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Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

  • Jiong Zhang (a1), Remco Duits (a2), Gonzalo Sanguinetti (a3) and Bart M. ter Haar Romeny (a1) (a4)

Abstract

Left-invariant PDE-evolutions on the roto-translation group SE(2)(and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) proposed by Citti & Sarti, and Petitot, and they include the direction process (for contour completion) proposed by Mumford. This paper presents a thorough study and comparison of the many numerical approaches, which, remarkably, are missing in the literature. Existing numerical approaches can be classified into 3 categories: Finite difference methods, Fourier based methods (equivalent to SE(2)-Fourier methods), and stochastic methods (Monte Carlo simulations). There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly (in the spatial Fourier domain) in previous works by Duits and van Almsick in 2005. Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches. We compute relative errors of all numerical approaches to the exact solutions, and the Fourier based methods show us the best performance with smallest relative errors. We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions, crucial in implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying stochastic processes that overcomes the singular behavior in the origin of time-integrated kernels. Finally, we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

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Corresponding author

*Corresponding and joint main authors.Email addresses:J.Zhang1@tue.nl (J. Zhang), R.Duits@tue.nl (R. Duits), g.r.sanguinetti@tue.nl (G. Sanguinetti), B.M.terhaarRomeny@tue.nl (B.-t.-H. Romeny)

