scholarly journals A Sweepline Algorithm for Generalized Delaunay Triangulations

1991 ◽  
Vol 20 (373) ◽  
Author(s):  
Sven Skyum
Keyword(s):  
Voronoi Diagram ◽  
Delaunay Triangulation ◽  
Distance Function ◽  
Euclidean Distance ◽  
Distance Functions ◽  
Minkowski Distance ◽  
Delaunay Triangulations ◽  
Strictly Convex ◽  
Euclidean Distance Function ◽  
Generalized Voronoi Diagram

We give a deterministic O(<em>n</em> log <em>n</em>) sweepline algorithm to construct the generalized Voronoi diagram for n points in the plane or rather its dual the generalized Delaunay triangulation. The algorithm uses no transformations and it is developed solely from the sweepline paradigm together with greediness. A generalized Delaunay triangulation can be based on an arbitrary strictly convex Minkowski distance function (including all L_p distance functions 1 &lt; p &lt; *) in contrast to ordinary Delaunay triangualations which are based on the Euclidean distance function.

scholarly journals Selective Image Segmentation Models Using Three Distance Functions

2021 ◽  
Vol 21 (No.1) ◽  
pp. 95-116
Author(s):  
Abdul Kadir Jumaat ◽  
Siti Aminah Abdullah
Keyword(s):  
Image Segmentation ◽  
Distance Function ◽  
Execution Time ◽  
Euclidean Distance ◽  
Operator Splitting ◽  
Splitting Method ◽  
Distance Functions ◽  
Similarity Coefficients ◽  
City Block ◽  
Euclidean Distance Function

Image segmentation can be defined as partitioning an image that contains multiple segments of meaningful parts for further processing. Global segmentation is concerned with segmenting the whole object of an observed image. Meanwhile, the selective segmentation model is focused on segmenting a specific object required to be extracted. The Convex Distance Selective Segmentation (CDSS) model, which uses the Euclidean distance function as the fitting term, was proposed in 2015. However, the Euclidean distance function takes time to compute. This paper proposed the reformulation of the CDSS minimization problem by changing the fitting term with three popular distance functions, namely Chessboard, City Block, and Quasi-Euclidean. The proposed models were CDSSNEW1, CDSSNEW2, and CDSSNEW3, which applied the Chessboard, City Block, and Quasi-Euclidean distance functions, respectively. In this study, the Euler-Lagrange (EL) equations of the proposed models were derived and solved using the Additive Operator Splitting method. Then, MATLAB coding was developed to implement the proposed models. The accuracy of the segmented image was evaluated using the Jaccard and Dice Similarity Coefficients. The execution time was recorded to measure the efficiency of the models. Numerical results showed that the proposed CDSSNEW1 model based on the Chessboard distance function could segment specific objects successfully for all grayscale images with the fastest execution time as compared to other models.

Improving Euclidean’s Consensus Degrees in Group Decision Making Problems Through a Uniform Extension

2021 ◽  
Author(s):  
J.M. Tapia ◽  
F. Chiclana ◽  
M.J. Del Moral ◽  
E. Herrera-Viedma
Keyword(s):  
Decision Making ◽  
Group Decision Making ◽  
Distance Function ◽  
Euclidean Distance ◽  
Group Decision ◽  
Distance Functions ◽  
Consensus Models ◽  
Interesting Task ◽  
Euclidean Distance Function ◽  
Uniform Extension

In a Group Decision Making problem, several people try to reach a single common decision by selecting one of the possible alternatives according to their respective preferences. So, a consensus process is performed in order to increase the level of accord amongst people, called experts, before obtaining the final solution. Improving the consensus degree as much as possible is a very interesting task in the process. In the evaluation of the consensus degree, the measurement of the distance representing disagreement among the experts’ preferences should be considered. Different distance functions have been proposed to implement in consensus models. The Euclidean distance function is one of the most commonly used. This paper analyzes how to improve the consensus degrees, obtained through the Euclidean distance function, when the preferences of the experts are slightly modified by using one of the properties of the Uniform distribution. We fulfil an experimental study that shows the betterment in the consensus degrees when the Uniform extension is applied, taking into account different number of experts and alternatives.

scholarly journals Selective Image Segmentation Models Using Three Distance Functions

2021 ◽  
Vol 21 (No.1) ◽  
pp. 95-116
Author(s):  
Abdul Kadir Jumaat ◽  
Siti Aminah Abdullah
Keyword(s):  
Image Segmentation ◽  
Distance Function ◽  
Execution Time ◽  
Euclidean Distance ◽  
Operator Splitting ◽  
Splitting Method ◽  
Distance Functions ◽  
Similarity Coefficients ◽  
City Block ◽  
Euclidean Distance Function

