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F-transform: Theoretical aspects and advanced applications
JID:FSS AID:6950 /EDI
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Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
F-transform: Theoretical aspects and advanced applications This special issue is the second one devoted to the theory and applications of the theory of F-transforms (FTs). We are glad that this theory attracts renowned researchers in the field of fuzzy sets, fuzzy logic and their applications. This volume discusses the various contributions to the aforementioned areas. The main theoretical development discussed in the included papers is the growing role of a fuzzy partition as a principal parameter of the FT. Specifically, if the inverse FT (iFT) is expected to be equal to the original function or robust with respect to a generating (kernel) function, then partitions for the FT and iFTs should be different. This conclusion and details of the construction of corresponding partitions can be found in the first two papers. Moreover, if the direct FT is expected to be flexible with respect to a diversity of raw data, then the corresponding fuzzy partition should be flexible as well. Such flexibility is achieved by omitting certain requirements in the notion of a fuzzy partition (e.g., the Ruspini condition, even in its generalized form). The definition of a generalized fuzzy partition is given in the fifth contribution. A brief summary of the content of this issue is provided below. 1. A new reconstruction from the F-transform components. Irina Perfilieva, Michal Holˇcapek, Vladik Kreinovich The first main result of this paper is that if a function fulfills the same conditions as in the standard sampling (Shannon) theorem, then it can be reconstructed from a set of its FT components. The second main result of this paper is that if a function comes with noise, then properly selecting a fuzzy partition and sample step leads to a significant reduction in noise if the function is also reconstructed from noisy samples twice. 2. An interval-valued inversion of the non-additive interval-valued F-transform: Use for upsampling a signal. Fares Graba, Olivier Strauss In this paper, a new signal upsampling method is proposed. This method relates to the iFT, which can also be regarded as one of the upsampling methods. The proposal extends both FTs and iFTs to the interval-valued mutually dual operators. Among practical aspects of this technique, its robustness with respect to knowledge on the sampling process is emphasized. 3. General approximation of fuzzy numbers by F-transform. Lucian Coroianu, Luciano Stefanini This paper discusses the approximation of general fuzzy numbers, which are difficult to handle due to a complex representation, by sequences of fuzzy numbers. The proposed approximation tool is the extended iFT which preserves the support and the quasi-concavity of a fuzzy number. A notable advantage of the proposed approach is the linear rate of uniform convergence, meaning that a desired error of approximation can be obtained with a lower computational burden. 4. A new fuzzy approximation method to Cauchy problems by fuzzy transform. Alireza Khastan, Irina Perfilieva, Zahra Alijani This contribution applies fuzzy methods to solutions of classical problems, especially to solving ordinary differential equations. Three new numeric schemes for solving the ordinary Cauchy problem are presented. All of these schemes are based on FTs of various degrees. The first method (Mid-FT) improves upon the previously proposed http://dx.doi.org/10.1016/j.fss.2015.11.016 0165-0114/© 2015 Elsevier B.V. All rights reserved.
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Editorial
FT-based Euler method by increasing the efficiency of the computation approximate derivatives (symmetric differences instead of directed differences). The two other methods use first- and second-degree FTs to compute the unknown function more accurately. All three methods provide better solutions than ordinary numeric methods (up to the second-order Runge–Kutta one). 5. Differentiation by the F-transform and application to edge detection. Irina Perfilieva, Petra Hod’áková, Petr Hurtík This study focuses on the extension of the theory of the F1 -transform for functions of two variables. The extension consists of using a generalized fuzzy partition. In this case, the quality of the approximation of partial derivatives by the F1 -transform components is estimated. The application to the edge detection problem is discussed and compared with the Canny edge detector. The advantages of the F1 -transform approach are emphasized. 6. Lattice fuzzy transforms from the perspective of mathematical morphology. Peter Sussner The definitions of lattice FTs are generalized from the original discrete version to the case of arbitrary universes of L-fuzzy sets. The paper demonstrates that one of the lattice FTs performs an erosion in the direct phase and a dilation in the inverse phase, whereas another lattice FT performs a dilation in the direct phase and an erosion in the inverse phase. Moreover, it is proven that every L-fuzzy erosion and every L-fuzzy dilation by a translation-invariant structuring element in L-fuzzy mathematical morphology can be written as a direct or inverse lattice FT. 7. Development of autofocusing algorithm based on fuzzy transforms. Seok-Beom Roh, Sung-Kwun Oh, Witold Pedrycz, Kisung Seo This contribution proposes a new autofocus measure operator, which is based on the FT technique. In contrast to many conventional approaches, this measure operator uses a localized version of an image variance method. Experimental comparative studies are conducted on test images using various focus measure operators. A comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods. At the end, I express my hope that all these contributions will be read with an interest and that they will be regarded as a notable step in the development of the theory of F-transforms. Irina Perfilieva University of Ostrava, Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic E-mail address: [email protected] 15 November 2015
