Bhu Dev Sharma

New operations over hesitant fuzzy sets

Hesitant fuzzy sets are considered to be the way to characterize vague phenomenon. Their study has opened a new area of research and applications. Set operations on them lead to a number of properties of these sets which are not evident in classical (crisp) sets make the area mathematically also very productive. Since these sets are defined in terms of functions and set of functions, which is not the case when the sets are crisp, it is possible to define several set operations. Such a study enriches the use of these sets. In this paper, four new operations are envisaged, defined and taken up to study a score of new identities on hesitant fuzzy sets.

Fuzzy Generalized Prioritized Weighted Average Operator and its Application to Multiple Attribute Decision Making

The prioritized weighted average (PWA) operator was originally introduced by Yager. The prominent characteristic of the PWA operator is that it takes into account prioritization among attributes and decision makers. By combining the idea of generalized mean and PWA operator, we propose a new prioritized aggregation operator called fuzzy generalized prioritized weighted average (FGPWA) operator for aggregating triangular fuzzy numbers. The properties of the new aggregation operator are studied out and their special cases are examined. Furthermore, based on the FGPWA operator, an approach to deal with multiple attribute group decision making problems under triangular fuzzy environments is developed. Finally, a practical example is provided to illustrate the multiple attribute group decision making process.

Trapezoid fuzzy linguistic prioritized weighted average operators and their application to multiple attribute group decision-making

The prioritized weighted average (PWA) operator was originally introduced by Yager. The prominent characteristic of the PWA operator is that it takes into account prioritization among attributes and decision makers. Motivated by the idea of PWA operator, we develop some prioritized weighted aggregation operators for aggregating trapezoid fuzzy linguistic information. The properties of the new aggregation operators are studied in detail. Furthermore, based on the proposed operators, some approaches to deal with multiple attribute group decision-making problems under trapezoid fuzzy linguistic environments are developed. Finally, a practical example is provided to illustrate the multiple attribute group decision-making process.

On intuitionistic fuzzy entropy of order-α

Using the idea of Renyis entropy, intuitionistic fuzzy entropy of order-α is proposed in the setting of intuitionistic fuzzy sets theory. This measure is a generalized version of fuzzy entropy of order-α proposed by Bhandari and Pal and intuitionistic fuzzy entropy defined by Vlachos and Sergiadis. Our study of the four essential and some other properties of the proposed measure clearly establishes the validity of the measure as intuitionistic fuzzy entropy. Finally, a numerical example is given to show that the proposed entropy measure for intuitionistic fuzzy set is reasonable by comparing it with other existing entropies.

Advanced free-form deformation and Kullback–Lieblier divergence measure for digital elevation model registration

Registration is the process of transforming various images of the same object, place, etc. to confer one coordinate system, be they multimodal, multi-temporal of the same place or images of different places having certain common characteristics. Many methods have been tried for image registration; however, only a handful of methods may be said to work for digital elevation model (DEM) registration. The motivation of the present work is to perform a robust and efficient algorithm to perform Elevation model image registration. In this paper, we present a new approach for DEM registration, particularly for multimodal or multi-temporal DEMs. For time efficient, robust and precise registration of DEMs, a novel idea has been proposed using rational DMS-spline volumes (Xu et al. in J Comput Sci Technol 23(5):862–873, 1996; McDonnell and Qin in J Gr Tools 12(3):24–41, 2007) (the term DMS is acronym for the three authors, namely Dahmen et al. in Math Comput 59(199):97–115, 1997). This is based on the usage of arbitrary topology lattices (MacCracken and Joy in 23rd annual conference on computer graphics and interactive techniques proceedings. ACM-SIGGRAPH, pp 181–188, 1996; Feng et al. in Vis Comput Int Comput Gr 22(1):28–42, 2005). Using free-form deformation has lead to the usage of global and local transformations for registration of candidate image domain to reference image domain. Also, a similarity measure based on Kullback–Leibler divergence (KLD) has been used for measuring robustness of the method so proposed. The task of registration is achieved by minimizing the cost function using rational DMS-spline functions for local registration. After experimentations, the results show that the registration process, of registration of candidate DEM to reference DEM, could be completed successfully. Comparison of similarity measurement methods such as mutual information, correlation coefficient and peak signal-to-noise ratio with that of KLD-based has been performed. Comparative study with existing works suggests that the presented scheme is better, when compared with respect to above mentioned parameters.

