Journal of Mathematical Imaging and Vision

, Volume 56, Issue 2, pp 352–366 | Cite as

Morphological Counterparts of Linear Shift-Invariant Scale-Spaces

  • Martin Schmidt
  • Joachim Weickert


It is well-known that there are striking analogies between linear shift-invariant systems and morphological systems for image analysis. So far, however, the relations between both system theories are mainly understood on a pure convolution / erosion level. A formal connection on the level of differential or pseudodifferential equations and their induced scale-spaces is still missing. The goal of our paper is to close this gap. We present a simple and fairly general dictionary that allows to translate any linear shift-invariant evolution equation into its morphological counterpart and vice versa. It is based on a scale-space representation by means of the symbol of its (pseudo)differential operator. Introducing a novel transformation, the Cramér–Fourier transform, puts us in a position to relate the symbol to the structuring function of a morphological scale-space of Hamilton–Jacobi type. As an application of our general theory, we derive the morphological counterparts of many linear shift-invariant scale-spaces, such as the Poisson scale-space, \(\alpha \)-scale-spaces, summed \(\alpha \)-scale-spaces, relativistic scale-spaces, and their anisotropic variants. Our findings are illustrated by experiments.


Scale-space Mathematical morphology Partial differential equations Pseudodifferential equations Convex analysis Cramér transform Infimal convolution 



Our research on this topic has been initiated by discussions between Joachim Weickert and Bernd Sturmfels (Berkeley) during the DEMAIN Conference in Münster (Sept. 30–Oct. 2, 2014). It is a pleasure to thank Angela Stevens (Münster) for organising this highly inspiring interdisciplinary conference.


