Skip to main content
SpringerLink
Log in

The Monogenic Scale-Space: A Unifying Approach to Phase-Based Image Processing in Scale-Space

  • Published:
Journal of Mathematical Imaging and Vision Aims and scope Submit manuscript

Abstract

In this paper we address the topics of scale-space and phase-based image processing in a unifying framework. In contrast to the common opinion, the Gaussian kernel is not the unique choice for a linear scale-space. Instead, we chose the Poisson kernel since it is closely related to the monogenic signal, a 2D generalization of the analytic signal, where the Riesz transform replaces the Hilbert transform. The Riesz transform itself yields the flux of the Poisson scale-space and the combination of flux and scale-space, the monogenic scale-space, provides the local features phase-vector and attenuation in scale-space. Under certain assumptions, the latter two again form a monogenic scale-space which gives deeper insight to low-level image processing. In particular, we discuss edge detection by a new approach to phase congruency and its relation to amplitude based methods, reconstruction from local amplitude and local phase, and the evaluation of the local frequency.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. J. Babaud, A.P.Witkin, M. Baudin, and R.O. Duda, “Uniqueness of the Gaussian kernel for scale-space filtering,” IEEE Transactions onPattern Analysis andMachine Intelligence,Vol. 8, No. 1, pp. 26–33, 1986.

    Google Scholar 

  2. J. Behar, M. Porat, and Y.Y. Zeevi. “Image reconstruction from localized phase,” IEEE Transactions on Signal Processing, Vol. 40, No. 4, pp. 736–743, 1992.

    Google Scholar 

  3. R.N. Bracewell, The Fourier Transform and its Applications, McGraw Hill, 1986.

  4. R.N. Bracewell, Two-Dimensional Imaging, Prentice Hall Signal Processing Series. Prentice Hall, Englewood Cliffs, 1995.

    Google Scholar 

  5. F. Brackx, R. Delanghe, and F. Sommen, Clifford Analysis, Pitman: Boston, 1982.

    Google Scholar 

  6. I. Bronstein, K. Semendjajew, G. Musiol, and H. Mühlig, Taschenbuch der Mathematik, Verlag Harri Deutsch, Frankfurt, 1993.

    Google Scholar 

  7. K. Burg, H. Haf, and F. Wille, Höhere Mathematik für Ingenieure, Band V Funktionalanalysis und Partielle Differentialgleichungen, Teubner Stuttgart, 1993.

  8. K. Burg, H. Haf, and F. Wille, Höhere Mathematik für Ingenieure, Band IV Vektoranalysis und Funktionentheorie, Teubner Stuttgart, 1994.

  9. P.J. Burt and E.H. Adelson, “The Laplacian pyramid as a compact image code,” IEEE Trans. Communications, Vol. 31, No. 4, pp. 532–540, 1983.

    Google Scholar 

  10. F. Catté, P.-L. Lions, J.-M. Morel, and T. Coll, “Image selective smoothing and edge detection by nonlinear diffusion,” SIAM J. Numer. Analysis, Vol. 32, pp. 1895–1909, 1992.

    Google Scholar 

  11. R. Duits, L.M.J. Florack, J. de Graaf, and B.M. ter Haar Romeny, “On the axioms of scale space theory,” Journal of Mathematical Imaging and Vision, 2002 (accepted).

  12. R. Duits, L.M.J. Florack, B. M. ter Haar Romeny, and J. de Graaf, “Scale-space axioms critically revisited,” in Signal and Image Processing, N. Younan (Ed.), IASTED, ACTA Press, Kauai, August 2002, pp. 304–309.

    Google Scholar 

  13. M. Evans, N. Hastings, and J.B. Peacock, Statistical Distributions, 3rd. ed., Wiley-Interscience, 2000.

  14. M. Felsberg, “Disparity from monogenic phase,” in 24. DAGM Symposium Mustererkennung, Zürich, L.V. Gool (Ed.), Vol. 2449 of Lecture Notes in Computer Science, Springer, Heidelberg, 2002, pp. 248–256.