R-norm entropy on intuitionistic fuzzy sets

Fuzzy sets have led to study of vague phenomena. Generalizations of fuzzy sets have led to deeper analysis of these types of studies. The problem that then arises is to finding quantitative measures for vagueness and other features of these phenomena. In the present paper, based on the concept of R-norm fuzzy entropy, an R-norm intuitionistic fuzzy entropy measure is proposed in the setting of intuitionistic fuzzy set theory. This measure is a generalized version of R-norm fuzzy entropy proposed by Hooda in 2004. Some properties of this measure are proved. Finally, a numerical example is given to show that the proposed entropy measure for intuitionistic fuzzy set is reasonable by comparing it with other existing intuitionistic fuzzy entropy measures.

Upper Bound on Correcting Partial Random Errors

Abstract Since coding has become a basic tool for practically all communication/electronic devices, it is important to carefully study the error patterns that actually occur. This allows correction of only partial errors rather than those which have been studied using Hamming distance, in non-binary cases. The paper considers a class of distances, SK-distances, in terms of which partial errors can be defined. Examining the sufficient condition for the existence of a parity check matrix for a given number of parity-checks, the paper contains an upper bound on the number of parity check digits for linear codes with property that corrects all partial random errors of an (n, k ) code with minimum SK-distance at least d. The result generalizes the rather widely used Varshamov-Gilbert bound, which follows from it as a particular case.

Codes Correcting Limited Patterns of Random Errors Using S-K Metric

Abstract Coding is essential in all communications and in all multi-operation devices, and errors do occur. For error control, the method in vogue is to use code words with redundant digits. The number of redundant digits is determined based on two things − the number of messages and the kind of errors that need to be controlled. For efficient coding the redundant digits have to be kept to the minimum. In this paper we introduce the idea of limited error patters while using the code alphabet {0,1, 2,..., 1},mod , q Z = q − q when q > 3. We define limitations of the errors in a position by substitution of the character there by a specified number of other characters, rather than by any other character. This is not possible through Hamming approach, because there a character in an error could be substituted by any other of the q-1 characters. The firm mathematical base is provided by use of a metric from the class of S-K metrics, Hamming metric being one of these. The paper gives upper bounds on the codeword lengths for various kinds of “random limited error patterns”. Examples and discussion bring out the tremendous improvement and generalization of Rao Hamming bound.

A Novel Non-rigid Free-form Deformation for Consistent Registration of Digital Elevation Models

The paper presents a novel non-rigid transformation for Digital Elevation Model (DEM) registration. This is based on symmetric diffeomorphic transformation which is inverse consistent as well. Registration allows for spatial alignments between the image pairs – reference and candidate images - under consideration and is necessary so that useful information can be integrated for further tasks. In general, DEM registrations task is to present a suitable transformation such that the candidate DEM becomes aligned to the reference DEM and that both may then be presented together. We have used modified B-spline based free-form deformation for non-rigid transformation using constrained parameters for making our model diffeomorphic and inverse consistent. It uses iterative learning for the formation of deformation model as per the proposed constraints. It has been able to map small scale as well as large scale deformations using the proposed technique and has been experimented with images having various terrain types. The proposed framework for the registration of DEMs has proven to be successful and gives favorable results when compared to existing techniques as well. The algorithm proposed in this literature has been tested for multimodal, multi-view, multi-temporal and multi-resolution images of the same area, and images having overlapping areas and also for existence of small holes and missing data. It has been extended to include not only a consitent single-reference-multiple-candidate registrations but also to find candidate DEMs for registration if not already provided.

Generalized Krawtchouk polynomials and the complete weight enumerator of the dual code

Abstract A well-known result in Coding Theory is that using the complete weight enumerator of a code, the complete weight enumerator of the dual code can be obtained. In this article, it is established that an associated matrix of coefficients is a generalized Hadamard matrix. The complete weight enumerator of the dual code can be expressed in terms of generalized Krawtchouk polynomials. Orthogonality conditions and recurrence relations satisfied by these polynomials are presented.