References

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[1]Abramowitz, M. and Stegun., I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, 1965.
[2]Agrachev, 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, 256:26212655, 2009.
[3]Ali, S.T., Antoine, J.P., and Gazeau., J.P.Coherent states, wavelets and their generalizations. Springer Verlag, New York, Berlin, Heidelberg, 1999.
[4]Aronszajn., N.Theory of reproducing kernels. Trans. Amer. Math. Soc, 68:337404, 1950.
[5]Aubin., T. A course in differential geometry. Graduate studies in mathematics. American Mathematical Society, 2001.
[6]August., J. The curve indicator random field. PhD thesis, Yale University, 2001.
[7]August, J. and Zucker., S.W.The curve indicator random field and Markov processes. IEEE-PAMI, Pattern Recognition and Machine Intelligence, 25(4), 2003.
[8]Barbieri, D., Citti, G., Cocci, G., and Sarti., A. A cortical-inspired geometry for contour perception and motion integration. Journal of Mathematical Imaging and Vision, 2014. Accepted and published digitally online.
[9]Barbieri, D., Citti, G., Sanguinetti, G., and Sarti., A.An uncertainty principle underlying the functional archtecture of V1. Journal Physiology Paris, 106:183193, 2012.
[10]Bekkers, E.J., Duits, R., Berendschot, T., and ter Haar Romeny., B. A multi-orientation analysis approach to retinal vessel tracking. Journal of Mathematical Imaging and Vision, pages 1-28, 2014.
[11]Ben-Yosef, G. and Ben-Shahar., O.A tangent bundle theory for visual curve completion. IEEE-PAMI, Pattern Recognition and Machine Intelligence, 34(7):12631280, 2012.
[12]Boscain, U., Chertovskih, R., Gauthier, J.P., and Remizov., A. Hypoelliptic diffusion and human vision: A semi-discrete new twist on the Petitot theory. SIAM Journal of Imaging, 2014.
[13]Boscain, U., Duits, R., Rossi, F., and Sachkov., Y. Curve cuspless reconstruction via sub-Riemannian geometry. SIAM Journal of Imaging. Accepted on ESAIM: Control, Optimization and Calculus of Variations (COCV), 2013.
[14]Boscain, U., Duplaix, J., Gautier, J.P., and Rossi., F.Anthropomorphic image reconstruction via hypoelliptic diffusion. SIAMJ. Control Optim, 50:13091336, 2012.
[15]Bosking, W. H., Zhang, Y., Schofield, B., and Fitzpatrick., D.Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. The Journal of Neuroscience, 17(6):21122127, March 1997.
[16]Casson, R.J., Chidlow, G., Wood, J.P., Crowston, J.G., and Goldberg, I.Definition of glaucoma: clinical and experimental concepts. Clinical and Experimental Ophthalmology, 40(4):341349, 2012.
[17]Chirikjian., G. S. Stochastic models, information theory, and Lie groups, volume 1 of Applied and Nuemrical Harmonic Analysis, 2013.
[18]Chirikjian, G.S. and Kyatkin., A.B. Engineering applications of noncommutative harmonic analysis: with emphasis on rotation and motion groups. Boca Raton CRC Press, 2001.
[19]Citti, G. and Sarti., A.A cortical based model of perceptional completion in the roto-translation space. Journal of Mathematical Imaging and Vision, 24(3):307326, 2006.
[20]Creusen, E.J., Duits, R., Vilanova, A., and Florack., M.J.Numerical schemes for linear and nonlinear enhancement of DW-MRI. Numer. Math. Theor. Meth. Appl, 6(1):138168, 2013.
[21]Duits., R. Perceptual organization in image analysis. PhD thesis, Eindhoven University of Technology, The Netherlands, Eindhoven. http://www.bmi2.bmt.tue.nl/Image-Analysis/People/RDuits/THESISRDUITS.pdf September 2005.
[22]Duits, R., Boscain, U., Rossi, F., and Sachkov., Y. Association fields via cuspless sub-Riemannian geodesics in SE(2). Journal of Mathematical Imaging and Vision, 2013. In Press.
[23]Duits, R. and Burgeth., B.Scale spaces on Lie groups. In Murli Sgallari and Paragios, editors, 1st International Conference on Scale Space and Variational Methods in Computer Vision, Lecture Notes on Computer Science., pages 300312. Springer-Verlag, Jun. 2007.
[24]Duits, R., Felsberg, M., Granlund, G., and ter Haar Romeny., B. M.Image analysis and reconstruction using a wavelet transform constructed from a reducible representation of the Euclidean motion group. International Journal of Computer Vision, 79(1):79102, 2007.
[25]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 Appl Math., A.M.S., 68:255292, 2010.
[26]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 diffusion equations on invertible orientation scores. Quarterly of Appl. Mathematics, A.M.S., 68:292331, 2010.
[27]Duits, R. and van Almsick., M. The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D-Euclidean motion group. Technical Report CASA-report, nr.43., Eindhoven University of Technology Dep. of mathematics and computer science. http://www.win.tue.nl/analysis/reports/rana05-43.pdf, 2005.
[28]Duits, R. and Franken., E. M.Left-invariant stochastic evolution equations on SE(2) and its applications to contour enhancement and contour completion via invertible orientation scores. Eindhoven University of Technology, CASA report, see http://arxiv.org/abs/07.11.0951 and http://www.win.tue.nl/casa/research/casareports/2007.html 35:179, 2007.