  1. 1.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover, New York (1974)MATHGoogle Scholar
  2. 2.
    Akian, M., Quadrat, J., Viot, M.: Bellman processes. ICAOS ’94: Discrete Event Systems. Lecture Notes in Control and Information Sciences, vol. 199, pp. 302–311. Springer, London (1994)Google Scholar
  3. 3.
    Alvarez, L., Guichard, F., Lions, P.L., Morel, J.M.: Axioms and fundamental equations in image processing. Arch. Ration. Mech. Anal. 123, 199–257 (1993)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Alvarez, L., Lions, P.L., Morel, J.M.: Image selective smoothing and edge detection by nonlinear diffusion II. SIAM J. Numer. Anal. 29, 845–866 (1992)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Angulo, J.: Pseudo-morphological image diffusion using the counter-harmonic paradigm. In: Blanc-Talon, J., Bone, D., Philips, W., Popescu, D., Scheunders, P. (eds.) Advanced Concepts for Intelligent Vision Systems. Lecture Notes in Computer Science, vol. 6474, pp. 426–437. Springer, Berlin (2010)CrossRefGoogle Scholar
  6. 6.
    Arehart, A.B., Vincent, L., Kimia, B.B.: Mathematical morphology: the Hamilton-Jacobi connection. In: Proceedings of Fourth International Conference on Computer Vision, pp. 215–219. IEEE Computer Society Press, Berlin (1993)Google Scholar
  7. 7.
    Baccelli, F., Cohen, G., Olsder, G.J., Quadrat, J.: Synchronization and Linearity: An Algebra for Discrete Event Systems. Wiley, Chichester (1992)MATHGoogle Scholar
  8. 8.
    Breuß, M., Burgeth, B., Weickert, J.: Anisotropic continuous-scale morphology. In: Martí, J., Benedí, J.M., Mendonça, A., Serrat, J. (eds.) Pattern Recognition and Image Analysis. Lecture Notes in Computer Science, vol. 4478, pp. 515–522. Springer, Berlin (2007)CrossRefGoogle Scholar
  9. 9.
    Brockett, R.W., Maragos, P.: Evolution equations for continuous-scale morphology. In: Proceedings of IEEE International Conference on Acoustics, Speech and Signal Processing, vol. 3, pp. 125–128. San Francisco, CA (1992)Google Scholar
  10. 10.
    Burgeth, B., Didas, S., Weickert, J.: The Bessel scale-space. In: Olsen, O., Florack, L., Kuijper, A. (eds.) Deep Structure, Singularities, and Computer Vision. Lecture Notes in Computer Science, vol. 3753, pp. 84–95. Springer, Berlin (2005)CrossRefGoogle Scholar
  11. 11.
    Burgeth, B., Didas, S., Weickert, J.: Relativistic scale-spaces. In: Kimmel, R., Sochen, N., Weickert, J. (eds.) Scale Space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3459, pp. 1–12. Springer, Berlin (2005)CrossRefGoogle Scholar
  12. 12.
    Burgeth, B., Weickert, J.: An explanation for the logarithmic connection between linear and morphological system theory. Int. J. Comput. Vis. 64(2/3), 157–169 (2005)CrossRefGoogle Scholar
  13. 13.
    Demetz, O., Weickert, J., Bruhn, A., Zimmer, H.: Optic flow scale-space. In: Bruckstein, A.M., ter Haar Romeny, B., Bronstein, A.M., Bronstein, M.M. (eds.) Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 6667, pp. 713–724. Springer, Berlin (2012)Google Scholar
  14. 14.
    Didas, S., Burgeth, B., Imiya, A., Weickert, J.: Regularity and scale-space properties of fractional high order linear filtering. In: Kimmel, R., Sochen, N., Weickert, J. (eds.) Scale Space and PDE Methods in Computer Vision. Lecture Notes in Computer Science, vol. 3459, pp. 13–25. Springer, Berlin (2005)CrossRefGoogle Scholar
  15. 15.
    Diop, E.H., Angulo, J.: Multiscale image analysis based on robust and adaptive morphological scale-spaces. Image Anal. Stereol. 34(1), 39–50 (2014)MathSciNetGoogle Scholar
  16. 16.
    Dorst, L., van den Boomgaard, R.: Morphological signal processing and the slope transform. Signal Process. 38, 79–98 (1994)CrossRefGoogle Scholar
  17. 17.
    Duits, R., Burgeth, B.: Scale spaces on Lie groups. In: Sgallari, F., Murli, A., Paragios, N. (eds.) Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 4485, pp. 300–312. Springer, Berlin (2007)CrossRefGoogle Scholar
  18. 18.
    Duits, R., DelaHaije, T., Creusen, E., Ghosh, A.: Morphological and linear scale spaces for fiber enhancement in DW-MRI. J. Math. Imaging Vis. 46(3), 326–368 (2013)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Duits, R., Florack, L., de Graaf, J., ter Haar Romeny, B.: On the axioms of scale space theory. J. Math. Imaging Vis. 20, 267–298 (2004)Google Scholar
