scholarly journals N-ary Mathematical Morphology

2016 ◽  
Vol 1 (1) ◽  
Author(s):  
Emmanuel Chevallier ◽  
Augustin Chevallier ◽  
Jesús Angulo
Keyword(s):  
Finite Number ◽  
Partial Order ◽  
Mathematical Morphology ◽  
Set Theory ◽  
Lattice Theory ◽  
Color Images ◽  
Binary Images ◽  
Grey Scale ◽  
Multivariate Images ◽  
Definition Of

AbstractMathematical morphology on binary images can be fully described by set theory. However, it is not sufficient to formulate mathematical morphology for grey scale images. This type of images requires the introduction of the notion of partial order of grey levels, together with the definition of sup and inf operators. More generally, mathematical morphology is now described within the context of the lattice theory. For a few decades, attempts are made to use mathematical morphology on multivariate images, such as color images, mainly based on the notion of vector order. However, none of these attempts has given fully satisfying results. Instead of aiming directly at the multivariate case we propose first an extension of binary mathematical morphology to an intermediary situation: images composed of a finite number of independent unordered labels. We propose then an second extension to a continuous case.

scholarly journals Quaternion Processing Techniques for Color Synthesized NDT Thermography

2021 ◽  
Vol 11 (2) ◽  
pp. 790
Author(s):  
Pablo Venegas ◽  
Rubén Usamentiaga ◽  
Juan Perán ◽  
Idurre Sáez de Ocáriz
Keyword(s):  
Signal To Noise Ratio ◽  
High Capacity ◽  
Synthetic Data ◽  
Relevant Information ◽  
Color Images ◽  
Destructive Testing ◽  
Application Procedure ◽  
Definition Of ◽  
Characterization Procedure ◽  
Processing Techniques

Infrared thermography is a widely used technology that has been successfully applied to many and varied applications. These applications include the use as a non-destructive testing tool to assess the integrity state of materials. The current level of development of this application is high and its effectiveness is widely verified. There are application protocols and methodologies that have demonstrated a high capacity to extract relevant information from the captured thermal signals and guarantee the detection of anomalies in the inspected materials. However, there is still room for improvement in certain aspects, such as the increase of the detection capacity and the definition of a detailed characterization procedure of indications, that must be investigated further to reduce uncertainties and optimize this technology. In this work, an innovative thermographic data analysis methodology is proposed that extracts a greater amount of information from the recorded sequences by applying advanced processing techniques to the results. The extracted information is synthesized into three channels that may be represented through real color images and processed by quaternion algebra techniques to improve the detection level and facilitate the classification of defects. To validate the proposed methodology, synthetic data and actual experimental sequences have been analyzed. Seven different definitions of signal-to-noise ratio (SNR) have been used to assess the increment in the detection capacity, and a generalized application procedure has been proposed to extend their use to color images. The results verify the capacity of this methodology, showing significant increments in the SNR compared to conventional processing techniques in thermographic NDT.

scholarly journals Final coalgebras as greatest fixed points in ZF set theory

1999 ◽  
Vol 9 (5) ◽  
pp. 545-567 ◽  
Author(s):  
LAWRENCE C. PAULSON
Keyword(s):  
Fixed Points ◽  
Set Theory ◽  
Theorem Prover ◽  
Infinite Length ◽  
Special Cases ◽  
Definition Of ◽  
Recursive Definitions ◽  
Ordered Pairs

A special final coalgebra theorem, in the style of Aczel (1988), is proved within standard Zermelo–Fraenkel set theory. Aczel's Anti-Foundation Axiom is replaced by a variant definition of function that admits non-well-founded constructions. Variant ordered pairs and tuples, of possibly infinite length, are special cases of variant functions. Analogues of Aczel's solution and substitution lemmas are proved in the style of Rutten and Turi (1993). The approach is less general than Aczel's, but the treatment of non-well-founded objects is simple and concrete. The final coalgebra of a functor is its greatest fixedpoint.Compared with previous work (Paulson, 1995a), iterated substitutions and solutions are considered, as well as final coalgebras defined with respect to parameters. The disjoint sum construction is replaced by a smoother treatment of urelements that simplifies many of the derivations.The theory facilitates machine implementation of recursive definitions by letting both inductive and coinductive definitions be represented as fixed points. It has already been applied to the theorem prover Isabelle (Paulson, 1994).

