Fuzzy Connectedness Research Articles

Degree, Neighbourhood, and Connectedness in Fuzzy Digraph

Objective: In the context of fuzzy digraphs, degree neighborhood, and connectedness play important roles in understanding the structure and relationships within the graph. Here are the objectives of degree, neighborhood, and connectedness in fuzzy digraphs: 1. Degree Neighborhood: The degree neighborhood of a vertex in a fuzzy digraph refers to the set of vertices that are directly connected to it. In fuzzy digraphs, the concept of degree neighborhood is extended to include the strength or degree of connection between vertices, which is represented by fuzzy values. 2. Connectedness: Connectedness in fuzzy digraphs refers to the existence of paths between any pair of vertices, considering the fuzzy nature of relationships. Methods: By using fuzzy vertex and fuzzy edges in fuzzy digraph to find degree (in-degree and out-degree), neighborhood (in-neighborhood and out-neighborhood), and to find connectedness in fuzzy digraph. Findings: In this study, some types of fuzzy digraphs are introduced, neighborhood (in-neighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraphs, and strong, weak, unilateral connectedness in fuzzy digraphs are introduced. Novelty: To show that let 𝐺𝐷 ∶ (𝜎, 𝜇) be a fuzzy digraph with fuzzy vertex set 𝜎 and fuzzy edge set 𝜇. In this paper, some types of fuzzy digraphs are introduced, neighborhood (inneighborhood and out-neighborhood) and degree (in-degree and out-degree) in fuzzy digraph and strong, weak, unilateral connectedness in fuzzy digraph are introduced. Furthermore, this paper establishes various results and theorems, and some properties of different fuzzy digraphs are discussed. Keywords: Fuzzy digraph; Degree in a fuzzy digraph; Neighborhood in a fuzzy digraph; Fuzzy diwalk; Fuzzy dicycle; Fuzzy diacyclic digraph

Breast tumor segmentation in ultrasound using distance-adapted fuzzy connectedness, convolutional neural network, and active contour

This study addresses computer-aided breast cancer diagnosis through a hybrid framework for breast tumor segmentation in ultrasound images. The core of the three-stage method is based on the autoencoder convolutional neural network. In the first stage, we prepare a hybrid pseudo-color image through multiple instances of fuzzy connectedness analysis with a novel distance-adapted fuzzy affinity. We produce different weight combinations to determine connectivity maps driven by particular image specifics. After the hybrid image is processed by the deep network, we adjust the segmentation outcome with the Chan-Vese active contour model. We find the idea of incorporating fuzzy connectedness into the input data preparation for deep-learning image analysis our main contribution to the study. The method is trained and validated using a combined dataset of 993 breast ultrasound images from three public collections frequently used in recent studies on breast tumor segmentation. The experiments address essential settings and hyperparameters of the method, e.g., the network architecture, input image size, and active contour setup. The tumor segmentation reaches a median Dice index of 0.86 (mean at 0.79) over the combined database. We refer our results to the most recent state-of-the-art from 2022–2023 using the same datasets, finding our model comparable in segmentation performance.

Self-adaptive two-stage density clustering method with fuzzy connectivity

Density Peak Clustering (DPC) was proposed in the journal Science in 2014 and has been widely applied in many fields due to its simplicity and effectiveness. However, there are few studies on the effectiveness of DPC algorithm and its variants on non-clean data sets. Inspired by the idea that DPC algorithm combines density and distance when determining clustering center, this paper creatively designs a two-stage density clustering method with fuzzy connectivity (TS-DCM). It could be used to distinguish different cluster partitions and further identify noise points and sample points. In addition, this paper also introduces a new clustering index: fuzzy connectivity, which could not only adjust the selection of DPC cutoff distance, but also provide a reference for adaptive adjustment of TS-DCM parameter selection, greatly improving the operating efficiency of the clustering algorithm. At the same time, a self-adaptive two-stage density clustering method (STS-DCM) is proposed to adjust the selection of parameters according to the feedback of clustering results. Finally, compared with other traditional and popular clustering algorithms, it is verified that the proposed algorithm has significant advantages in speed and accuracy. Moreover, for non-clean data sets, the algorithm is robust and effective.

