Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist
Abstract
This paper presents a semidiscrete alternative to the theory of neurogeometry of vision, due to Citti, Petitot, and Sarti. We propose a new ingredient, namely, working on the group of translations and discrete rotations $SE(2,N)$. The theoretical side of our study relates the stochastic nature of the problem with the Moore group structure of $SE(2,N)$. Harmonic analysis over this group leads to very simple finite dimensional reductions. We then apply these ideas to the inpainting problem which is reduced to the integration of a completely parallelizable finite set of Mathieu-type diffusions (indexed by the dual of $SE(2,N)$ in place of the points of the Fourier plane, which is a drastic reduction). The integration of the the Mathieu equations can be performed by standard numerical methods for elliptic diffusions and leads to a very simple and efficient class of inpainting algorithms. We illustrate the performances of the method on a series of deeply corrupted images.
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References
1.
A. A. Abramov and V. B. Andreev, On the application of the method of successive substitution to the determination of periodic solutions of differential and differences equations, Zh. Vychisl. Mat. Mat. Fiz., 3 (1963), pp. 377--381.
2.
Y. Achdou and N. Tchou, A finite difference scheme on a non commutative group, Numer. Math., 89 (2001), pp. 401--424.
3.
A. Agrachev, U. Boscain, J.-P. Gauthier, and F. Rossi, The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups, J. Funct. Anal., 256 (2009), pp. 2621--2655.
4.
N. U. Ahmed and T. E. Dabbous, Nonlinear filtering of systems governed by Ito differential equations with jump parameters, J. Math. Anal. Appl., 115 (1986), pp. 76--92.
5.
J. August and S. W. Zucker, The curve indicator random field and Markov processes, IEEE Trans. Pattern Anal. Mach. Intell., 25 (2003), pp. 387--400.
6.
D. Ballard, G. Hinton, and T. Sejnowski, Parallel visual computation, Nature, 306 (1983), pp. 21--26.
7.
A. Bellaiche, The tangent space in sub-Riemannian geometry, in Sub-Riemannian Geometry, A. Bellaiche and J.-J. Risler, eds., Progr. Math. 144, Birkhäuser, Basel, 1996, pp. 1--78.
8.
O. Ben-Shahar, P. Huggins, T. Izo, and S. W. Zucker, Cortical connections and early visual function: Intra- and inter-columnar processing, J. Physiol. Paris, 97 (2003), pp. 191--208.
9.
U. Boscain, G. Charlot, and F. Rossi, Existence of planar curves minimizing length and curvature, Proc. Steklov Inst. Math., 270 (2010), pp. 43--56.
10.
U. Boscain, R. Duits, F. Rossi, and Yu. L. Sachkov, Curve cuspless reconstruction via sub-Riemannian geometry, ESAIM Control Optim. Calc. Var., to appear; available online from \burlhttps://journals.cambridge.org/action/displayJournal?jid=COV.
11.
U. Boscain, J. Duplaix, J.-P. Gauthier, and F. Rossi, Anthropomorphic image reconstruction via hypoelliptic diffusion, SIAM J. Control Optim., 50 (2012), pp. 1309--1336.
12.
P. C. Bressloff, J. D. Cowan, M. Golubitsky, P. J. Thomas, and M. C. Wiener, Geometric visual hallucinations, Euclidean symmetry, and the functional architecture of striate cortex, Phil. Trans. R. Soc. Lond. B, 356 (2001), pp. 299--330.
13.
H. Chu, Compactification and duality of topological groups, Trans. Amer. Math. Soc., 123 (1966), pp. 310--324.
14.
G. Citti and A. Sarti, A cortical based model of perceptual completion in the roto-translation space, J. Math. Imaging Vision, 24 (2006), pp. 307--326.
15.
C. Corduneanu, N. Gheorghiu, and V. Barbu, Almost Periodic Functions, 2nd ed., Chelsea, New York, 1989.
16.
J. Dixmier, Les $C^{\ast}$-algèbres et leurs représentations, 2nd ed., Gauthier--Villars, Paris, 1969.
17.
M. Duflo and C. C. Moore, On the regular representation of a nonunimodular locally compact group, J. Functional Analysis, 21 (1976), pp. 209--243.
18.
R. Duits, U. Boscain, F. Rossi, and Yu. L. Sachkov, Association fields via cuspless sub-Riemannian geodesics in SE(2), J. Math. Imaging Vision, to appear; available online from \burlhttp://link.springer.com/article/10.1007% 2Fs10851-013-0475-y.
19.
R. Duits, M. Felsberg, G. Granlund, and B. M. ter Haar Romeny, Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the Euclidean motion group, Int. J. Comput. Vis., 72 (2007), pp. 79--102.
20.
R. Duits and E. M. Franken, Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part \textupI: Linear left-invariant diffusion equations on SE(2), Quart. Appl. Math., 68 (2010), pp. 255--292.
21.
R. Duits and E. M. Franken, Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part \textupII: Nonlinear left-invariant diffusions on invertible orientation scores, Quart. Appl. Math., 68 (2010), pp. 293--331.
22.
