Showing papers by "Daniel Potts published in 2001"

Fast Fourier transforms for nonequispaced data: a tutorial

Abstract: In this chapter we consider approximativemethods for the fast computation of multivariate discrete Fourier transforms for nonequispaced data (NDFT) in the time domain and in the frequency domain. In particularwe are interested in the approximation error as function of the arithmetic complexity of the algorithm. We discuss the robustness of NDFTiaalgorithms with respect to roundoff errors and applyNDFTalgorithms for the fast computation of Besseltransforms.

A new linogram algorithm for computerized tomography

Abstract: We propose a new linogram algorithm for the high quality Fourier reconstruction of digital N x N images from their Radon transform. The algorithm is based on univariate fast Fourier transforms for nonequispaced data in the time domain and in the frequency domain. The algorithm requires only O(N 2 log N) arithmetic operations and preserves the good reconstruction quality of the filtered backprojection.

Spherical Wavelets with an Application in Preferred Crystallographic Orientation

Abstract: Several useful representations of a function f : Ωp 7→ IR exist which are usually related to specific purposes: (i) series expansion into spherical harmonics to do mathematics, (ii) series expansion into (unimodal) radial basis functions to do probability and statistics, (iii) series expansion into spline functions to do numerics. In many practical applications the common problem is to reconstruct an approximation of f from sampled data (ri, f(ri)), i = 1, . . . , n, with some convenient properties using one of the above representations. Their critical parameter, e.g. (i) the degree of the harmonic series expansion, (ii) the spherical dispersion of unimodal radial functions, (iii) the choice of the knots, may to some extent be adjusted to the total number and/or the geometric arrangement of the measurement locations. However, these representations are in no way involved in the sampling process itself. After briefly reviewing the basics of wavelets and the specifics of spherical wavelets, another representation of f in terms of spherical wavelets is introduced. It will be shown that spherical wavelets are well suited to render functions defined on a sphere. Moreover, it will be demonstrated that wavelets are well apt to allow for locally varying spatial resolution, thus providing a digital device to zoom into those spherical areas where the function f is of special interest. Such a device seems to be required to increase the spatial resolution by a factor of 1000 or greater locally. Thus, spherical wavelets provide the means to control the sampling process to gradually adapt automatically to a local refinement of the spatial resolution. In particular, it is shown that spherical wavelets apply to X–ray pole intensity data as well as to crystallographic orientation density functions, and that the multiscale resolution easily transfers from pole spheres to orientation space.

Circulant preconditioners for indefinite Toeplitz systems

Abstract: In recent papers circulant preconditioners were proposed for ill-conditioned Hermitian Toeplitz matrices generated by 2π-periodic continuous functions with zeros of even order. It was shown that the spectra of the preconditioned matrices are uniformly bounded except for a finite number of outliers and therefore the conjugate gradient method, when applied to solving these circulant preconditioned systems, converges very quickly. In this paper, we consider indefinite Toeplitz matrices generated by 2π-periodic continuous functions with zeros of odd order. In particular, we show that the singular values of the preconditioned matrices are essentially bounded. Numerical results are presented to illustrate the fast convergence of CGNE, MINRES and QMR methods.