International Journal of Modern Education and Computer Science, 2020
In this work we present two methods of vector graphic objects retrieval based on a fuzzy descript... more In this work we present two methods of vector graphic objects retrieval based on a fuzzy description of their shapes. Both methods enable the retrieval of vector images resembling to a given fuzzy pattern. The basic method offers a geometrical interpretation of a fuzziness measure as a radius of a circle with the center in each vertex of a given candidate object. It enables the representation of uncertain information about a pattern object defined by its “fuzzy” vertices. The advanced method generalizes this approach by considering an ellipse instead of a circle. The basic method can be used for the comparison of polygons and other primitives in vector images. The advanced method can be used for complex shapes retrieval. To enable saving a “fuzzy” image as a file, the modification of the SVG format with a new attribute “fuzziness” is proposed for both methods. The advanced method practical implementation is illustrated by the retrieval of medical images, namely, heart computer tomog...
Several operators on intuitionistic fuzzy sets, such as union, intersection, sum and product, hav... more Several operators on intuitionistic fuzzy sets, such as union, intersection, sum and product, have been defined using common triangular norms and conorms. In this paper we introduce and analyse the properties of a generalised union and a generalised intersection of intuitionistic fuzzy sets using a general t-norm and t-conorm. In particular we will investigate properties such as commutativity, associativity, distributivity, idempotency, Morgan-laws, and absorption laws.
Classical database systems have been introduced in the late 50’s and have proved their usefulness... more Classical database systems have been introduced in the late 50’s and have proved their usefulness in various domains. However, their incompetence to deal with vague and imprecise information, has lead to new data base designs. On the other hand the use of linguistic terms has also shown its usefulness. The assignment of linguistic terms to phenomena in order to describe the characteristics or properties of objects is very natural. People make such assignments every day. A drawback of most new database designs is that often the natural aspect of making assignment is lost. In this paper we introduce a new database model based on quasi-order relations (reflexive and symmetric). The proposed model describes the mathematical background of the assignment of values to database attributes, using the theory of evaluation problems and sets. The constructed model offers an interesting new approach to the theory of database design in combination with linguistic terms.
In this paper we present a new method to solve the fuzzy linear system in the Fuzzy Finite Elemen... more In this paper we present a new method to solve the fuzzy linear system in the Fuzzy Finite Element Method (FFEM). We make use of an approach presented by A. Vroman, G. Deschrijver and E.E. Kerre. [1] to solve general fuzzy linear systems. This method is based on parametric functions and has an exponential time complexity in the number of fuzzy numbers in the fuzzy linear system. By taking the reccurrence of fuzzy variables, sometimes under a different form, in the linear system within the FFEM into account, we can optimize this method depending on the level of reccurrence of those fuzzy variables. The method remains exponentially in time but the exponent depends on the level of reccurrence of the fuzzy variables. The base is greater than the base in the original method, which was 2, because of the loss of monotonicity in the parametric functions, but the cost to construct those parametric functions from the fuzzy linear system is higher in our modified approach. This is compensated ...
We argue that fuzzy equivalences, and in particular fuzzy equalities, in general are not suitable... more We argue that fuzzy equivalences, and in particular fuzzy equalities, in general are not suitable to model approximate equality due to the notion of transitivity. Using them for this purpose therefore leads to counter-intuitive results, as we illustrate in detail in a fuzzy relationbased framework for the representation of linguistic modifiers. We solve the problem by choosing resemblance relations.
TWMS Journal of Applied and Engineering Mathematics
This paper presents axiomatic as well as constructive definitions of intuitionistic fuzzy bi-impl... more This paper presents axiomatic as well as constructive definitions of intuitionistic fuzzy bi-implicators based on intuitionistic fuzzy t-norms and their intuitionistic fuzzy residual implicators. The interrelationship among different proposed classes is presented along with a detailed study of the properties of one of these intuitionistic fuzzy bi-implicators called the intuitionistic fuzzy β−bi-implicator operator constructed using Lukasiewicz intuitionistic fuzzy t-norm and its R-implicator.
