Abstract
The focus of this article is on exploring the concept of abstract homogeneity in relation to partially ordered sets. Our study delves into various properties of this notion, including self-homogeneity. We then apply these concepts to the practical area of image processing, demonstrating their relevance. Our application of abstract homogeneous functions in computer vision surpasses classical edge detection methods and approaches state-of-the-art results, highlighting their potential showing that a suitable family of functions for this task are the abstract homogeneous ones. Additionally, we improve these results by creating consensus feature images, which aggregate features to enhance the effect of abstract homogeneity. This method offers a subtle yet effective improvement in image processing tasks.
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Notes
Kulisch-Miranker order is the most used partial order on \(\mathbb {I}(\mathbb {R})\). It is defined as: \(X \ {\preceq }_{KM} \ Y \ \text {iff} \ \underline{X} \le \underline{Y}\) and \(\overline{X} \le \overline{Y}\) Kulisch and Miranker (1981); Román-Flores et al. (2013). Some other examples of interval partial orders can be found in Han and Hu (2015).
Bustince et al. introduced it inspired by the lexicographical order of points in \(\mathbb {R}^2\) (see Bustince et al. (2013)).
Note that \(X-X = \varvec{0}\) iff X is a degenerated interval.
The minimum function \(\min \), takes a n-tuple as input and returns the smallest coordinate. Likewise, the maximum function, \(\max \), takes an n-tuple as input and returns the greatest coordinate.
Given a partially ordered set \(\langle A,\preceq _A \rangle \), a binary function \(f: A^2 \rightarrow A\) is said to be bisymmetric if \(f(f(x,y), f(u,v)) = f(f(x,u), f(y,v))\) for all \(x,y,u,v \in A\) (cf. Ovchinnikov (1993)).
\(F_{AM_{n}}\) is the IV-arithmetic mean and \(P_2\) is the IV-product function, both defined in Example 4
After references.
After references.
References
Arbeláez P, Maire M, Fowlkes C, Malik J (2011) Contour detection and hierarchical image segmentation. IEEE Trans Pattern Anal Mach Intell 33(5):898–916. https://doi.org/10.1109/TPAMI.2010.161
Bedregal BC, Santiago RHN (2013) Interval representations, łukasiewicz implicators and smets-magrez axioms. Inf Sci 221:192–200. https://doi.org/10.1016/j.ins.2012.09.022
Bedregal B, Bustince H, Palmeira E, Dimuro G, Fernandez J (2017) Generalized interval-valued OWA operators with interval weights derived from interval-valued overlap functions. Int J Approx Reason 90:1–16. https://doi.org/10.1016/j.ijar.2017.07.001
Beliakov G, Pradera A, Calvo T (2007) Aggregation Functions: A Guide for Practitioners vol. 221, 1st edn. Springer, Berlin. https://doi.org/10.1007/978-3-540-73721-6
Bezdek JC, Chandrasekhar R, Attikouzel Y (1998) A geometric approach to edge detection. IEEE Trans Fuzzy Syst 6(1):52–75. https://doi.org/10.1109/91.660808
Brickell F (1967) A theorem on homogeneous functions. Journal of the London Mathematical Society s1-42(1), 325–329 https://doi.org/10.1112/jlms/s1-42.1.325
Bustince H, Pagola M, Mesiar R, Hullermeier E, Herrera F (2012) Grouping, overlap, and generalized bientropic functions for fuzzy modeling of pairwise comparisons. IEEE Trans Fuzzy Syst 20(3):405–415. https://doi.org/10.1109/TFUZZ.2011.2173581
Bustince H, Fernandez J, Kolesárová A, Mesiar R (2013) Generation of linear orders for intervals by means of aggregation functions. Fuzzy Sets Syst 220:69–77. https://doi.org/10.1016/j.fss.2012.07.015
Bustince H, Fernandez J, Mesiar R, Montero J, Orduna R (2010) Overlap functions. Nonlinear Anal Theory Methods Appl 72(3):1488–1499. https://doi.org/10.1016/j.na.2009.08.033
