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Erosion und Dilatation

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Morphologische Bildverarbeitung

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Zusammenfassung

In der Morphologie werden Bilder als Mengen betrachtet. Eine Mengendarstellung eines Grauwertbildes erhält man entweder durch die Betrachtung seines Untergraphen oder seiner aufeinanderfolgenden Querschnitte. Morphologische Operatoren haben die Extraktion relevanter Bildstrukturen zum Ziel. Dies kann durch das Proben des Bildes mit einer anderen Menge bekannter Form, die strukturierendes Element (SE) genannt wird, erreicht werden. Die Form des SEs wird normalerweise anhand von a priori-Wissen über die relevanten und irrelevanten Bildstrukturen ausgewählt. Unter irrelevanten Strukturen wollen wir entweder Rauschen oder andere Objekte verstehen, die wir unterdrücken möchten. Abbildung 3.1 veranschaulicht den morphologischen Ansatz zur Bildverarbeitung. Die Abbildungen 3.1a und b zeigen Binär- und Grauwertbilder. Abbildung 3.1c wird ausschließlich für Übergabezwecke benutzt: Sie gibt eine Schattierung und Schraffierung von Abb. 3.1b wieder, um so seine topographische Darstellung zu erzeugen. Beispiele für strukturierende Elemente finden sich in Abb. 3.1d: eine Scheibe, ein Sechseck, ein Quadrat, eine Raute (d. h., ein Quadrat mit um 45° geneigten Seiten), ein horizontales Liniensegment und ein Punktepaar. Benutzt man ein vertikales SE als Probe, so können alle vertikalen Strukturen der Binär- und Grauwertbilder (Abbn. 3.1e und f) extrahiert werden.

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Literatur

  • Adams, R. (1993), ‘Radial decomposition of discs and spheres.’, Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 55(5), 325–332.

    Google Scholar 

  • Banon, G. & Barrera, J. (1991), ‘Minimal representation for translation-invariant set mappings by mathematical morphology’, SIAM J. Appl. Math. 51, 1782–1798.

    Article  MathSciNet  MATH  Google Scholar 

  • Beucher, S. (1990), Segmentation d’images et morphologie mathématique, PhD thesis, Ecole des Mines de Paris.

    Google Scholar 

  • Breen, E. & Soille, P. (1993), Generalization of van Herk recursive erosion/dilation algorithm to lines at arbitrary angles, in K. Fung &; A. Ginige, eds, ‘Proc. DICTA’93: Digital Image Computing: Techniques and Applications’, APRS, Sydney, pp. 549–555.

    Google Scholar 

  • Carr, J. & Benzer, W. (1991), ‘On the practice of estimating fractal dimension’, Mathematical Geology 23(7), 945–958.

    Article  Google Scholar 

  • Chaudhuri, B. (1990), ‘An efficient algorithm for running window pel gray level ranking in 2-D images’, Pattern Recognition Letters 11(2), 77–80.

    Article  MATH  Google Scholar 

  • Dubuc, B., Quiniou, J.-F., Roques-Carmes, C., Tricot, C. & Zucker, S. (1989), ‘Evaluating the fractal dimension of profiles’, Physical Review A 39(3), 1500–1512.

    Article  MathSciNet  Google Scholar 

  • Freeman, H. (1961), ‘On the encoding of arbitrary geometric configurations’, IRE Transactions on Electronic Computers 10, 260–268.

    Article  MathSciNet  Google Scholar 

  • Gambotto, J. (1982), Algorithms for region description and modifications based on chain code transformations, in ‘Proc. ICASSP’, Paris, pp. 1920–1923.

    Google Scholar 

  • Gil, J. & Werman, M. (1993), ‘Computing 2-D min, median and max filters’, IEEE Transactions on Pattern Analysis and Machine Intelligence 15(5), 504–507

    Article  Google Scholar 

  • Hadwiger, H. (1950), ‘Minkowskische Addition und Subtraktion beliebiger Punktmengen und die Theoreme von Erhard Schmidt’, Mathematische Zeitschrift 53, 210–218.

    Article  MathSciNet  MATH  Google Scholar 

  • Haralick, R. (1984), ‘Digital step edges from zero crossing of second directional derivatives’, IEEE Transactions on Pattern Analysis and Machine Intelligence 6(1), 58–68.

    Article  Google Scholar 

  • Heijmans, H. (1993), ‘A note on the umbra transform in gray-scale morphology’, Pattern Recognition Letters 14, 877–881.

    Article  MATH  Google Scholar 

  • Heijmans, H. (1994), Morphological image operators, Advances in Electronics and Electron Physics, Academic Press.

    Google Scholar 

  • Heijmans, H. (1995), ‘Mathematical morphology: a modern approach in image processing based on algebra and geometry’, SIAM Review 37(1), 1–36.