International Journal of Computational Intelligence Systems
This paper comprises a new iterative method for multi-person decision making based on multiplicat... more This paper comprises a new iterative method for multi-person decision making based on multiplicative consistency with incomplete reciprocal preference relations (IRPRs). Additionally, multiplicative transitivity property of reciprocal preference relation (RPR) is used at the first level to estimate the unknown preference values and get the complete preference relation, then it is confirmed to be multiplicative consistent by using transitive closure formula. Following this, expert's weights are evaluated by merging consistency and trust weights. The consistency weights against the experts are evaluated through multiplicative consistency investigation of the preferences given by each expert, while trust weights play the role to measure the level of trust for an expert. The consensus process determines whether the selection procedure should start or not. If it results in negative, the feedback mechanism is used to enhance the consensus degree. At the end, a numerical example is given to demonstrate the efficiency and practicality of the proposed method.
During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research uni... more During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms.
In this paper we investigate additive generators in Atanassov’s intuitionistic fuzzy and interval... more In this paper we investigate additive generators in Atanassov’s intuitionistic fuzzy and interval-valued fuzzy set theory. Starting from generalized arithmetic operators satisfying some axioms we define additive generators and we characterize continuous generators which map exact elements to exact elements in terms of generators on the unit interval. We give a necessary and sufficient condition under which a generator actually generates a t-norm and we show that the generated t-norm belongs to particular classes of t-norms depending on the arithmetic operators involved in the definition of the generator.
International Journal of Computers Communications & Control, 2015
This paper presents a new method to estimate the unknown values in incomplete interval-valued fuz... more This paper presents a new method to estimate the unknown values in incomplete interval-valued fuzzy preference relations (IVFPRs). The method is based on the min-consistency and is used to develop the algorithm for group decision making (GDM) dealing with incomplete IVFPRs.
Mashhour et al. (1) introduced the notions of P1-paracompactness and P2-paracom pactness of topol... more Mashhour et al. (1) introduced the notions of P1-paracompactness and P2-paracom pactness of topological spaces in terms of preopen sets. In this paper, we introduce and investigate a weaker form of paracompactness which is called P3-paracompact. We obtain various characterizations, properties, examples, and counterexamples concerning it and its relationships with other types of spaces. In particular, we show that if
Similarity measures, originally introduced to express the degree of comparison between two fuzzy ... more Similarity measures, originally introduced to express the degree of comparison between two fuzzy sets, can be applied in several ways to digital images. In [1–3] we have shown how similarity measures were used for the construction of image quality evaluation measures. In this paper we illustrate how fuzzy similarity measures can be useful for the comparison of histograms of digital images. We will show that similarity measures can be applied to two different types of histograms: normalized histograms and ordered normalized histograms.
We consider discrete possibilistic systems for which the available information is given by one-st... more We consider discrete possibilistic systems for which the available information is given by one-step transition possibilities and initial possibilities. These systems can be represented by a collection of variables satisfying a possibilistic counterpart of the Markov condition. This means that, given the values assumed by a selection of variables, the possibility that a subsequent variable assumes some value is only dependent on the value taken by the most recent variable of the selection. The one-step transition possibilities are recovered by computing the conditional possibility of any two consecutive variables. Under the behavioural interpretation as marginal betting rates against events these ‘conditional’ possibilities and the initial possibilities should satisfy the rationality criteria of ‘avoiding sure loss’ and ‘coherence’. We show that this is indeed the case when the conditional possibilities are defined using Dempster’s conditioning rule.
