Higher Degree F-transforms Based on B-splines of Two Variables
The paper deals with the higher degree fuzzy transforms (F-transforms with polynomial components) for functions of two variables in the case when two-dimensional generalized fuzzy partition is given by B-splines of two variables. We investigate properties
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Department of Mathematics, University of Latvia, Zellu 25, Riga 1002, Latvia {martins.kokainis,svetlana.asmuss}@lu.lv 2 Institute of Mathematics and Computer Science, University of Latvia, Raina Bulvaris 29, Riga 1459, Latvia
Abstract. The paper deals with the higher degree fuzzy transforms (F-transforms with polynomial components) for functions of two variables in the case when two-dimensional generalized fuzzy partition is given by B-splines of two variables. We investigate properties of the direct and inverse F-transform in this case and prove that using B-splines as basic functions of fuzzy partition allows us to improve the quality of approximation.
Keywords: F-transform mation error · B-splines
1
· Two-dimensional fuzzy partition · Approxi-
Introduction
The concept of fuzzy transform (F -transform or F 0 -transform) was introduced in 2001 [11] (see also the key paper [9]) and generalized to the case of higher degree (F m -transform, m ≥ 1) in 2011 [12] by I. Perfilieva with co-authors. Initially, the technique of F -transform was described for functions of one variable. The extension for the two-dimensional case (i.e. for functions of two variables) has been introduced in [17] and developed in [7,8]. There is a number of papers dealing with fuzzy transforms with respect to a fuzzy partition with specially designed basic functions including fuzzy partitions based on splines (see, e.g., [1,5]). In both mentioned cases the ordinary F -transforms (i.e. with the classical components-numbers) for functions of one variable have been considered. The construction of higher degree fuzzy transforms with respect to a fuzzy partition given by central, odd degree B-splines of one variable was presented by the authors at the previous FUZZ-IEEE conference (IEEE International Conference on Fuzzy Systems) in Istanbul in 2015 (see [6]). At the current stage our main focus area refers to the case of two variables. Let us note that in last years exactly two-dimensional F-transforms have been widely applied in image processing for solving problems of image compression, reconstruction, denoising, fusion, edge detection (see, e.g., [2,3,13,14]). Our motivation to develop and investigate the technique of spline-based F-transform c Springer International Publishing Switzerland 2016 J.P. Carvalho et al. (Eds.): IPMU 2016, Part I, CCIS 610, pp. 648–659, 2016. DOI: 10.1007/978-3-319-40596-4 54
Higher Degree F-transforms Based on B-splines of Two Variables
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for the case of several variables has been caused by the rapid development of such applications. Our research focuses on the F m -transform with respect to a generalized uniform fuzzy partition given by B-splines of two variables of degree 2k1 − 1 and 2k2 − 1 respectively. We generalize for the two-dimensional case the main result of [6] and obtain that in this case the inverse F m -transform is precise for polynomials of two variables of degree r1 ≤ 2k1 − 1 and r2 ≤ 2k2 − 1 when r1 + r2 ≤ 2m + 1. On the basis of this result we obtain error estimations for approximation b
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