    Article  MathSciNet  MATH  Google Scholar 

  • Heijmans, H. & Vincent, L. (1993), Graph morphology in image analysis, in E. Dougherty, ed., ‘Mathematical morphology in image processing’, Marcel Dekker, chapter 6, pp. 171–203.

    Google Scholar 

  • Heijmans, H., Nacken, P., Toet, A. & Vincent, L. (1992), ‘Graph morphology’, Journal of Visual Communication and Image Representation 3(2), 24–38.

    Article  Google Scholar 

  • Huang, T., ed. (1981), Two-dimensional digital signal processing II: transforms and median filters, Springer-Verlag.

    Google Scholar 

  • Huang, T., Yang, G. & Tang, G. (1979), ‘A fast two-dimensional median filtering algorithm’, IEEE Transactions on Acoustics, Speech and Signal Processing 27(1), 13–18.

    Article  Google Scholar 

  • Jackway, P. (1994), ‘Properties of multiscale morphological smoothing by poweroids’, Pattern Recognition Letters 15, 135–140.

    Article  MATH  Google Scholar 

  • Jeulin, D. & Kurdy, M. (1992), Directional mathematical morphology for oriented image restoration and segmentation, in ‘Acta Stereologica’, Vol. 11, pp. 545–550.

    Google Scholar 

  • Ji, L., Piper, J. & Tang, J.-Y. (1989), ‘Erosion and dilation of binary images by arbitrary structuring elements using interval coding’, Pattern Recognition Letters 9, 201–209.

    Article  MATH  Google Scholar 

  • Jones, R. & Soille, P. (1996a), Periodic lines and their applications to granulometries, in P. Maragos, W. Schafer & M. Butt, eds, ‘Mathematical Morphology and its Applications to Image and Signal Processing’, Kluwer Academic Publishers, pp. 264–272.

    Google Scholar 

  • Jones, R. & Soille, P. (1996b), ‘Periodic lines: Definition, cascades, and application to granulometries’, Pattern Recognition Letters 17(10), 1057–1063.

    Article  Google Scholar 

  • Jones, R. & Svalbe, I. (1994), Basis algorithms in mathematical morphology, in ‘Advances in Electronics and Electron Physics’, Vol. 89, Academic Press, pp. 325–390.

    Google Scholar 

  • Kurdy, M. & Jeulin, D. (1989), Directional mathematical morphology operations, in ‘Acta Stereologica’, Vol. 8/2, pp. 473–480.

    Google Scholar 

  • Mandelbrot, B. (1967), ‘How long is the coast of Great-Britain? Statistical self- similarity and fractional dimension’, Science 155, 636–638.

    Article  Google Scholar 

  • Mandelbrot, B. (1983), The fractal geometry of nature, W.H. Freemann and Company, New York.

    Google Scholar 

  • Mandelbrot, B. (1991), Die fraktale Geometric der Natur, Birkhäuser Verlag, Basel.

    Google Scholar 

  • Maragos, P. (1993), Fractal signal analysis using mathematical morphology, in P. Hawkes & B. Kazan, eds, ‘Advances in electronics and electron physics’, Academic Press.

    Google Scholar 

  • Maragos, P. & Sun, F.-K. (1993), ‘Measuring the fractal dimension of signals: morphological covers and iterative optimization’, IEEE Transactions on Signal Processing 41(1), 108–121.

    Article  MATH  Google Scholar 

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

    Google Scholar 

  • Matheron, G. (1975), Random sets and integral geometry, Wiley.

    MATH  Google Scholar 

  • Mazille, J. (1989), ‘Mathematical morphology and convolutions’, Journal of Microscopy 156(Ptl), 3–13.

    Article  Google Scholar 

  • Meyer, F. (1992), ‘Mathematical morphology: from 2D to 3D’, Journal of Microscopy 165, Pt 1, 5–28.

    Article  Google Scholar 

  • Minkowski, H. (1901), ‘Über die Begriffe Länge, Oberfläche und Volumen’, Jahresbericht der Deutschen Mathematiker Vereinigung 9, 115–121.

    MATH  Google Scholar 

  • Minkowski, H. (1903), ‘Volumen und Oberfläche’, Math. Ann. 57, 447–495.

    Article  MathSciNet  MATH  Google Scholar 

  • Nagakawa, Y. & Rosenfeld, A. (1978), ‘A note on the use of local min and max operations in digital picture processing’, IEEE Transactions on Systems, Man and Cybernetics 8, 632–635.

    Article  Google Scholar 

  • Pecht, J. (1985), ‘Speeding up successive Minkowski operations’, Pattern Recognition Letters 3(2), 113–117.

    Article  MATH  Google Scholar 

  • Ragnemalm, I. (1992), ‘Fast erosion and dilation by contour processing and thresholding of distance maps’, Pattern Recognition Letters 13, 161–166.