In order to accelerate the acquisition process in multiple-coil Magnetic Resonance scanners, para... more In order to accelerate the acquisition process in multiple-coil Magnetic Resonance scanners, parallel techniques were developed. These techniques reduce the acquisition time via a sub-sampling of the k-space and a reconstruction process. From a signal and noise perspective, the use of a acceleration techniques modify the structure of the noise within the image. In the most common algorithms, like SENSE, the final magnitude image after the reconstruction is known to follow a Rician distribution for each pixel, just like single coil systems. However, the noise is spatially non-stationary, i.e. the variance of noise becomes xdependent. This effect can also be found in magnitude images due to other processing inside the scanner. In this work we propose a method to adapt well-known noise filtering techniques initially designed to deal with stationary noise to the case of spatially variant Rician noise. The method copes with inaccurate estimates of variant noise patterns in the image, showing its robustness in realistic cases. The method employs a consensus strategy in conjunction with a set of aggregation functions and a penalty function. Multiple possible outputs are generated for each pixel assuming different unknown input parameters. The consensus approach merges them into a unique filtered image. As a filtering technique, we have selected the Linear Minimum Mean Square Error (LMMSE) estimator for Rician data, which has been used to test our methodology due to its simplicity and robustness. Results with synthetic and in vivo data confirm the good behavior of our approach.
Abstract—There has been a long history of using fuzzy lan-guage equivalence to compare the behavi... more Abstract—There has been a long history of using fuzzy lan-guage equivalence to compare the behavior of fuzzy systems, but the comparison at this level is too coarse. Recently, a finer behavioral measure, bisimulation, has been introduced to fuzzy finite automata. However, the results obtained are applicable only to finite-state systems. In this paper, we consider bisimulation for general fuzzy systems which may be infinite-state or infinite-event, by modeling them as fuzzy transition systems. To help understand and check bisimulation, we characterize it in three ways by enumerating whole transitions, comparing individual transitions, and using a monotonic function. In addition, we address composition operations, subsystems, quotients, and ho-momorphisms of fuzzy transition systems and discuss their properties connected with bisimulation. The results presented here are useful for comparing the behavior of general fuzzy systems. In particular, this makes it possible to relate an infin...
Comptes Rendus De L Academie Bulgare Des Sciences, 2017
This paper introduces a new intuitionistic fuzzy multicriteria decision making method of evaluati... more This paper introduces a new intuitionistic fuzzy multicriteria decision making method of evaluation based on degree of inclusion of two intuitionistic fuzzy sets. We have called the new technique TOPIIS (Technique to Order Preference by Inclusion of Ideal Solution). The technique is applied to develop an effective employee performance appraisal.
International Journal of Intelligent Systems, 2014
This paper presents a new method to estimate the unknown values in an incomplete interval-valued ... more This paper presents a new method to estimate the unknown values in an incomplete interval-valued fuzzy preference relations (IVFPRs). The method is based on the min-consistency and is used to develop the algorithm for group decision making GDM dealing with incomplete IVFPRs.
The concept of an image of a fuzzy set under a fuzzy relation has proved to be a very powerful to... more The concept of an image of a fuzzy set under a fuzzy relation has proved to be a very powerful tool in fuzzy set theoretical applications. In this paper, we explain how it can be used to model linguistic expressions. For the representation of expressions, such as "at least middle-aged", "brighter than average", we will use fuzzy ordering relations, while resemblance relations will be suitable to model linguistic terms, such as "more or less expensive" and "very tall." We will show how these representations can be smoothly integrated in approximate reasoning schemes using the compositional rule of inference.
The purpose of this paper is two-fold. Firstly, a general concept of closedness of fuzzy sets und... more The purpose of this paper is two-fold. Firstly, a general concept of closedness of fuzzy sets under fuzzy preorderings is proposed and investigated along with the corresponding opening and closure operators. Secondly, the practical impact of this notion is demonstrated by applying it to the analysis of ordering-based modifiers.
Proceedings of the 5th International Conference on Fuzzy Logic and Applications, 2003
In this paper, we will show how fuzzy similarity measures are used in establishing measures for i... more In this paper, we will show how fuzzy similarity measures are used in establishing measures for image quality evaluation. Similarity measures, originally introduced to express the degree of comparison between two fuzzy sets, can be applied to digital images. We will show how neighbourhood-based similarity measures and histogram similarity measures can be combined in order to improve the perceptive behaviour of these similarity measures. In this way, we obtained several new image quality measures, which outperform the Mean Squared Error in the sense of image quality evaluation because the results of the new measures coincide better with human perception.
