Abstract
Fuzzy mathematical morphology has been extensively used in many different applications such as edge detection, noise reduction and shape and pattern recognition. The fundamentals of this morphology are based on an appropriate selection of the operators involved, namely the conjunction and implication. In this work we investigate the use of the Mayor-Torrens family of t-norms, from both theoretical and practical point of view. The results suggest that competitive results can be obtained by using the t-norms of this family.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
It can be downloaded from ftp://figment.csee.usf.edu/pub/ROC/edge_comparison_dataset.tar.gz.
- 2.
This image database can be downloaded from http://sipi.usc.edu/database/misc.tar.gz.
References
M. Baczyński, B. Jayaram, Fuzzy Implications, vol. 231 (Berlin Heidelberg, Studies in Fuzziness and Soft Computing (Springer, 2008)
M. Baczyński, B. Jayaram, S. Massanet, J. Torrens, Fuzzy implications: past, present, and future, in Springer Handbook of Computational Intelligence, ed. by J. Kacprzyk, W. Pedrycz (Springer, Berlin Heidelberg, 2015), pp. 183–202
I. Bloch, H. Maître, Fuzzy mathematical morphologies: a comparative study. Pattern Recognit. 28, 1341–1387 (1995)
R. Bock, J. Meier, L.G. Nyúl, J. Hornegger, G. Michelson, Glaucoma risk index: automated glaucoma detection from color fundus images. Med. Image Anal. 14(3), 471–481 (2010)
K. Bowyer, C. Kranenburg, S. Dougherty, Edge detector evaluation using empirical ROC curves, in IEEE Conference on Computer Vision and Pattern Recognition (CVPR ’99), vol. 1, pp. 354–359 (1999)
A. Budai, J. Odstricilik, R. Kollar, J. Jan, T. Kubena, G. Michelson, A public database for the evaluation of fundus image segmentation algorithms, in Proceedings of The Association of Research in Vision and Ophthalmology (ARVO) Annual Meeting, Vancouver, Canada, pp. 1345–1345 (2011)
J. Canny, A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8(6), 679–698 (1986)
B. De Baets, Fuzzy morphology: a logical approach, in Uncertainty Analysis in Engineering and Science: Fuzzy Logic, Statistics, and Neural Network Approach, ed. by B.M. Ayyub, M.M. Gupta (Kluwer Academic Publishers, Norwell, 1997), pp. 53–68
B. De Baets, Generalized idempotence in fuzzy mathematical morphology, in Fuzzy Techniques in Image Processing, vol. 52. Studies in Fuzziness and Soft Computing, E.E. Kerre, M. Nachtegael Chap. 2. (Physica, New York, 2000), pp. 58–75
Diabetic Retinopathy Study Research Group and others, Photocoagulation treatment of proliferative diabetic retinopathy: clinical application of Diabetic Retinopathy Study (DRS) findings, DRS Report Number 8. Ophthalmology 88(7), 583–600 (1981)
J. Fodor, M. Roubens, Fuzzy Preference Modelling and Multicriteria Decision Support. Knowledge Engineering and Problem Solving (Kluwer Academic Publishers, Dordrecht, 1994) (System Theory)
M. González, D. Ruiz-Aguilera, J. Torrens, Algebraic properties of fuzzy morphological operators based on uninorms, in Artificial Intelligence Research and Development. Frontiers in Artificial Intelligence and Applications, vol. 100 (IOS Press, Amsterdam, 2003), pp. 27–38
M. González-Hidalgo, S. Massanet, Closing and opening based on discrete t-norms. Applications to natural image analysis, in Proceedings of the 7th Conference of the European Society for Fuzzy Logic and Technology, EUSFLAT 2011, Aix-Les-Bains, France, 18–22 July 2011, pp 358–365 (2011)
M. González-Hidalgo, S. Massanet, A fuzzy mathematical morphology based on discrete t-norms: fundamentals and applications to image processing. Soft Comput. 18(11), 2297–2311 (2014)
M. González-Hidalgo, S. Massanet, A. Mir, D. Ruiz-Aguilera, A fuzzy filter for high-density salt and pepper noise removal, in Advances in Artificial Intelligence, vol. 8109, Lecture Notes in Computer Science, ed. by C. Bielza, et al. (Springer, Berlin, 2013), pp. 70–79
M. González-Hidalgo, S. Massanet, A. Mir, D. Ruiz-Aguilera, High-density impulse noise removal using fuzzy mathematical morphology, in Proceedings of the 8th Conference of the European Society of Fuzzy Logic and Technology Conference (EUSFLAT 2013), ed. by G. Pasi, J. Montero, D. Ciucci (Atlantis Press, Milano, Italy, 2013), pp. 728–735
M. González-Hidalgo, S. Massanet, A. Mir, D. Ruiz-Aguilera, On the choice of the pair conjunction-implication into the fuzzy morphological edge detector. IEEE Trans. Fuzzy Syst. 23(4), 872–884 (2015)
M. González-Hidalgo, A. Mir-Torres, D. Ruiz-Aguilera, J. Torrens, Image analysis applications of morphological operators based on uninorms, in Proceedings of the IFSA-EUSFLAT 2009 Conference, Lisbon, Portugal, pp. 630–635 (2009)