  20. 20.
    Felsberg, M., Sommer, G.: Scale-adaptive filtering derived from the Laplace equation. In: Radig, B., Florczyk, S. (eds.) Pattern Recognition. Lecture Notes in Computer Science, vol. 2032, pp. 95–106. Springer, Berlin (2001)Google Scholar
  21. 21.
    Florack, L.: Image Structure, Computational Imaging and Vision, vol. 10. Kluwer, Dordrecht (1997)Google Scholar
  22. 22.
    Florack, L.M.J., Maas, R., Niessen, W.J.: Pseudo-linear scale-space theory. Int. J. Comput. Vis. 31(2/3), 247–259 (1999)CrossRefGoogle Scholar
  23. 23.
    Guichard, F., Morel, J.M.: Partial differential equations and image iterative filtering. In: Duff, I.S., Watson, G.A. (eds.) The State of the Art in Numerical Analysis. IMA Conference Series (New Series), vol. 63, pp. 525–562. Clarendon Press, Oxford (1997)Google Scholar
  24. 24.
    Heijmans, H.J.A.M.: Morphological scale-spaces, scale-invariance and Lie groups. In: H. Talbot, R. Beare (eds.) International Symposium on Mathematical Morphology, pp. 253–264 (2002)Google Scholar
  25. 25.
    Heijmans, H.J.A.M., Maragos, P.: Lattice calculus of the morphological slope transform. Signal Process. 59(1), 17–42 (1997)CrossRefMATHGoogle Scholar
  26. 26.
    Iijima, T.: Basic theory on normalization of pattern (in case of typical one-dimensional pattern). Bull. Electrotech. Lab. 26, 368–388 (1962). In JapaneseGoogle Scholar
  27. 27.
    Iijima, T.: Basic theory on normalization of two-dimensional visual pattern. Stud. Inf. Control (IECE, Japan) 1, 15–22 (1963). Pattern Recognition Issue. In JapaneseGoogle Scholar
  28. 28.
    Iijima, T.: Basic equation of figure and observational transformation. Syst. Comput. Controls 2(4), 70–77 (1971)Google Scholar
  29. 29.
    Jackway, P.T.: Properties of multiscale morphological smoothing by poweroids. Pattern Recognit. Lett. 15(2), 135–140 (1994)CrossRefMATHGoogle Scholar
  30. 30.
    Jackway, P.T.: On dimensionality in multiscale morphological scale-space with elliptic poweroid structuring functions. J. Vis. Commun. Image Represent. 6(2), 189–195 (1995)CrossRefGoogle Scholar
  31. 31.
    Jackway, P.T., Deriche, M.: Scale-space properties of the multiscale morphological dilation-erosion. IEEE Trans. Pattern Anal. Mach. Intell. 18, 38–51 (1996)CrossRefGoogle Scholar
  32. 32.
    Kanters, F., Florack, L., Duits, R., Platel, B., Haar Romeny, B.: ScaleSpaceViz: \(\alpha \)-scale spaces in practice. Pattern Recognit. Image Anal. 17(1), 106–116 (2007)Google Scholar
  33. 33.
    Kimia, B.B., Siddiqi, K.: Geometric heat equation and non-linear diffusion of shapes and images. Comput. Vis. Image Underst. 64, 305–322 (1996)CrossRefGoogle Scholar
  34. 34.
    Koenderink, J.J.: The structure of images. Biol. Cybernet. 50, 363–370 (1984)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Landström, A.: An approach to adaptive quadratic structuring functions based on the local structure tensor. In: Benediktsson, J.A., Chanussot, J., Najman, L., Talbot, H. (eds.) Mathematical Morphology and Its Applications to Signal and Image Processing. Lecture Notes in Computer Science, vol. 9082, pp. 729–740. Springer, Berlin (2015)CrossRefGoogle Scholar
  36. 36.
    Lindeberg, T.: Scale-Space Theory in Computer Vision. Kluwer, Boston (1994)CrossRefMATHGoogle Scholar
  37. 37.
    Lindeberg, T.: Generalized Gaussian scale-space axiomatics comprising linear scale-space, affine scale-space and spatio-temporal scale-space. J. Math. Imaging Vis. 40, 36–81 (2011)MathSciNetCrossRefMATHGoogle Scholar
  38. 38.
    Lions, P.L.: Generalized Solutions of Hamilton-Jacobi Equations. Research Notes In Mathematics, vol. 69. Pitman, London (1992)Google Scholar
  39. 39.
    Lowe, D.L.: Distinctive image features from scale-invariant keypoints. Int. J. Comput. Vis. 60(2), 91–110 (2004)CrossRefGoogle Scholar
  40. 40.
    Lyons, R.G.: Understanding Digital Signal Processing. Prentice Hall, Englewood Cliffs (2004)Google Scholar
  41. 41.
    Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. AMS, Providence (2015)Google Scholar
  42. 42.
