Modelling Numerical and Spatial Uncertainty in GrayscaleImage Capture Using Fuzzy Set Theory
Mike Nachtegae∗, Peter Sussner∗∗, Tom Mélange∗,
and Etienne E. Kerre∗
∗Dept. of Applied Mathematics and Computer Science, Ghent University, Fuzziness and Uncertainty Modelling Research Unit
Krijgslaan 281 - S9, 9000 Gent, Belgium
∗∗Dept. of Applied Mathematics, University of Campinas, Campinas, SP 13083 859, Brazil
In this paper, we will discuss interval-valued and intuitionistic fuzzy sets as a model for grayscale images, taking into account the uncertainty regarding the measured grayscale values, which in some cases is also related to the uncertainty regarding the spatial position of an object in an image. We will demonstrate the practical potential of this image model by introducing an interval-valued morphological theory and by illustrating its application with some examples. The results show that the uncertainty that is present during the image capture not only can be modelled, but can also be propagated such that the information regarding the uncertainty is never lost.
and Etienne E. Kerre, “Modelling Numerical and Spatial Uncertainty in GrayscaleImage Capture Using Fuzzy Set Theory,” J. Adv. Comput. Intell. Intell. Inform., Vol.13, No.5, pp. 529-536, 2009.
-  L. Zadeh, “Fuzzy Sets,” in: Information Control, Vol.8, pp. 338-353, 1965.
-  S. Schulte, M. Nachtegael, V. DeWitte, D. Van der Weken, and E. E. Kerre, “Fuzzy impulse noise reduction methods for color images,” in: Proc. of FUZZY DAYS 2006 (Int. Conf. on Computational Intelligence), Dortmund (Germany), pp. 711-720.
-  S. Schulte, V. De Witte, M. Nachtegael, T. Mélange, and E. E. Kerre, “A new fuzzy additive noise reduction method,” in: Lecture Notes in Computer Science, Vol.4633 (Image Analysis and Recognition — Proc. of ICIAR 2007), pp. 12-23, 2007, ISBN 978-3-540-74258-6.
-  E. Vansteenkiste, D. Van der Weken, W. Philips, and E. E. Kerre, “Evaluation of the perceptual performance of fuzzy image quality measures,” in: Lecture Notes in Computer Science, Vol.4251, pp. 623-630, 2006.
-  J. Serra, “Image analysis and mathematical morphology,” Academic Press Inc, London, 1982.
-  B. De Baets, “Fuzzy morphology: a logical approach,” in: B. M. Ayyub and M. M. Gupta (Eds.), Uncertainty Analysis in Engineering and Sciences: Fuzzy Logic, Statistics, and Neural Network Approach, Kluwer Academic Publishers, Boston, pp. 53-67, 1997.
-  M. Nachtegael and E. E. Kerre, “Connections between binary, grayscale and fuzzy mathematical morphologies,” in: Fuzzy Sets and Systems, Vol.124, No.1, pp. 73-86, 2001.
-  P. Sussner and M. E. Valle, “Classification of Fuzzy Mathematical Morphologies Based on Concepts of Inclusion Measure and Duality,” in: Journal of Mathematical Imaging and Vision, accepted for publication, 2008.
-  I. Bloch, “Mathematical Morphology on Bipolar Fuzzy Sets,” in: Proc. of ISMM 2007 (Int. Symposium on Mathematical Morphology), pp. 3-4, 2007.
-  I. Bloch, “Dilation and Erosion of Spatial Bipolar Fuzzy Sets,” in: Lecture Notes in Artificial Intelligence, Vol.4578 (Proc. of WILF 2007), pp. 385-393, 2007.
-  M. Nachtegael, P. Sussner, T. Mélange, and E. E. Kerre, “Some Aspects of Interval-valued and Intuitionistic Fuzzy Mathematical Morphology,” accepted for IPCV 2008 — Int. Conf. on Image Processing, Computer Vision and Pattern Recognition (July 14-17, 2008, Las Vegas, US).
