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

Abstract

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 Pratt's 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 KleeneDienes 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.

On the Generalization of the Fuzzy Morphological Operators for Edge Detection

“…As internal operators T and I into both the generalized and usual morphological operators, we have considered the nilpotent minimum t-norm T nM and the Kleene-Dienes fuzzy implication I KD (x, y) = max{1 − x, y}. Note that the pair (T nM , I KD ) is the best configuration of the usual morphological gradient derived from t-norms for edge detection purposes (see [11]). Finally, as external operators, t-normT and t-conormŜ, we have considered the t-norms of Table 1 except the drastic t-norm whose expression is not adequate to detect edges and their dual t-conorms.…”

“…In fact, the fuzzy mathematical morphology based on (T LK , I LK ) is closely related to the umbra approach towards grey-level mathematical morphology as shown by Sussner and Valle in [10]. Nevertheless, there are more t-norms and implications that can be used to define a morphological gradient in edge detection in the fuzzy mathematical morphology framework with notable improvements in its performance [11] since not all the algebraic properties are necessary for edge detection. Especially, the configuration (T nM , I KD ), where T nM is the nilpotent minimum and I KD is the Kleene-Dienes implication, has shown better results than the configuration (T LK , I LK ).…”

Edge Detection Algorithm of a Symmetric Difference Kernel SAR Image Based on the GAN Network Model

“…Inspired by a two-person zero-sum game, Goodfellow et al proposed a generative antagonism network GAN, which consists of a generator G and a discriminator D. A zero-sum game is based on the idea that the sum of two people's interests in the game is zero; that is, one party's income is the other party's loss [12]. Generator G captures the distribution model of real data samples first, and then generates new data samples using the captured distribution model.…”

Mathematical Morphology on the Spherical CIELab Quantale with an Application in Color Image Boundary Detection

“…We first computed the re-scaled and complemented achromatic morphological gradient (g) using a color morphological approach. Then, like González-Hidalgo et al [18], we converted the gray-scale gradient image (g) into a binary image β by applying the non-maximum suppression method followed by an hysteresis threshold [11,31]. For simplicity, we fixed the threshold values at T 1 = 0.01 and T 2 = 0.2 in our experiments.…”