On the Dynamic Isotropy of Mechanisms With Two Degrees of Freedom

2005 ◽  
Author(s):  
Raffaele Di Gregorio ◽  
Alessandro Cammarata ◽  
Rosario Sinatra
Keyword(s):  
Finite Number ◽  
Degrees Of Freedom ◽  
Point Of View ◽  
Closed Curves ◽  
Special Cases ◽  
Dynamic Performances ◽  
Definition Of ◽  
Geometric Locus

The comparison of mechanisms with different topology or with different geometry, but with the same topology, is a necessary operation during the design of a machine sized for a given task. Therefore, tools that evaluate the dynamic performances of a mechanism are welcomed. This paper deals with the dynamic isotropy of 2-dof mechanisms starting from the definition introduced in a previous paper. In particular, starting from the condition that identifies the dynamically isotropic configurations, it shows that, provided some special cases are not considered, 2-dof mechanisms have at most a finite number of isotropic configurations. Moreover, it shows that, provided the dynamically isotropic configurations are excluded, the geometric locus of the configuration space that collects the points associated to configurations with the same dynamic isotropy is constituted by closed curves. This results will allow the classification of 2-dof mechanisms from the dynamic-isotropy point of view, and the definition of some methodologies for the characterization of the dynamic isotropy of these mechanisms. Finally, examples of applications of the obtained results will be given.

scholarly journals Fuzzy Soft Multiset Theory

2012 ◽  
Vol 2012 ◽  
pp. 1-20 ◽  
Author(s):  
Shawkat Alkhazaleh ◽  
Abdul Razak Salleh
Keyword(s):  
Decision Making ◽  
Set Theory ◽  
Fuzzy Set ◽  
Soft Set ◽  
Mathematical Tool ◽  
Basic Properties ◽  
Definition Of

In 1999 Molodtsov introduced the concept of soft set theory as a general mathematical tool for dealing with uncertainty. Alkhazaleh et al. in 2011 introduced the definition of a soft multiset as a generalization of Molodtsov's soft set. In this paper we give the definition of fuzzy soft multiset as a combination of soft multiset and fuzzy set and study its properties and operations. We give examples for these concepts. Basic properties of the operations are also given. An application of this theory in decision-making problems is shown.

Optimization Analysis on Dynamic Reduction Algorithm

2018 ◽  
Vol 6 (5) ◽  
pp. 447-458
Author(s):  
Yizhou Chen ◽  
Jiayang Wang
Keyword(s):  
Set Theory ◽  
Rough Set ◽  
Rough Set Theory ◽  
Reduction Process ◽  
New Method ◽  
Reduction Algorithm ◽  
Optimization Analysis ◽  
Sampling Process ◽  
Dynamic Methods ◽  
Definition Of

Abstract On the basis of rough set theory, the strengths of dynamic reduction are elaborated compared with traditional non-dynamic methods. A systematic concept of dynamic reduction from sampling process to the generation of the reduct set is presented. A new method of sampling is created to avoid the defects of being too subjective. And in order to deal with the over-sized time consuming problem in traditional dynamic reduction process, a quick algorithm is proposed within the constraint conditions. We have also proved that dynamic core possesses the essential characteristics of a reduction core on the basis of the formalized definition of the multi-layered dynamic core.

Uncertainty Analysis Using Fuzzy Random Set Theory

2016 ◽  
pp. 384-418
Author(s):  
Debabrata Datta
Keyword(s):  
Uncertainty Analysis ◽  
Set Theory ◽  
Practical Problem ◽  
Epistemic Uncertainty ◽  
Uncertainty Modeling ◽  
Random Set ◽  
Uncertain Variables ◽  
Definition Of ◽  
Hydrological Applications ◽  
Fuzzy Random

