A Learning Framework for Morphological Operators Using Counter–Harmonic Mean

  • Jonathan Masci
  • Jesús Angulo
  • Jürgen Schmidhuber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7883)


We present a novel framework for learning morphological operators using counter-harmonic mean. It combines concepts from morphology and convolutional neural networks. A thorough experimental validation analyzes basic morphological operators dilation and erosion, opening and closing, as well as the much more complex top-hat transform, for which we report a real-world application from the steel industry. Using online learning and stochastic gradient descent, our system learns both the structuring element and the composition of operators. It scales well to large datasets and online settings.


mathematical morphology convolutional networks online learning machine learning 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Angulo, J.: Pseudo-morphological image diffusion using the counter-harmonic paradigm. In: Blanc-Talon, J., Bone, D., Philips, W., Popescu, D., Scheunders, P. (eds.) ACIVS 2010, Part I. LNCS, vol. 6474, pp. 426–437. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  2. 2.
    Bullen, P.: Handbook of Means and Their Inequalities, 2nd edn. Springer (1987)Google Scholar
  3. 3.
    Cireşan, D.C., Meier, U., Masci, J., Schmidhuber, J.: A committee of neural networks for traffic sign classification. In: International Joint Conference on Neural Networks, IJCNN 2011 (2011)Google Scholar
  4. 4.
    Cireşan, D.C., Meier, U., Masci, J., Schmidhuber, J.: Flexible, high performance convolutional neural networks for image classification. In: International Joint Conference on Artificial Intelligence, IJCA I2011 (2011)Google Scholar
  5. 5.
    Ciresan, D.C., Giusti, A., Gambardella, L.M., Schmidhuber, J.: Deep neural networks segment neuronal membranes in electron microscopy images. In: NIPS (2012)Google Scholar
  6. 6.
    Ciresan, D.C., Meier, U., Gambardella, L.M., Schmidhuber, J.: Convolutional neural network committees for handwritten character classification. In: ICDAR, pp. 1250–1254 (2011)Google Scholar
  7. 7.
    Harvey, N.R., Marshall, S.: The use of genetic algorithms in morphological filter design. Signal Processing: Image Communication 8(1), 55–71 (1996)CrossRefGoogle Scholar
  8. 8.
    Hubel, D.H., Wiesel, T.N.: Receptive fields and functional architecture of monkey striate cortex. The Journal of Physiology 195(1), 215–243 (1968)Google Scholar
  9. 9.
    LeCun, Y.: Une procédure d’apprentissage pour réseau à seuil asymétrique. In: Proceedings of Cognitiva, Paris, vol. 85, pp. 599–604 (1985)Google Scholar
  10. 10.
    Masci, J., Meier, U., Cireşan, D.C., Fricout, G., Schmidhuber, J.: Steel defect classification with max-pooling convolutional neural networks. In: International Joint Conference on Neural Networks (2012)Google Scholar
  11. 11.
    Nakashizuka, M., Takenaka, S., Iiguni, Y.: Learning of structuring elements for morphological image model with a sparsity prior. In: IEEE International Conference on Image Processing, ICIP 2010, pp. 85–88 (2010)Google Scholar
  12. 12.
    Pessoa, L.F.C., Maragos, P.: Mrl-filters: a general class of nonlinear systems and their optimal design for image processing. IEEE Transactions on Image Processing 7(7), 966–978 (1998)CrossRefGoogle Scholar
  13. 13.
    Salembier, P.: Adaptive rank order based filters. Signal Processing 27, 1–25 (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    Salembier, P.: Structuring element adaptation for morphological filters. J. of Visual Communication and Image Representation 3(2), 115–136 (1992)CrossRefGoogle Scholar
  15. 15.
    Scherer, D., Müller, A., Behnke, S.: Evaluation of pooling operations in convolutional architectures for object recognition. In: Diamantaras, K., Duch, W., Iliadis, L.S. (eds.) ICANN 2010, Part III. LNCS, vol. 6354, pp. 92–101. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  16. 16.
    Serra, J.: Image analysis and mathematical morphology. Academic Press, London (1982)zbMATHGoogle Scholar
  17. 17.
    Soille, P.: Morphological Image Analysis: Principles and Applications. Springer (1999)Google Scholar
  18. 18.
    Turaga, S.C., Murray, J.F., Jain, V., Roth, F., Helmstaedter, M., Briggman, K., Denk, W., Seung, H.S.: Convolutional networks can learn to generate affinity graphs for image segmentation. Neural Comput. 22(2), 511–538 (2010)zbMATHCrossRefGoogle Scholar
  19. 19.
    van Vliet, L.J.: Robust local max-min filters by normalized power-weighted filtering. In: Proc. of IEEE ICPR 2004, vol. 1, pp. 696–699 (2004)Google Scholar
  20. 20.
    Werbos, P.J.: Beyond Regression: New Tools for Prediction and Analysis in the Behavioral Sciences. Ph.D. thesis, Harvard University (1974)Google Scholar
  21. 21.
    Wilson, S.S.: Training structuring elements in morphological networks. In: Mathematical Morphology in Image Processing, ch. 1, pp. 1–42. Marcel Dekker (1993)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Jonathan Masci
    • 1
  • Jesús Angulo
    • 2
  • Jürgen Schmidhuber
    • 1
  1. 1.IDSIA – USI – SUPSIManno–LuganoSwitzerland
  2. 2.CMM-Centre de Morphologie Mathématique, Mathématiques et SystèmesMINES ParisTechFrance

Personalised recommendations