International Journal of Modern Education and Computer Science, 2020
In this work we present two methods of vector graphic objects retrieval based on a fuzzy descript... more In this work we present two methods of vector graphic objects retrieval based on a fuzzy description of their shapes. Both methods enable the retrieval of vector images resembling to a given fuzzy pattern. The basic method offers a geometrical interpretation of a fuzziness measure as a radius of a circle with the center in each vertex of a given candidate object. It enables the representation of uncertain information about a pattern object defined by its “fuzzy” vertices. The advanced method generalizes this approach by considering an ellipse instead of a circle. The basic method can be used for the comparison of polygons and other primitives in vector images. The advanced method can be used for complex shapes retrieval. To enable saving a “fuzzy” image as a file, the modification of the SVG format with a new attribute “fuzziness” is proposed for both methods. The advanced method practical implementation is illustrated by the retrieval of medical images, namely, heart computer tomog...
Several operators on intuitionistic fuzzy sets, such as union, intersection, sum and product, hav... more Several operators on intuitionistic fuzzy sets, such as union, intersection, sum and product, have been defined using common triangular norms and conorms. In this paper we introduce and analyse the properties of a generalised union and a generalised intersection of intuitionistic fuzzy sets using a general t-norm and t-conorm. In particular we will investigate properties such as commutativity, associativity, distributivity, idempotency, Morgan-laws, and absorption laws.
Classical database systems have been introduced in the late 50’s and have proved their usefulness... more Classical database systems have been introduced in the late 50’s and have proved their usefulness in various domains. However, their incompetence to deal with vague and imprecise information, has lead to new data base designs. On the other hand the use of linguistic terms has also shown its usefulness. The assignment of linguistic terms to phenomena in order to describe the characteristics or properties of objects is very natural. People make such assignments every day. A drawback of most new database designs is that often the natural aspect of making assignment is lost. In this paper we introduce a new database model based on quasi-order relations (reflexive and symmetric). The proposed model describes the mathematical background of the assignment of values to database attributes, using the theory of evaluation problems and sets. The constructed model offers an interesting new approach to the theory of database design in combination with linguistic terms.
In this paper we present a new method to solve the fuzzy linear system in the Fuzzy Finite Elemen... more In this paper we present a new method to solve the fuzzy linear system in the Fuzzy Finite Element Method (FFEM). We make use of an approach presented by A. Vroman, G. Deschrijver and E.E. Kerre. [1] to solve general fuzzy linear systems. This method is based on parametric functions and has an exponential time complexity in the number of fuzzy numbers in the fuzzy linear system. By taking the reccurrence of fuzzy variables, sometimes under a different form, in the linear system within the FFEM into account, we can optimize this method depending on the level of reccurrence of those fuzzy variables. The method remains exponentially in time but the exponent depends on the level of reccurrence of the fuzzy variables. The base is greater than the base in the original method, which was 2, because of the loss of monotonicity in the parametric functions, but the cost to construct those parametric functions from the fuzzy linear system is higher in our modified approach. This is compensated ...
We argue that fuzzy equivalences, and in particular fuzzy equalities, in general are not suitable... more We argue that fuzzy equivalences, and in particular fuzzy equalities, in general are not suitable to model approximate equality due to the notion of transitivity. Using them for this purpose therefore leads to counter-intuitive results, as we illustrate in detail in a fuzzy relationbased framework for the representation of linguistic modifiers. We solve the problem by choosing resemblance relations.
TWMS Journal of Applied and Engineering Mathematics
This paper presents axiomatic as well as constructive definitions of intuitionistic fuzzy bi-impl... more This paper presents axiomatic as well as constructive definitions of intuitionistic fuzzy bi-implicators based on intuitionistic fuzzy t-norms and their intuitionistic fuzzy residual implicators. The interrelationship among different proposed classes is presented along with a detailed study of the properties of one of these intuitionistic fuzzy bi-implicators called the intuitionistic fuzzy β−bi-implicator operator constructed using Lukasiewicz intuitionistic fuzzy t-norm and its R-implicator.
