Yingxin Wu, Masahiro Takatsuka · Proceedings of SPIE, the International Society for Optical Engineering/Proceedings of SPIE · 2005
Self-Organizing map (SOM) is a widely used tool to find clustering and also to visualize high dimensional data. Several spherical SOMs have been proposed to create a more accurate representation of the data by removing the “border effect”. In this paper, we compare several spherical lattices for the purpose of implementation of a SOM. We then introduce a 2D rectangular grid data structure for representing the geodesic dome. This new approach improves the neighborhood searching process in the spherical gird. The new Geodesic SOM and its data structure are tested using socio-demographic data. In the experiments, we try to create a notion of direction in the Geodesic SOM. The direction facilitates more consistent visual comparison of different datasets as well as to assist viewers building their mental maps.
Cited by 13
Rômulo Magalhães De Sousa, Oliveira Júnior · 2012
The Geodesic Self-Organizing Map (GeoSOM) is a variation of traditional SOM, which uses an icosahedron-based tessellation as spherical lattice to eliminate the border effect to minimize the distortion in the reduction of high-dimensional spaces. Border effect is a problem intrinsic of low-dimensional neural grid, where neurons in the border have a less possibility to have its synaptic weights updated. The almost perfect regularity of a tessellated icosahedron projection onto a sphere solves this problem, reducing in two thirds of the distortion of 2D SOM. However, two problems appear resulting from complex shape of this Platonic polyhedron. First, the growth curve of lattice sizing follows a strong upward tendency that means a loss of control over the lattice sizing. Second, an overall visualization of topographic map is only possible with a geodesic projection of prototype vector positions from the surface of the sphere to a 2D plane that results in a elliptical map, flat on top and bottom, that avoids an orthogonal alignment of the data in the left and right sides, causing some distortion in the presentation of results and avoid an intuitive visualization of the map. This work proposes a geodesic self-organizing map, called 4HSOM, which uses a tessellated tetrahedron as lattice to eliminate the border effect, maximizing the control over the lattice sizing, with an easier overall visualization of topographic map without any geodetic projection, resulting from the minimalist geometric structure of the tetrahedron, although the tessellated tetrahedron has the greatest irregularity among the Platonic polyhedra. The work presents a comparative analysis between the results achieved by 4HSOM and GeoSOM.
Cited by 3
Katrin Heimann (6454412), Sebo Uithol (6454415), Marta Calbi (532253) et al. · Figshare · 2019
<p>Electrodes circled in red indicate central electrode-clusters of interest chosen to assess rolandic mu-rhythm modulation (electrodes 30, 31, 36, 37, 41,42, 53, 54 in left and 79, 80, 86, 87, 93, 103, 104, 105 in right). Electrodes circled in green indicate electrode-clusters of interest chosen to assess whether recordings in central areas might be affected by posterior activity (electrodes 69, 70, 73, 74 in left and electrodes 82, 83, 88, 89 in right).</p>
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Alexey V. Bolsinov, Božidar Jovanović · Annals of Global Analysis and Geometry · 2003
Cited by 75
Robin W. Tucker, Timothy J. Walton · Fundamental theories of physics · 2025
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Ian McIntosh, Draper, Christopher · White Rose Research Online (University of Leeds, The University of Sheffield, University of York) · 2016
For an oriented isometric immersed submanifold of the n-sphere, the spherical Gauss map is the Legendrian immersion of its unit normal bundle into the unit sphere subbundle of the tangent bundle of the sphere, and the geodesic Gauss map projects this into the manifold of oriented geodesics (the Grassmannian of oriented 2-planes in Euclidean n+1-space), giving a Lagrangian immersion of the unit normal bundle into a Kaehler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of the immersion, and show that when it has conformal shape form this depends only on its mean curvature. In particular we deduce that the geodesic Gauss map of every minimal surface in the n-sphere is minimal Lagrangian. We also give simple proofs that: deformations of the immersion always correspond to Hamiltonian deformations of its geodesic Gauss map; the mean curvature vector of the geodesic Gauss map is always a Hamiltonian vector field. This extends work of Palmer on the case when the immersion is a hypersurface.
Cited by 1Open access
Rick O. Gilmore (3166008), Amanda L. Thomas (788786), Jeremy Fesi (3166011) · Figshare · 2016
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Alexey V. Bolsinov, Božidar Jovanović · ArXiv.org · 2001
The purpose of this paper is to discuss the relationship between commutative and non-commutative integrability of Hamiltonian systems and to construct new examples of integrable geodesic flows on Riemannian manifolds. In particular, we prove that the geodesic flow of the bi-invariant metric on any bi-quotient of a compact Lie group is integrable in non-commutative sense by means of polynomial integrals, and therefore, in classical commutative sense by means of $C^\infty$--smooth integrals.
Cited by 18Open access
Chris Draper, Ian McIntosh · Communications in Analysis and Geometry · 2016
For an oriented isometric immersion f : M S n the spherical Gauss map is the Legendrian immersion of its unit normal bundle UM into the unit sphere subbundle of T S n , and the geodesic Gauss map projects this into the manifold of oriented geodesics in S n (the Grassmannian of oriented 2-planes in R n+1 ), giving a Lagrangian immersion of UM into a Khler-Einstein manifold. We give expressions for the mean curvature vectors for both the spherical and geodesic Gauss maps in terms of the second fundamental form of f , and show that when f has conformal shape form this depends only on the mean curvature of f . In particular we deduce that the geodesic Gauss map of every minimal surface in S n is minimal Lagrangian. We also give simple proofs that: deformations of f always correspond to Hamiltonian deformations of ; the mean curvature vector of is always a Hamiltonian vector field.
