This page demo is based on :
"Cheng-Yuna Lious and Yen-Ting Kuo, ¡§Conformal self-organizing map for a genus-zero manifold¡¨The Visual Computer, 1432-8726 (Online)"
"C.-Y. Liou and W.-P. Tai, ¡§Conformality in the self-organization network¡¨ Artificial Intelligence, vol. 116, page 265-286, 2000"
Wen-Pin Tai and Cheng-Yuan Liou (2000), Image representation by self-organizing conformal network, The Visual Computer, vol. 16, Issue 2, page 91-105, SCI&EI.
"C.-Y. Liou and W.-P. Tai, ¡§Conformal self-organization for continuity on a feature map¡¨ Neural Networks, vol. 12, page 893-905, 1999"
Introduction
1. Conventional SOM model arrange neurons on Euclidean space. It has two spaces: 1.(inner) Network space, the neurons are invariable in this space. 2.(outer) Input space, neuron compete with each other to learn the input patterns. See the Figure below.
2. 2D Euclidean mesh cannot wrap genus-0 manifold without seam. For example, see below, a 2D mesh cannot wrap the rabbit model.
3. For topological equivalent, the network space is extended to a sphere. For arranging neurons uniformly on the sphere surface, geodesic dome is used.
(a) Icosahedron with 20 faces (b)~(f) interpolation form (a)
4 . By using
conformal mapping between the network space and input space. The relative
location of the projection from given pattern in the input space can be mapped
backed to the network space.
SC map defines a conformal mapping from a polygon to a unit disk in complex; for example, left triangle to middle disk. By the intermediate unit disk, triangle to triangle conformal mapping can be get.
5. The learning process is illustrated by the Figure below.
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Demo
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<Demo movie, mpg 1.7MB | 6.7MB>
(a)~(b) Conformal spherical self-organizing map result for Venus head model, it contains 5762 neurons. (c)~(d) Spherical SOM result of the same parameter.
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(a)~(c) Conformal spherical self-organizing map result for Venus head model, it contains 5762 neurons. (d)~(f) Spherical SOM result of the same parameter.
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Right diagram shows the difference between these two models by conformality metric defined in CSM.
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<Demo movie, mpg 3.5MB>
Morphing procedure between two heads.
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(a) Original model (b) Trim the patterns beside its mouth (c) CSSM result
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(a) Male model with noise (b) CSSM result without showing the lines (c) CSSM result with showing the lines
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(c) Input data (d) CSSM result with variance of neighborhood function set to 0.2~0.1 (e) CSSM result with variance of neighborhood function set to 0.2~0.01
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If you have any question, feel free to contact us.
Cheng-Yuan
Liou Yen-Ting
Kuo
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