R. Duits, H. Führ, B. Janssen, M. Bruurmijn, L. Florack, and H. van Assen, Evolution equations on Gabor transforms and their applications, Appl. Comput. Harmon. Anal., 35 (2013), pp. 483--526.
23.
R. Duits and M. van Almsick, The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group, Quart. Appl. Math., 66 (2008), pp. 27--67.
24.
P. E. Forssen, Low and Medium Level Vision Using Channel Representations, Ph.D. thesis, Linköping University, Linköping, Sweden, 2004.
25.
E. M. Franken, Enhancement of Crossing Elongated Structures in Images, Ph.D. thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 2008; available online from \burlhttp://alexandria.tue.nl/extra2/200910002.pdf.
26.
E. M. Franken and R. Duits, Crossing-preserving coherence-enhancing diffusion on invertible orientation scores, Int. J. Comput. Vis., 85 (2009), pp. 253--278.
27.
H. Heyer, Dualität über lokal kompakter gruppen, Springer, Berlin, 1970.
28.
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 Vision, 36 (2010), pp. 1--27.
29.
W. C. Hoffman, The visual cortex is a contact bundle, Appl. Math. Comput., 32 (1989), pp. 137--167.
30.
D. H. Hübel and T. N. Wiesel, Receptive fields of single neurones in the cat's striate cortex, J. Physiol., 148 (1959), pp. 574--591.
31.
R. Léandre, Majoration en temps petit de la densité d'une diffusion dégénérée, Probab. Theory Related Fields, 74 (1987), pp. 289--294.
32.
R. Léandre, Minoration en temps petit de la densité d'une diffusion dégénérée, J. Funct. Anal., 74 (1987), pp. 399--414.
33.
T. S. Lee, Image representation using 2D Gabor wavelets, IEEE Trans. Pattern Anal. Mach. Intell., 18 (1996), pp. 959--971.
34.
G. I. Marchuk, Methods of Numerical Mathematics, Appl. Math. 2, Springer-Verlag, New York, Berlin, 1982.
35.
A. P. Mashtakov, A. A. Ardentov, and Yu. L. Sachkov, Parallel algorithm and software for image inpainting via sub-Riemannian minimizers on the group of rototranslations, Numer. Math. Theory Methods Appl., 6 (2013), pp. 95--115.
36.
R. Montgomery, A Tour of Sub-Riemannian Geometries, Their Geodesics and Applications, Math. Surveys Monogr. 91, American Mathematical Society, Providence, RI, 2002.
37.
D. Mumford, Elastica and computer vision, in Algebraic Geometry and Its Applications, Springer, New York, 1994, pp. 491--506.
38.
J. Petitot, Neurogéométrie de la vision: Modèles mathématiques et physiques des architectures fonctionnelles, Éditions de l' École Polytechnique, Palaiseau, France, 2008.
39.
J. Petitot and Y. Tondut, Vers une neuro-geometrie. Fibrations corticales, structures de contact et contours subjectifs modaux, Math. Inform. Sci. Humaines, 145 (1999), pp. 5--101.
40.
L. Rifford and E. Trélat, Morse-Sard type results in sub-Riemannian geometry, Math. Ann., 332 (2005), pp. 145--159.
41.
Yu. L. Sachkov, Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane, ESAIM Control Optim. Calc. Var., 16 (2010), pp. 1018--1039.
42.
Yu. L. Sachkov, Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane, ESAIM Control Optim. Calc. Var., 17 (2011), pp. 293--321.
43.
G. Sanguinetti, G. Citti, and A. Sarti, Image completion using a diffusion driven mean curvature flow in a sub-Riemannian space, in Proceedings of the 3rd International Conference on Computer Vision Theory and Applications (VISAPP 2008), Vol. 2, Funchal, Portugal, 2008, pp. 46--53.
44.
N. V. Swindale, D. Shoham, A. Grinvald, T. Bonhoeffer, and M. Höbener, Visual cortex maps are optimized for uniform coverage, Nat. Neurosci., 3 (2000), pp. 822--826.
45.
N. Ya. Vilenkin, Special Functions and the Theory of Group Representations, Transl. Math. Monogr. 22, AMS, Providence, RI, 1968.
46.
G. Warner, Harmonic Analysis on Semi-simple Groups, Vol. 1, Springer-Verlag, New York, 1972.
47.
G. Warner, Harmonic Analysis on Semi-simple Groups, Vol. 2, Springer-Verlag, New York, 1972.
48.
A. Weil, L'intégration dans les groupes topologiques et ses applications, Actual. Sci. Ind. 869, Hermann et Cie, Paris, 1940; reprinted, Princeton, NJ, 1941.
49.
L. R. Williams and D. W. Jacobs, Stochastic completion fields: A neural model of illusory contour shape and salience, Neural Comput., 9 (1997), pp. 837--858.
50.
J. W. Zweck and L. R. Williams, Euclidean group invariant computation of stochastic completion fields using shiftable-twistable functions, J. Math. Imaging Vision, 21 (2004), pp. 135--154.
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© 2014, Society for Industrial and Applied Mathematics.
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Submitted: 12 June 2013
Accepted: 17 January 2014
Published online: 24 April 2014
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