International Journal of Computational Intelligence Systems
This paper comprises a new iterative method for multi-person decision making based on multiplicat... more This paper comprises a new iterative method for multi-person decision making based on multiplicative consistency with incomplete reciprocal preference relations (IRPRs). Additionally, multiplicative transitivity property of reciprocal preference relation (RPR) is used at the first level to estimate the unknown preference values and get the complete preference relation, then it is confirmed to be multiplicative consistent by using transitive closure formula. Following this, expert's weights are evaluated by merging consistency and trust weights. The consistency weights against the experts are evaluated through multiplicative consistency investigation of the preferences given by each expert, while trust weights play the role to measure the level of trust for an expert. The consensus process determines whether the selection procedure should start or not. If it results in negative, the feedback mechanism is used to enhance the consensus degree. At the end, a numerical example is given to demonstrate the efficiency and practicality of the proposed method.
During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research uni... more During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms.
In this paper we investigate additive generators in Atanassov’s intuitionistic fuzzy and interval... more In this paper we investigate additive generators in Atanassov’s intuitionistic fuzzy and interval-valued fuzzy set theory. Starting from generalized arithmetic operators satisfying some axioms we define additive generators and we characterize continuous generators which map exact elements to exact elements in terms of generators on the unit interval. We give a necessary and sufficient condition under which a generator actually generates a t-norm and we show that the generated t-norm belongs to particular classes of t-norms depending on the arithmetic operators involved in the definition of the generator.
International Journal of Computers Communications & Control, 2015
This paper presents a new method to estimate the unknown values in incomplete interval-valued fuz... more This paper presents a new method to estimate the unknown values in incomplete interval-valued fuzzy preference relations (IVFPRs). The method is based on the min-consistency and is used to develop the algorithm for group decision making (GDM) dealing with incomplete IVFPRs.
Mashhour et al. (1) introduced the notions of P1-paracompactness and P2-paracom pactness of topol... more Mashhour et al. (1) introduced the notions of P1-paracompactness and P2-paracom pactness of topological spaces in terms of preopen sets. In this paper, we introduce and investigate a weaker form of paracompactness which is called P3-paracompact. We obtain various characterizations, properties, examples, and counterexamples concerning it and its relationships with other types of spaces. In particular, we show that if
Similarity measures, originally introduced to express the degree of comparison between two fuzzy ... more Similarity measures, originally introduced to express the degree of comparison between two fuzzy sets, can be applied in several ways to digital images. In [1–3] we have shown how similarity measures were used for the construction of image quality evaluation measures. In this paper we illustrate how fuzzy similarity measures can be useful for the comparison of histograms of digital images. We will show that similarity measures can be applied to two different types of histograms: normalized histograms and ordered normalized histograms.
We consider discrete possibilistic systems for which the available information is given by one-st... more We consider discrete possibilistic systems for which the available information is given by one-step transition possibilities and initial possibilities. These systems can be represented by a collection of variables satisfying a possibilistic counterpart of the Markov condition. This means that, given the values assumed by a selection of variables, the possibility that a subsequent variable assumes some value is only dependent on the value taken by the most recent variable of the selection. The one-step transition possibilities are recovered by computing the conditional possibility of any two consecutive variables. Under the behavioural interpretation as marginal betting rates against events these ‘conditional’ possibilities and the initial possibilities should satisfy the rationality criteria of ‘avoiding sure loss’ and ‘coherence’. We show that this is indeed the case when the conditional possibilities are defined using Dempster’s conditioning rule.
