• Communications for Statistical Applications and Methods
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• v.22 no.4
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• pp.321-331
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• 2015

The K-means clustering algorithm is a popular and widely used method for clustering. For covariance matrices, we consider a geodesic clustering algorithm based on the K-means clustering framework in consideration of symmetric positive definite matrices as a Riemannian (non-Euclidean) manifold. This paper considers a geodesic clustering algorithm for data consisting of symmetric positive definite (SPD) matrices, utilizing the Riemannian geometric structure for SPD matrices and the idea of a K-means clustering algorithm. A K-means clustering algorithm is divided into two main steps for which we need a dissimilarity measure between two matrix data points and a way of computing centroids for observations in clusters. In order to use the Riemannian structure, we adopt the geodesic distance and the intrinsic mean for symmetric positive definite matrices. We demonstrate our proposed method through simulations as well as application to real financial data.

• Honam Mathematical Journal
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• v.30 no.2
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• pp.335-341
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• 2008

In this paper, we use the convexity of distance function between geodesics in a singular Hadamard space to generalize Hadamard-Cartan theorem for 2-dimensional metric spaces. We also determine a neighborhood of a closed geodesic where no other closed geodesic exists in a complete space of nonpositive curvature.

• Proceedings of the Korean Information Science Society Conference
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• 2005.07b
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• pp.748-750
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• 2005

Isomap [1] is a manifold learning algorithm, which extends classical multidimensional scaling (MDS) by considering approximate geodesic distance instead of Euclidean distance. The approximate geodesic distance matrix can be interpreted as a kernel matrix, which implies that Isomap can be solved by a kernel eigenvalue problem. However, the geodesic distance kernel matrix is not guaranteed to be positive semidefinite. In this paper we employ a constant-adding method, which leads to the Mercer kernel-based Isomap algorithm. Numerical experimental results with noisy 'Swiss roll' data, confirm the validity and high performance of our kernel Isomap algorithm.

• Journal of Korea Multimedia Society
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• v.19 no.12
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• pp.1927-1935
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• 2016

This paper propose a framework for human body parts in RGB-D image. We conduct tasks of obtaining person area, finding candidate areas and local detection in order to detect hand, foot and head which have features of long accumulative geodesic distance. A person area is obtained with background subtraction and noise removal by using depth image which is robust to illumination change. Finding candidate areas performs construction of graph model which allows us to measure accumulative geodesic distance for the candidates. Instead of raw depth map, our approach constructs graph model with segmented regions by quadtree structure to improve searching time for the candidates. Local detection uses HOG based SVM for each parts, and head is detected for the first time. To minimize false detections for hand and foot parts, the candidates are classified with upper or lower body using the head position and properties of geodesic distance. Then, detect hand and foot with the local detectors. We evaluate our algorithm with datasets collected Kinect v2 sensor, and our approach shows good performance for head, hand and foot detection.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
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• v.17 no.2
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• pp.137-143
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• 1999

The straight spatial distance or geodesic distance on WGS84 ellipsoid measured by GPS should be reduced on geoid to be used in Korean Geodetic System. The factors for this reduction are geoidal height and mean radius of the earth. On this study the effects of these factors on reducing distance were analyzed and the result showed that mean geoidal height should be multiplied by $1.6\times{10}^{-7}$ per unit distance for reducing geodesic distance on reference ellipsoid to that on geoid. Condsidering that the geoidal height on Bessel ellipsoid in Korea is -45 m in northeast and -75 m in middle west. It also showed that the difference of geodesic distance between on reference ellipsoid and on geoid is about 7-12 mm per km.

• Journal of KIISE:Software and Applications
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• v.36 no.6
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• pp.479-490
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• 2009

A number of temporal abstraction approaches have been suggested so far to handle the high computational complexity of Markov decision problems (MDPs). Although the structure of temporal abstraction can significantly affect the efficiency of solving the MDP, to our knowledge none of current temporal abstraction approaches explicitly consider the relationship between topology and efficiency. In this paper, we first show that a topological measurement from complex network literature, mean geodesic distance, can reflect the efficiency of solving MDP. Based on this, we build an incremental method to systematically build temporal abstractions using a network model that guarantees a small mean geodesic distance. We test our algorithm on a realistic 3D game environment, and experimental results show that our model has subpolynomial growth of mean geodesic distance according to problem size, which enables efficient solving of resulting MDP.

• The Journal of the Korea Contents Association
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• v.8 no.5
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• pp.94-103
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• 2008

In this paper, we adapt the $\kappa$-means clustering technique to segmenting a given 3D mesh. In order to avoid the locally minimal convergence and speed up the computing time, first we extract sharp vertices from the mesh by analysing its curvature and convexity that respectively reflect the local and global geometric characteristics from the viewpoint of cognitive science. Next the sharp vertices are partitioned into $\kappa$ clusters by iterated converging with the $\kappa$-means clustering method based on the geodesic distance instead of the Euclidean distance between each pair of the sharp vertices. For obtaining the effective result of $\kappa$-means clustering method, it is crucial to assign an initial value to $\kappa$ appropriately. Hence, we automatically compute a reasonable number of clusters as an initial value of $\kappa$. Finally the mesh segmentation is completed by merging other vertices except the sharp vertices into the nearest cluster by geodesic distance.

• The Journal of the Convergence on Culture Technology
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• v.9 no.3
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• pp.81-94
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• 2023

The study aims to compare and analyze the social network structures of Qatar Airways,s Singapore Airlines, Emirates Airlines, and ANA Airlines, recording the top 1 to 4, and Korean Air in ninth by Skytrax's airline evaluations in 2022. This study uses NodeXL, a social network analysis program, to analyze the social networks of 5 airlines, Vertex, Unique Edges, Single-Vertex Connected Components, Maximum Geodesic Distance, Average Geodesic Distance, Average Degree Centrality, Average Closeness Centrality, and Average Betweenness Centrality as indicators to compare the differences in these social networks of the airlines. As a result, Singapore's social network has a better network structure than the other airlines' social networks in terms of sharing information and transmitting resources. In addition, Qatar Airways and Singapore Airlines are superior to the other airlines in playing roles and powers of influencers who affect the flow of information and resources and the interaction within the airline's social network. The study suggests some implications to enhance the usefulness of social networks for marketing.

• Bulletin of the Korean Mathematical Society
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• v.47 no.1
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• pp.73-79
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• 2010

A pair of point vortices of the same strength but opposite sign is called a vortex dipole. We consider the limiting case where two vortices approach infinitely close while the ratio of the strength to the distance kept constant. The motion of such point vortex dipole on the ellipsoid of revolution is investigated geometrically to conclude that the trajectory draws a geodesic up to the leading order of perturbation, whose direction is determined by the initial orientation of the dipole. Related issues are also remarked.

• International Journal of Contents
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• v.5 no.4
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• pp.55-61
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• 2009

In this paper, we propose a hierarchical method for segmenting a given 3D mesh, which hierarchically clusters sharp vertices of the mesh using the metric of geodesic distance among them. Sharp vertices are extracted from the mesh by analyzing convexity that reflects global geometry. As well as speeding up the computing time, the sharp vertices of this kind avoid the problem of local optima that may occur when feature points are extracted by analyzing the convexity that reflects local geometry. For obtaining more effective results, the sharp vertices are categorized according to the priority from the viewpoint of cognitive science, and the reasonable number of clusters is automatically determined by analyzing the geometric features of the mesh.