B. De Baets

Residual operators of uninorms.

Abstract Uninorms are an important generalization of t-norms and t-conorms, having a neutral element lying anywhere in the unit interval. A uninorm shows a typical block structure and is built from a t-norm, a t-conorm and a mean operator. Two important classes of uninorms are characterized, corresponding to the use of the minimum operator (the class Umin) and maximum operator (the class Umax) as mean operator. The characterization of representable uninorms, i.e. uninorms with an additive generator, and of left-continuous and right-continuous idempotent uninorms is recalled. Two residual operators are associated with a uninorm and it is characterized when they yield an implicator and coimplicator. The block structure of the residual implicator of members of the class Umin and of the residual coimplicator of members of the class Umax is investigated. Explicit expressions for the residual implicator and residual coimplicator of representable uninorms and of certain left-continuous or right-continuous idempotent uninorms are given. Additional properties such as contrapositivity are discussed.

A comparative study of similarity measures

Abstract The purpose of this paper is twofold. Firstly, we would like to comment on the study of similarity measures carried out by Pappis and Karacapilidis (1993). Their definition of ‘approximate equality’ of fuzzy sets is modified and relevant properties related to this correlated definition are listed. Secondly, a new class of similarity measures, extracted from the work of Bandler and Kohout (1980) on fuzzy power sets, is presented. The properties of the concept of approximate equality corresponding to these similarity measures are discussed, and involve a study of implication operators.

Knowledge-based versus data-driven fuzzy habitat suitability models for river management

Aquatic habitat suitability models have increasingly received attention due to their wide management applications. Ecological expert knowledge has been frequently incorporated in such models to link environmental conditions to the quantitative habitat suitability of aquatic species. Since the formalisation of problem-specific human expert knowledge is often difficult and tedious, data-driven machine learning techniques may be helpful to extract knowledge from ecological datasets. In this paper, both expert knowledge-based and data-driven fuzzy habitat suitability models were developed and the performance of these models was compared. For the data-driven models, a hill-climbing optimisation algorithm was applied to derive ecological knowledge from the available data. Based on the available ecological expert knowledge and on biological samples from the Zwalm river basin (Belgium), habitat suitability models were generated for the mayfly Baetis rhodani (Pictet 1843). Data-driven models appeared to outperform expert knowledge-based models substantially, while a step-forward model selection procedure indicated that physical habitat variables adequately described the mayfly habitat suitability in the studied area. This study has important implications on the application of expert knowledge in ecological studies, especially if this knowledge is extrapolated to other areas. The results suggest that data-driven models can complement expert knowledge-based approaches and hence improve model reliability.

Quantitative error measures for edge detection

How to measure the performance of an edge detection method is an open problem. Many performance measures have been presented in the literature. However, there seems to be no agreement on the best option, and there exist no practical studies of the features of some of the most recent measures. In this work we make a comprehensive overview of the different proposals, followed by a practical comparison of the most representative measures on synthetic as well as natural edge images.

On the existence and construction of T -transitive closures

In this paper, it is shown that any fuzzy relation R on an arbitrary universe X has a T - transitive closure. The triangular norm T involved is not subject to any conditions. This existential result can be turned into an explicit expression in the case of a left-continuous triangular norm T . Furthermore, this continuity restriction vanishes in the case of a finite universe X . In that case, the explicit expression can also be operationalized.

Accurate particle size distribution determination by nanoparticle tracking analysis based on 2-D Brownian dynamics simulation

A physical model is presented to simulate the average step length distribution during nanoparticle tracking analysis experiments as a function of the particle size distribution and the distribution of the number of steps within the tracks. Considering only tracks of at least five steps, numerical simulation could be replaced by a normal distribution approximation. Based on this model, simulation of a step length distribution allows obtaining a much more reliable estimation of the particle size distribution, thereby reducing the artificial broadening of the distribution, as is typically observed by direct conversion of step length to particle size data. As this fitting procedure also allowed including data from particles that were followed for a relatively low number of steps, the measurement time could be reduced for particles that are known to be monodisperse. Whereas the inversion is less sensitive towards the particle size distribution width, still similar values were obtained for both the average diameter and standard deviation of a polystyrene latex sample irrespective of the track length, provided that the latter included at least five steps.

A gravitational approach to edge detection based on triangular norms

We study the method of Sun et al. for edge detection based on the Law of Universal Gravity. We analyze the effect of the substitution of the product operation by other triangular norms in the calculation of the gravitational forces. We treat edges as fuzzy sets for which membership degrees are extracted from the resulting gravitational force on each pixel. We consider several prototypical triangular norms and experimentally show that their features determine the kind of edges detected. The new method is tested on the Berkeley Segmentation Dataset, showing to be competitive compared to the Canny method.

Effect of granule size on autotrophic nitrogen removal in a granular sludge reactor

Autotrophic nitrogen removal through sequential partial nitritation and anammox reactions can be achieved in biofilm reactors by controlling the oxygen concentration in the bulk liquid in such a way that nitrite oxidizers are outcompeted by anammox bacteria. In the case of granular sludge reactors, the granule size may influence the optimal range of oxygen concentration, as has been confirmed in the present study by means of numerical simulations. The range of oxygen concentrations for which combined partial nitritation and anammox conversion is established becomes broader for larger particles and with increasing influent ammonium concentrations. At the same time the likelihood of nitrite accumulation in the reactor effluent also increases.

Fuzzy relational compositions

Abstract This paper investigates the fuzzy relational compositions defined by Bandler and Kohout. It is indicated that these compositions have to be modified in order to capture the desired interpretation. Several alternative definitions are suggested and discussed. The paper concludes with an overview of the properties of these improved fuzzy relational compositions.

Construction of Interval-Valued Fuzzy Relations With Application to the Generation of Fuzzy Edge Images

In this paper, we present a new construction method for interval-valued fuzzy relations (interval-valued fuzzy images) from fuzzy relations (fuzzy images) by vicinity. This construction method is based on the concepts of triangular norm (t-norm) and triangular conorm (t-conorm). We analyze the effect of using different t-norms and t -conorms. Furthermore, we examine the influence of different sizes of the submatrix around each element of a fuzzy relation on the interval-valued fuzzy relation. Finally, we apply our construction method to image processing, and we compare the results of our approach with those obtained by means of other, i.e., fuzzy and nonfuzzy, techniques.