Interval-valued and intuitionistic fuzzy mathematical morphologies as special cases of L-fuzzy mathematical morphology

Mathematical morphology (MM) offers a wide range of tools for image processing and computer vision. MM was originally conceived for the processing of binary images and later extended to gray-scale morphology. Extensions of classical binary morphology to gray-scale morphology include approaches based on fuzzy set theory that give rise to fuzzy mathematical morphology (FMM). From a mathematical point of view, FMM relies on the fact that the class of all fuzzy sets over a certain universe forms a complete lattice. Recall that complete lattices provide for the most general framework in which MM can be conducted. The concept of L-fuzzy set generalizes not only the concept of fuzzy set but also the concepts of interval-valued fuzzy set and Atanassov’s intuitionistic fuzzy set. In addition, the class of L-fuzzy sets forms a complete lattice whenever the underlying set L constitutes a complete lattice. Based on these observations, we develop a general approach towards L-fuzzy mathematical morphology in this paper. Our focus is in particular on the construction of connectives for interval-valued and intuitionistic fuzzy mathematical morphologies that arise as special, isomorphic cases of L-fuzzy MM. As an application of these ideas, we generate a combination of some well-known medical image reconstruction techniques in terms of interval-valued fuzzy image processing

Minimal sufficient positive-operator valued measure on a separable Hilbert space

We introduce a concept of a minimal sufficient positive-operator valued measure (POVM), which is the least redundant POVM among the POVMs that have the equivalent information about the measured quantum system. Assuming the system Hilbert space to be separable, we show that for a given POVM a sufficient statistic called a Lehmann-Scheff\'{e}-Bahadur statistic induces a minimal sufficient POVM. We also show that every POVM has an equivalent minimal sufficient POVM and that such a minimal sufficient POVM is unique up to relabeling neglecting null sets. We apply these results to discrete POVMs and information conservation conditions proposed by the author.Comment: 25 pages. The main result is improved, and a new appendix is adde

A Formal Approach based on Fuzzy Logic for the Specification of Component-Based Interactive Systems

Formal methods are widely recognized as a powerful engineering method for the specification, simulation, development, and verification of distributed interactive systems. However, most formal methods rely on a two-valued logic, and are therefore limited to the axioms of that logic: a specification is valid or invalid, component behavior is realizable or not, safety properties hold or are violated, systems are available or unavailable. Especially when the problem domain entails uncertainty, impreciseness, and vagueness, the appliance of such methods becomes a challenging task. In order to overcome the limitations resulting from the strict modus operandi of formal methods, the main objective of this work is to relax the boolean notion of formal specifications by using fuzzy logic. The present approach is based on Focus theory, a model-based and strictly formal method for componentbased interactive systems. The contribution of this work is twofold: i) we introduce a specification technique based on fuzzy logic which can be used on top of Focus to develop formal specifications in a qualitative fashion; ii) we partially extend Focus theory to a fuzzy one which allows the specification of fuzzy components and fuzzy interactions. While the former provides a methodology for approximating I/O behaviors under imprecision, the latter enables to capture a more quantitative view of specification properties such as realizability.Comment: In Proceedings FESCA 2015, arXiv:1503.0437

Enhancement of dronogram aid to visual interpretation of target objects via intuitionistic fuzzy hesitant sets

In this paper, we address the hesitant information in enhancement task often caused by differences in image contrast. Enhancement approaches generally use certain filters which generate artifacts or are unable to recover all the objects details in images. Typically, the contrast of an image quantifies a unique ratio between the amounts of black and white through a single pixel. However, contrast is better represented by a group of pix- els. We have proposed a novel image enhancement scheme based on intuitionistic hesi- tant fuzzy sets (IHFSs) for drone images (dronogram) to facilitate better interpretations of target objects. First, a given dronogram is divided into foreground and background areas based on an estimated threshold from which the proposed model measures the amount of black/white intensity levels. Next, we fuzzify both of them and determine the hesitant score indicated by the distance between the two areas for each point in the fuzzy plane. Finally, a hyperbolic operator is adopted for each membership grade to improve the pho- tographic quality leading to enhanced results via defuzzification. The proposed method is tested on a large drone image database. Results demonstrate better contrast enhancement, improved visual quality, and better recognition compared to the state-of-the-art methods.Web of Science500866