    Maragos, P.: Morphological systems: slope transforms and max-min difference and differential equations. Signal Process. 38(1), 57–77 (1994)CrossRefMATHGoogle Scholar
  43. 43.
    Maragos, P., Schafer, R.: Morphological filters-Part I: their set-theoretic analysis and relations to linear shift-invariant filters. IEEE Trans. Acoust. Speech Signal Process. 35(8), 1153–1169 (1987)Google Scholar
  44. 44.
    Nielsen, M., Florack, L., Deriche, R.: Regularization, scale-space and edge detection filters. J. Math. Imaging Vis. 7, 291–307 (1997)MathSciNetCrossRefGoogle Scholar
  45. 45.
    Olsen, O., Florack, L., Kuijper, A. (eds.): Deep Structure, Singularities, and Computer Vision. Lecture Notes in Computer Science, vol. 3753. Springer, Berlin (2005)Google Scholar
  46. 46.
    Oppenheim, A.V., Schafer, R.W., Buck, J.R.: Discrete-Time Signal Processing, 2nd edn. Prentice Hall, Englewood Cliffs (1999)Google Scholar
  47. 47.
    Otsu, N.: Mathematical studies on feature extraction in pattern recognition. Tech. Rep. 818 (PhD Thesis), Electrotechnical Laboratory, Tsukuba, Japan (1981). (In Japanese)Google Scholar
  48. 48.
    Pauwels, E.J., Van Gool, L.J., Fiddelaers, P., Moons, T.: An extended class of scale-invariant and recursive scale space filters. IEEE Trans. Pattern Anal. Mach. Intell. 17, 691–701 (1995)CrossRefGoogle Scholar
  49. 49.
    Perona, P., Malik, J.: Scale space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)CrossRefGoogle Scholar
  50. 50.
    Poggio, T., Voorhees, H., Yuille, A.: A regularized solution to edge detection. J. Complex. 4(2), 106–123 (1988)MathSciNetCrossRefMATHGoogle Scholar
  51. 51.
    Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)CrossRefMATHGoogle Scholar
  52. 52.
    Sapiro, G., Tannenbaum, A.: Affine invariant scale-space. Int. J. Comput. Vis. 11, 25–44 (1993)CrossRefMATHGoogle Scholar
  53. 53.
    Scherzer, O., Weickert, J.: Relations between regularization and diffusion filtering. J. Math. Imaging Vis. 12(1), 43–63 (2000)MathSciNetCrossRefMATHGoogle Scholar
  54. 54.
    Schmidt, M., Weickert, J.: The morphological equivalents of relativistic and alpha-scale-spaces. In: Aujol, J., Nikolova, M., Papadakis, N. (eds.) Scale Space and Variational Methods in Computer Vision. Lecture Notes in Computer Science, vol. 9087, pp. 28–39. Springer, Berlin (2015)Google Scholar
  55. 55.
    Sporring, J., Nielsen, M., Florack, L., Johansen, P. (eds.): Gaussian Scale-Space Theory. Computational Imaging and Vision, vol. 8. Kluwer, Dordrecht (1997)Google Scholar
  56. 56.
    Taylor, M.: Partial Differential Equations II: Qualitative Studies of Linear Equations. Applied Mathematical Sciences. Springer, New York (2010)Google Scholar
  57. 57.
    van den Boomgaard, R.: Mathematical morphology: Extensions towards computer vision. Ph.D. thesis, University of Amsterdam, The Netherlands (1992)Google Scholar
  58. 58.
    van den Boomgaard, R.: The morphological equivalent of the Gauss convolution. Nieuw Archief Voor Wiskunde 10(3), 219–236 (1992)MathSciNetMATHGoogle Scholar
  59. 59.
    van den Boomgaard, R., Smeulders, A.: The morphological structure of images: the differential equations of morphological scale-space. IEEE Trans. Pattern Anal. Mach. Intell. 16, 1101–1113 (1994)CrossRefGoogle Scholar
  60. 60.
    Weickert, J.: Anisotropic Diffusion in Image Processing. Teubner, Stuttgart (1998)MATHGoogle Scholar
  61. 61.
    Weickert, J., Ishikawa, S., Imiya, A.: Linear scale-space has first been proposed in Japan. J. Math. Imaging Vis. 10(3), 237–252 (1999)MathSciNetCrossRefMATHGoogle Scholar
  62. 62.
    Welk, M.: Families of generalised morphological scale spaces. In: Griffin, L., Lillholm, M. (eds.) Scale Space Methods in Computer Vision. Lecture Notes in Computer Science, vol. 2695, pp. 770–784. Springer, Berlin (2003)CrossRefGoogle Scholar
  63. 63.
    Witkin, A.P.: Scale-space filtering. In: Proceedings of Eighth International Joint Conference on Artificial Intelligence, vol. 2, pp. 945–951. Karlsruhe, West Germany (1983)Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Mathematical Image Analysis Group, Faculty of Mathematics and Computer ScienceSaarland UniversitySaarbrückenGermany

Personalised recommendations