‎On Generalized Mixture Functions

Transactions on Fuzzy Sets and Systems

مقاله 11، دوره 1، شماره 2، بهمن 2022، صفحه 99-128   

نوع مقاله: Original Article

شناسه دیجیتال (DOI): 10.30495/tfss.2022.1962834.1041

نویسندگان

Antonio Diego Silva Farias¹؛ Valdigleis da Silva Costa²؛ Luiz Ranyer A. Lopes³؛ Regivan Hugo Nunes Santiago⁴؛ Benjamin Bedregal^{* 4}

¹Department of Exact and Natural Sciences, Federal Rural University of Semi-Arid–UFERSA Pau dos Ferros-RN, Brazil

²Collegiate of Computer Science Federal University of Vale do Sao Francisco–UNIVASF Salgueiro-PE, Brazil

³Federal Institute of Rio Grande Norte–IFRN Natal-RN, Brazil

⁴Department of Informatics and Applied Mathematics Federal University of Rio Grande do Norte-UFRN Natal-RN, Brazil

چکیده

In the literature it is very common to see problems in which it is necessary to aggregate a set of data into a single one‎. ‎An important tool able to deal with these issues is the aggregation functions‎, ‎which we can highlight as the OWA functions‎. ‎However‎, ‎there are other functions that are also capable of performing these tasks‎, ‎such as the preaggregation function and mixture functions‎. ‎In this paper we investigate two special types of functions‎, ‎the Generalized Mixture functions and Bounded Generalized Mixture functions‎, ‎which generalize both OWA and Mixture functions‎. ‎We also prove some properties‎, ‎constructions and examples of these functions‎. ‎Both the Generalized and Bounded Generalized Mixture functions are developed in such a way that the weight vectors are variables that depend on the input vector‎, ‎which generalizes the aggregation functions‎: ‎ Minimum‎, ‎Maximum‎, Arithmetic Mean and Median, ‎and are extensively used in image processing‎. ‎Finally‎, ‎we propose a Generalized Mixture function‎, ‎denoted by $\mathbf{H}$‎, ‎and we show that $\mathbf{H}$ satisfies a series of properties in order to apply this function in an illustrative example of application‎: ‎The image reduction process‎.

کلیدواژه‌ها

Aggregation functions‎؛ ‎Preaggregation functions‎؛ ‎OWA functions‎؛ ‎Generalized Mixture functions‎؛ ‎Image reduction‎

مراجع

[1] M. Baczynski and B. Jayaram, Fuzzy Implications, Vol. 231 of Studies in Fuzziness and Soft Computing, Springer, Berlin, (2008). doi: 10.1007/978-3-540-69082-5.
[2] T. V. V. Batista, B. Bedregal and R. M. Moraes, Constructing multi-layer classi er ensembles using the Choquet integral based on overlap and quasi-overlap functions, Neurocomputing, 500 (2022), 413-421.
[3] G. Beliakov, A. Pradera and T. Calvo, Aggregation Functions: A Guide for Practitioners, Vol. 221 of Studies in Fuzziness and Soft Computing, Springer, Berlin, (2007).