Fermatean Fuzzy CODAS Approach with Topology and Its Application to Sustainable Supplier Selection

A Fermatean fuzzy set (FFS) is a reliable method for representing uncertainty in “multi-criteria decision-making” (MCDM). This research seeks to examine the topological properties of FFSs and to establish the notion of “Fermatean fuzzy topology” (FFT). An FFT is the generalisation of existing fuzzy topologies. Several aspects of FFT are examined and various novel concepts are proposed, which include Fermatean fuzzy α-continuity between FFTSs and Fermatean fuzzy connectedness. To deal multiple challenges in sustainable supply chain management, a Fermatean fuzzy “combinative distance-based assessment” (CODAS) method was developed. The proposed FF CODAS technique involves various key features for MCDM. Firstly, a known reputation vector or equal expert weights is determined based on the reputation, experience and qualifications of the experts. Secondly, the Fermatean fuzzy direct rating approach is used to establish the relative relevance of criteria based on the expert group’s evaluation preferences. Thirdly, the Fermatean fuzzy CODAS approach is used to construct alternative orderings based on their assessment scores. Finally, an application is developed to show the benefit of the suggested supplier selection approach. Additionally, the symmetry of an optimal decision in application is carried out by a comparison analysis of the suggested models with some existing models.

Connectivity indices of m-polar fuzzy network model, with an application to a product manufacturing problem

Connectivity is among the most essential concerns in graph theory and its applications. We consider this issue in a framework that stems from the combination of m-polar fuzzy set theory with graphs. We introduce two measurements of connectedness of m-polar fuzzy graphs that we call their connectivity and average connectivity indices. Examples are given, and the theoretical performance of these concepts is investigated. Particularly, we are concerned with the effect of deleting a vertex or an edge from an m-polar fuzzy graph, on its connectivity and average connectivity indices. We also establish bounding expressions for the connectivity index in complete m-polar fuzzy graphs, complete bipartite m-polar fuzzy graphs, and wheel m-polar fuzzy graphs. Moreover, we introduce some special types of vertices called m-polar fuzzy connectivity reducing vertices, m-polar fuzzy connectivity enhancing vertices, and m-polar fuzzy connectivity neutral vertices. Our theoretical contribution is applied to a product manufacturing problem that takes advantage of multi-polar uncertain information. The justification for our application is systematized using an algorithm. Finally, we compare the proposed method to existing methodologies to demonstrate its feasibility and applicability.

Atlas-Based Segmentation in Extraction of Knee Joint Bone Structures from CT and MR.

The main goal of the approach proposed in this study, which is dedicated to the extraction of bone structures of the knee joint (femoral head, tibia, and patella), was to show a fully automated method of extracting these structures based on atlas segmentation. In order to realize the above-mentioned goal, an algorithm employed automated image-matching as the first step, followed by the normalization of clinical images and the determination of the 11-element dataset to which all scans in the series were allocated. This allowed for a delineation of the average feature vector for the teaching group in the next step, which automated and streamlined known fuzzy segmentation methods (fuzzy c-means (FCM), fuzzy connectedness (FC)). These averaged features were then transmitted to the FCM and FC methods, which were implemented for the testing group and correspondingly for each scan. In this approach, two features are important: the centroids (which become starting points for the fuzzy methods) and the surface area of the extracted bone structure (protects against over-segmentation). This proposed approach was implemented in MATLAB and tested in 61 clinical CT studies of the lower limb on the transverse plane and in 107 T1-weighted MRI studies of the knee joint on the sagittal plane. The atlas-based segmentation combined with the fuzzy methods achieved a Dice index of 85.52-89.48% for the bone structures of the knee joint.

Novel Concepts of q -Rung Orthopair Fuzzy Topology and WPM Approach for Multicriteria Decision-Making

A q -rung orthopair fuzzy set (q-ROFS) is a robust approach for fuzzy modeling, computational intelligence, and multicriteria decision-making (MCDM) problems. The aim of this paper is to study the topological structure on q-ROFSs and define the idea of q -rung orthopair fuzzy topology (q-ROF topology). The characterization of q-ROF α -continuous mappings between q-ROF topological spaces and q-ROF connectedness is investigated. Some relationships of different types of q -rung orthopair fuzzy connectedness are also investigated. Additionally, the “ q -rung orthopair fuzzy weighted product model” (q-ROF WPM) is developed for MCDM of a hierarchical healthcare system. Due to limited and insufficient resources, a hierarchical healthcare system (HHS) is very effective to deal with the increasing problems of healthcare. Recognizing the stage of a disease with the symptoms, ranking the critical condition of patients, and referring patients to feasible hospitals are key features of HHS. A HHS will provide healthcare services in three levels, a primary health centers for initial stage of disease, secondary hospitals for secondary stage of disease, and tertiary hospital for the third-order stage. A numerical example is illustrated to demonstrate the efficiency of q-ROF WPM and advantages of HHS.