M. González-Hidalgo, S. Massanet, A. Mir, D. Ruiz-Aguilera, A fuzzy morphological hit-or-miss transform for grey-level images: a new approach. Fuzzy Sets Syst. 286, 30-65 (2016)
E. Kerre, M. Nachtegael, Fuzzy Techniques in Image Processing, vol. 52 (Studies in Fuzziness and Soft Computing (Springer, New York, 2000)
E. Klement, R. Mesiar, E. Pap, Triangular Norms (Kluwer Academic Publishers, London, 2000)
P.D. Kovesi, MATLAB and Octave functions for computer vision and image processing. Centre for Exploration Targeting, School of Earth and Environment, The University of Western Australia. Retrieved from: http://www.csse.uwa.edu.au/_pk/research/matlabfns/ in 1994
D. Lesage, E.D. Angelini, I. Bloch, G. Funka-Lea, A review of 3D vessel lumen segmentation techniques: models, features and extraction schemes. Med. Image Anal. 13(6), 819–845 (2009)
C. Lopez-Molina, B. De Baets, H. Bustince, Quantitative error measures for edge detection. Pattern Recognit. 46(4), 1125–1139 (2013)
M. Mas, M. Monserrat, J. Torrens, E. Trillas, A survey on fuzzy implication functions. IEEE Trans. Fuzzy Syst. 15(6), 1107–1121 (2007)
G. Mayor, J. Torrens, On a family of t-norms. Fuzzy Sets Syst. 41, 161–166 (1991)
R. Medina-Carnicer, R. Muoz-Salinas, E. Yeguas-Bolivar, L. Diaz-Mas, A novel method to look for the hysteresis thresholds for the Canny edge detector. Pattern Recognit. 44(6), 1201–1211 (2011)
M. Nachtegael, E. Kerre, Classical and fuzzy approaches towards mathematical morphology, in Fuzzy Techniques in Image Processing, vol. 52. E.E. Kerre, M. Nachtegael. Studies in Fuzziness and Soft Computing, Chap. 1 (Physica, New York, 2000), pp. 3–57
N. Otsu, A threshold selection method from gray-level histograms. IEEE Trans. Syst. Man Cybern. 9, 62–66 (1979)
G. Papari, N. Petkov, Edge and line oriented contour detection: state of the art. Image Vis. Comput. 29(2–3), 79–103 (2011)
W.K. Pratt, Digital Image Processing, 4th edn. (Wiley-Interscience, 2007)
C. Rijsbergen, Information Retrieval (Butterworths, 1979)
S. Schulte, V. De Witte, M. Nachtegael, D. Van der Weken, E. Kerre, Fuzzy random impulse noise reduction method. Fuzzy Sets Syst. 158(3), 270–283 (2007)
J. Serra, Image Analysis and Mathematical Morphology, vols. 1, 2 (Academic Press, London, 1982)
J.V. Soares, J.J. Leandro, R.M. Cesar Jr., H.F. Jelinek, M.J. Cree, Retinal vessel segmentation using the 2-D Gabor wavelet and supervised classification. IEEE Trans. Med. Imaging 25(9), 1214–1222 (2006)
J. Staal, M.D. Abràmoff, M. Niemeijer, M. Viergever, B. Van Ginneken et al., Ridge-based vessel segmentation in color images of the retina. IEEE Trans. Med. Imaging 23(4), 501–509 (2004)
Z. Wang, A.C. Bovik, H.R. Sheikh, E.P. Simoncelli, Image quality assessment: from error visibility to structural similarity. IEEE Trans. Image Process. 13(4), 600–612 (2004)
C. Wilkinson, F.L. Ferris, R.E. Klein, P.P. Lee, C.D. Agardh, M. Davis, D. Dills, A. Kampik, R. Pararajasegaram, J.T. Verdaguer et al., Proposed international clinical diabetic retinopathy and diabetic macular edema disease severity scales. Ophthalmology 110(9), 1677–1682 (2003)
F. Zana, J.-C. Klein, Segmentation of vessel-like patterns using mathematical morphology and curvature evaluation. IEEE Trans. Image Process. 10(7), 1010–1019 (2001)
D. Ze-Feng, Y. Zhou-Ping, X. You-Lun, High probability impulse noise-removing algorithm based on mathematical morphology. IEEE Signal Process. Lett. 14(1), 31–34 (2007)
K. Zuiderveld, Contrast limited adaptive histogram equalization, in Graphics Gems IV (Academic Press Professional Inc, 1994), pp. 474–485
Acknowledgments
This project was partially supported by the Spanish project TIN 2013-42795-P. P. Bibiloni also benefited from a fellowship of the Conselleria d’Educaci, Cultura i Universitats of the Govern de les Illes Balears under an operational program co-financed by the European Social Fund.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
Bibiloni, P., González-Hidalgo, M., Massanet, S., Mir, A., Ruiz-Aguilera, D. (2016). Mayor-Torrens t-norms in the Fuzzy Mathematical Morphology and Their Applications. In: Calvo Sánchez, T., Torrens Sastre, J. (eds) Fuzzy Logic and Information Fusion. Studies in Fuzziness and Soft Computing, vol 339. Springer, Cham. https://doi.org/10.1007/978-3-319-30421-2_13
Download citation
DOI: https://doi.org/10.1007/978-3-319-30421-2_13
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-30419-9
Online ISBN: 978-3-319-30421-2
eBook Packages: EngineeringEngineering (R0)