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Symmetry, Volume 9, Issue 11 (November 2017)

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Cover Story (view full-size image) CPT and Lorentz symmetry are generally assumed to hold exactly in nature. The possibility of their [...] Read more.
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Open AccessArticle Analytical Treatment of Higher-Order Graphs: A Path Ordinal Method for Solving Graphs
Symmetry 2017, 9(11), 288; https://doi.org/10.3390/sym9110288
Received: 13 October 2017 / Revised: 17 November 2017 / Accepted: 17 November 2017 / Published: 22 November 2017
Cited by 1 | Viewed by 923 | PDF Full-text (3286 KB) | HTML Full-text | XML Full-text
Abstract
Analytical treatment of the composition of higher-order graphs representing linear relations between variables is developed. A path formalism to deal with problems in graph theory is introduced. It is shown how paths in the composed graph representing individual contributions to variables relation can
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Analytical treatment of the composition of higher-order graphs representing linear relations between variables is developed. A path formalism to deal with problems in graph theory is introduced. It is shown how paths in the composed graph representing individual contributions to variables relation can be enumerated and represented by ordinals. The method allows for one to extract partial information and gives an alternative to classical graph approach. Full article
(This article belongs to the Special Issue Graph Theory)
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Open AccessArticle The Choice of Gadolinium-Based Contrast Agents: A Radiologist’s Responsibility between Pharmaceutical Equivalence and Bioethical Issues
Symmetry 2017, 9(11), 287; https://doi.org/10.3390/sym9110287
Received: 24 October 2017 / Revised: 14 November 2017 / Accepted: 17 November 2017 / Published: 22 November 2017
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Abstract
Contrast Agents (CA) are among the most commonly prescribed drugs worldwide, and are used, with a variety of techniques, to increase and intensify the differences between body tissues and to help radiologist make diagnoses in a fast and precise way. In recent decades,
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Contrast Agents (CA) are among the most commonly prescribed drugs worldwide, and are used, with a variety of techniques, to increase and intensify the differences between body tissues and to help radiologist make diagnoses in a fast and precise way. In recent decades, advancements in research have resulted in significant improvements in their composition, and have made them safer and better-tolerated by patients; this notwithstanding, although the currently available CA are generally considered to be safe, their use is not completely without risk. The use of CA faces the radiologist with economic considerations, bioethical dilemmas, and possible profiles of professional responsibility. In fact, to achieve the best results in diagnostic imaging, radiologists have to focus on making an appropriate choice of CA, in consideration of efficacy, safety and appropriateness. Moreover, besides by cost/benefit models widely introduced in health management, radiologists are also influenced by their responsibility of appropriate use for the various diagnostic tests and, finally, the choice of best CA to utilise for each individual patient. Thus, the dilemma of choosing between the best and the most cost-effective tests and procedures is occurring more frequently every day. Different variables, such as the patient, examinations, and technology available, can affect the choice of CA in terms of obtaining the highest diagnostic quality, minimum impact on higher-risk patients, and optimisation of used volumes and injection flows. Full article
(This article belongs to the Special Issue Medical Imaging and Imaging Modalities)
Open AccessArticle Fishmeal Supplier Evaluation and Selection for Aquaculture Enterprise Sustainability with a Fuzzy MCDM Approach
Symmetry 2017, 9(11), 286; https://doi.org/10.3390/sym9110286
Received: 29 October 2017 / Revised: 11 November 2017 / Accepted: 14 November 2017 / Published: 21 November 2017
Cited by 3 | Viewed by 851 | PDF Full-text (1335 KB) | HTML Full-text | XML Full-text
Abstract
In the aquaculture industry, feed that is of poor quality or nutritionally imbalanced can cause problems including low weight, poor growth, poor palatability, and increased mortality, all of which can induce a decrease in aquaculture production. Fishmeal is considered a better source of
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In the aquaculture industry, feed that is of poor quality or nutritionally imbalanced can cause problems including low weight, poor growth, poor palatability, and increased mortality, all of which can induce a decrease in aquaculture production. Fishmeal is considered a better source of protein and its addition as an ingredient in the aquafeed makes aquatic animals grow fast and healthy. This means that fishmeal is the most important feed ingredient in aquafeed for the aquaculture industry. For the aquaculture industry in Taiwan, about 144,000 ton/USD $203,245,000 of fishmeal was imported, mostly from Peru, in 2016. Therefore, the evaluation and selection of fishmeal suppliers is a very important part of the decision-making process for a Taiwanese aquaculture enterprise. This study constructed a multiple criteria decision-making evaluation model for the selection of fishmeal suppliers using the VlseKriterijumska Optimizacija I Kompromisno Resenje (VIKOR) approach based on the weights obtained with the entropy method in a fuzzy decision-making environment. This hybrid approach could effectively and conveniently measure the comprehensive performance of the main Peruvian fishmeal suppliers for practical applications. In addition, the results and processes described herein function as a good reference for an aquaculture enterprise in making decisions when purchasing fishmeal. Full article
(This article belongs to the Special Issue Fuzzy Techniques for Decision Making) Printed Edition available
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Open AccessArticle Optimized Charging Scheduling with Single Mobile Charger for Wireless Rechargeable Sensor Networks
Symmetry 2017, 9(11), 285; https://doi.org/10.3390/sym9110285
Received: 7 October 2017 / Revised: 15 November 2017 / Accepted: 16 November 2017 / Published: 21 November 2017
Cited by 1 | Viewed by 954 | PDF Full-text (4215 KB) | HTML Full-text | XML Full-text
Abstract
Due to the rapid development of wireless charging technology, the recharging issue in wireless rechargeable sensor network (WRSN) has been a popular research problem in the past few years. The weakness of previous work is that charging route planning is not reasonable. In
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Due to the rapid development of wireless charging technology, the recharging issue in wireless rechargeable sensor network (WRSN) has been a popular research problem in the past few years. The weakness of previous work is that charging route planning is not reasonable. In this work, a dynamic optimal scheduling scheme aiming to maximize the vacation time ratio of a single mobile changer for WRSN is proposed. In the proposed scheme, the wireless sensor network is divided into several sub-networks according to the initial topology of deployed sensor networks. After comprehensive analysis of energy states, working state and constraints for different sensor nodes in WRSN, we transform the optimized charging path problem of the whole network into the local optimization problem of the sub networks. The optimized charging path with respect to dynamic network topology in each sub-network is obtained by solving an optimization problem, and the lifetime of the deployed wireless sensor network can be prolonged. Simulation results show that the proposed scheme has good and reliable performance for a small wireless rechargeable sensor network. Full article