Automated segmentation of epidermis in high-frequency ultrasound of pathological skin using a cascade of DeepLab v3+ networks and fuzzy connectedness.

This study proposes a novel, fully automated framework for epidermal layer segmentation in different skin diseases based on 75MHz high-frequency ultrasound (HFUS) image data. A robust epidermis segmentation is a vital first step to detect changes in thickness, shape, and intensity and therefore support diagnosis and treatment monitoring in inflammatory and neoplastic skin lesions. Our framework links deep learning and fuzzy connectedness for image analysis. It consists of a cascade of two DeepLab v3+models with a ResNet-50 backbone and a fuzzy connectedness analysis module for fine segmentation. Both deep models are pre-trained on the ImageNet dataset and subjected to transfer learning using our HFUS database of 580 images with atopic dermatitis, psoriasis and non-melanocytic skin tumors. The first deep model is used to detect the appropriate region of interest, while the second stands for the main segmentation procedure. We use the softmax layer of the latter twofold to prepare the input data for fuzzy connectedness analysis: as a reservoir of seed points and a direct contribution to the input image. In the experiments, we analyze different configurations of the framework, including region of interest detection, deep model backbones and training loss functions, or fuzzy connectedness analysis with parameter settings. We also use the Dice index and epidermis thickness to compare our results to state-of-the-art approaches. The Dice index of 0.919 yielded by our model over the entire dataset (and exceeding 0.93 in inflammatory diseases) proves its superiority over the other methods.

Bipolar intuitionistic fuzzy graph over Cayley groups

In this article, we discuss the notion of Cayley bipolar intuitionistic fuzzy graph and proved some results and properties on it. We study and discuss many results and properties of bipolar intuitionistic fuzzy graph with respect of algebraic structures. On the other hand we study Cayley bipolar intuitionistic fuzzy graph connectedness.

An Over view on intuitionistic fuzzy topological spaces

We present a brief overview some fundamental results on the intuitionistic fuzzy topological spaces, and give some introductory results about fuzzy open set, fuzzy closed set, fuzzy neighborhood, fuzzy interior set, fuzzy continuity, fuzzy compactness and fuzzy connectedness in these spaces.

Dynamic and metabolic quantification of nuclear medicine images in the PET/CT modality

Imaging using nuclear medicine is one of the most common procedures in medical centers. Its great advantage is its capacity to analyze the metabolism of the patient, resulting in early diagnosis. However, quantification in nuclear medicine is complicated by many factors, including degradation due to attenuation, scattering, reconstruction algorithms, and assumed models. This project seeks to improve the accuracy and the precision of quantification in PET/CT images. We developed a framework, comprising consecutively interlinked steps initiated with the simulation of 3D anthropomorphic phantoms. These phantoms were used to generate realistic PET/CT projections by applying the Geant4 Application for Tomography Emission platform using Monte Carlo simulation. Then, a 3D image reconstruction was created, followed by an Anscombe/Wiener filter and a fuzzy connectedness segmentation process. After defining the region of interest, input activity and response activity curves were generated as excitation functions of the compartment model to enable metabolic quantification of the selected organ or structure. Finally, real PET/CT images provided by the Heart Institute of Hospital das Clinicas, School of Medicine of the University of Sao Paulo were analyzed using the method. Metabolic parameters of the three-compartment model based on the MASH anthropomorphic phantom and real PET images were computed for each of the approaches used in this project; the results were similar to the theoretically characteristic values. The three-dimensional filtering step using the Ascombe/Wiener filter was preponderant and had a high impact on the metabolic quantification process and on other important stages of the whole project.

