Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-05T15:50:28.794Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Part of: Multivariate analysis Miscellaneous topics - Partial differential equations Stochastic analysis Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  15 February 2016

Jiong Zhang ,
Remco Duits ,
Gonzalo Sanguinetti  and
Bart M. ter Haar Romeny
Jiong Zhang*
Affiliation:
Department of Biomedical Engineering, Biomedical Image Analysis (BMIA)
Remco Duits*
Affiliation:
Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Gonzalo Sanguinetti
Affiliation:
Department of Mathematics and Computer Science, Image Science and Technology (IST/e)
Bart M. ter Haar Romeny
Affiliation:
Department of Biomedical Engineering, Biomedical Image Analysis (BMIA) Northeastern University, Shenyang, China
*
*Corresponding and joint main authors.Email addresses:[email protected] (J. Zhang), [email protected] (R. Duits), [email protected] (G. Sanguinetti), [email protected] (B.-t.-H. Romeny)
*Corresponding and joint main authors.Email addresses:[email protected] (J. Zhang), [email protected] (R. Duits), [email protected] (G. Sanguinetti), [email protected] (B.-t.-H. Romeny)
Get access

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.

Keywords

Brownian motionEuclidean motion groupPDE's on SE(2)Mathieu operatorscontour completioncontour enhancementretinal imaging

MSC classification

Secondary: 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc. 65M80: Fundamental solutions, Green's function methods, etc. 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 62H35: Image analysis 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 1 , February 2016 , pp. 1 - 50
Copyright
Copyright © Global-Science Press 2016 

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]Abramowitz, M. and Stegun., I. A. Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, 1965.CrossRefGoogle Scholar
[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, 256:2621–2655, 2009.CrossRefGoogle Scholar
[3]Ali, S.T., Antoine, J.P., and Gazeau., J.P.Coherent states, wavelets and their generalizations. Springer Verlag, New York, Berlin, Heidelberg, 1999.Google Scholar
[4]Aronszajn., N.Theory of reproducing kernels. Trans. Amer. Math. Soc, 68:337–404, 1950.CrossRefGoogle Scholar
[5]Aubin., T. A course in differential geometry. Graduate studies in mathematics. American Mathematical Society, 2001.CrossRefGoogle Scholar
[6]August., J. The curve indicator random field. PhD thesis, Yale University, 2001.Google Scholar
[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.Google Scholar
[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.CrossRefGoogle Scholar
[9]Barbieri, D., Citti, G., Sanguinetti, G., and Sarti., A.An uncertainty principle underlying the functional archtecture of V1. Journal Physiology Paris, 106:183–193, 2012.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[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):1263–1280, 2012.CrossRefGoogle ScholarPubMed
[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.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[14]Boscain, U., Duplaix, J., Gautier, J.P., and Rossi., F.Anthropomorphic image reconstruction via hypoelliptic diffusion. SIAMJ. Control Optim, 50:1309–1336, 2012.CrossRefGoogle Scholar
[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):2112–2127, March 1997.CrossRefGoogle ScholarPubMed
[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):341–349, 2012.CrossRefGoogle ScholarPubMed
[17]Chirikjian., G. S. Stochastic models, information theory, and Lie groups, volume 1 of Applied and Nuemrical Harmonic Analysis, 2013.CrossRefGoogle Scholar
[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.CrossRefGoogle Scholar
[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):307–326, 2006.CrossRefGoogle Scholar
[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):138–168, 2013.CrossRefGoogle Scholar
[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.Google Scholar
[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.CrossRefGoogle Scholar
[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 300–312. Springer-Verlag, Jun. 2007.Google Scholar
[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):79–102, 2007.CrossRefGoogle Scholar
[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., 68:255–292, 2010.CrossRefGoogle Scholar
[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., 68:292–331, 2010.Google Scholar
[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.Google Scholar
[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 35:1–79, 2007.Google Scholar
[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., 66:27–67, 2008.CrossRefGoogle Scholar
[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.Google Scholar
[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, 10:376–388, 1992.CrossRefGoogle Scholar
[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.Google Scholar
[33]Franken, E.M. and Duits., R.Crossing preserving coherence-enhancing diffusion on invertible orientation scores. International Journal of Computer Vision (IJCV), 85(3): 253–278, 2009.CrossRefGoogle Scholar
[34]H., Führ.Abstract Harmonic Analysis of Continuous Wavelet Transforms Springer Heidelberg-New York, 2005.Google Scholar
[35]Gaveau., B.Principe de moindre action, propagation de la chaleur et estimees sous elliptiques sur certains groupes nilpotents. Acta Mathematica, 139:96–153, 1977.CrossRefGoogle Scholar
[36]Grossmann, A., Morlet, J., and Paul., T.Integral transforms associated to square integrable representations. J. Math. Phys., 26:2473–2479, 1985.CrossRefGoogle Scholar
[37]L., Hörmander.Hypoellptic second order differential equations. Acta Mathematica, 119:147–171, 1968.Google Scholar
[38]Hubel, D.H. and Wiesel., T.N.Receptive fields of single neurons in the cat's striate cortex. Journal of Physiology, 148:574–591, 1959.CrossRefGoogle Scholar
[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):125–136, 2013.CrossRefGoogle ScholarPubMed
[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.Google Scholar
[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. 77–88. Springer, 1997.CrossRefGoogle Scholar
[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.pdfGoogle Scholar
[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):95–115, 2013.Google Scholar
[44]Meixner, J. and Schaefke., F.W.Mathieusche Funktionen und Sphaeroidfunktionen. Springer, 1954.CrossRefGoogle Scholar
[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.Google Scholar
[46]Mumford., D.Elastica and computer vision. Algebraic Geometry and Its Applications. Springer-Verlag, pages 491–506, 1994.CrossRefGoogle Scholar
[47]Petitot., J. Neurogéometrie de la vision-Modeles mathématiques et physiques des architectures fonctionelles. Les Éditions de l'École Poly technique. 2008.Google Scholar
[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.Google Scholar
[49]Sachkov., Y.Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane. ESAIM: COCV, 16(4):1018–1039, 2010.Google Scholar
[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.pdfGoogle Scholar
[51]Sarti, A. and Citti., G. Neuromathematics of vision, volume 1 of Springer: Lecture Notes in Morphogenesis.Google Scholar
[52]Sharma, U. and Duits., R. Left-invariant evolutions of wavelet transforms on the similitude group. Applied Computational Harmonic Analysis, 2014. Under Review.CrossRefGoogle Scholar
[53]ter Elst, A.F.M. and Robinson., D.W.Weighted subcoercive operators on Lie groups. Journal of Functional Analysis, 157:88–163, 1998.CrossRefGoogle Scholar
[54]Thomas, G.B. and Finney., R.L.Calculus and analytic geometry (9th ed.). Addison Wesley, 1996.Google Scholar
[55]Unser, M., Aldroubi, A., and Eden., M.B-spline signal processing: Part 1-theory. IEEE Transcations on Signal Processing, 41(2):821–832, 1993.CrossRefGoogle Scholar
[56]van Almsick., M. Context models of lines and contours. PhD thesis, Eindhoven University of Technology, 2007.Google Scholar
[57]Yosida., K.Functional Analysis. Springer, 1995.CrossRefGoogle Scholar
[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):135–154, 2004.CrossRefGoogle Scholar