    Google Scholar 

  15. M. Felsberg, Low-Level Image Processing with the Structure Multivector, Ph.D. thesis, Institute of Computer Science and Applied Mathematics, Christian-Albrechts-University of Kiel, 2002. TR no. 0203, available at http://www.informatik.unikiel. de/reports/2002/0203.html.

  16. M. Felsberg, R. Duits, and L. Florack, “The monogenic scale space on a bounded domain and its applications,” in Scale Space Conference, 2003 (accepted).

  17. M. Felsberg and G. Sommer, “A new extension of linear signal processing for estimating local properties and detecting features,” in 22. DAGMSymposium Mustererkennung, G. Sommer, N. Krüger, and C. Perwass (Eds.), Springer, Heidelberg, Kiel, 2000, pp. 195–202.

    Google Scholar 

  18. M. Felsberg and G. Sommer, “The monogenic signal,” IEEE Transactions on Signal Processing, Vol. 49, No. 12, pp. 3136–3144, 2001.

    Google Scholar 

  19. M. Felsberg and G. Sommer, “Scale adaptive filtering derived from the Laplace equation,” in 23. DAGM Symposium Mustererkennung, B. Radig and S. Florczyk (Eds.),Vol. 2191 of Lecture Notes in Computer Science, Springer, Heidelberg, München, 2001, pp. 124–131.

  20. M. Felsberg and G. Sommer, “The Poisson scale-space: A uni-fied approach to phase-based image processing in scale-space,” Tech. Rep. LiTH-ISY-R-2453, Dept. EE, Linköping University, SE-581 83 Linköping, Sweden, 2002.

  21. M. Felsberg and G. Sommer, “The structure multivector,” in Applied Geometrical Algebras in Computer Science and Engineering, Birkhäuser, Boston, 2002, pp. 437–448.

    Google Scholar 

  22. L. Florack, Image Structure, Vol. 10 of Computational Imaging and Vision, Kluwer Academic Publishers, 1997.

  23. L. Florack and A. Kuijper, “The topological structure of scalespace images,” Journal of Mathematical Imaging and Vision, Vol. 12, No. 1, pp. 65–79, 2000.

    Google Scholar 

  24. G.H. Granlund, “In search of a general picture processing operator,” Computer Graphics and Image Processing, Vol. 8, pp. 155–173, 1978.

    Google Scholar 

  25. G.H. Granlund and H. Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers: Dordrecht, 1995.

    Google Scholar 

  26. S.L. Hahn, Hilbert Transforms in Signal Processing, Artech House: Boston, London, 1996.

    Google Scholar 

  27. D. Hestenes, “Multivector calculus,” J. Math. Anal. and Appl., Vol. 24, No. 2, pp. 313–325, 1968.

    Google Scholar 

  28. R.A. Hummel, “Representations based on zero-crossings in scale space,” in Proc. IEEE Comp. Soc. Conf. Computer Vision and Pattern Recognition, Miami Beach, 1986, pp. 204–209.

  29. T. Iijima, “Basic theory of pattern observation,” in Papers of Technical Group on Automata and Automatic Control, IECE, Japan, December 1959.

  30. T. Iijima, “Basic theory on normalization of pattern (In case of a typical one-dimensional pattern),” Bulletin of the Electrotechnical Laboratory, Vol. 26, pp. 368–388, 1962.

    Google Scholar 

  31. T. Iijima, “Observation theory of two-dimensional visual patterns,” in Papers of Technical Group on Automata and Automatic Control, IECE, Japan, October 1962.

  32. B. Jähne, Digitale Bildverarbeitung, Springer: Berlin, 1997.

    Google Scholar 

  33. J.J. Koenderink, “The structure of images,” Biological Cybernetics, Vol. 50, pp. 363–370, 1984.

    Google Scholar 

  34. P. Kovesi, “Image features from phase information,” Videre: Journal of Computer Vision Research, Vol. 1, No. 3, 1999.