[4] G. Beliakov, H. Bustince and D. Paternain, Image reduction using means on discrete product lattices, IEEE Trans. Image Process., 21 (2012), 1070-1083.
[5] G. Beliakov, T. Calvo and T. Wilkin, On the weak monotonicity of Gini means and other mixture functions, Inf. Sci., 300 (2015), 70-84.
[6] G. Beliakov, H. Bustince and T. Calvo, A Practical Guide to Averaging Functions, Vol. 329 of Studies in Fuzziness and Soft Computing, Springer, Berlin, (2016).
[7] G. Beliakov, G. Das, H. Q. Vu, T. Wilkin and Y. Xiang, Fuzzy connectives for ecient image reduction and speeding up image analysis, IEEE Access, 6 (2018), 68403-68414.
[8] H. Bustince, M. Galar, B. Bedregal, A. Kolesarova and R. Mesiar, A new approach to interval-valued Choquet integrals and the problem of ordering in interval-valued fuzzy set applications, IEEE Trans. Fuzzy Syst., 21 (2013), 1150-1162.
[9] H. Bustince, J. Fernandez, A. Kolesarova and R. Mesiar, Directional monotonicity of fusion functions, Eur. J. Oper. Res., 244 (2015), 300-308.
[10] H. Bustince, R. Mesiar, J. Fernandez, M. Galar, D. Paternain, A. H. Altalhi, G. P. Dimuro, B. R. C. Bedregal and Z. Takac, d-Choquet integrals: Choquet integrals based on dissimilarities, Fuzzy Sets Syst., 414 (2021), 1-27.
[11] H. Bustince, B. Bedregal, M. J. Campion, I.A. da Silva, J. Fernandez, E. Indurain, A. Raventos-Pujol, R. H. N. Santiago, Aggregation of Individual Rankings Through Fusion Functions: Criticism and Optimality Analysis, IEEE Trans. Fuzzy Syst., 30(3) (2022), 638-648.
[12] C.-H. Cheng and J.-R. Chang, MCDM aggregation model using situational ME-OWA and ME-OWGA operators, Int. J. Uncertain. Fuzziness Knowl. Based Syst, 14 (2006), 421-443.
[13] V. S. Costa, A. D. S. Farias, B. R. C. Bedregal, R. H. N. Santiago and A. M. de P. Canuto, Combining multiple algorithms in classi er ensembles using generalized mixture functions, Neurocomputing, 313 (2018), 402-414.
[14] S. C. Dighe and R. Shriram, Dental biometrics for human identi cation based on dental work and image properties in Periapical radiographs, TENCON 2012 - 2012 IEEE Region 10 Conference, (2012), 1-6. doi: 10.1109/TENCON.2012.6412216.
[15] G. P. Dimuro, J. Fernandez, B. R. C. Bedregal, R. Mesiar, J. A. Sanz, G. Lucca and H. Bustince, The state-of-art of the generalizations of the Choquet integral: From aggregation and pre-aggregation to ordered directionally monotone functions, Inf. Fusion, 57 (2020), 27-43.
[16] D. Dubois and H. Prade, Fundamentals of Fuzzy Sets, Vol. 7 of The Handbooks of Fuzzy Sets, 1 ed., Springer, New York, (2000).
[17] D. Dubois and H. Prade, On the use of aggregation operations in information fusion processes, Fuzzy Sets Syst., 142 (2004), 143-161.
[18] A. D. S. Farias, V. S. Costa, R. H. N. Santiago and B. R. C. Bedregal, The image reduction process based on generalized mixture functions, 2016 Annual Conference of the North American Fuzzy Information Processing Society (NAFIPS), 31 October 2016-04 November 2016, El Paso, TX, USA , (2016), 1-6. doi: 10.1109/NAFIPS.2016.7851591.

[19] A. D. S. Farias, L. R. A. Lopes, B. C. Bedregal and R. H. N. Santiago, Closure properties for fuzzy recursively enumerable languages and fuzzy recursive languages, J. Intell. Fuzzy Syst., 31 (2016), 1795-1806.
[20] A. D. S. Farias, R. H. N. Santiago and B. R. C. Bedregal, Some properties of generalized mixture functions, 2016 IEEE International Conference on Fuzzy Systems, FUZZ-IEEE, July 24-29, 2016, Vancouver, BC, Canada, (2016), 288-293. doi: 10.1109/FUZZ-IEEE.2016.7737699.
[21] A. D. S. Farias, C. Callejas, R. H. N. Santiago and B. R. C. Bedregal, Directional and ordered directional monotonicity of generalized and bounded generalized mixture functions, 2018 IEEE International Conference on Fuzzy Systems, FUZZ-IEEE, July 8-13, 2018, Rio de Janeiro, Brazil, (2018), 1-7. doi:10.1109/FUZZ-IEEE.2018.8491541.
[22] M. Grabisch, J. L. Marichal, R. Mesiar and E. Pap, Aggregation Functions, Vol. 127 of Encyclopedia of Mathematics and its Applications, University Press Cambridge, Cambridge, (2009).
[23] R. C. Gonzales and R. E. Woods, Digital Image Processing, 3rd ed., Pearson, New Jersey, (2008).
[24] E. Hancer, B. Xue, M. Zhang, D. Karaboga and B. Akay, A multi-objective arti cial bee colony approach to feature selection using fuzzy mutual information, 2015 IEEE Congress on Evolutionary Computation (CEC), (2015), 2420{2427. doi: 10.1109/CEC.2015.7257185.
[25] G. Jaffino, A. Banumathi, U. Gurunathan, J. P. Jose, Dental work extraction for di erent radiographic images in human forensic identi cation, 2014 International Conference on Communication and Network Technologies (ICCNT), (2014), 52-56. doi: 10.1109/CNT.2014.7062724.
[26] R. P. Joseph, C. S. Singh and M. Manikandan, Brain tumor MRI image segmentation and detection in image processing, Int. J. Res. Technol., 3 (2014), 1-5.
[27] A. Jurio, M. Pagola, R. Mesiar, G. Beliakov and H. Bustince, Image magni cation using interval information, IEEE Trans. Image Process., 20 (2011), 3112-3123.
[28] R. Keys, Cubic convolution interpolation for digital image processing, IEEE Trans. Acoust. Speech Signal Process., 29 (1981).
[29] J. H. Kim, I. J. Ahn, W. H. Nam and J. B. Ra, An e ective post- ltering framework for 3-D PET image denoising based on noise and sensitivity characteristics, IEEE Trans. Nucl. Sci., 62 (2015), 137-147.
[30] T. M. Lehmann, C. Gonner and K. Spitzer, Survey: interpolation methods in medical image processing, IEEE Trans. Medical Imaging, 18 (1999), 1049-1075.
[31] X. Liang and W. Xu, Aggregation method for motor drive systems, Electric Power Syst. Research, 117 (2014), 27-35.
[32] L. Lingling, Z. Xian, H. Pengju and L. Zhigang, The research on the method of fuzzy information processing, 2012 3rd International Conference on System Science, Engineering Design and Manufacturing Informatization (ICSEM), 2 (2012), 47-50. doi: 10.1109/ICSSEM.2012.6340804.
[33] I. Lizasoain and C. Moreno, OWA operators de ned on complete lattices, Fuzzy Sets Syst., 224 (2013), 36-52.
[34] G. Lucca, J. A. Sanz, G. P. Dimuro, B. Bedregal, R. Mesiar, A. Kolesarova and H. Bustince, Preaggregation functions: Construction and an application, IEEE Trans. Fuzzy Syst., 24 (2016), 260-272.

