Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group

Authors: Remco Duits and Markus van Almsick
Journal: Quart. Appl. Math. 66 (2008), 27-67
MSC (2000): Primary 22E25, 37L05, 68U10; Secondary 34B30, 47D06
DOI: https://doi.org/10.1090/S0033-569X-07-01066-0
Published electronically: December 12, 2007
MathSciNet review: 2396651
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Abstract | References | Similar Articles | Additional Information

Abstract: We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green's functions which we derive in explicit form. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant basis of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green's functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, second order stochastic evolution equations.

References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I. A. Stegun, editors.
    Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables.
    Dover Publications, Inc., New York, 1965.
    Originally published by the National Bureau of Standards in 1964.
  • 2. M. A. van Almsick.
    Context Models of Lines and Contours
    PhD thesis, Eindhoven University of Technology, Department of Biomedical Engineering, Eindhoven, The Netherlands, 2007.
  • 3. J.P. Antoine.
    Directional wavelets revisited: Cauchy wavelets and symmetry detection in patterns.
    Applied and Computational Harmonic Analysis, 6:314-345, 1999. MR 1685408 (2000b:42025)
  • 4. N. Aronszajn.
    Theory of reproducing kernels,
    Trans. A.M.S., vol. 68, pp. 337-404, 1950. MR 0051437 (14:479c)
  • 5. J. August.
    The Curve Indicator Random Field.
    PhD thesis, Yale University, 2001.
  • 6. J. August and S.W. Zucker.
    The curve indicator random field and markov processes.
    IEEE-PAMI, Pattern Recognition and Machine Intelligence, 25, 2003.
    Number 4.
  • 7. G. Blanch and D. S. Clemm.
    The double points of Mathieu's equation.
    Math.Comp., 23:97-108, 1969. MR 0239727 (39:1084)
  • 8. W.H. Bosking, Y. Zhang, B. Schofield, and D. Fitzpatrick.
    Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex.
    The Journal of Neuroscience, 17(6):2112-2127, March 1997.
  • 9. G. Citti and A. Sarti.
    A cortical based model of perceptional completion in the roto-translation space. Journal of Mathematical Imaging and Vision, 24:307-326, 2006.
  • 10. M. Duits.
    A functional Hilbert space approach to frame transforms and wavelet transforms.
    September 2004.
    Master thesis in Applied Analysis. Dep. Mathematics and Computer Science, Eindhoven University of Technology
  • 11. M. Duits and R. Duits.
    Unitary wavelet transforms based on reducible representations of the affine group.
    In preparation.
  • 12. R. Duits, E.M. Franken and M. van Almsick.
    Contour Enhancement via linear and non-linear evolution equations on the Euclidean Motion Group.
    In preparation.
  • 13. R. Duits.
    Perceptual Organization in Image Analysis.
    PhD thesis, Eindhoven University of Technology, Department of Biomedical Engineering, The Netherlands, 2005.
    A digital version is available at URL: http:// www.bmi2.bmt.tue.nl/Image-Analysis/People/RDuits/THESISRDUITS.pdf.
  • 14. R. Duits and M. van Almsick,
    The explicit solutions of the left-invariant evolution equations on the Euclidean motion group.
    Technical Report CASA 05-43, Department of Mathematics and Computer science, Eindhoven University of Technolgy, The Netherlands, December 2005.
    A digital version is available on the web at URL: http://yp.bmt.tue.nl/pdfs/6321.pdf.
  • 15. 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.
    International Journal of Computer Vision.
    Accepted for publication. To appear in Volume 72, issue 1, April 2007.
  • 16. R. Duits, M. Duits, M. van Almsick and B.M. ter Haar Romeny.
    Invertible Orientation Scores as an Application of Generalized Wavelet Theory.
