatika i Telemekhanika, 2009, No. 4, 78–88. (English translation in Automation and remote control.)Google Scholar
[22] Euler, L., Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive Solutio problematis isoperimitrici latissimo sensu accepti, Lausanne, Geneva, 1744.Google Scholar
[23] Mumford, D., Elastica and computer vision, Algebraic Geometry and Its Applications, pp. 491–506, 1994.Google Scholar
[24] Hladky, R. K. and Pauls, S. D., Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model, Journal of Mathematical Imaging and Vision, 36(1) (2010), 1–27.Google Scholar
[25] Langer, M., Computational Perception, Lecture Notes, 2008, http://www.cim.mcgill.ca/īanger/646.html Google Scholar
[26] Marr, D. and Hildreth, E., Theory of edge detection, Proceedings of the Royal Society of London. Series B, Biological Sciences, Vol. 207, No. 1167, 1980, pp. 187–217.Google Scholar
[27] Peichl, L. and Wässle, H., Size, scatter and coverage of ganglion cell receptive field centres in the cat retina, J. Physiol., 291 (1979), 117–141.Google Scholar
[28] Boscain, U., Duplaix, J., Gauthier, J. P and Rossi, F., Anthropomorphic image reconstruction via hypoelliptic diffusion. To appear on SIAM Journal of Control and Optimization.Google Scholar
[29] Cao, F., Gousseau, Y., Masnou, S. and Perez, P., Geometrically guided exemplar-based inpainting, SIAM Journal on Imaging Sciences, 4(4) (2011), 1143–1179.Google Scholar
[30] Rothschild, L. P. and Stein, E. M., Hypoelliptic differential operators and nilpotent groups, Acta Math., 137 (1976), 247–320.Google Scholar
[31] Nagel, A., Stein, E. and Wainger, S., Balls and metrics defined by vector fields I: Basic Properties, Acta Math., 155 (1985), 130–147.Google Scholar
[32] Hormander, L., Hypoelliptic second order differential equations, Acta Math., 119 (1968), 147–171.Google Scholar
[33] Damon, J., Local Morse theory for solutions to the heat equation and Gaussian blurring, JDE, 115(1995), 368–401.CrossRefGoogle Scholar