In order to accelerate the acquisition process in multiple-coil Magnetic Resonance scanners, para... more In order to accelerate the acquisition process in multiple-coil Magnetic Resonance scanners, parallel techniques were developed. These techniques reduce the acquisition time via a sub-sampling of the k-space and a reconstruction process. From a signal and noise perspective, the use of a acceleration techniques modify the structure of the noise within the image. In the most common algorithms, like SENSE, the final magnitude image after the reconstruction is known to follow a Rician distribution for each pixel, just like single coil systems. However, the noise is spatially non-stationary, i.e. the variance of noise becomes xdependent. This effect can also be found in magnitude images due to other processing inside the scanner. In this work we propose a method to adapt well-known noise filtering techniques initially designed to deal with stationary noise to the case of spatially variant Rician noise. The method copes with inaccurate estimates of variant noise patterns in the image, showing its robustness in realistic cases. The method employs a consensus strategy in conjunction with a set of aggregation functions and a penalty function. Multiple possible outputs are generated for each pixel assuming different unknown input parameters. The consensus approach merges them into a unique filtered image. As a filtering technique, we have selected the Linear Minimum Mean Square Error (LMMSE) estimator for Rician data, which has been used to test our methodology due to its simplicity and robustness. Results with synthetic and in vivo data confirm the good behavior of our approach.
Abstract—There has been a long history of using fuzzy lan-guage equivalence to compare the behavi... more Abstract—There has been a long history of using fuzzy lan-guage equivalence to compare the behavior of fuzzy systems, but the comparison at this level is too coarse. Recently, a finer behavioral measure, bisimulation, has been introduced to fuzzy finite automata. However, the results obtained are applicable only to finite-state systems. In this paper, we consider bisimulation for general fuzzy systems which may be infinite-state or infinite-event, by modeling them as fuzzy transition systems. To help understand and check bisimulation, we characterize it in three ways by enumerating whole transitions, comparing individual transitions, and using a monotonic function. In addition, we address composition operations, subsystems, quotients, and ho-momorphisms of fuzzy transition systems and discuss their properties connected with bisimulation. The results presented here are useful for comparing the behavior of general fuzzy systems. In particular, this makes it possible to relate an infin...
Comptes Rendus De L Academie Bulgare Des Sciences, 2017
This paper introduces a new intuitionistic fuzzy multicriteria decision making method of evaluati... more This paper introduces a new intuitionistic fuzzy multicriteria decision making method of evaluation based on degree of inclusion of two intuitionistic fuzzy sets. We have called the new technique TOPIIS (Technique to Order Preference by Inclusion of Ideal Solution). The technique is applied to develop an effective employee performance appraisal.
International Journal of Intelligent Systems, 2014
This paper presents a new method to estimate the unknown values in an incomplete interval-valued ... more This paper presents a new method to estimate the unknown values in an incomplete interval-valued fuzzy preference relations (IVFPRs). The method is based on the min-consistency and is used to develop the algorithm for group decision making GDM dealing with incomplete IVFPRs.
The concept of an image of a fuzzy set under a fuzzy relation has proved to be a very powerful to... more The concept of an image of a fuzzy set under a fuzzy relation has proved to be a very powerful tool in fuzzy set theoretical applications. In this paper, we explain how it can be used to model linguistic expressions. For the representation of expressions, such as "at least middle-aged", "brighter than average", we will use fuzzy ordering relations, while resemblance relations will be suitable to model linguistic terms, such as "more or less expensive" and "very tall." We will show how these representations can be smoothly integrated in approximate reasoning schemes using the compositional rule of inference.
The purpose of this paper is two-fold. Firstly, a general concept of closedness of fuzzy sets und... more The purpose of this paper is two-fold. Firstly, a general concept of closedness of fuzzy sets under fuzzy preorderings is proposed and investigated along with the corresponding opening and closure operators. Secondly, the practical impact of this notion is demonstrated by applying it to the analysis of ordering-based modifiers.
Proceedings of the 5th International Conference on Fuzzy Logic and Applications, 2003
In this paper, we will show how fuzzy similarity measures are used in establishing measures for i... more In this paper, we will show how fuzzy similarity measures are used in establishing measures for image quality evaluation. Similarity measures, originally introduced to express the degree of comparison between two fuzzy sets, can be applied to digital images. We will show how neighbourhood-based similarity measures and histogram similarity measures can be combined in order to improve the perceptive behaviour of these similarity measures. In this way, we obtained several new image quality measures, which outperform the Mean Squared Error in the sense of image quality evaluation because the results of the new measures coincide better with human perception.
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Papers by Etienne Kerre