Bipolar intuitionistic fuzzy graph over Cayley groups

In this article, we discuss the notion of Cayley bipolar intuitionistic fuzzy graph and proved some results and properties on it. We study and discuss many results and properties of bipolar intuitionistic fuzzy graph with respect of algebraic structures. On the other hand we study Cayley bipolar intuitionistic fuzzy graph connectedness.

Fuzzy rough sets with a fuzzy ideal

In this paper, we defined the fuzzy upper, fuzzy lower, and fuzzy boundary sets of a rough fuzzy set λ in a fuzzy approximation space (X,R). Based on λ and R, we introduced the fuzzy ideal approximation interior operator intlambdaR and the fuzzy ideal approximation closure operator text {cl}_{lambda }^{R}. We joined the fuzzy ideal notion with the fuzzy approximation spaces, and then introduced the fuzzy ideal approximation closure and interior operators associated to a rough fuzzy set λ. Fuzzy ideal approximation connectedness and the fuzzy ideal approximation continuity between fuzzy ideal approximation spaces are introduced.

Multiple sclerosis segmentation method in magnetic resonance imaging using fuzzy connectedness, binarization, mathematical morphology, and 3D reconstruction

IntroductionMagnetic resonance imaging (MRI) is the most used medical modality for diagnosis and monitoring of multiple sclerosis (MS). A segmentation process is an important task to quantify lesion and its progression. However, manual segmentation of 3D images is tedious, time-consuming, and often not reproducible. The state of the art presents results with room for improvements. Consequently, a semiautomatic segmentation process is proposed and described in this study.MethodsThe method consists on a 3D segmentation semiautomatic process for MS lesions in MRI. It initiates by firstly carrying out a preprocessing stage; thus, contrast adjustment is applied to enhance sclerosis regions from other brain information. Secondly, a feature extraction block based on fuzzy connectedness is performed so as to isolate sclerosis lesions from other brain regions. Finally, 3D brain reconstruction is executed along with sclerosis to provide a useful 3D information.ResultsThe robustness of this approach is demonstrated by high correlation between the results and their corresponding gold standard. The results were also obtained by computing parameters of accuracy of image segmentation, as well as overlap Dice. The proposed method reached true positive of 75.61%, false positive of 16.37%, and DICE of 78.23%.ConclusionThe high correlation between specialist and proposed approach outcome, a better monitoring of the disease, is provided; the specialist can understand the patient’s symptoms, thereby increasing the patient’s quality of life.

Optimum Cuts in Graphs by General Fuzzy Connectedness with Local Band Constraints

The goal of this work is to describe an efficient algorithm for finding a binary segmentation of an image such that the indicated object satisfies a novel high-level prior, called local band, LB, constraint; the returned segmentation is optimal, with respect to an appropriate graph-cut measure, among all segmentations satisfying the given LB constraint. The new algorithm has two stages: expanding the number of edges of a standard edge-weighted graph of an image; applying to this new weighted graph an algorithm known as an oriented image foresting transform, OIFT. In our theoretical investigation, we prove that OIFT algorithm belongs to a class of general fuzzy connectedness algorithms and so has several good theoretical properties, like robustness for seed placement. The extension of the graph constructed in the first stage ensures, as we prove, that the resulted object indeed satisfies the given LB constraint. We also notice that this graph construction is flexible enough to allow combining it with other high-level constraints. Finally, we experimentally demonstrate that the LB constraint gives competitive results as compared to geodesic star convexity, boundary band, and hedgehog shape prior, all implemented within OIFT framework and applied to various scenarios involving natural and medical images.

Fuzzy volumetric delineation of brain tumor and survival prediction

A novel three-dimensional detailed delineation algorithm is introduced for Glioblastoma multiforme tumors in MRI. It efficiently delineates the whole tumor, enhancing core, edema and necrosis volumes using fuzzy connectivity and multi-thresholding, based on a single seed voxel. While the whole tumor volume delineation uses FLAIR and T2 MRI channels, the outlining of the enhancing core, necrosis and edema volumes employs the T1C channel. Discrete curve evolution is initially applied for multi-thresholding, to determine intervals around significant (visually critical) points, and a threshold is determined in each interval using bi-level Otsu’s method or Li and Lee’s entropy. This is followed by an interactive whole tumor volume delineation using FLAIR and T2 MRI sequences, requiring a single user-defined seed. An efficient and robust whole tumor extraction is executed using fuzzy connectedness and dynamic thresholding. Finally, the segmented whole tumor volume in T1C MRI channel is again subjected to multi-level segmentation, to delineate its sub-parts, encompassing enhancing core, necrosis and edema. This was followed by survival prediction of patients using the concept of habitats. Qualitative and quantitative evaluation, on FLAIR, T2 and T1C MR sequences of 29 GBM patients, establish its superiority over related methods, visually as well as in terms of Dice scores, Sensitivity and Hausdorff distance.

