Abstract. The Radon transform Rf of functions f on SO(3) has recently been applied extensively in texture analysis, i.e. the analysis of preferred crystallographic orientation. In practice one has to determine the orientation probability density function f ∈ L 2 (SO(3)) from Rf ∈ L 2 (S 2 × S 2 ) which is known only on a discrete set of points. Since one has only partial information about Rf the inversion of the Radon transform becomes an ill-posed inverse problem. Motivated by this problem we define a new notion of the Radon transform Rf of functions f on general compact Lie groups and introduce two approximate inversion algorithms which utilize our previously developed generalized variational splines on manifolds. Our new algorithms fit very well to the application of Radon transform on SO(3) to texture analysis.
The paper at hand is concerned with creating a flexible wavelet theory on the three sphere S 3 and the rotation group SO(3). The theory of zonal functions and reproducing kernels will be used to develop conditions for an admissible wavelet. After explaining some preliminaries on group actions and some basics on approximation theory, we will prove reconstruction formulas of linear and bilinear wavelet transformed L 2 -functions on S 3 . Moreover, specific examples will be constructed and visualized. Second, we deal with the construction of wavelets on the rotation group SO(3). It will be shown that the Radon transform of a wavelet packet on SO(3) gives a wavelet packet on S 2 for every fixed detection direction.
The aim of this exposition is to explain basic ideas behind the concept of
diffusive wavelets on spheres in the language of representation theory of Lie
groups and within the framework of the group Fourier transform given by
Peter-Weyl decomposition of $L^2(G)$ for a compact Lie group $G$.
After developing a general concept for compact groups and their homogeneous
spaces we give concrete examples for tori -which reflect the situation on
$R^n$- and for spheres $S^2$ and $S^3$.Comment: 20 pages, 3 figure
In this paper we study the solutions to the diffusion equation on some conformally flat cylinders and on the n-torus. Using the Clifford algebra calculus with an appropriate Witt basis, the solutions can be expressed as multiperiodic eigensolutions to the parabolic Dirac operator. We study their fundamental properties, give representation formulas of all these solutions and develop some integral representation formulas. In particular we set up a Green type formula for the solutions to the homogeneous diffusion equation on cylinders and tori. Then we also treat the inhomogeneous diffusion equation diffusion with prescribed boundary conditions in Lipschitz domains on these manifolds. As main application, we construct well localized diffusion wavelets on this class of cylinders and tori by means of multiperiodic eigensolutions to the parabolic Dirac operator. We round off with presenting some concrete numerical simulations for the three dimensional case.
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