[35] G. Lucca, J. A. Sanz, G. P. Dimuro, B. R. C. Bedregal and H. Bustince, A proposal for tuning the parameter in C C-integrals for application in fuzzy rule-based classi cation systems, Nat. Comput., 19 (2020), 533-546.
[36] N. V. Manokar, V. Manokar, Rinesh, K. P. Sridhar and L. M. Patnaik, Wavelets based decomposition and classi cation of diseased fMRI brain images for inter racial disease types of Alzheimer's vs tumors using SOFM and enhancement by LVQ neural networks, 2012 2nd IEEE International Conference on Parallel Distributed and Grid Computing (PDGC), (2012), 822-827. doi: 10.1109/PDGC.2012.6449929.
[37] J. M. Merigo and A. M. Gil-Lafuente, The induced generalized OWA operator, Inf. Sci., 179 (2009), 729-741.
[38] R. Mesiar and J. Spirkova, Weighted means and weighting functions, Kybernetika, 42 (2006), 151-160.
[39] R. Mesiar, J. Spirkova and L. Vavrikova, Weighted aggregation operators based on minimization, Inf. Sci., 178 (2008), 1133-1140.
[40] J. Mihailovic, A. Savic, J. Bogdanovic-Pristov and K. Radotic, MRI brain tumors images by using independent component analysis, 2011 IEEE 9th International Symposium on Intelligent Systems and Informatics, (2011), 433-435. doi: 10.1109/SISY.2011.6034366.
[41] T. Milfont, I. Mezzomo, B. Bedregal, E. Mansilla and H. Bustince, Aggregation functions on ndimensional ordered vectors equipped with an admissible order and an application in multi-criteria group decision-making, Int. J. Approx. Reason., 137 (2021), 34-50.
[42] N. Mustafa, S. A. Khan, J. P. Li, M. Khalil, K. Kumar and M. Giess, Medical image de-noising schemes using wavelet transform with  xed form thresholding, 2014 11th International Computer Conference on Wavelet Active Media Technology and Information Processing (ICCWAMTIP), (2014), 397-402. doi:10.1109/ICCWAMTIP.2014.7073435.
[43] D. Nolasco, F. Costa, E. Palmeira, D. Alves, B. Bedregal, T. Rocha, R. Ribeiro and J. Silva, Wavelet-fuzzy power quality diagnosis system with inference method based on overlap functions: Case study in an AC microgrid, Eng. Appl. Artif. Intell., 85 (2019), 284-294.
[44] D. Paternain, A. Jurio, E. Barrenechea, H. Bustince, B. Bedregal and E. Szmidt, An alternative to fuzzy methods in decision-making problems, Expert Syst. Appl., 39 (2012), 7729-7735.
[45] D. Paternain, J. Fernandez, H. Bustince, R. Mesiar and G. Beliakov, Construction of image reduction operators using averaging aggregation functions, Fuzzy Sets Syst., 261 (2015), 87-111.
[46] R. A. M. Pereira and G. Pasi, On non-monotonic aggregation: mixture operators, Proc. 4th Meeting of the EURO Working Group on Fuzzy Sets (EUROFUSE'99) and 2nd Internat. Conf. on Soft and Intelingent Computing (SIC'99), Budapest, Hungary, (1999).
[47] R. A. M. Pereira, The orness of mixture operators: the exponential case, Proc. 8th Internat. Conf. on Information Processing and Management of Uncertainty in Knowledge-Based Systems (IPMU'2000), Madrid, Spain, (2000).
[48] R. A. M. Pereira and R. A. Ribeiro, Aggregation with generalized mixture operators using weighting functions, Fuzzy Sets Syst., 137 (2003), 43-58.
[49] R. A. Ribeiro and R. A. M. Pereira, Generalized mixture operators using weighting functions: A comparative study with WA and OWA, Eur. J. Oper. Res., 145 (2003), 329-342.