A Novel Framework for Improving Pulse-Coupled Neural Networks With Fuzzy Connectedness for Medical Image Segmentation

A pulse-coupled neural network (PCNN) is a promising image segmentation approach that requires no training. However, it is challenging to successfully apply a PCNN to medical image segmentation due to common but difficult scenarios such as irregular object shapes, blurred boundaries, and intensity inhomogeneity. To improve this situation, a novel framework incorporating fuzzy connectedness ( FC ) is proposed. First, a comparative study of the traditional PCNN models is carried out to analyze the framework and firing mechanism. Then, the characteristic matrix of fuzzy connectedness ( CMFC ) is presented for the first time. The CMFC can provide more intensity information and spatial relationships at the pixel level, which is helpful for producing a more reasonable firing mechanism in the PCNN models. Third, by integrating the CMFC into the PCNN framework models, a construction scheme of FC-PCNN models is designed. To illustrate this concept, a general solution that can be applied to different PCNN models is developed. Next, the segmentation performances of the proposed FC-PCNN models are evaluated by comparison with the traditional PCNN models, the traditional segmentation methods, and deep learning methods. The test images include synthetic and real medical images from the Internet and three public medical image datasets. The quantitative and visual comparative analysis demonstrates that the proposed FC-PCNN models outperform the traditional PCNN models and the traditional segmentation methods and achieve competitive performance to the deep learning methods. In addition, the proposed FC-PCNN models have favorable capability to eliminate inference from surrounding artifacts.

Fracturing Proppant Quality Estimation Based on Fuzzy Connectedness and Shape Similarity

To improve the accuracy and efficiency of measuring fracturing proppant sphericity and roundness, a measurement scheme is designed by combining fuzzy connectedness and shape similarity. Firstly, in the proppant images collected by the microscope, according to the color characteristics of the test placement plane’s hole and proppant, the region of each proppant and that of the hole without proppant are marked separately. Secondly, based on the feature of holes without proppant, the statistical method of labeling connected components is used to only reserve all the proppant regions. Thirdly, select a part of the region of each proppant as the seed region to calculate the fuzzy connectedness map between seed regions and other regions. Fourthly, select the appropriate connectedness threshold for each proppant based on histogram of fuzzy connectedness map of each proppant region to obtain the accurate region of each proppant. Finally, the shape similarity measurement method based on Hu moment is used to determine the sphericity and roundness of each proppant by measuring the similarity between the region of each proppant and that of 20 particles in the standard template. The experimental results show that the proposed scheme is consistent with manual measurement flow, and can avoid the human labor, meanwhile, the accuracy of measurement is basically consistent with the results of manual measurement.

Fuzzy roughness via ideals

In this paper, we join the notion of fuzzy ideal to the notion of fuzzy approximation space to define the notion of fuzzy ideal approximation spaces. We introduce the fuzzy ideal approximation interior operator int Φ λ and the fuzzy ideal approximation closure operator cl Φ λ , and moreover, we define the fuzzy ideal approximation preinterior operator p int Φ λ and the fuzzy ideal approximation preclosure operator p cl Φ λ with respect to that fuzzy ideal defined on the fuzzy approximation space (X, R) associated with some fuzzy set λ ∈ IX. Also, we define fuzzy separation axioms, fuzzy connectedness and fuzzy compactness in fuzzy approximation spaces and in fuzzy ideal approximation spaces as well, and prove the implications in between.

R-Fuzzy Rs-compactness and r-fuzzy Rs-connectedness in the sense of Sostak’s

The purpose of this paper is to introduce the concepts of fuzzy regular semi compactness, fuzzy regular semi connectedness, fuzzy regular semi strongly connectedness and fuzzy regular semi-C5-connectedness. Some interesting properties of these notions are studied. In this connection, interrelations are discussed. Examples are provided wherever necessary