[m3SC+; v1.219; Prn:25/11/2015; 15:55] P.1 (1-2)
Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
F-transform: Theoretical aspects and advanced applications This special issue is the second one devoted to the theory and applications of the theory of F-transforms (FTs). We are glad that this theory attracts renowned researchers in the field of fuzzy sets, fuzzy logic and their applications. This volume discusses the various contributions to the aforementioned areas. The main theoretical development discussed in the included papers is the growing role of a fuzzy partition as a principal parameter of the FT. Specifically, if the inverse FT (iFT) is expected to be equal to the original function or robust with respect to a generating (kernel) function, then partitions for the FT and iFTs should be different. This conclusion and details of the construction of corresponding partitions can be found in the first two papers. Moreover, if the direct FT is expected to be flexible with respect to a diversity of raw data, then the corresponding fuzzy partition should be flexible as well. Such flexibility is achieved by omitting certain requirements in the notion of a fuzzy partition (e.g., the Ruspini condition, even in its generalized form). The definition of a generalized fuzzy partition is given in the fifth contribution. A brief summary of the content of this issue is provided below. 1. A new reconstruction from the F-transform components. Irina Perfilieva, Michal Holˇcapek, Vladik Kreinovich The first main result of this paper is that if a function fulfills the same conditions as in the standard sampling (Shannon) theorem, then it can be reconstructed from a set of its FT components. The second main result of this paper is that if a function comes with noise, then properly selecting a fuzzy partition and sample step leads to a significant reduction in noise if the function is also reconstructed from noisy samples twice. 2. An interval-valued inversion of the non-additive interval-valued F-transform: Use for upsampling a signal. Fares Graba, Olivier Strauss In this paper, a new signal upsampling method is proposed. This method relates to the iFT, which can also be regarded as one of the upsampling methods. The proposal extends both FTs and iFTs to the interval-valued mutually dual operators. Among practical aspects of this technique, its robustness with respect to knowledge on the sampling process is emphasized. 3. General approximation of fuzzy numbers by F-transform. Lucian Coroianu, Luciano Stefanini This paper discusses the approximation of general fuzzy numbers, which are difficult to handle due to a complex representation, by sequences of fuzzy numbers. The proposed approximation tool is the extended iFT which preserves the support and the quasi-concavity of a fuzzy number. A notable advantage of the proposed approach is the linear rate of uniform convergence, meaning that a desired error of approximation can be obtained with a lower computational burden. 4. A new fuzzy approximation method to Cauchy problems by fuzzy transform. Alireza Khastan, Irina Perfilieva, Zahra Alijani This contribution applies fuzzy methods to solutions of classical problems, especially to solving ordinary differential equations. Three new numeric schemes for solving the ordinary Cauchy problem are presented. All of these schemes are based on FTs of various degrees. The first method (Mid-FT) improves upon the previously proposed http://dx.doi.org/10.1016/j.fss.2015.11.016 0165-0114/© 2015 Elsevier B.V. All rights reserved.
JID:FSS AID:6950 /EDI
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Editorial
FT-based Euler method by increasing the efficiency of the computation approximate derivatives (symmetric differences instead of directed differences). The two other methods use first- and second-degree FTs to compute the unknown function more accurately. All three methods provide better solutions than ordinary numeric methods (up to the second-order Runge–Kutta one). 5. Differentiation by the F-transform and application to edge detection. Irina Perfilieva, Petra Hod’áková, Petr Hurtík This study focuses on the extension of the theory of the F1 -transform for functions of two variables. The extension consists of using a generalized fuzzy partition. In this case, the quality of the approximation of partial derivatives by the F1 -transform components is estimated. The application to the edge detection problem is discussed and compared with the Canny edge detector. The advantages of the F1 -transform approach are emphasized. 6. Lattice fuzzy transforms from the perspective of mathematical morphology. Peter Sussner The definitions of lattice FTs are generalized from the original discrete version to the case of arbitrary universes of L-fuzzy sets. The paper demonstrates that one of the lattice FTs performs an erosion in the direct phase and a dilation in the inverse phase, whereas another lattice FT performs a dilation in the direct phase and an erosion in the inverse phase. Moreover, it is proven that every L-fuzzy erosion and every L-fuzzy dilation by a translation-invariant structuring element in L-fuzzy mathematical morphology can be written as a direct or inverse lattice FT. 7. Development of autofocusing algorithm based on fuzzy transforms. Seok-Beom Roh, Sung-Kwun Oh, Witold Pedrycz, Kisung Seo This contribution proposes a new autofocus measure operator, which is based on the FT technique. In contrast to many conventional approaches, this measure operator uses a localized version of an image variance method. Experimental comparative studies are conducted on test images using various focus measure operators. A comparative analysis demonstrates that the proposed algorithm is superior to the conventional methods. At the end, I express my hope that all these contributions will be read with an interest and that they will be regarded as a notable step in the development of the theory of F-transforms. Irina Perfilieva University of Ostrava, Institute for Research and Applications of Fuzzy Modelling, NSC IT4Innovations, 30. dubna 22, 701 03 Ostrava 1, Czech Republic E-mail address: [email protected] 15 November 2015