Cabrera GG, Ehrgott M, Mason A, Philpott A (2014) Multi-objective optimisation of positively homogeneous functions and an application in radiation therapy. Oper Res Lett 42(4):268–272
Canny J (1986) A computational approach to edge detection. IEEE Trans Pattern Anal Mach Intell PAMI 8(6):679–698. https://doi.org/10.1109/TPAMI.1986.4767851
Carmichael R (1852) XX. Homogeneous functions, and their index symbol. Lon Edinburgh Dublin Philos Mag J Sci 3(16):129–140. https://doi.org/10.1080/14786445208646968
Chen B-Y (2014) Solutions to homogeneous Monge-Ampère equations of homothetic functions and their applications to production models in economics. J Math Anal Appl 411(1):223–229. https://doi.org/10.1016/j.jmaa.2013.09.029
Ciak R, Shafei B, Steidl G (2013) Homogeneous penalizers and constraints in convex image restoration. J Math Imag Vis 47:210–230. https://doi.org/10.1007/s10851-012-0392-5
Davey BA, Priestley HA (2002) Introduction to lattices and order. Cambridge University Press, Cambridge. https://doi.org/10.1017/CBO9780511809088
Dollár P, Zitnick CL (2014) Fast edge detection using structured forests. IEEE Trans Pattern Anal Mach Intell 37(8):1558–1570
Dominicy Y, Heikkilä M, Ilmonen P, Veredas D (2020) Flexible multivariate Hill estimators. J Econom 217(2):398–410. https://doi.org/10.1016/j.jeconom.2019.12.010
Ebanks BR (1998) Quasi-homogeneous associative functions. Int J Math Math Sci 21:351–357. https://doi.org/10.1155/S0161171298000489
Ermilov MM, Surkova LE, Samoletov RV (2021). In: Bogoviz AV, Suglobov AE, Maloletko AN, Kaurova OV, Lobova SV (eds) Mathematical modeling of consumer behavior, taking into account entropy. Springer, Cham. https://doi.org/10.1007/978-3-030-57831-2_28
Estrada FJ, Jepson AD (2009) Benchmarking Image Segmentation Algorithms. Int J Comput Vis 85(2):167–181. https://doi.org/10.1007/s11263-009-0251-z
Frisch R (1949) On the zeros of homogeneous functions. Econometrica (pre-1986) 17(1):28
Han J, Hu BQ (2015) Generation of partial orders for intervals by means of the slope function. Fuzzy Sets Syst 266:67–83. https://doi.org/10.1016/j.fss.2014.06.007
Kulisch UW, Miranker WL (1981) Computer arithmetic in theory and practice. Academic Press, New York. https://doi.org/10.1016/C2013-0-11018-5
Lewis JP (1999) Homogeneous functions and Euler’s Theorem, pp. 297–303. Palgrave Macmillan UK, London. https://doi.org/10.1007/978-1-349-15324-4_22
Lima L, Bedregal B, Bustince H, Barrenechea E, Rocha M (2016) An interval extension of homogeneous and pseudo-homogeneous t-norms and t-conorms. Inf Sci 355–356:328–347. https://doi.org/10.1016/j.ins.2015.11.031
Lima L, Bedregal B, Rocha M, Castillo-Lopez A, Fernandez J, Bustince H (2022) On some classes of nullnorms and h-pseudo homogeneity. Fuzzy Sets Syst 427:23–36. https://doi.org/10.1016/j.fss.2020.12.007
Lopez-Molina C, Bustince H, Fernandez J, Couto P, De Baets B (2010) A gravitational approach to edge detection based on triangular norms. Pattern Recogn 43(11):3730–3741. https://doi.org/10.1016/j.patcog.2010.05.035
Lopez-Molina C, De Baets B, Bustince H (2013) Quantitative error measures for edge detection. Pattern Recogn 46(4):1125–1139. https://doi.org/10.1016/j.patcog.2012.10.027
Lopez-Molina C, De Baets B, Bustince H (2014) A framework for edge detection based on relief functions. Inf Sci 278:127–140. https://doi.org/10.1016/j.ins.2014.03.028
Marco-Detchart C, Bustince H, Fernández J, Mesiar R, Lafuente J, Barrenechea E, Pintor JM (2021) Ordered directional monotonicity in the construction of edge detectors. Fuzzy Sets Syst 421:111–132
Martin DR, Fowlkes CC, Malik J (2004) Learning to detect natural image boundaries using local brightness, color, and texture cues. IEEE Trans Pattern Anal Mach Intell 26(5):530–549. https://doi.org/10.1109/TPAMI.2004.1273918