Uncertainty analysis of any physical model is always an essential task from the point of decision making analysis. Two kinds of uncertainties exist: (1) aleatory uncertainty which is due to randomness of the parameters of models of interest and (2) the epistemic uncertainty which is due to fuzziness of the parameters of the same models. So far both these uncertainties are addressed independently; however since in any practical problem both the types of uncertain variables present, it is required to address them jointly. In order to solve practical problems on uncertainty modeling, it is required to replace the abstract definition of hybrid set by fuzzy random set. Since uncertainty modeling using fuzzy random set has not been carried out so far, the present chapter will address the utility of fuzzy random set for uncertainty modeling on geotechnical and hydrological applications. This chapter will present the fundamentals of fuzzy random set and their application in uncertainty analysis.

scholarly journals On bisimple semigroups generated by a finite number of idempotents

1982 ◽  
Vol 33 (1) ◽  
pp. 92-101 ◽  
Author(s):  
Karl Byleen
Keyword(s):  
Finite Number ◽  
Partial Order ◽  
Natural Partial Order ◽  
Rees Matrix Semigroups ◽  
Matrix Semigroups ◽  
Completely Simple

AbstractNon-completely simple bisimple semigroups S which are generated by a finite number of idempotents are studied by means of Rees matrix semigroups over local submonoids eSe, e = e2 ∈ S. If under the natural partial order on the set Es of idempotents of such a semigroup S the sets ω(e) = {ƒ ∈ Es: ƒ ≤ e} for each e ∈ Es are well-ordered, then S is shown to contain a subsemigroup isomorphic to Sp4, the fundamental four-spiral semigroup. A non-completely simple hisimple semigroup is constructed which is generated by 5 idempotents but which does not contain a subsemigroup isomorphic to Sp4.

Evaluation of Quantified Statements Using Gradual Numbers

2011 ◽  
pp. 246-269 ◽  
Author(s):  
Ludovic Liétard ◽  
Daniel Rocacher
Keyword(s):  
Decision Making ◽  
Expert Systems ◽  
Set Theory ◽  
Fuzzy Set ◽  
Fuzzy Set Theory ◽  
Relational Databases ◽  
Fuzzy Numbers ◽  
Theoretical Background ◽  
Definition Of ◽  
Flexible Querying

This chapter is devoted to the evaluation of quantified statements which can be found in many applications as decision making, expert systems, or flexible querying of relational databases using fuzzy set theory. Its contribution is to introduce the main techniques to evaluate such statements and to propose a new theoretical background for the evaluation of quantified statements of type “Q X are A” and “Q B X are A.” In this context, quantified statements are interpreted using an arithmetic on gradual numbers from Nf, Zf, and Qf. It is shown that the context of fuzzy numbers provides a framework to unify previous approaches and can be the base for the definition of new approaches.

scholarly journals SQUARES, ASCENT PATHS, AND CHAIN CONDITIONS

2018 ◽  
Vol 83 (04) ◽  
pp. 1512-1538 ◽  
Author(s):  
CHRIS LAMBIE-HANSON ◽  
PHILIPP LÜCKE
Keyword(s):  
Partial Order ◽  
Set Theory ◽  
Consistency Strength ◽  
Chain Conditions ◽  
Regular Cardinals ◽  
Aronszajn Tree ◽  
Square Principles ◽  
Image Position ◽  
Aronszajn Trees ◽  
Consistency Strengths

AbstractWith the help of various square principles, we obtain results concerning the consistency strength of several statements about trees containing ascent paths, special trees, and strong chain conditions. Building on a result that shows that Todorčević’s principle $\square \left( {\kappa ,\lambda } \right)$ implies an indexed version of $\square \left( {\kappa ,\lambda } \right)$, we show that for all infinite, regular cardinals $\lambda < \kappa$, the principle $\square \left( \kappa \right)$ implies the existence of a κ-Aronszajn tree containing a λ-ascent path. We then provide a complete picture of the consistency strengths of statements relating the interactions of trees with ascent paths and special trees. As a part of this analysis, we construct a model of set theory in which ${\aleph _2}$-Aronszajn trees exist and all such trees contain ${\aleph _0}$-ascent paths. Finally, we use our techniques to show that the assumption that the κ-Knaster property is countably productive and the assumption that every κ-Knaster partial order is κ-stationarily layered both imply the failure of $\square \left( \kappa \right)$.

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