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Open AccessArticle Granular Structure of Type-2 Fuzzy Rough Sets over Two Universes
Symmetry 2017, 9(11), 284; https://doi.org/10.3390/sym9110284
Received: 24 October 2017 / Revised: 14 November 2017 / Accepted: 15 November 2017 / Published: 21 November 2017
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Abstract
Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory,
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Granular structure plays a very important role in the model construction, theoretical analysis and algorithm design of a granular computing method. The granular structures of classical rough sets and fuzzy rough sets have been proven to be clear. In classical rough set theory, equivalence classes are basic granules, and the lower and upper approximations of a set can be computed by those basic granules. In the theory of fuzzy rough set, granular fuzzy sets can be used to describe the lower and upper approximations of a fuzzy set. This paper discusses the granular structure of type-2 fuzzy rough sets over two universes. Definitions of type-2 fuzzy rough sets over two universes are given based on a wavy-slice representation of type-2 fuzzy sets. Two granular type-2 fuzzy sets are deduced and then proven to be basic granules of type-2 fuzzy rough sets over two universes. Then, the properties of lower and upper approximation operators and these two granular type-2 fuzzy sets are investigated. At last, several examples are given to show the applications of type-2 fuzzy rough sets over two universes. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle Hierarchical Meta-Learning in Time Series Forecasting for Improved Interference-Less Machine Learning
Symmetry 2017, 9(11), 283; https://doi.org/10.3390/sym9110283
Received: 24 October 2017 / Revised: 1 November 2017 / Accepted: 1 November 2017 / Published: 18 November 2017
Cited by 1 | Viewed by 1579 | PDF Full-text (3809 KB) | HTML Full-text | XML Full-text
Abstract
The importance of an interference-less machine learning scheme in time series prediction is crucial, as an oversight can have a negative cumulative effect, especially when predicting many steps ahead of the currently available data. The on-going research on noise elimination in time series
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The importance of an interference-less machine learning scheme in time series prediction is crucial, as an oversight can have a negative cumulative effect, especially when predicting many steps ahead of the currently available data. The on-going research on noise elimination in time series forecasting has led to a successful approach of decomposing the data sequence into component trends to identify noise-inducing information. The empirical mode decomposition method separates the time series/signal into a set of intrinsic mode functions ranging from high to low frequencies, which can be summed up to reconstruct the original data. The usual assumption that random noises are only contained in the high-frequency component has been shown not to be the case, as observed in our previous findings. The results from that experiment reveal that noise can be present in a low frequency component, and this motivates the newly-proposed algorithm. Additionally, to prevent the erosion of periodic trends and patterns within the series, we perform the learning of local and global trends separately in a hierarchical manner which succeeds in detecting and eliminating short/long term noise. The algorithm is tested on four datasets from financial market data and physical science data. The simulation results are compared with the conventional and state-of-the-art approaches for time series machine learning, such as the non-linear autoregressive neural network and the long short-term memory recurrent neural network, respectively. Statistically significant performance gains are recorded when the meta-learning algorithm for noise reduction is used in combination with these artificial neural networks. For time series data which cannot be decomposed into meaningful trends, applying the moving average method to create meta-information for guiding the learning process is still better than the traditional approach. Therefore, this new approach is applicable to the forecasting of time series with a low signal to noise ratio, with a potential to scale adequately in a multi-cluster system due to the parallelized nature of the algorithm. Full article
(This article belongs to the Special Issue Novel Machine Learning Approaches for Intelligent Big Data)
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Open AccessArticle An Appraisal Model Based on a Synthetic Feature Selection Approach for Students’ Academic Achievement
Symmetry 2017, 9(11), 282; https://doi.org/10.3390/sym9110282
Received: 14 October 2017 / Revised: 11 November 2017 / Accepted: 13 November 2017 / Published: 18 November 2017
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Abstract
Obtaining necessary information (and even extracting hidden messages) from existing big data, and then transforming them into knowledge, is an important skill. Data mining technology has received increased attention in various fields in recent years because it can be used to find historical
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Obtaining necessary information (and even extracting hidden messages) from existing big data, and then transforming them into knowledge, is an important skill. Data mining technology has received increased attention in various fields in recent years because it can be used to find historical patterns and employ machine learning to aid in decision-making. When we find unexpected rules or patterns from the data, they are likely to be of high value. This paper proposes a synthetic feature selection approach (SFSA), which is combined with a support vector machine (SVM) to extract patterns and find the key features that influence students’ academic achievement. For verifying the proposed model, two databases, namely, “Student Profile” and “Tutorship Record”, were collected from an elementary school in Taiwan, and were concatenated into an integrated dataset based on students’ names as a research dataset. The results indicate the following: (1) the accuracy of the proposed feature selection approach is better than that of the Minimum-Redundancy-Maximum-Relevance (mRMR) approach; (2) the proposed model is better than the listing methods when the six least influential features have been deleted; and (3) the proposed model can enhance the accuracy and facilitate the interpretation of the pattern from a hybrid-type dataset of students’ academic achievement. Full article
(This article belongs to the Special Issue Information Technology and Its Applications)
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Open AccessArticle Time Series Seasonal Analysis Based on Fuzzy Transforms
Symmetry 2017, 9(11), 281; https://doi.org/10.3390/sym9110281
Received: 27 September 2017 / Revised: 23 October 2017 / Accepted: 11 November 2017 / Published: 17 November 2017
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Abstract
We define a new seasonal forecasting method based on fuzzy transforms. We use the best interpolating polynomial for extracting the trend of the time series and generate the inverse fuzzy transform on each seasonal subset of the universe of discourse for predicting the