[50] S. Sahu, N. Loya and A. G. Keskar, Restoration and enhancement of impulse noise image for human visual system, 2015 IEEE International Conference on Electronics, Computing and Communication Technologies (CONECCT), (2015), 1-6. doi: 10.1109/CONECCT.2015.7383913.
[51] B. Z. Shaick and L. Yaroslavsky, Image reduction for object recognition, 4th EURASIP-IEEE Region 8 International Symposium on Video/Image Processing and Multimedia Communications VIPromCom, (2002), 333-338. doi: 10.1109/VIPROM.2002.1026678.
[52] A. J. Solanki, K. R. Jain and N. P. Desai, ISEF based identi cation of RCT/ lling in dental caries of decayed tooth, Int. J. Image Process., 7 (2013), 149-162.
[53] Z.-X. Su, G.-P. Xia, M.-Y. Chen and L. Wang, Induced generalized intuitionistic fuzzy OWA operator for multi-attribute group decision making, Expert Syst. Appl., 39 (2012), 1902-1910.
[54] Suprijanto, Gianto, E. Juliastuti, Azhari and L. Epsilawati, Image contrast enhancement for  lm-based dental panoramic radiography, 2012 International Conference on System Engineering and Technology (ICSET), (2012), 1-5. doi:10.1109/ICSEngT.2012.6339321.
[55] P. Thevenaz, T. Blu and M. Unser, Interpolation revisited, IEEE Trans. Medical Imaging, 19 (2000), 739-758.
[56] T. Wilkin, Image reduction operators based on non-monotonic averaging functions, 2013 IEEE Int. Conf. on Fuzzy Systems, (FUZZ), (2013), 1-8. doi: 10.1109/FUZZ-IEEE.2013.6622458.
[57] T. Wilkin and G. Beliakov, Weakly monotonic averaging functions, Int. J. Intell. Syst., 30(2) (2015), 144-169.
[58] S. K. Woo, K. M. Kim, T. S. Lee, J. H. Jung, J. G. Kim, J. S. Kim, T. H. Choi, G. I. An and G. J. Cheon, Registration method for the detection of tumors in lung and liver using multimodal small animal imaging, IEEE Trans. Nucl. Sci., 56 (2009), 1454-1458.
[59] B. Y. Wu and X. Q. Sheng, A complex image reduction technique using genetic algorithm for the MoM solution of half-space MPIE, IEEE Trans. Antennas Propag., 63 (2015), 3727-3731.
[60] R. R. Yager, On ordered weighted averaging aggregation operators in multicriteria decisionmaking, IEEE Trans. Syst. Man Cybern., 18 (1988), 183-190.
[61] R. R. Yager, Centered OWA operators, Soft Comput., 11 (2006), 631-639.
[62] J. Yang, J. Wright, T. Huang and Y. Ma, Image super-resolution as sparse representation of raw image patches, IEEE Conference on Computer Vision and Pattern Recognition, CVPR 2008, (2008), 1-8. doi: 10.1109/CVPR.2008.4587647.
[63] J. Yang, J. Wright, T. S. Huang and Y. Ma, Image super-resolution via sparse representation, IEEE Trans. Image Process., 19 (2010), 2861-2873.
[64] S.-M. Zhou, F. Chiclana, R. I. John and J. M. Garibaldi, Type-1 OWA operators for aggregating uncertain information with uncertain weights induced by type-2 linguistic quanti ers, Fuzzy Sets Syst., 159 (2008), 3281-3296.