Arnau Mir

On the Choice of the Pair Conjunction–Implication Into the Fuzzy Morphological Edge Detector

In this paper, the fuzzy morphological gradients from the fuzzy mathematical morphologies based on t-norms and conjunctive uninorms are deeply analyzed in order to establish which pair of conjunction and fuzzy implications are optimal, in accordance with their performance in edge detection applications. A novel three-step algorithm based on the fuzzy morphology is proposed. The comparison is performed by means of the so-called Pratts figure of merit. In addition, a statistical analysis is carried out to study the relationship between the different configurations and to establish a classification of the conjunctions and implications considered. Both the objective measure and the statistical analysis conclude that the pairs nilpotent minimum t-norm and the Kleene-Dienes implication, and the idempotent uninorm obtained with the classical negation as a generator and its residual implication, are the best configurations in this approach, because they also obtain competitive results with respect to other approaches.

Cophenetic metrics for phylogenetic trees, after Sokal and Rohlf

BackgroundPhylogenetic tree comparison metrics are an important tool in the study of evolution, and hence the definition of such metrics is an interesting problem in phylogenetics. In a paper in Taxon fifty years ago, Sokal and Rohlf proposed to measure quantitatively the difference between a pair of phylogenetic trees by first encoding them by means of their half-matrices of cophenetic values, and then comparing these matrices. This idea has been used several times since then to define dissimilarity measures between phylogenetic trees but, to our knowledge, no proper metric on weighted phylogenetic trees with nested taxa based on this idea has been formally defined and studied yet. Actually, the cophenetic values of pairs of different taxa alone are not enough to single out phylogenetic trees with weighted arcs or nested taxa.ResultsFor every (rooted) phylogenetic tree T, let its cophenetic vectorφ(T) consist of all pairs of cophenetic values between pairs of taxa in T and all depths of taxa in T. It turns out that these cophenetic vectors single out weighted phylogenetic trees with nested taxa. We then define a family of cophenetic metrics dφ,p by comparing these cophenetic vectors by means of Lp norms, and we study, either analytically or numerically, some of their basic properties: neighbors, diameter, distribution, and their rank correlation with each other and with other metrics.ConclusionsThe cophenetic metrics can be safely used on weighted phylogenetic trees with nested taxa and no restriction on degrees, and they can be computed in O(n2) time, where n stands for the number of taxa. The metrics dφ,1 and dφ,2 have positive skewed distributions, and they show a low rank correlation with the Robinson-Foulds metric and the nodal metrics, and a very high correlation with each other and with the splitted nodal metrics. The diameter of dφ,p, for p⩾1, is in O(n(p+2)/p), and thus for low p they are more discriminative, having a wider range of values.

A Fuzzy Filter for High-Density Salt and Pepper Noise Removal

In this paper, a novel filter for high-density salt and pepper noise removal based on the fuzzy mathematical morphology using t-norms is proposed. This filter involves two phases, namely, a detection step of the corrupted pixels and the restoration of the image using a specialized regularization method using fuzzy open-close and close-open sequences. The experimental results show that the proposed algorithm outperforms other nonlinear filtering methods both from the visual point of view and the values of some objective performance measures for images corrupted up to 90% of noise.

High-density impulse noise removal using fuzzy mathematical morphology

This paper proposes a filtering method for highdensity impulse noise removal based on the fuzzy mathematical morphology using t-norms. The method is a two phased method. In the first phase, an impulse noise detector based on the fuzzy tophat transforms is used to identify pixels which are likely to be contaminated by noise. In the second phase, the image is restored using a specialized regularization method using fuzzy open-close or fuzzy close-open sequences applied only to those selected contaminated pixels and applying then a block smart erase algorithm. Experimental results show that the proposed algorithm presents a better performance in terms of edge preservation and noise suppression than other nonlinear filtering methods, including the presented in [1], in which this method is based on.

Exact formulas for the variance of several balance indices under the Yule model

One of the main applications of balance indices is in tests of null models of evolutionary processes. The knowledge of an exact formula for a statistic of a balance index, holding for any number

A tool for analytical simulation of B-splines surface deformation

n

Mayor-Torrens t-norms in the Fuzzy Mathematical Morphology and Their Applications

of leaves, is necessary in order to use this statistic in tests of this kind involving trees of any size. In this paper we obtain exact formulas for the variance under the Yule model of the Sackin, the Colless and the total cophenetic indices of binary rooted phylogenetic trees with