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We define a new seasonal forecasting method based on fuzzy transforms. We use the best interpolating polynomial for extracting the trend of the time series and generate the inverse fuzzy transform on each seasonal subset of the universe of discourse for predicting the value of an assigned output. In the first example, we use the daily weather dataset of the municipality of Naples (Italy) starting from data collected from 2003 to 2015 making predictions on mean temperature, max temperature and min temperature, all considered daily. In the second example, we use the daily mean temperature measured at the weather station “Chiavari Caperana” in the Liguria Italian Region. We compare the results with our method, the average seasonal variation, Auto Regressive Integrated Moving Average (ARIMA) and the usual fuzzy transforms concluding that the best results are obtained under our approach in both examples. In addition, the comparison results show that, for seasonal time series that have no consistent irregular variations, the performance obtained with our method is comparable with the ones obtained using Support Vector Machine- and Artificial Neural Networks-based models. Full article
(This article belongs to the Special Issue Symmetry in Fuzzy Sets and Systems)
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Open AccessArticle How Objective a Neutral Word Is? A Neutrosophic Approach for the Objectivity Degrees of Neutral Words
Symmetry 2017, 9(11), 280; https://doi.org/10.3390/sym9110280
Received: 9 October 2017 / Revised: 12 November 2017 / Accepted: 15 November 2017 / Published: 17 November 2017
Cited by 3 | Viewed by 1174 | PDF Full-text (396 KB) | HTML Full-text | XML Full-text
Abstract
In the latest studies concerning the sentiment polarity of words, the authors mostly consider the positive and negative constructions, without paying too much attention to the neutral words, which can have, in fact, significant sentiment degrees. More precisely, not all the neutral words
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In the latest studies concerning the sentiment polarity of words, the authors mostly consider the positive and negative constructions, without paying too much attention to the neutral words, which can have, in fact, significant sentiment degrees. More precisely, not all the neutral words have zero positivity or negativity scores, some of them having quite important nonzero scores for these polarities. At this moment, in the literature, a word is considered neutral if its positive and negative scores are equal, which implies two possibilities: (1) zero positive and negative scores; (2) nonzero, but equal positive and negative scores. It is obvious that these cases represent two different categories of neutral words that must be treated separately by a sentiment analysis task. In this paper, we present a comprehensive study about the neutral words applied to English as is developed with the aid of SentiWordNet 3.0: the publicly available lexical resource for opinion mining. We designed our study in order to provide an accurate classification of the so-called “neutral words” described in terms of sentiment scores and using measures from neutrosophy theory. The intended scope is to fill the gap concerning the neutrality aspect by giving precise measurements for the words’ objectivity. Full article
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Open AccessArticle Novel Integrated Multi-Criteria Model for Supplier Selection: Case Study Construction Company
Symmetry 2017, 9(11), 279; https://doi.org/10.3390/sym9110279
Received: 12 October 2017 / Revised: 4 November 2017 / Accepted: 9 November 2017 / Published: 17 November 2017
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Abstract
Supply chain presents a very complex field involving a large number of participants. The aim of the complete supply chain is finding an optimum from the aspect of all participants, which is a rather complex task. In order to ensure optimum satisfaction for
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Supply chain presents a very complex field involving a large number of participants. The aim of the complete supply chain is finding an optimum from the aspect of all participants, which is a rather complex task. In order to ensure optimum satisfaction for all participants, it is necessary that the beginning phase consists of correct evaluations and supplier selection. In this study, the supplier selection was performed in the construction company, on the basis of a new approach in the field of multi-criteria model. Weight coefficients were obtained by DEMATEL (Decision Making Trial and Evaluation Laboratory) method, based on the rough numbers. Evaluation and the supplier selection were made on the basis of a new Rough EDAS (Evaluation based on Distance from Average Solution) method, which presents one of the latest methods in this field. In order to determine the stability of the model and the applicability of the proposed Rough EDAS method, an extension of the COPRAS and MULTIMOORA method by rough numbers was also performed in this study, and the findings of the comparative analysis were presented. Besides the new approaches based on the extension by rough numbers, the results are also compared with the Rough MABAC (MultiAttributive Border Approximation area Comparison) and Rough MAIRCA (MultiAttributive Ideal-Real Comparative Analysis). In addition, in the sensitivity analysis, 18 different scenarios were formed, the ones in which criteria change their original values. At the end of the sensitivity analysis, SCC (Spearman Correlation Coefficient) of the obtained ranges was carried out, confirming the applicability of the proposed approaches. Full article
(This article belongs to the Special Issue Civil Engineering and Symmetry)
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Open AccessArticle Conflicting Information Fusion Based on an Improved DS Combination Method
Symmetry 2017, 9(11), 278; https://doi.org/10.3390/sym9110278
Received: 30 October 2017 / Revised: 14 November 2017 / Accepted: 15 November 2017 / Published: 16 November 2017
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Abstract
An effective and reliable fusion method for conflicting information is proposed in this paper. Compared with a single-sensor system, a multi-sensor fusion system can comprehensively combine the redundancy and complementarity of multi-sensor information to obtain better system performance. Hence, the multi-sensor fusion system
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An effective and reliable fusion method for conflicting information is proposed in this paper. Compared with a single-sensor system, a multi-sensor fusion system can comprehensively combine the redundancy and complementarity of multi-sensor information to obtain better system performance. Hence, the multi-sensor fusion system has become one of the research hotspots. However, due to lack knowledge about the measurement environment and limited sensor accuracy, the multi-sensor system inevitably appears to have imperfect, uncertain and inconsistent information. To solve the problem, we introduce one powerful uncertainty reasoning method: Dempster–Shafer theory (DS theory). With convincing measurement and a forceful combination of uncertain information, DS theory is widely applied in various fields, like decision-making, expert systems, target tracking, monitoring systems, etc. Nevertheless, DS theory will produce counter-intuitive fusion results when the pieces of evidence are highly conflicting. To address this issue, we raise an improved DS combination method for conflicting information fusion in this paper. First of all, the modified Minkowski distance function and the betting-commitment distance function are separately employed to revise potentially conflicting pieces of evidence. The procedure availably solves the conflicting situations caused by unreliable and imprecise evidence sources, which enhances the consistency among pieces of evidence. Then, based on two revised pieces of evidence, a conflicting redistribution strategy based on locally conflicting analyses is put forward. The approach dexterously combines two revised pieces of evidence to avoid conflicting situations caused by compulsive normalization, which further improves the accuracy and convergence speed of the multi-sensor fusion system. Finally, two experimental analyses with consistent information and conflicting information reveal the remarkable effectiveness and priority of the proposed algorithm for the multi-sensor fusion system. Consequently, this paper has certain value for the multi-sensor fusion system. Full article