    Image Processing, Analysis, Recognition, and Understanding.
    Volume 17, Nr. 1: pp. 42-75 , 2007.
  • 17. R. Duits, L.M.J. Florack, J. de Graaf, and B. ter Haar Romeny.
    On the axioms of scale space theory.
    Journal of Mathematical Imaging and Vision, 20:267-298, May 2004. MR 2060148 (2005k:94005)
  • 18. R. Duits, M. van Almsick, M. Duits, E. Franken, and L.M.J. Florack.
    Image processing via shift-twist invariant operations on orientation bundle functions.
    In Niemann Zhuralev et al. Geppener, Gurevich, editor, 7th International Conference on Pattern Recognition and Image Analysis: New Information Technologies, pages 193-196, St. Petersburg, October 2004.
  • 19. R. Duits, B. Burgeth.
    Scale Spaces on Lie groups.
    In the proceedings of SSVM 2007, 1st international conference on scale space and variational methods in computer vision, Lecture Notes on Computer Science, Springer-Verlag, 2007, p. 300-312, Ischia, Italy, June 2007.
  • 20. N. Dungey, A. F. M. ter Elst, and D. W. Robinson.
    Analysis on Lie groups with polynomial growth, volume 214.
    Birkhauser-Progress in Mathematics, Boston, 2003. MR 2000440 (2004i:22010)
  • 21. M. Felsberg, P.-E. Forssén, and H. Scharr.
    Efficient robust smoothing of low-level signal features.
    Technical Report LiTH-ISY-R-2619, SE-581, 83 Linkoping, Sweden, August 2004.
  • 22. M. Felsberg, P.-E. Forssén, and H. Scharr.
    Channel smoothing: Efficient robust smoothing of low-level signal features.
    IEEE Transactions on Pattern Analysis and Machine Intelligence, 2005.
  • 23. G. Floquet.
    Sur les équations différentielles linéaires à coefficients périodiques.
    Ann. École Norm. Sup., 12(47), 1883.
  • 24. P.-E. Forssén and G. H. Granlund.
    Sparse feature maps in a scale hierarchy.
    In G. Sommer and Y.Y. Zeevi, editors, Proc. Int. Workshop on Algebraic Frames for the Perception-Action Cycle, volume 1888 of Lecture Notes in Computer Science, Kiel, Germany, September 2000. Springer, Heidelberg.
  • 25. P.E. Forssen.
    Low and Medium Level Vision using Channel Representations.
    PhD thesis, Linkoping University, Dept. EE, Linkoping, Sweden, March 2004.
  • 26. E. Franken, M. van Almsick, P. Rongen, L.M.J. Florack and B.M. ter Haar Romeny.
    An Efficient Method for Tensor Voting using Steerable Filters.
    Proceedings European Congress on Computer Vision 2006, 288-240, 2006.
  • 27. E. Franken, R. Duits and B.M. ter Haar Romeny.
    Nonlinear Diffusion on the Euclidean Motion Group.
    In the proc. of the 1st international conference on scale space and variational methods in computer vision, Lecture Notes on Computer Science, Springer-Verlag, 2007, p. 461-472, Ischia, Italy, June 2007.
  • 28. H. Führ.
    Abstract Harmonic Analysis of Continuous Wavelet Transforms.
    Springer, Heidelberg-New York, 2005. MR 2130226 (2006m:43003)
  • 29. A. Grossmann, J. Morlet, and T. Paul.
    Integral transforms associated to square integrable representations.
    J. Math. Phys., 26:2473-2479, 1985. MR 803788 (86k:22013)
  • 30. W. Hebisch.
    Estimates on the semigroups generated by left-invariant operators on Lie groups.
    Journal fuer die Reine und Angewandte Mathematik, 423:1-45, 1992. MR 1142482 (93d:22008)
  • 31. G. W. Hill.
    On the part of motion of the lunar perigee, which is a function of the mean motions of the sun and the moon.
    Acta Mathematica, 1, 1886.
  • 32. L. Hormander.
    Hypoellptic second order differential equations.
    Acta Mathematica, 119:147-171, 1968. MR 0222474 (36:5526)
  • 33. C. Hunter and B. Guerrieri.
    The eigenvalues of Mathieu's equation and their branch points.
    Studies in Applied Mathematics, 64:113-141, 1981. MR 608595 (82c:34030)
  • 34. P. E. T. Jorgensen.
    Representations of differential operators on a Lie group.