    Article  Google Scholar 

  • Rao, A. (1990), A taxonomy for texture description and identification, Springer-Verlag, New York.

    MATH  Google Scholar 

  • Rigaut, J.-P. (1988), ‘Automated image segmentation by mathematical morphology and fractal geometry’, Journal of Microscopy 150(Pt 1), 21–30.

    Article  Google Scholar 

  • Rivest, J.-F., Soille, P. & Beucher, S. (1993), ‘Morphological gradients’, Journal of Electronic Imaging 2(4), 326–336.

    Article  Google Scholar 

  • Salembier, P. (1992), ‘Structuring element adaptation for morphological filters’, Journal of Visual Communication and Image Representation 3(2), 115–136.

    Article  Google Scholar 

  • Schmitt, M. (1989), Des algorithmes morphologiques à 1’intelligence artificielle, PhD thesis, Ecole des Mines de Paris.

    Google Scholar 

  • Serra, J. (1982), Image analysis and mathematical morphology, Academic Press, London.

    MATH  Google Scholar 

  • Shih, F. & Mitchell, O. (1992), ‘A mathematical morphology approach to Euclidean distance transformation’, IEEE Transactions on Image processing 2(1), 197–204.

    Article  Google Scholar 

  • Shih, F. & Wu, H. (1992), ‘Optimization on Euclidean distance transformation using grayscale morphology’, Journal of Visual Communication and Image Representation 3(2), 104–114.

    Article  Google Scholar 

  • Soille, P. & Rivest, J.-F. (1996), ‘On the validity of fractal dimension measurements in image analysis’, Journal of Visual Communication and Image Representation 7(3), 217–229.

    Article  Google Scholar 

  • Soille, P., Breen, E. & Jones, R. (1996), ‘Recursive implementation of erosions and dilations along discrete lines at arbitrary angles’, IEEE Transactions on Pattern Analysis and Machine Intelligence 18(5), 562–567.

    Article  Google Scholar 

  • Sternberg, S. (1982), Cellular computers and biomedical image processing, in J. Sklansky &; J. Bisconte, eds, ‘Biomedical Images and Computers’, Vol. 17 of Lecture Notes in Medical Informatics, Springer-Verlag, Berlin, pp. 294–319.

    Google Scholar 

  • Sternberg, S. (1986), ‘Grayscale morphology’, Computer Graphics and Image Processing 35, 333–355.

    Article  Google Scholar 

  • van den Boomgaard, R. & van Baien, R. (1992), ‘Methods for-fast morphological image transforms using bitmapped binary images’, Computer Vision, Graphics, and Image Processing: Graphical Models and Image Processing 54(3), 252–258.

    Google Scholar 

  • Van Droogenbroeck, M. & Talbot, H. (1996), ‘Fast computation of morphological operations with arbitrary structuring elements’, Pattern Recognition Letters 17(14), 1451–1460.

    Article  Google Scholar 

  • van Herk, M. (1992), ‘A fast algorithm for local minimum and maximum filters on rectangular and octogonal kernels’, Pattern Recognition Letters 13, 517–521.

    Article  Google Scholar 

  • van Vliet, L. & Verwer, B. (1988), ‘A contour processing method for fast binary neighbourhood operations’, Pattern Recognition Letters 7, 27–36.

    Article  Google Scholar 

  • van Vliet, L., Young, I. & Beckers, G. (1989), ‘A nonlinear Laplace operator as edge detector in noisy images’, Computer Vision, Graphics, and Image Processing 45(2), 167–195.

    Article  Google Scholar 

  • Vincent, L. (1989), ‘Graphs and mathematical morphology’, Signal Processing 16, 365–388.

    Article  MathSciNet  Google Scholar 

  • Vincent, L. (1991), ‘Morphological transformations of binary images with arbitrary structuring elements’, Signal Processing 22(1), 3–23.

    Article  Google Scholar 

  • Vincent, L. (1993), Morphological algorithms, in E. Dougherty, ed., ‘Mathematical morphology in image processing’, Marcel Dekker, chapter 8, pp. 255–288.

    Google Scholar 

  • Zamperoni, P. (1980), ‘Dilatation und Erosion von konturcodierten Binärbildern’, Microscopica Acta Suppl. 4, 245–249.

    Google Scholar 

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Authors and Affiliations

  1. LGI2P, Ecole des Mines d’Alès-EERIE, Parc Scientifique G.Besse, F-30000, Nîmes, France

    Dr. Pierre Soille

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  1. Dr. Pierre Soille
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© 1998 Springer-Verlag Berlin Heidelberg

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Soille, P. (1998). Erosion und Dilatation. In: Morphologische Bildverarbeitung. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-72190-8_3

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  • DOI: https://doi.org/10.1007/978-3-642-72190-8_3

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-72191-5

  • Online ISBN: 978-3-642-72190-8

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