Hypoelliptic Diffusion and Human Vision: A Semidiscrete New Twist

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.

[1]  Constantin Corduneanu,et al.  Almost periodic functions , 1968 .

[2]  J. Gauthier,et al.  The intrinsic hypoelliptic Laplacian and its heat kernel on unimodular Lie groups , 2008, 0806.0734.

[3]  Michael Felsberg,et al.  Image Analysis and Reconstruction using a Wavelet Transform Constructed from a Reducible Representation of the Euclidean Motion Group , 2007, International Journal of Computer Vision.

[4]  James Damon,et al.  Generic Structure of Two-Dimensional Images Under Gaussian Blurring , 1998, SIAM J. Appl. Math..

[5]  E. Franken.  Enhancement of crossing elongated structures in images , 2008 .

[6]  Alexey Pavlovich Mashtakov.  Parallel Algorithm and Software for Image Inpainting via Sub-Riemannian Minimizers on the Group of Rototranslations , 2013 .

[7]  A. A. Abramov,et al.  On the application of the method of successive substitution to the determination of periodic solutions of differential and difference equations , 1963 .

[8]  W. Hoffman.  The visual cortex is a contact bundle , 1989 .

[9]  Ohad Ben-Shahar,et al.  Cortical connections and early visual function: intra- and inter-columnar processing , 2003, Journal of Physiology-Paris.

[10]  Giovanna Citti,et al.  Image Completion Using a Diffusion Driven Mean Curvature Flowin A Sub-Riemannian Space , 2008, VISAPP.

[11]  N. U. Ahmed,et al.  Nonlinear filtering of systems governed by Ito differential equations with jump parameters , 1986 .

[12]  L. Florack,et al.  Evolution equations on Gabor transforms and their applications , 2011, 1110.6087.

[13]  Per-Erik Forssén,et al.  Low and Medium Level Vision Using Channel Representations , 2004 .

[14]  J. Petitot,et al.  Vers une neurogéométrie. Fibrations corticales, structures de contact et contours subjectifs modaux , 1999 .

[15]  Yuri L. Sachkov.  Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane , 2010 .

[16]  D. Mumford.  Elastica and Computer Vision , 1994 .

[17]  Geoffrey E. Hinton,et al.  Parallel visual computation , 1983, Nature.

[18]  R. Léandre,et al.  Minoration en temps petit de la densité d'une diffusion dégénérée , 1987 .

[19]  R. Duits,et al.  The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group , 2007 .

[20]  J. Dixmier.  Les C*-algèbres et leurs représentations .. , 1964 .

[21]  R. Léandre,et al.  Majoration en temps petit de la densité d'une diffusion dégénérée , 1987 .

[22]  R. Duits,et al.  Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part II: Non-linear left-invariant diffusions on invertible orientation scores , 2010 .

[23]  Herbert Heyer.  Dualität lokalkompakter Gruppen , 1970 .

[24]  Hsin Chu,et al.  Compactification and duality of topological groups , 1966 .

[25]  Francesco Rossi,et al.  Existence of planar curves minimizing length and curvature , 2009, 0906.5290.

[26]  Lance R. Williams,et al.  Euclidean Group Invariant Computation of Stochastic Completion Fields Using Shiftable-Twistable Functions , 2000, Journal of Mathematical Imaging and Vision.

[27]  Calvin C. Moore,et al.  On the regular representation of a nonunimodular locally compact group , 1976 .

[28]  Giovanna Citti,et al.  A Cortical Based Model of Perceptual Completion in the Roto-Translation Space , 2006, Journal of Mathematical Imaging and Vision.

[29]  André Weil,et al.  L'integration dans les groupes topologiques et ses applications , 1951 .

[30]  D. Hubel,et al.  Receptive fields of single neurones in the cat's striate cortex , 1959, The Journal of physiology.

[31]  Jean-Paul Gauthier,et al.  Anthropomorphic Image Reconstruction via Hypoelliptic Diffusion , 2010, SIAM J. Control. Optim..

[32]  Remco Duits,et al.  Association Fields via Cuspless Sub-Riemannian Geodesics in SE(2) , 2013, Journal of Mathematical Imaging and Vision.

[33]  Scott D. Pauls,et al.  Minimal Surfaces in the Roto-Translation Group with Applications to a Neuro-Biological Image Completion Model , 2009, Journal of Mathematical Imaging and Vision.

[34]  N. Vilenkin.  Special Functions and the Theory of Group Representations , 1968 .

[35]  Tai Sing Lee,et al.  Image Representation Using 2D Gabor Wavelets , 1996, IEEE Trans. Pattern Anal. Mach. Intell..

[36]  Remco Duits,et al.  Curve cuspless reconstruction via sub-Riemannian geometry , 2012, 1203.3089.

[37]  R. Duits,et al.  Left-invariant parabolic evolutions on SE(2) and contour enhancement via invertible orientation scores. Part I: Linear left-invariant diffusion equations on SE(2) , 2010 .

[38]  G. Marchuk.  Methods of Numerical Mathematics , 1982 .

[39]  Andrei Agrachev,et al.  Sub-Riemannian structures on 3D lie groups , 2010, 1007.4970.

[40]  Y. Sachkov,et al.  Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane , 2011 .

[41]  Dusa McDuff,et al.  HARMONIC ANALYSIS ON SEMI‐SIMPLE LIE GROUPS—I , 1974 .

[42]  Amiram Grinvald,et al.  Visual cortex maps are optimized for uniform coverage , 2000, Nature Neuroscience.

[43]  Steven W. Zucker,et al.  Sketches with Curvature: The Curve Indicator Random Field and Markov Processes , 2003, IEEE Trans. Pattern Anal. Mach. Intell..

[44]  Patrick Pérez,et al.  Geometrically Guided Exemplar-Based Inpainting , 2011, SIAM J. Imaging Sci..

[45]  M. Golubitsky,et al.  Geometric visual hallucinations, Euclidean symmetry and the functional architecture of striate cortex. , 2001, Philosophical transactions of the Royal Society of London. Series B, Biological sciences.

[46]  Lance R. Williams,et al.  Stochastic Completion Fields: A Neural Model of Illusory Contour Shape and Salience , 1995, Neural Computation.

[47]  L. Hörmander.  Hypoelliptic second order differential equations , 1967 .

[48]  A. Agrachev,et al.  Control Theory from the Geometric Viewpoint , 2004 .

[49]  Yves Achdou,et al.  A finite difference scheme on a non commutative group , 2001, Numerische Mathematik.

[50]  Remco Duits,et al.  Crossing-Preserving Coherence-Enhancing Diffusion on Invertible Orientation Scores , 2009, International Journal of Computer Vision.