-  M. Nachtegael, P. Sussner, T. Mélange, and E. E. Kerre, “An Interval-valued Fuzzy Morphological Model based on Lukasiewicz-Operators,” accepted for ACIVS 2008 — Int. Conf. on Advanced Concepts for Intelligent Vision Systems (October 21-24, 2008, Juan-les-Pins, France).
-  J. M. Mendel, “Computing With Words: Zadeh, Turing, Popper and Occam,” in: IEEE Computational Intelligence Magazine, Vol.2, No.4, pp. 10-17, 2007.
-  O. Castillo and P. Melin, “Intelligent Systems with Interval Type-2 Fuzzy Logic,” in: Int. Journal of Innovative Computing, Information and Control, Vol.4, No.4, pp. 771-783, 2008.
-  G. Deschrijver and E. E. Kerre, “On the Relationship Between some Extensions of Fuzzy Set Theory,” in: Fuzzy Sets and Systems, Vol.133, pp. 227-235, 2003.
-  I. B. Turksen and Z. Zhong, “An Approximate Analogical Reasoning Schema based on Similarity Measures and Interval-Valued Fuzzy Sets,” in: Fuzzy Sets and Systems, Vol.34, No.3, pp. 323-346, 1990.
-  S. D. Cabrera, K. Iyer, G. Xiang, and V. Kreinovich, “On Inverse Halftoning: Computational Complexity and Interval Computations,” in: Proc. of CISS 2005 (39th Conf. on Information Sciences and Systems), The John Hopkins University, paper 164, 2005.
-  A. E. Brito and O. Kosheleva, “Interval + Image =Wavelet: For Image Processing under Interval Uncertainty, Wavelets are Optimal,” in: Reliable Computing, Vol.4, No.3, pp. 291-301, 1998.
-  E. Barrenechea, “Image Processing with interval-valued Fuzzy Sets- Edge Detection - Contrast,” Ph.D. thesis, Public University of Navarra, 2005.
-  K. Atanassov, “Intuitionistic Fuzzy Sets,” Physica Verlag, Heidelberg (Germany), 1999.
-  I. K. Vlachos and D. G. Sergiadis, “Towards Intuitionistic Fuzzy Image Processing,” in: Proc. of the Int. Conf. on Computational Intelligence for Modelling, Control and Automation and Int. Conf. on Intelligent Agents, Web Technologies and Internet Commerce, pp. 2-7, 2005.
-  I. K. Vlachos and D. G. Sergiadis, “Intuitionistic Fuzzy Information — Applications to Pattern Recognition,” in: Pattern Recognition Letters, Vol.28, pp. 197-206, 2006.
-  I. K. Vlachos and D. G. Sergiadis, “Hesitancy Histogram Equalization,” in: Proc. of FUZZ-IEEE 2007, pp. 1-6, 2007.
-  J. Goguen, “L-fuzzy sets,” in: Journal of Mathematical Analysis and Applications, Vol.18, pp. 145-174, 1967.
-  R. M. Haralick, S. R. Sternberg, and X. Zhuang, “Image analysis using mathematical morphology,” in: IEEE Transactions on Pattern Analysis and Machine Intelligence, Vol.9, No.4, pp. 532-550, 1987.
-  C. Cornelis, G. Deschrijver, and E. E. Kerre, “Implication in Intuitionistic and Interval-valued Fuzzy Set Theory: Construction, Classification, Application,” in: Int. Journal of Approximate Reasoning, Vol.35, pp. 55-95, 2004.
-  G. Deschrijver, C. Cornelis, and E. E. Kerre, “On the Representation of Intuitionistic Fuzzy t-norms and t-conorms,” in: IEEE Transactions on Fuzzy Systems, Vol.12, No.1, pp. 45-61, 2004.
-  G. Deschrijver and C. Cornelis, “Representability in Interval-valued Fuzzy Set Theory,” in: Int. Journal of Uncertainty, Fuzziness and Knowledge-Based Systems, Vol.15, No.3, pp. 345-361, 2007.
-  G. Deschrijver, “Arithmetic Operators in Interval-valued Fuzzy Set Theory,” in: Information Sciences, Vol.177, No.14, pp. 2906-2924, 2007.