[29]Duits, R. and van Almsick., M.The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group. Quart. Appl. Math., 66:2767, 2008.
[30]Dungey, N., ter Elst, A.F.M., and Robinson., D.W.Analysis on Lie groups with polynomial growth. Birkhauser-Progress in Mathematics, 214, 2003.
[31]Florack, L.M.J., ter Haar Romeny, B.M., Koenderink, J.J., and Viergever., A.Scale and the differential structure of images. Image and Vision Computing, 10:376388, 1992.
[32]Franken., E.M. Enhancement of Crossing Elongated Structures in Images. PhD Thesis, Eindhoven University of Technology, Netherlands. http://www.bmia.bmt.tue.nl/people/EFranken/PhDThesisErikFranken.pdf October 2008.
[33]Franken, E.M. and Duits., R.Crossing preserving coherence-enhancing diffusion on invertible orientation scores. International Journal of Computer Vision (IJCV), 85(3): 253278, 2009.
[34]H., Führ.Abstract Harmonic Analysis of Continuous Wavelet Transforms Springer Heidelberg-New York, 2005.
[35]Gaveau., B.Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents. Acta Mathematica, 139:96153, 1977.
[36]Grossmann, A., Morlet, J., and Paul., T.Integral transforms associated to square integrable representations. J. Math. Phys., 26:24732479, 1985.
[37]L., Hörmander.Hypoellptic second order differential equations. Acta Mathematica, 119:147171, 1968.
[38]Hubel, D.H. and Wiesel., T.N.Receptive fields of single neurons in the cat's striate cortex. Journal of Physiology, 148:574591, 1959.
[39]Ikram, M.K., Ong, Y.T., Cheung, C.Y., and Wong., T.Y.Retinal vascular caliber measurements: clinical significance, current knowledge and future perspectives. Ophthalmologica, 229(3):125136, 2013.
[40]Jones, W.B. and Thron., W.J.The method of fractional steps: the solution of problems of mathematical physics in several variables. Encyclopedia of Mathematics and its Applications (Book 11). Massachusetts: Addison-Wesley 1980.
[41]Kalitzin, S.N., ter Haar Romeny, B.M., and Viergever., M.A.Invertible orientation bundles on 2D scalar images. In Scale-Space Theories in Computer Vision, pp. 7788. Springer, 1997.
[42]Martens., F.J.L. Spaces of analytic functions on inductive/projective limits of Hilbert Spaces PhD Thesis, University of Technology Eindhoven, Eindhoven, Netherlands, 1988. http://alexandria.tue.nl/extra3/proefschrift/PRF6A/8810117.pdf
[43]Mashtakov, A.P., Ardentov, A.A., and Sachkov., Y.L.Parallel algorithm and software for image inpainting via sub-Riemannian minimizers on the group of rototranslations. Numerical Methods: Theory and Applications, 6(1):95115, 2013.
[44]Meixner, J. and Schaefke., F.W.Mathieusche Funktionen und Sphaeroidfunktionen. Springer, 1954.
[45]MomayyezSiakhal, P. and Siddiqi, K.. 3D stochastic completion fields for mapping connectivity in diffusion MRI. IEEE-PAMI, Pattern Recognition and Machine Intelligence, 35(4), 2013.
[46]Mumford., D.Elastica and computer vision. Algebraic Geometry and Its Applications. Springer-Verlag, pages 491506, 1994.
[47]Petitot., J. Neurogéometrie de la vision-Modeles mathématiques et physiques des architectures fonctionelles. Les Éditions de l'École Poly technique. 2008.
[48]Skibbe, H. and Reisert., M. Left-invariant diffusion on the motion group in terms of the irreducible representations of SO(3). Preprint on arXiv:1202.5414v1, see http://arxiv.org/pdf/1202.5414v1.pdf. 2012.
[49]Sachkov., Y.Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV, 16(4):10181039, 2010.
[50]Sanguinetti., G. Invariant models of vision between phenomenology, image statistics and neurosciences. PhD thesis, Universidad de la Republica, Montevideo, Uruguay, 2011. https://www.colibri.udelar.edu.uy/bitstream/123456789/2902/1/San11.pdf
[51]Sarti, A. and Citti., G. Neuromathematics of vision, volume 1 of Springer: Lecture Notes in Morphogenesis.
[52]Sharma, U. and Duits., R. Left-invariant evolutions of wavelet transforms on the similitude group. Applied Computational Harmonic Analysis, 2014. Under Review.
[53]ter Elst, A.F.M. and Robinson., D.W.Weighted subcoercive operators on Lie groups. Journal of Functional Analysis, 157:88163, 1998.
[54]Thomas, G.B. and Finney., R.L.Calculus and analytic geometry (9th ed.). Addison Wesley, 1996.
[55]Unser, M., Aldroubi, A., and Eden., M.B-spline signal processing: Part 1-theory. IEEE Transcations on Signal Processing, 41(2):821832, 1993.
[56]van Almsick., M. Context models of lines and contours. PhD thesis, Eindhoven University of Technology, 2007.
[57]Yosida., K.Functional Analysis. Springer, 1995.
[58]Zweck, J. and Williams., L.R.Euclidean group invariant computation of stochastic completion fields using shiftable-twistable functions. Journal of Mathematical Imaging and Vision, 21(2):135154, 2004.

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Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

  • Jiong Zhang (a1), Remco Duits (a2), Gonzalo Sanguinetti (a3) and Bart M. ter Haar Romeny (a1) (a4)

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