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Open AccessArticle Malignant and Benign Mass Segmentation in Mammograms Using Active Contour Methods
Symmetry 2017, 9(11), 277; https://doi.org/10.3390/sym9110277
Received: 1 October 2017 / Revised: 10 November 2017 / Accepted: 12 November 2017 / Published: 16 November 2017
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Abstract
The correct segmentation of tumours can simplify formulate the diagnostic hypothesis, particularly in cases of irregular shapes, with fuzzy margins or spicules growing into the surrounding tissue, which are more likely to be malignant. In this study, the following active contour methods were
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The correct segmentation of tumours can simplify formulate the diagnostic hypothesis, particularly in cases of irregular shapes, with fuzzy margins or spicules growing into the surrounding tissue, which are more likely to be malignant. In this study, the following active contour methods were used to segment the masses: an edge–based active contour model using an inflation/deflation force with a damping coefficient (EM), a geometric active contour model (GAC) and an active contour without edges (ACWE). The preprocessing techniques presented in this publication are to reduce noise and at the same time amplify uniform areas of images in order to improve segmentation results. In addition, the use of image sampling by bicubic interpolation was tested to shorten the evolution time of active contour methods. The experiments used a test set composed of 100 cases taken from two publicly available databases: Digital Database for Screening Mammography (DDSM) and Mammographic Image Analysis Society (MIAS) database. The qualitative assessment concerned the ability to formulate an adequate diagnostic hypothesis and, for the individual methods (malignant and benign cases together), it amounted to at least: 81% (EM), 76% (GAC), and 69% (ACWE). The quantitative test consisted of measuring the following indexes: overlap value (OV) and extra fraction (EF). The OV of the segmentation for malignant and benign cases had the following average values: 0.81 ∓ 0.10 (EM), 0.79 ∓ 0.09 (GAC), 0.76 ∓ 0.18 (ACWE). The average values of the EF index, in turn, amounted to: 0.07 ∓ 0.06 (EM), 0.07 ∓ 0.05 (GAC) 0.34 ∓ 0.32 (ACWE). The qualitative and quantitative results obtained are the best for EM and are comparable or better than for other methods presented in the literature. Full article
(This article belongs to the Special Issue Advances in Medical Image Segmentation)
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Open AccessArticle Retinal Vessel Segmentation via Structure Tensor Coloring and Anisotropy Enhancement
Symmetry 2017, 9(11), 276; https://doi.org/10.3390/sym9110276
Received: 23 October 2017 / Revised: 9 November 2017 / Accepted: 10 November 2017 / Published: 14 November 2017
Cited by 3 | Viewed by 1505 | PDF Full-text (7025 KB) | HTML Full-text | XML Full-text
Abstract
Retinal vessel segmentation is one of the preliminary tasks for developing diagnosis software systems related to various retinal diseases. In this study, a fully automated vessel segmentation system is proposed. Firstly, the vessels are enhanced using a Frangi Filter. Afterwards, Structure Tensor is
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Retinal vessel segmentation is one of the preliminary tasks for developing diagnosis software systems related to various retinal diseases. In this study, a fully automated vessel segmentation system is proposed. Firstly, the vessels are enhanced using a Frangi Filter. Afterwards, Structure Tensor is applied to the response of the Frangi Filter and a 4-D tensor field is obtained. After decomposing the Eigenvalues of the tensor field, the anisotropy between the principal Eigenvalues are enhanced exponentially. Furthermore, this 4-D tensor field is converted to the 3-D space which is composed of energy, anisotropy and orientation and then a Contrast Limited Adaptive Histogram Equalization algorithm is applied to the energy space. Later, the obtained energy space is multiplied by the enhanced mean surface curvature of itself and the modified 3-D space is converted back to the 4-D tensor field. Lastly, the vessel segmentation is performed by using Otsu algorithm and tensor coloring method which is inspired by the ellipsoid tensor visualization technique. Finally, some post-processing techniques are applied to the segmentation result. In this study, the proposed method achieved mean sensitivity of 0.8123, 0.8126, 0.7246 and mean specificity of 0.9342, 0.9442, 0.9453 as well as mean accuracy of 0.9183, 0.9442, 0.9236 for DRIVE, STARE and CHASE_DB1 datasets, respectively. The mean execution time of this study is 6.104, 6.4525 and 18.8370 s for the aforementioned three datasets respectively. Full article
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Open AccessArticle Neutrosophic Duplet Semi-Group and Cancellable Neutrosophic Triplet Groups
Symmetry 2017, 9(11), 275; https://doi.org/10.3390/sym9110275
Received: 14 October 2017 / Revised: 9 November 2017 / Accepted: 10 November 2017 / Published: 14 November 2017
Cited by 27 | Viewed by 1080 | PDF Full-text (1298 KB) | HTML Full-text | XML Full-text
Abstract
The notions of the neutrosophic triplet and neutrosophic duplet were introduced by Florentin Smarandache. From the existing research results, the neutrosophic triplets and neutrosophic duplets are completely different from the classical algebra structures. In this paper, we further study neutrosophic duplet sets, neutrosophic
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The notions of the neutrosophic triplet and neutrosophic duplet were introduced by Florentin Smarandache. From the existing research results, the neutrosophic triplets and neutrosophic duplets are completely different from the classical algebra structures. In this paper, we further study neutrosophic duplet sets, neutrosophic duplet semi-groups, and cancellable neutrosophic triplet groups. First, some new properties of neutrosophic duplet semi-groups are funded, and the following important result is proven: there is no finite neutrosophic duplet semi-group. Second, the new concepts of weak neutrosophic duplet, weak neutrosophic duplet set, and weak neutrosophic duplet semi-group are introduced, some examples are given by using the mathematical software MATLAB (MathWorks, Inc., Natick, MA, USA), and the characterizations of cancellable weak neutrosophic duplet semi-groups are established. Third, the cancellable neutrosophic triplet groups are investigated, and the following important result is proven: the concept of cancellable neutrosophic triplet group and group coincide. Finally, the neutrosophic triplets and weak neutrosophic duplets in BCI-algebras are discussed. Full article
(This article belongs to the Special Issue Neutrosophic Theories Applied in Engineering)
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Open AccessFeature PaperArticle Operations on Oriented Maps
Symmetry 2017, 9(11), 274; https://doi.org/10.3390/sym9110274
Received: 31 July 2017 / Revised: 7 November 2017 / Accepted: 11 November 2017 / Published: 14 November 2017
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Abstract
A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as