Image segmentation can be defined as partitioning an image that contains multiple segments of meaningful parts for further processing. Global segmentation is concerned with segmenting the whole object of an observed image. Meanwhile, the selective segmentation model is focused on segmenting a specific object required to be extracted. The Convex Distance Selective Segmentation (CDSS) model, which uses the Euclidean distance function as the fitting term, was proposed in 2015. However, the Euclidean distance function takes time to compute. This paper proposed the reformulation of the CDSS minimization problem by changing the fitting term with three popular distance functions, namely Chessboard, City Block, and Quasi-Euclidean. The proposed models were CDSSNEW1, CDSSNEW2, and CDSSNEW3, which applied the Chessboard, City Block, and Quasi-Euclidean distance functions, respectively. In this study, the Euler-Lagrange (EL) equations of the proposed models were derived and solved using the Additive Operator Splitting method. Then, MATLAB coding was developed to implement the proposed models. The accuracy of the segmented image was evaluated using the Jaccard and Dice Similarity Coefficients. The execution time was recorded to measure the efficiency of the models. Numerical results showed that the proposed CDSSNEW1 model based on the Chessboard distance function could segment specific objects successfully for all grayscale images with the fastest execution time as compared to other models.

scholarly journals Geometric applications of critical point theory to submanifolds of complex projective space

1974 ◽  
Vol 55 ◽  
pp. 5-31 ◽  
Author(s):  
Thomas E. Cecil
Keyword(s):  
Projective Space ◽  
Critical Point ◽  
Distance Function ◽  
Complex Projective Space ◽  
Euclidean Distance ◽  
Point Theory ◽  
Euclidean Distance Function ◽  
Complex Projective ◽  
Class Of Functions

In a recent paper, [6], Nomizu and Rodriguez found a geometric characterization of umbilical submanifolds Mn ⊂ Rn+p in terms of the critical point behavior of a certain class of functions Lp, p ⊂ Rn+p, on Mn. In that case, if p ⊂ Rn+p, x ⊂ Mn, then Lp(x) = (d(x,p))2, where d is the Euclidean distance function.

Identifying the Mastery Concepts in Linear Algebra by Using FCM-CM Algorithm

2010 ◽  
Vol 44-47 ◽  
pp. 3897-3901
Author(s):  
Hsiang Chuan Liu ◽  
Yen Kuei Yu ◽  
Jeng Ming Yih ◽  
Chin Chun Chen
Keyword(s):  
Objective Function ◽  
Linear Algebra ◽  
Fuzzy Clustering ◽  
Distance Function ◽  
Mahalanobis Distance ◽  
Euclidean Distance ◽  
Clustering Algorithm ◽  
Clustering Algorithms ◽  
Euclidean Distance Function ◽  
Fuzzy C Means Algorithm

Euclidean distance function based fuzzy clustering algorithms can only be used to detect spherical structural clusters. Gustafson-Kessel (GK) clustering algorithm and Gath-Geva (GG) clustering algorithm were developed to detect non-spherical structural clusters by employing Mahalanobis distance in objective function, however, both of them need to add some constrains for Mahalanobis distance. In this paper, the authors’ improved Fuzzy C-Means algorithm based on common Mahalanobis distance (FCM-CM) is used to identify the mastery concepts in linear algebra, for comparing the performances with other four partition algorithms; FCM-M, GG, GK, and FCM. The result shows that FCM-CM has better performance than others.

scholarly journals Round Randomized Learning Vector Quantization for Brain Tumor Imaging

2016 ◽  
Vol 2016 ◽  
pp. 1-19 ◽  
Author(s):  
Siti Norul Huda Sheikh Abdullah ◽  
Farah Aqilah Bohani ◽  
Baher H. Nayef ◽  
Shahnorbanun Sahran ◽  
Omar Al Akash ◽  
...  
Keyword(s):  
Brain Tumor ◽  
Magnetic Resonance ◽  
Vector Quantization ◽  
Distance Function ◽  
Euclidean Distance ◽  
Brain Magnetic Resonance Imaging ◽  
Learning Vector Quantization ◽  
Magnetic Resonance Images ◽  
Class Distribution ◽  
Euclidean Distance Function

Brain magnetic resonance imaging (MRI) classification into normal and abnormal is a critical and challenging task. Owing to that, several medical imaging classification techniques have been devised in which Learning Vector Quantization (LVQ) is amongst the potential. The main goal of this paper is to enhance the performance of LVQ technique in order to gain higher accuracy detection for brain tumor in MRIs. The classical way of selecting the winner code vector in LVQ is to measure the distance between the input vector and the codebook vectors using Euclidean distance function. In order to improve the winner selection technique, round off function is employed along with the Euclidean distance function. Moreover, in competitive learning classifiers, the fitting model is highly dependent on the class distribution. Therefore this paper proposed a multiresampling technique for which better class distribution can be achieved. This multiresampling is executed by using random selection via preclassification. The test data sample used are the brain tumor magnetic resonance images collected from Universiti Kebangsaan Malaysia Medical Center and UCI benchmark data sets. Comparative studies showed that the proposed methods with promising results are LVQ1, Multipass LVQ, Hierarchical LVQ, Multilayer Perceptron, and Radial Basis Function.

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