    Journal of Functional Analysis, 20:105-135, 1975. MR 0383469 (52:4350)
  • 35. S. N. Kalitzin, B. M. ter Haar Romeny, and M. A. Viergever.
    Invertible apertured orientation filters in image analysis.
    Int. Journal of Computer Vision, 31(2/3):145-158, April 1999.
  • 36. T. S. Lee.
    Image representation using 2D gabor wavelets.
    IEEE-Transactions on Pattern Analysis and Machine Inteligence, 18(10):959-971, 1996.
  • 37. W. Magnus and S. Winkler.
    Hill's equation.
    Dover, New York, 1979. MR 559928 (80k:34001)
  • 38. Gérard Medioni, Mi-Suen Lee, and Chi-Keung Tang.
    A Computational Framework for Segmentation and Grouping.
    Elsevier, Amsterdam.
  • 39. J. Meixner and F. W. Schaefke.
    Mathieusche Funktionen und Sphaeroidfunktionen.
    Springer-Verlag, Berlin-Gotingen-Heidelberg, 1954.
  • 40. D. Mumford.
    Elastica and computer vision.
    Algebraic Geometry and Its Applications. Springer-Verlag, pages 491-506, 1994. MR 1272050 (95a:92026)
  • 41. B. Øksendahl.
    Stochastic differential equations: an introduction with applications.
    Springer, Berlin, 1998. MR 1619188 (99c:60119)
  • 42. N. Petkov and M. Kruizinga.
    Computational models of visual measures specialized in the detection of periodic and aperiodic orientation visual stimuli and grating cells.
    Biological Cybernetics, 76:83-96, 1997.
  • 43. Suigiura, M.
    Unitary representations and harmonic analysis.
    North-Holland, 2nd Edition, Mathematical Library, 44, Amsterdam, Kodansha, Tokyo, 1990. MR 1049151 (91c:22028)
  • 44. K.K. Thornber and L.R. Williams.
    Analytic solution of stochastic completion fields.
    Biological Cybernetics, 75:141-151, 1996.
  • 45. K.K. Thornber and L.R. Williams.
    Characterizing the Distribution of Completion Shapes with Corners Using a Mixture of Random Processes.
    Pattern Recognition, 33:543-553, 2000.
  • 46. D. Y. Ts'0, R. D. Frostig, E. E. Lieke, and A. Grinvald.
    Functional organization of primate visual cortex revealed by high resolution optical imaging.
    Science, 249:417-20, 1990.
  • 47. M. A. van Almsick, R. Duits, E. Franken, and B.M. ter Haar Romeny.
    From stochastic completion fields to tensor voting.
    In Proceedings DSSCC-workshop on Deep Structure Singularities and Computer Vision, pages 124-134, Maastricht, The Netherlands, June 9-10, 2005. Springer-Verlag.
  • 48. V.S. Varadarajan.
    Lie Groups, Lie-Algebras and Their Representations.
    Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1984. MR 746308 (85e:22001)
  • 49. H. Volkmer.
    On the growth of convergence radii for the eigenvalues of the Mathieu equation.
    Math. Nachr., 192:239-253, 1998. MR 1626348 (2000a:34053)
  • 50. L. R. Williams and J.W. Zweck.
    A rotation and translation invariant saliency network.
    Biological Cybernetics, 88:2-10, 2003.
  • 51. J. Zweck and L. R. Williams.
    Euclidean group invariant computation of stochastic completion fields using shiftable-twistable functions.
    Journal of Mathematical Imaging and Vision, 21(2):135-154, 2004. MR 2090129 (2005d:68131)

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Additional Information

Remco Duits
Affiliation: Department of Mathematics/Computer Science and Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: R.Duits@tue.nl

Markus van Almsick
Affiliation: Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: M.v.Almsick@tue.nl

DOI: https://doi.org/10.1090/S0033-569X-07-01066-0
Keywords: Lie groups, stochastic evolution equations, image analysis, direction process, completion field
Received by editor(s): May 2, 2006
Published electronically: December 12, 2007
Additional Notes: The Netherlands Organization for Scientific Research is gratefully acknowledged for financial support.
Article copyright: © Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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