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A map on a closed surface is a two-cell embedding of a finite connected graph. Maps on surfaces are conveniently described by certain trivalent graphs, known as flag graphs. Flag graphs themselves may be considered as maps embedded in the same surface as the original graph. The flag graph is the underlying graph of the dual of the barycentric subdivision of the original map. Certain operations on maps can be defined by appropriate operations on flag graphs. Orientable surfaces may be given consistent orientations, and oriented maps can be described by a generating pair consisting of a permutation and an involution on the set of arcs (or darts) defining a partially directed arc graph. In this paper we describe how certain operations on maps can be described directly on oriented maps via arc graphs. Full article
(This article belongs to the Special Issue Polyhedral Structures) Printed Edition available
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Open AccessArticle A Hybrid Fuzzy DEA/AHP Methodology for Ranking Units in a Fuzzy Environment
Symmetry 2017, 9(11), 273; https://doi.org/10.3390/sym9110273
Received: 2 October 2017 / Revised: 2 November 2017 / Accepted: 6 November 2017 / Published: 14 November 2017
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Abstract
In this paper, a novel approach combining fuzzy data envelopment analysis (DEA) and the analytical hierarchical process (AHP) is proposed to rank units with multiple fuzzy criteria. The hybrid fuzzy DEA/AHP approach derives the AHP pairwise comparisons by fuzzy DEA and utilizes AHP
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In this paper, a novel approach combining fuzzy data envelopment analysis (DEA) and the analytical hierarchical process (AHP) is proposed to rank units with multiple fuzzy criteria. The hybrid fuzzy DEA/AHP approach derives the AHP pairwise comparisons by fuzzy DEA and utilizes AHP to fully rank units. It shows that the proposed approach generates a logical ranking of units that has perfect compatibility with fuzzy DEA ranking and there is no any form of subjective analysis engaged within the methodology. A study on the facility layout design in manufacturing systems is provided to illustrate the superiority of the proposed approach and show the compatibility between the proposed approach and fuzzy DEA ranking. Full article
(This article belongs to the Special Issue Fuzzy Sets Theory and Its Applications)
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Open AccessArticle Function-Oriented Networking and On-Demand Routing System in Network Using Ant Colony Optimization Algorithm
Symmetry 2017, 9(11), 272; https://doi.org/10.3390/sym9110272
Received: 10 October 2017 / Revised: 2 November 2017 / Accepted: 3 November 2017 / Published: 10 November 2017
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Abstract
In this paper, we proposed and developed Function-Oriented Networking (FON), a platform for network users. It has a different philosophy as opposed to technologies for network managers of Software-Defined Networking technology, OpenFlow. It is a technology that can immediately reflect the demands of
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In this paper, we proposed and developed Function-Oriented Networking (FON), a platform for network users. It has a different philosophy as opposed to technologies for network managers of Software-Defined Networking technology, OpenFlow. It is a technology that can immediately reflect the demands of the network users in the network, unlike the existing OpenFlow and Network Functions Virtualization (NFV), which do not reflect directly the needs of the network users. It allows the network user to determine the policy of the direct network, so it can be applied more precisely than the policy applied by the network manager. This is expected to increase the satisfaction of the service users when the network users try to provide new services. We developed FON function that performs on-demand routing for Low-Delay Required service. We analyzed the characteristics of the Ant Colony Optimization (ACO) algorithm and found that the algorithm is suitable for low-delay required services. It was also the first in the world to implement the routing software using ACO Algorithm in the real Ethernet network. In order to improve the routing performance, several algorithms of the ACO Algorithm have been developed to enable faster path search-routing and path recovery. The relationship between the network performance index and the ACO routing parameters is derived, and the results are compared and analyzed. Through this, it was possible to develop the ACO algorithm. Full article
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Open AccessArticle A New Multi-Attribute Decision-Making Method Based on m-Polar Fuzzy Soft Rough Sets
Symmetry 2017, 9(11), 271; https://doi.org/10.3390/sym9110271
Received: 19 October 2017 / Revised: 1 November 2017 / Accepted: 1 November 2017 / Published: 10 November 2017
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Abstract
We introduce notions of soft rough m-polar fuzzy sets and m-polar fuzzy soft rough sets as novel hybrid models for soft computing, and investigate some of their fundamental properties. We discuss the relationship between m-polar fuzzy soft rough approximation operators
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We introduce notions of soft rough m-polar fuzzy sets and m-polar fuzzy soft rough sets as novel hybrid models for soft computing, and investigate some of their fundamental properties. We discuss the relationship between m-polar fuzzy soft rough approximation operators and crisp soft rough approximation operators. We also present applications of m-polar fuzzy soft rough sets to decision-making. Full article
(This article belongs to the Special Issue Fuzzy Techniques for Decision Making) Printed Edition available
Open AccessArticle Multi-Attribute Decision-Making Based on Prioritized Aggregation Operator under Hesitant Intuitionistic Fuzzy Linguistic Environment
Symmetry 2017, 9(11), 270; https://doi.org/10.3390/sym9110270
Received: 26 October 2017 / Revised: 3 November 2017 / Accepted: 5 November 2017 / Published: 9 November 2017
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Abstract
A hesitant intuitionistic fuzzy linguistic set (HIFLS) that integrates both qualitative and quantitative evaluations is an extension of the linguistic set, intuitionistic fuzzy set (IFS), hesitant fuzzy set (HFS) and hesitant intuitionistic fuzzy set (HIFS). It can describe the qualitative evaluation information given
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A hesitant intuitionistic fuzzy linguistic set (HIFLS) that integrates both qualitative and quantitative evaluations is an extension of the linguistic set, intuitionistic fuzzy set (IFS), hesitant fuzzy set (HFS) and hesitant intuitionistic fuzzy set (HIFS). It can describe the qualitative evaluation information given by the decision-makers (DMs) and reflect their uncertainty. In this article, we defined some new operational laws and comparative method for HIFLSs. Then, based on these operations, we propose two prioritized aggregation (PA) operators for HIFLSs: prioritized weighted averaging operator for HIFLSs (HIFLPWA) and prioritized weighted geometric operator for HIFLSs (HIFLPWG). Based on these aggregation operators, an approach for multi-attribute decision-making (MADM) is developed under the environment of HIFLSs. Finally, a practical example is given to show the practicality and effectiveness of the developed approach by comparing with the other representative methods. Full article
Open AccessArticle Fast and Efficient Data Forwarding Scheme for Tracking Mobile Targets in Sensor Networks
Symmetry 2017, 9(11), 269; https://doi.org/10.3390/sym9110269
Received: 23 October 2017 / Revised: 5 November 2017 / Accepted: 6 November 2017 / Published: 9 November 2017
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Abstract
Transferring emergent target tracking data to sinks is a major challenge in the Industrial Internet of Things (IIoT), because inefficient data transmission can cause significant personnel and property loss. For tracking a constantly moving mobile target, sensing data should be delivered to sinks
