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Spherical harmonics in texture analysis |
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| Authors: | Helmut Schaeben K Gerald van den Boogaart |
| Institution: | Geoscience Mathematics and Informatics, Freiberg University of Mining and Technology, Gustav-Zeuner-Str. 12, Freiberg D-09596, Germany |
| Abstract: | The objective of this contribution is to emphasize the fundamental role of spherical harmonics in constructive approximation on the sphere in general and in texture analysis in particular. The specific purpose is to present some methods of texture analysis and pole-to-orientation probability density inversion in a unifying approach, i.e. to show that the classic harmonic method, the pole density component fit method initially introduced as a distinct alternative, and the spherical wavelet method for high-resolution texture analysis share a common mathematical basis provided by spherical harmonics. Since pole probability density functions and orientation probability density functions are probability density functions defined on the sphere Ω3 Therefore, first a basic and simplified method to derive real symmetrized spherical harmonics with the mathematical property of providing a representation of rotations or orientations, respectively, is presented. Then, standard orientation or pole probability density functions, respectively, are introduced by summation processes of harmonic series expansions of |
| Keywords: | Real-valued spherical harmonics Real-valued harmonics for the rotation group Harmonic series expansions Spectral decomposition of the X-ray transform Summability Spherical singular integral Spherical radial basis functions Spherical wavelets Methods of texture analysis |
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