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Double-Sided Probing by Map of Asplund’s Distances Using Logarithmic Image Processing in the Framework of Mathematical Morphology

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Mathematical Morphology and Its Applications to Signal and Image Processing (ISMM 2017)

Part of the book series: Lecture Notes in Computer Science ((LNIP,volume 10225))

Abstract

We establish the link between Mathematical Morphology and the map of Asplund’s distances between a probe and a grey scale function, using the Logarithmic Image Processing scalar multiplication. We demonstrate that the map is the logarithm of the ratio between a dilation and an erosion of the function by a structuring function: the probe. The dilations and erosions are mappings from the lattice of the images into the lattice of the positive functions. Using a flat structuring element, the expression of the map of Asplund’s distances can be simplified with a dilation and an erosion of the image; these mappings stays in the lattice of the images. We illustrate our approach by an example of pattern matching with a non-flat structuring function.

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References

  1. Asplund, E.: Comparison between plane symmetric convex bodies and parallelograms. Math. Scand. 8, 171–180 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  2. Banon, G.J.F., Faria, S.D.: Morphological approach for template matching. In: Proceedings X Brazilian Symposium on Computer Graphics and Image Processing, pp. 171–178, October 1997

    Google Scholar 

  3. Banon, G.J.F., Barrera, J.: Decomposition of mappings between complete lattices by mathematical morphology, part I. General lattices. Signal Process. 30(3), 299–327 (1993). http://www.sciencedirect.com/science/article/pii/0165168493900153

    Article  MATH  Google Scholar 

  4. Barat, C., Ducottet, C., Jourlin, M.: Virtual double-sided image probing: a unifying framework for non-linear grayscale pattern matching. Pattern Recogn. 43(10), 3433–3447 (2010). http://www.sciencedirect.com/science/article/pii/S0031320310001962

    Article  MATH  Google Scholar 

  5. Birkhoff, G.: Lattice Theory, American Mathematical Society Colloquium Publications, vol. 25, 3rd edn. American Mathematical Society, Providence (1967)

    Google Scholar 

  6. Brailean, J., Sullivan, B., Chen, C., Giger, M.: Evaluating the EM algorithm for image processing using a human visual fidelity criterion. In: International Conference on Acoustics, Speech, and Signal Processing, ICASSP 1991, vol. 4, pp. 2957–2960, April 1991

    Google Scholar 

  7. Grünbaum, B.: Measures of symmetry for convex sets. In: Proceedings of Symposia in Pure Mathematics, vol. 7, pp. 233–270 (1963)

    Google Scholar 

  8. Hadwiger, H.: Vorlesungen Über Inhalt, Oberfläche und Isoperimetrie. Grundlehren der Mathematischen Wissenschaften. Springer, Heidelberg (1957)

    Book  MATH  Google Scholar 

  9. Heijmans, H., Ronse, C.: The algebraic basis of mathematical morphology I. Dilations and erosions. Comput. Vis. Graph. Image Process. 50(3), 245–295 (1990). http://www.sciencedirect.com/science/article/pii/0734189X9090148O

    Article  MATH  Google Scholar 

  10. Jourlin, M.: Chapter one - gray-level LIP model. Notations, recalls, and first applications. In: Jourlin, M. (ed.) Logarithmic Image Processing: Theory and Applications. Advances in Imaging and Electron Physics, vol. 195, pp. 1–26. Elsevier, Amsterdam (2016). http://www.sciencedirect.com/science/article/pii/S1076567016300313

    Google Scholar 

  11. Jourlin, M.: Chapter three - metrics based on logarithmic laws. In: Jourlin, M. (ed.) Logarithmic Image Processing: Theory and Applications. Advances in Imaging and Electron Physics, vol. 195, pp. 61–113. Elsevier, Amsterdam (2016). http://www.sciencedirect.com/science/article/pii/S1076567016300337

    Google Scholar 

  12. Jourlin, M., Breugnot, J., Itthirad, F., Bouabdellah, M., Closs, B.: Chapter 2 - logarithmic image processing for color images. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 168, pp. 65–107. Elsevier, Amsterdam (2011)

    Google Scholar 

  13. Jourlin, M., Carré, M., Breugnot, J., Bouabdellah, M.: Chapter 7 - logarithmic image processing: additive contrast, multiplicative contrast, and associated metrics. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 171, pp. 357–406. Elsevier, Amsterdam (2012)

    Google Scholar 

  14. Jourlin, M., Couka, E., Abdallah, B., Corvo, J., Breugnot, J.: Asplünd’s metric defined in the logarithmic image processing (LIP) framework: a new way to perform double-sided image probing for non-linear grayscale pattern matching. Pattern Recogn. 47(9), 2908–2924 (2014)

    Article  Google Scholar 

  15. Jourlin, M., Pinoli, J.: A model for logarithmic image processing. J. Microsc. 149(1), 21–35 (1988)

    Article  Google Scholar 

  16. Jourlin, M., Pinoli, J.: Logarithmic image processing: the mathematical and physical framework for the representation and processing of transmitted images. In: Hawkes, P.W. (ed.) Advances in Imaging and Electron Physics, vol. 115, pp. 129–196. Elsevier, Amsterdam (2001)

    Google Scholar 

  17. Jourlin, M., Pinoli, J.C.: Image dynamic range enhancement and stabilization in the context of the logarithmic image processing model. Signal Process. 41(2), 225–237 (1995). http://www.sciencedirect.com/science/article/pii/0165168494001026