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Transferring emergent target tracking data to sinks is a major challenge in the Industrial Internet of Things (IIoT), because inefficient data transmission can cause significant personnel and property loss. For tracking a constantly moving mobile target, sensing data should be delivered to sinks continuously and quickly. Although there is some related research, the end to end tracking delay is still unsatisfying. In this paper, we propose a Fast and Efficient Data Forwarding (FEDF) scheme for tracking mobile targets in sensor networks to reduce tracking delay and maintain a long lifetime. Innovations of the FEDF scheme that differ from traditional scheme are as follows: firstly, we propose a scheme to transmit sensing data through a Quickly Reacted Routing (QRR) path which can reduce delay efficiently. Duty cycles of most nodes on a QRR path are set to 1, so that sleep delay of most nodes turn 0. In this way, end to end delay can be reduced significantly. Secondly, we propose a perfect method to build QRR path and optimize it, which can make QRR path work more efficiently. Target sensing data routing scheme in this paper belongs to a kind of trail-based routing scheme, so as the target moves, the routing path becomes increasingly long, reducing the working efficiency. We propose a QRR path optimization algorithm, in which the ratio of the routing path length to the optimal path is maintained at a smaller constant in the worst case. Thirdly, it has a long lifetime. In FEDF scheme duty cycles of nodes near sink in a QRR path are the same as that in traditional scheme, but duty cycles of nodes in an energy-rich area are 1. Therefore, not only is the rest energy of network fully made use of, but also the network lifetime stays relatively long. Finally, comprehensive performance analysis shows that the FEDF scheme can realize an optimal end to end delay and energy utilization at the same time, reduce end to end delay by 87.4%, improve network energy utilization by 2.65%, and ensure that network lifetime is not less than previous research. Full article
(This article belongs to the Special Issue Advances in Future Internet and Industrial Internet of Things)
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Open AccessArticle New Operations of Picture Fuzzy Relations and Fuzzy Comprehensive Evaluation
Symmetry 2017, 9(11), 268; https://doi.org/10.3390/sym9110268
Received: 7 October 2017 / Revised: 31 October 2017 / Accepted: 2 November 2017 / Published: 8 November 2017
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Abstract
In this paper, some new operations and basic properties of picture fuzzy relations are intensively studied. First, a new inclusion relation (called type-2 inclusion relation) of picture fuzzy relations is introduced, as well as the corresponding type-2 union, type-2 intersection and type-2 complement
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In this paper, some new operations and basic properties of picture fuzzy relations are intensively studied. First, a new inclusion relation (called type-2 inclusion relation) of picture fuzzy relations is introduced, as well as the corresponding type-2 union, type-2 intersection and type-2 complement operations. Second, the notions of anti-reflexive kernel, symmetric kernel, reflexive closure and symmetric closure of a picture fuzzy relation are introduced and their properties are explored. Moreover, a new method to solve picture fuzzy comprehensive evaluation problems is proposed by defining the new composition operation of picture fuzzy relations, and the picture fuzzy comprehensive evaluation model is built. Finally, an application example (about investment risk) of picture fuzzy comprehensive evaluation is given, and the effective experiment results are obtained. Full article
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Open AccessArticle Fuzzy-System-Based Detection of Pupil Center and Corneal Specular Reflection for a Driver-Gaze Tracking System Based on the Symmetrical Characteristics of Face and Facial Feature Points
Symmetry 2017, 9(11), 267; https://doi.org/10.3390/sym9110267
Received: 17 October 2017 / Revised: 31 October 2017 / Accepted: 2 November 2017 / Published: 6 November 2017
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Abstract
Recently, many studies have actively dealt with the issue of driver-gaze tracking for monitoring the forward gaze and physical condition. Driver-gaze tracking is an effective method of measuring a driver’s inattention that is one of the major causes of traffic accidents. Among many
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Recently, many studies have actively dealt with the issue of driver-gaze tracking for monitoring the forward gaze and physical condition. Driver-gaze tracking is an effective method of measuring a driver’s inattention that is one of the major causes of traffic accidents. Among many gaze-tracking methods, the corneal specular reflection (SR)-based method becomes ineffective, unlike in an indoor environment, when a driver’s head rotates, which makes SR disappear from input images or disperses SR in the lachrymal gland or eyelid, thereby increasing the gaze-tracking error. Besides, since a driver’s eyes in a vehicle environment need to be captured in a wide range covering his head rotation, the eye region is captured in a relatively low resolution compared to face-only images taken in indoor environments at the same resolution, making pupil and corneal SR difficult to detect accurately. To solve these problems, we propose a fuzzy-system-based method for detecting a driver’s pupil and corneal SR for gaze tracking in a vehicle environment. Unlike existing studies detecting pupil and corneal SR in both eyes, the method proposed in this research uses the results of a fuzzy system based on two features considering the symmetrical characteristics of face and facial feature points to determine the status of a driver’s head rotation. Based on the output of the fuzzy system, the proposed method excludes the eye region, which is very likely to have a high error rate of detection due to excessive head rotation, from the detection process of the pupil and corneal SR. Accordingly, the proposed method detects pupil and corneal SR only in the eye region that apparently has a low detection error rate, thereby achieving accurate detection. We use 20,654 images capturing 15 subjects (including subjects wearing glasses), who gaze at pre-set fifteen regions in a vehicle, to measure the detection accuracy of the pupil and corneal SR for each region and the gaze tracking accuracy. Our experimental results show that the proposed method performs better than existing methods. Full article
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Open AccessArticle The Development of Improved Incremental Models Using Local Granular Networks with Error Compensation
Symmetry 2017, 9(11), 266; https://doi.org/10.3390/sym9110266
Received: 1 October 2017 / Revised: 25 October 2017 / Accepted: 1 November 2017 / Published: 5 November 2017
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Abstract
In this paper, we use the fundamental idea of the incremental model (IM) and develop the design framework. The design method of IM is composed of two steps. In the first step, we perform a linear regression (LR) as the global model. In
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In this paper, we use the fundamental idea of the incremental model (IM) and develop the design framework. The design method of IM is composed of two steps. In the first step, we perform a linear regression (LR) as the global model. In the second step, the errors obtained by the global model are predicted by fuzzy if-then rules generated through a local linguistic model. Although the effectiveness of IM has been demonstrated in various prediction examples, we propose an improved incremental model (IIM) to deal with complex nonlinear characteristics. For this purpose, we employ adaptive neuro-fuzzy networks (ANFN) or radial basis function networks (RBFN) to create local granular networks in the design of IIM. Furthermore, we use quadratic regression (QR) as a global model, because linear relationship of LR may not hold in many settings. Numerical studies concern four datasets (automobile data, energy efficiency data, Boston housing data and computer hardware data). The experimental results demonstrate that IIM outperformed the previous models. Full article