Mayor G, Mesiar R, Torrens J (2008) On quasi-homogeneous copulas. Kybernetika 44(6):745–756
Medina-Carnicer R, Madrid-Cuevas FJ, Carmona-Poyato A, Muñoz-Salinas R (2009) On candidates selection for hysteresis thresholds in edge detection. Pattern Recogn 42(7):1284–1296. https://doi.org/10.1016/j.patcog.2008.10.027
Moore EH (1887) Algebraic surfaces of which every plane-section is unicursal in the light of n-dimensional geometry. Am J Math 10(1):17–28. https://doi.org/10.2307/2369399
Ovchinnikov S (1993) On some bisymmetric functions on closed intervals. In: [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems, pp. 1137–11392. https://doi.org/10.1109/FUZZY.1993.327355
Philip WE (1900) A slight extension of Euler’s theorem on homogeneous functions. Proc Edinb Math Soc 18:101–102. https://doi.org/10.1017/S0013091500029424
Qiao J, Hu BQ (2019) On homogeneous, quasi-homogeneous and pseudo-homogeneous overlap and grouping functions. Fuzzy Sets Syst 357:58–90. https://doi.org/10.1016/j.fss.2018.06.001
Román-Flores H, Chalco-Cano Y, Silva GN (2013) A note on Gronwall type inequality for interval-valued functions. In: 2013 Joint IFSA World Congress and NAFIPS Annual Meeting (IFSA/NAFIPS), pp. 1455–1458. https://doi.org/10.1109/IFSA-NAFIPS.2013.6608616
Santiago R, Bedregal B, Dimuro GP, Fernandez J, Bustince H, Fardoun HM (2022) Abstract homogeneous functions and consistently influenced/disturbed multi-expert decision making. IEEE Trans Fuzzy Syst 30(9):3447–3459. https://doi.org/10.1109/TFUZZ.2021.3117438
Santiago R, Sesma-Sara M, Fernandez J, Takáč Z, Mesiar R, Bustince H (2022) \( \cal{F} \)-homogeneous functions and a generalization of directional monotonicity. Int J Intell Syst 37(9):5949–5970. https://doi.org/10.1002/int.22823
Sobel I, Feldman G (1973) A \(3 \times 3\) isotropic gradient operator for image processing. Pattern Classification and Scene Analysis, 271–272
Su Y, Zong W, Mesiar R (2022) Characterization of homogeneous and quasi-homogeneous binary aggregation functions. Fuzzy Sets Syst 433:96–107. https://doi.org/10.1016/j.fss.2021.04.020
Wang Y, Hu BQ (2022) Constructing overlap and grouping functions on complete lattices by means of complete homomorphisms. Fuzzy Sets Syst 427:71–95. https://doi.org/10.1016/j.fss.2021.03.015
Xie A, Su Y, Liu H (2016) On pseudo-homogeneous triangular norms, triangular conorms and proper uninorms. Fuzzy Sets Syst 287:203–212. https://doi.org/10.1016/j.fss.2014.11.026
Zapata H, Bustince H, Montes S, Bedregal B, Dimuro GP, Takáč Z, Baczyński M, Fernandez J (2017) Interval-valued implications and interval-valued strong equality index with admissible orders. Int J Approx Reason 88:91–109. https://doi.org/10.1016/j.ijar.2017.05.009
Acknowledgements
We thank the support for this work received from: (a) the National Council for Scientific and Technological Development (CNPq-Brazil) through the projects 312899/2021-1 and 403167/2022-1; (b) Ministerio de Ciencia y Inovacción through the projects PID2020-119478GB-I00, PID2022-136627NB-I00, TED2021-131295B-C32 and PID2021-123673OB-C31, supported by MCIN/AEI/10.13039/501100011033/FEDER,UE; (c) the Slovak Academy of Sciences through the projects APVV-20-0069 and VEGA 1/0036/23; (d) PROMETEO grant CIPROM/2021/077 from the Conselleria de Innovación, Universidades, Ciencia y Sociedad Digital - Generalitat Valenciana; (e) Early Research Project grant PAID-06-23 by the Vice Rectorate Office for Research from Universitat Politècnica de València (UPV).
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Monteiro, A.S., Santiago, R., Papčo, M. et al. Abstract homogeneity on partially ordered Sets. Comp. Appl. Math. 44, 191 (2025). https://doi.org/10.1007/s40314-025-03156-4
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DOI: https://doi.org/10.1007/s40314-025-03156-4