    Article  MATH  Google Scholar 

  18. Khosravi, M., Schafer, R.W.: Template matching based on a grayscale hit-or-miss transform. IEEE Trans. Image Process. 5(6), 1060–1066 (1996)

    Article  Google Scholar 

  19. Matheron, G.: Eléments pour une théorie des milieux poreux. Masson, Paris (1967)

    Google Scholar 

  20. Minkowski, H.: Volumen und oberfläche. Math. Ann. 57, 447–495 (1903). http://eudml.org/doc/158108

    Article  MathSciNet  MATH  Google Scholar 

  21. Noyel, G., Angulo, J., Jeulin, D.: Morphological segmentation of hyperspectral images. Image Anal. Stereol. 26(3), 101–109 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. Noyel, G., Angulo, J., Jeulin, D., Balvay, D., Cuenod, C.A.: Multivariate mathematical morphology for DCE-MRI image analysis in angiogenesis studies. Image Anal. Stereol. 34(1), 1–25 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  23. Noyel, G., Jourlin, M.: Asplünd’s metric defined in the logarithmic image processing (LIP) framework for colour and multivariate images. In: IEEE International Conference on Image Processing (ICIP), pp. 3921–3925, September 2015

    Google Scholar 

  24. Noyel, G., Jourlin, M.: Spatio-colour Asplünd ’s metric and logarithmic image processing for colour images (LIPC). In: CIARP2016 - XXI IberoAmerican Congress on Pattern Recognition. International Association for Pattern Recognition (IAPR), Lima, Peru, November 2016. https://hal.archives-ouvertes.fr/hal-01316581

  25. Odone, F., Trucco, E., Verri, A.: General purpose matching of grey level arbitrary images. In: Arcelli, C., Cordella, L.P., di Baja, G.S. (eds.) IWVF 2001. LNCS, vol. 2059, pp. 573–582. Springer, Heidelberg (2001). doi:10.1007/3-540-45129-3_53

    Chapter  Google Scholar 

  26. Serra, J.: Image Analysis and Mathematical Morphology: Theoretical Advances, vol. 2. Academic Press, London (1988)

    Google Scholar 

  27. Serra, J., Cressie, N.: Image Analysis and Mathematical Morphology, vol. 1. Academic Press, London (1982)

    Google Scholar 

  28. Soille, P.: Morphological Image Analysis: Principles and Applications, 2nd edn. Springer, New York (2003)

    MATH  Google Scholar 

  29. Verdú-Monedero, R., Angulo, J., Serra, J.: Anisotropic morphological filters with spatially-variant structuring elements based on image-dependent gradient fields. IEEE Trans. Image Process. 20(1), 200–212 (2011)

    Article  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. International Prevention Research Institute, 95 cours Lafayette, 69006, Lyon, France

    Guillaume Noyel & Michel Jourlin

  2. Lab. H. Curien, UMR CNRS 5516, 18 rue Pr. B. Lauras, 42000, St-Etienne, France

    Michel Jourlin

Authors
  1. Guillaume Noyel
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  2. Michel Jourlin
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Corresponding author

Correspondence to Guillaume Noyel .

Editor information

Editors and Affiliations

  1. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Jesús Angulo

  2. MINES ParisTech, PSL-Research University , Fontainebleau, France

    Santiago Velasco-Forero

  3. MINES ParisTech, PSL-Research University, Fontainebleau, France

    Fernand Meyer

A Appendix

A Appendix

Le us demonstrate that the Aplünd’s metric is a metric in the space of equivalence classes . In order to be a metric on , must satisfy the four following properties:

  1. 1.

    (positivity): , \(\forall x \in D\), (Def. 1), because

    \(\Rightarrow \lambda > \mu \) because is an ordered set with the order \(\le \)

    , .

  2. 2.

    (Axiom of separation):

    (A.1)

    Reciprocally:

    (A.2)

    Equations A.1 and A.2 \(\Rightarrow \) .

  3. 3.

    (Triangle inequality): Let us define: and . We have

    (A.3)
    (A.4)

    In the same way:

    $$\begin{aligned} \mu _1 \mu _2 \ge \mu _3 \text {, with } \mu _1, \mu _2, \mu _3 > 0 \end{aligned}$$
    (A.5)

    Equations A.3, A.4 and A.5 \(\Rightarrow \) \(\frac{\lambda _1 \lambda _2}{\mu _1 \mu _2} \ge \frac{\lambda _3}{\mu _3}\)

    .

  4. 4.

    (Axiom of symmetry): Let us define: .

    Def. 1 \(\Rightarrow \) , because \(k>0\)

    .

    In the same way, we have \(\frac{1}{\mu _1} = \lambda _2\).

    Therefore, .

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Noyel, G., Jourlin, M. (2017). Double-Sided Probing by Map of Asplund’s Distances Using Logarithmic Image Processing in the Framework of Mathematical Morphology. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds) Mathematical Morphology and Its Applications to Signal and Image Processing. ISMM 2017. Lecture Notes in Computer Science(), vol 10225. Springer, Cham. https://doi.org/10.1007/978-3-319-57240-6_33

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  • DOI: https://doi.org/10.1007/978-3-319-57240-6_33

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-57239-0

  • Online ISBN: 978-3-319-57240-6

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