(This article belongs to the Special Issue Symmetry in Fuzzy Sets and Systems)
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Open AccessArticle A Recourse-Based Type-2 Fuzzy Programming Method for Water Pollution Control under Uncertainty
Symmetry 2017, 9(11), 265; https://doi.org/10.3390/sym9110265
Received: 29 September 2017 / Revised: 29 October 2017 / Accepted: 31 October 2017 / Published: 4 November 2017
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Abstract
In this study, a recourse-based type-2 fuzzy programming (RTFP) method is developed for supporting water pollution control of basin systems under uncertainty. The RTFP method incorporates type-2 fuzzy programming (TFP) within a two-stage stochastic programming with recourse (TSP) framework to handle uncertainties expressed
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In this study, a recourse-based type-2 fuzzy programming (RTFP) method is developed for supporting water pollution control of basin systems under uncertainty. The RTFP method incorporates type-2 fuzzy programming (TFP) within a two-stage stochastic programming with recourse (TSP) framework to handle uncertainties expressed as type-2 fuzzy sets (i.e., a fuzzy set in which the membership function is also fuzzy) and probability distributions, as well as to reflect the trade-offs between conflicting economic benefits and penalties due to violated policies. The RTFP method is then applied to a real case of water pollution control in the Heshui River Basin (a rural area of China), where chemical oxygen demand (COD), total nitrogen (TN), total phosphorus (TP), and soil loss are selected as major indicators to identify the water pollution control strategies. Solutions of optimal production plans of economic activities under each probabilistic pollutant discharge allowance level and membership grades are obtained. The results are helpful for the authorities in exploring the trade-off between economic objective and pollutant discharge decision-making based on river water pollution control. Full article
(This article belongs to the Special Issue Fuzzy Techniques for Decision Making) Printed Edition available
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Open AccessArticle The Selection of Wagons for the Internal Transport of a Logistics Company: A Novel Approach Based on Rough BWM and Rough SAW Methods
Symmetry 2017, 9(11), 264; https://doi.org/10.3390/sym9110264
Received: 30 September 2017 / Revised: 30 October 2017 / Accepted: 1 November 2017 / Published: 4 November 2017
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Abstract
The rationalization of logistics activities and processes is very important in the business and efficiency of every company. In this respect, transportation as a subsystem of logistics, whether internal or external, is potentially a huge area for achieving significant savings. In this paper,
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The rationalization of logistics activities and processes is very important in the business and efficiency of every company. In this respect, transportation as a subsystem of logistics, whether internal or external, is potentially a huge area for achieving significant savings. In this paper, the emphasis is placed upon the internal transport logistics of a paper manufacturing company. It is necessary to rationalize the movement of vehicles in the company’s internal transport, that is, for the majority of the transport to be transferred to rail transport, because the company already has an industrial track installed in its premises. To do this, it is necessary to purchase at least two used wagons. The problem is formulated as a multi-criteria decision model with eight criteria and eight alternatives. The paper presents a new approach based on a combination of the Simple Additive Weighting (SAW) method and rough numbers, which is used for ranking the potential solutions and selecting the most suitable one. The rough Best–Worst Method (BWM) was used to determine the weight values of the criteria. The results obtained using a combination of these two methods in their rough form were verified by means of a sensitivity analysis consisting of a change in the weight criteria and comparison with the following methods in their conventional and rough forms: the Analytic Hierarchy Process (AHP), Technique for Ordering Preference by Similarity to Ideal Solution (TOPSIS) and MultiAttributive Border Approximation area Comparison (MABAC). The results show very high stability of the model and ranks that are the same or similar in different scenarios. Full article
(This article belongs to the Special Issue Civil Engineering and Symmetry)
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Open AccessArticle Deep Learning-Based Iris Segmentation for Iris Recognition in Visible Light Environment
Symmetry 2017, 9(11), 263; https://doi.org/10.3390/sym9110263
Received: 8 October 2017 / Revised: 27 October 2017 / Accepted: 1 November 2017 / Published: 4 November 2017
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Abstract
Existing iris recognition systems are heavily dependent on specific conditions, such as the distance of image acquisition and the stop-and-stare environment, which require significant user cooperation. In environments where user cooperation is not guaranteed, prevailing segmentation schemes of the iris region are confronted
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Existing iris recognition systems are heavily dependent on specific conditions, such as the distance of image acquisition and the stop-and-stare environment, which require significant user cooperation. In environments where user cooperation is not guaranteed, prevailing segmentation schemes of the iris region are confronted with many problems, such as heavy occlusion of eyelashes, invalid off-axis rotations, motion blurs, and non-regular reflections in the eye area. In addition, iris recognition based on visible light environment has been investigated to avoid the use of additional near-infrared (NIR) light camera and NIR illuminator, which increased the difficulty of segmenting the iris region accurately owing to the environmental noise of visible light. To address these issues; this study proposes a two-stage iris segmentation scheme based on convolutional neural network (CNN); which is capable of accurate iris segmentation in severely noisy environments of iris recognition by visible light camera sensor. In the experiment; the noisy iris challenge evaluation part-II (NICE-II) training database (selected from the UBIRIS.v2 database) and mobile iris challenge evaluation (MICHE) dataset were used. Experimental results showed that our method outperformed the existing segmentation methods. Full article
(This article belongs to the Special Issue Deep Learning-Based Biometric Technologies)
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Open AccessArticle A Dual Hesitant Fuzzy Rough Pattern Recognition Approach Based on Deviation Theories and Its Application in Urban Traffic Modes Recognition
Symmetry 2017, 9(11), 262; https://doi.org/10.3390/sym9110262
Received: 26 September 2017 / Revised: 18 October 2017 / Accepted: 27 October 2017 / Published: 3 November 2017
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Abstract
In this paper, the dual hesitant fuzzy rough set (DHFRS) is studied from the viewpoint of assessment deviations. Firstly, according to the relationship between intuitionistic fuzzy set and vague set, the DHFRS is transferred into a fuzzy set, where the membership of any