  35. S.G. Krantz, Handbook of Complex Variables, Birkhäuser: Boston, 1999.

    Google Scholar 

  36. G. Krieger and C. Zetzsche, “Nonlinear image operators for the evaluation of local intrinsic dimensionality,” IEEE Transactions on Image Processing, Vol. 5, No. 6, pp. 1026–1041, 1996.

    Google Scholar 

  37. T. Lindeberg, Scale-Space Theory in Computer Vision, The Kluwer International Series in Engineering and Computer Science. Kluwer Academic Publishers: Boston, 1994.

    Google Scholar 

  38. T. Lindeberg, “Linear spatio-temporal scale-space,” in Scale-Space Theory in Computer Vision, Vol. 1252 of Lecture Notes in Computer Science, Springer: Utrecht, Netherlands, 1997.

    Google Scholar 

  39. T. Lindeberg, On the Axiomatic Foundations of Linear Scale-Space: Combining Semi-Group Structure with Causality vs. Scale Invariance, Ch. 6, Kluwer Academic, 1997.

  40. A. Papoulis, The Fourier Integral and its Applications, McGraw-Hill: New York, 1962.

    Google Scholar 

  41. A. Papoulis, Probability, Random Variables and Stochastic Processes, McGraw-Hill, 1965.

  42. E.J. Pauwels, L.J. Van Gool, P. Fiddelaers, and T. Moons, “An extended class of scale-invariant and recursive scale space filters,” IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol. 17, No. 7, pp. 691–701, 1995.

    Google Scholar 

  43. P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” IEEE Trans. Pattern Analysis and Machine Intelligence, Vol. 12, No. 7, pp. 629–639, 1990.

    Google Scholar 

  44. D. Reisfeld, “The constrained phase congruency feature detector: Simultaneous localization, classification and scale determination,” Pattern Recognition Letters, Vol. 17, pp. 1161–1169, 1996.

    Google Scholar 

  45. J.L. Schiff, The Laplace Transform, Undergraduate Texts in Mathematics. Springer: New York, 1999.

    Google Scholar 

  46. N. Sochen, R. Kimmel, and R. Malladi, “A geometrical framework for low level vision,” IEEE Trans. on Image Processing, Special Issue on PDE Based Image Processing, Vol. 7, No. 3, pp. 310–318, 1998.

    Google Scholar 

  47. E. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press: New Jersey, 1971.

    Google Scholar 

  48. J. Weickert, “Anisotropic siffusion in image processing,” Ph.D. thesis, Faculty of Mathematics, University of Kaiserslautern, 1996.

  49. J. Weickert, “A review of nonlinear diffusion filtering,” in Scale-Space Theory in Computer Vision, B. ter Haar Romeny, L. Florack, J. Koenderink, and M. Viergever (Eds.), Vol. 1252 of LNCS, Springer: Berlin, 1997, pp. 260–271.

    Google Scholar 

  50. J. Weickert, S. Ishikawa, and A. Imiya, “Scale-space has first been proposed in Japan,” Mathematical Imaging and Vision, Vol. 10, pp. 237–252, 1999.

    Google Scholar 

  51. A.P. Witkin, “Scale-space filtering,” in Proc. 8th Int. Joint Conf. Art. Intell., 1983, pp. 1019–1022.

  52. A.L. Yuille and T. Poggio, “Scaling theorems for zerocrossings,” IEEE Trans. Pattern Analysis and Machine Intell., Vol. 8, pp. 15–25, 1986.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Electrical Engineering, Linköping University, Sweden

    M. Felsberg

  2. Institute of Computer Science and Applied Mathematics, Christian-Albrechts-University of Kiel, Germany

    G. Sommer

Authors
  1. M. Felsberg
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Sommer
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

About this article

Cite this article

Felsberg, M., Sommer, G. The Monogenic Scale-Space: A Unifying Approach to Phase-Based Image Processing in Scale-Space. Journal of Mathematical Imaging and Vision 21, 5–26 (2004). https://doi.org/10.1023/B:JMIV.0000026554.79537.35

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1023/B:JMIV.0000026554.79537.35

Search

Navigation