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Yingxin Wu, Masahiro Takatsuka · Neural Networks · 2006
Cited by 63
G Fanizza, M Gasperini, G Marozzi et al. · Journal of Cosmology and Astroparticle Physics · 2013
The remarkable properties of the recently proposed geodesic light-cone (GLC) gauge allow to explicitly solve the geodetic-deviation equation, and thus to derive an exact expression for the Jacobi map J^A_B(s,o) connecting a generic source s to a geodesic observer o in a generic space time. In this gauge J^A_B factorizes into the product of a local quantity at s times one at o, implying similarly factorized expressions for the area and luminosity distance. In any other coordinate system J^A_B is simply given by expressing the GLC quantities in terms of the corresponding ones in the new coordinates. This is explicitly done, at first and second order, respectively, for the synchronous and Poisson gauge-fixing of a perturbed, spatially-flat cosmological background, and the consistency of the two outcomes is checked. Our results slightly amend previous calculations of the luminosity-redshift relation and suggest a possible non-perturbative way for computing the effects of inhomogeneities on observations based on light-like signals.
Cited by 77Open access
Yingxin Wu, Masahiro Takatsuka · 2008
Abstract- In order to remove the “border effect”, several spherical Self-Organizing Maps (SOM) based on the geodesic dome have been proposed. However, existing neighborhood searching methods on the geodesic dome are much more time-consuming than searching on the normal rectangular/hexagonal grid. In this paper, we present detailed descriptions of the algorithms used in training the Geodesic SOM (GeoSOM), which we previously proposed. Experimental results show that the GeoSOM runs almost at the same speed of the conventional 2D SOM and it also represents the data more accurately. In visualizing the GeoSOM, a 2D data map can be created by projecting the spherical SOM onto 2D the plane. The user is able to select any point of interest to be the center of the 2D map. No retraining is required when changing the center point. Key words- data structure, geodesic dome, Self-Organizing Map, visualization 1
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Josué Meléndez, M. Alvarez-Ramírez, Antonio García · Communications in Nonlinear Science and Numerical Simulation · 2020
Cited by 1
Amie Wilkinson · Ergodic Theory and Dynamical Systems · 1998
We prove that the time-one map of the geodesic flow for a closed, negatively curved surface is stably ergodic.
Cited by 61
Roy L. Adler, Leopold Flatto · Contemporary mathematics - American Mathematical Society · 1984
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Sang Wook Yoo, Joon‐Kyung Seong, Minhyuk Sung et al. · IEEE Transactions on Visualization and Computer Graphics · 2012
This paper addresses the problem of computing the geodesic distance map from a given set of source vertices to all other vertices on a surface mesh using an anisotropic distance metric. Formulating this problem as an equivalent control theoretic problem with Hamilton-Jacobi-Bellman partial differential equations, we present a framework for computing an anisotropic geodesic map using a curvature-based speed function. An ordered upwind method (OUM)-based solver for these equations is available for unstructured planar meshes. We adopt this OUM-based solver for surface meshes and present a triangulation-invariant method for the solver. Our basic idea is to explore proximity among the vertices on a surface while locally following the characteristic direction at each vertex. We also propose two speed functions based on classical curvature tensors and show that the resulting anisotropic geodesic maps reflect surface geometry well through several experiments, including isocontour generation, offset curve computation, medial axis extraction, and ridge/valley curve extraction. Our approach facilitates surface analysis and processing by defining speed functions in an application-dependent manner.
Cited by 8
Wei Quan, Bogdan J. Matuszewski, Lik‐Kwan Shark · Lecture notes in computer science · 2015
Cited by 1
Yingxin Wu, Masahiro Takatsuka · 2006
The two-dimensional (2D) Self-Organizing Map (SOM) has a well-known “border effect”. Several spherical SOMs which use lattices of the tessellated icosahedron have been proposed to solve this problem. However, existing data structures for such SOMs are either not space efficient or are time consuming when searching the neighborhood. We introduce a 2D rectangular grid data structure to store the icosahedron-based geodesic dome. Vertices relationships are maintained by their positions in the data structure rather than by immediate neighbor pointers or an adjacency list. Increasing the number of neurons can be done efficiently because the overhead caused by pointer updates is reduced. Experiments show that the spherical SOM using our data structure, called a GeoSOM, runs with comparable speed to the conventional 2D SOM. The GeoSOM also reduces data distortion due to removal of the boundaries. Furthermore, we developed an interface to project the GeoSOM onto the 2D plane using a cartographic approach, which gives users a global view of the spherical data map. Users can change the center of the 2D data map interactively. In the end, we compare the GeoSOM to the other spherical SOMs by space complexity and time complexity.
Cited by 1
Qiang Gao · Bulletin of the London Mathematical Society · 2023
Abstract Let be a holomorphic quadratic differential of finite norm in the unit disk . Let be equipped with the metric . In this paper, we prove that a geodesic‐preserving map of the unit disk into itself is affine. A map is called geodesic‐preserving if the image of each geodesic is a geodesic. We make no assumptions on the continuity of such a map.
Cited by 2Open access
Pablo D. Carrasco, Túlio Vales · Discrete and Continuous Dynamical Systems · 2020
<p style='text-indent:20px;'>In this note we consider a symmetric Random Walk defined by a <inline-formula><tex-math id="M1">\begin{document}$ (f, f^{-1}) $\end{document}</tex-math></inline-formula> Kalikow type system, where <inline-formula><tex-math id="M2">\begin{document}$ f $\end{document}</tex-math></inline-formula> is the time-one map of the geodesic flow corresponding to an hyperbolic manifold. We provide necessary and sufficient conditions for the existence of an stationary measure for the walk that is equivalent to the volume in the corresponding unit tangent bundle. Some dynamical consequences for the Random Walk are deduced in these cases.
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