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In this paper, the dual hesitant fuzzy rough set (DHFRS) is studied from the viewpoint of assessment deviations. Firstly, according to the relationship between intuitionistic fuzzy set and vague set, the DHFRS is transferred into a fuzzy set, where the membership of any given element to it has multi-grouped values. By the idea of bootstrap sampling, a group of four sets are generated to describe the membership degree on DHFRS, where the elements of the aforementioned sets are all considered as assessment values. Secondly, the generated sets are dealt with by assessment deviation theories, and specifically, two variables are proposed to describe the systematic and random deviations of the sets. Thirdly, the true-value of the membership degree of any elements to the set is estimated by a deviation-based dual hesitant fuzzy rough weighted aggregating operator. Fourthly, a dual hesitant fuzzy rough pattern recognition approach based on assessment deviation theories is proposed. Finally, an urban traffic modes recognition example is given to illustrate the validity of the proposed theories on DHFRSs. Full article
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Open AccessArticle Multiple Attribute Decision-Making Methods Based on Normal Intuitionistic Fuzzy Interaction Aggregation Operators
Symmetry 2017, 9(11), 261; https://doi.org/10.3390/sym9110261
Received: 17 October 2017 / Revised: 25 October 2017 / Accepted: 1 November 2017 / Published: 3 November 2017
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Abstract
Normal intuitionistic fuzzy numbers (NIFNs), which combine the normal fuzzy number (NFN) with intuitionistic number, can easily express the stochastic fuzzy information existing in real decision making, and power-average (PA) operator can consider the relationships of different attributes by assigned weighting vectors which
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Normal intuitionistic fuzzy numbers (NIFNs), which combine the normal fuzzy number (NFN) with intuitionistic number, can easily express the stochastic fuzzy information existing in real decision making, and power-average (PA) operator can consider the relationships of different attributes by assigned weighting vectors which depend upon the input arguments. In this paper, we extended PA operator to process the NIFNs. Firstly, we defined some basic operational rules of NIFNs by considering the interaction operations of intuitionistic fuzzy sets (IFSs), established the distance between two NIFNs, and introduced the comparison method of NIFNs. Then, we proposed some new aggregation operators, including normal intuitionistic fuzzy weighted interaction averaging (NIFWIA) operator, normal intuitionistic fuzzy power interaction averaging (NIFPIA) operator, normal intuitionistic fuzzy weighted power interaction averaging (NIFWPIA) operator, normal intuitionistic fuzzy generalized power interaction averaging (NIFGPIA) operator, and normal intuitionistic fuzzy generalized weighted power interaction averaging (NIFGWPIA) operator, and studied some properties and some special cases of them. Based on these operators, we developed a decision approach for multiple attribute decision-making (MADM) problems with NIFNs. The significant characteristics of the proposed method are that: (1) it is easier to describe the uncertain information than the existing fuzzy sets and stochastic variables; (2) it used the interaction operations in part of IFSs which could overcome the existing weaknesses in operational rules of NIFNs; (3) it adopted PA operator which could relieve the influence of unreasonable data given by biased decision makers; and (4) it made the decision-making results more flexible and reliable because it was with generalized parameter which could be regard as the risk attitude value of decision makers. Finally, an illustrative example is given to verify its feasibility, and to compare with the existing methods. Full article
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Open AccessArticle Image Recovery of an Infrared Sub-Imaging System Based on Compressed Sensing
Symmetry 2017, 9(11), 260; https://doi.org/10.3390/sym9110260
Received: 26 September 2017 / Revised: 12 October 2017 / Accepted: 23 October 2017 / Published: 2 November 2017
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Abstract
An infrared (IR) sub-imaging system is composed of an optical scanning device and a single IR detector, which provides the target location information to the servo system. Currently, further improvement of positioning accuracy and imaging quality in the traditional rosette scanning guidance mode
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An infrared (IR) sub-imaging system is composed of an optical scanning device and a single IR detector, which provides the target location information to the servo system. Currently, further improvement of positioning accuracy and imaging quality in the traditional rosette scanning guidance mode is experiencing a bottleneck. The emergence of the compressed sensing (CS) technique provides a new solution for this problem as it can recover a high-resolution IR image including richer information with fewer sampling points. In this paper, the complementarity of the CS framework and IR rosette sub-imaging system was analyzed. A new method to improve the resolution of reconstructed IR images, multi-frame joint compressive imaging (MJCI), was proposed. The simulation results revealed the potential of the CS technique when applied to the IR sub-imaging system and demonstrated that the proposed method performed well for reconstruction. Full article
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Open AccessArticle Correlation Coefficients of Probabilistic Hesitant Fuzzy Elements and Their Applications to Evaluation of the Alternatives
Symmetry 2017, 9(11), 259; https://doi.org/10.3390/sym9110259
Received: 25 September 2017 / Revised: 28 October 2017 / Accepted: 29 October 2017 / Published: 2 November 2017
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Abstract
Correlation coefficient is one of the broadly use indexes in multi-criteria decision-making (MCDM) processes. However, some important issues related to correlation coefficient utilization within probabilistic hesitant fuzzy environments remain to be addressed. The purpose of this study is introduced a MCDM method based
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Correlation coefficient is one of the broadly use indexes in multi-criteria decision-making (MCDM) processes. However, some important issues related to correlation coefficient utilization within probabilistic hesitant fuzzy environments remain to be addressed. The purpose of this study is introduced a MCDM method based on correlation coefficients utilize probabilistic hesitant fuzzy information. First, the covariance and correlation coefficient between two PHFEs is introduced, the properties of the proposed covariance and correlation coefficient are discussed. In addition, the northwest corner rule to obtain the expected mean related to the multiply of two PHFEs is introduced. Second, the weighted correlation coefficient is proposed to make the proposed MCDM method more applicable. And the properties of the proposed weighted correlation coefficient are also discussed. Finally, an illustrative example is demonstrated the practicality and effectiveness of the proposed method. An illustrative example is presented to demonstrate the correlation coefficient propose in this paper lies in the interval [−1, 1], which not only consider the strength of relationship between the PHFEs but also whether the PHFEs are positively or negatively related. The advantage of this method is it can avoid the inconsistency of the decision-making result due to the loss of information. Full article
(This article belongs to the Special Issue Fuzzy Techniques for Decision Making) Printed Edition available
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