Thesis to the dissertation examination

Supervisor: doc. Ing. Miloˇs Drutarovsk´y, CSc. ”If we save even one life, we have been cost effective.”

Abstract not found

The spectral analysis of non uniform sampled data sequences using Fourier Periodogram method is the classical approach.In view of data fitting and computational standpoints why the Least squares periodogram (LSP) method is preferable than the “classical ” Fourier periodogram and as well as to the frequentlyused form of LSP due to Lomb and Scargle is explained. Then a new method of spectral analysis of nonuniform data sequences can be interpreted as an iteratively weighted LSP that makes use of a data-dependent weighting matrix built from the most recent spectral estimate. It is iterative and it makes use of an adaptive (i.e., data-dependent) weighting, we refer to it as the iterative adaptive approach (IAA).LSP and IAA are nonparametric methods that can be used for the spectral analysis of general data sequences with both continuous and discrete spectra. However, they are most suitable for data sequences with discrete spectra (i.e., sinusoidal data), which is the case we emphasize in this paper. Of the existing methods for nonuniform sinusoidal data, Welch, MUSIC and ESPRIT methods appear to be the closest in spirit to the IAA proposed here. Indeed, all these methods make use of the estimated covariance matrix that is computed in the first iteration of IAA from LSP. Comparative study of LSP with MUSIC and ESPRIT methods are discussed.

Abstract not found

We introduce a general purpose algorithm for rapidly computing certain types of oscillatory integrals which frequently arise in problems connected to wave propagation and general hyperbolic equations. The problem is to evaluate numerically a so-called Fourier integral operator (FIO) of the form ∫ e 2πiΦ(x,ξ) a(x, ξ) ˆ f(ξ)dξ at points given on a Cartesian grid. Here, ξ is a frequency variable, ˆ f(ξ) is the Fourier transform of the input f, a(x, ξ) is an amplitude and Φ(x, ξ) is a phase function, which is typically as large as |ξ|; hence the integral is highly oscillatory at high frequencies. Because an FIO is a dense matrix, a naive matrix vector product with an input given on a Cartesian grid of size N by N would require O(N 4) operations. This paper develops a new numerical algorithm which requires O(N 2.5 log N) operations, and as low as O ( √ N) in storage space. It operates by localizing the integral over polar wedges with small angular aperture in the frequency plane. On each wedge, the algorithm factorizes the kernel e 2πiΦ(x,ξ) a(x, ξ) into two components: 1) a diffeomorphism which is handled by means of a nonuniform FFT and 2) a residual factor which is handled by numerical separation of the spatial and frequency variables. The key to the complexity and accuracy estimates is that the separation rank of the residual kernel is provably independent of the problem size. Several numerical examples demonstrate the efficiency and accuracy of the proposed methodology. We also discuss the potential of our ideas for various applications such as reflection seismology.

We present a combinatorial method for solving a certain system of polynomial equations of Vandermonde type in 2N variables by reducing it to the problem of solving two special linear systems of size N and rooting a single univariate polynomial of degree N. Over C, all solutions can be found with fixed precision using, up to polylogarithmic factors, O(N 2) bitwise operations in the worst case. Furthermore, if the data is well conditioned, then this can be reduced to O(N) bit operations, up to polylogarithmic factors. As an application, we show how this can be used to fit data to a complex exponential sum with N terms in the same, nearly optimal, time.

Non-Cartesian sampling is widely used for fast magnetic resonance imaging (MRI). Accurate and fast image reconstruction from non-Cartesian k-space data becomes a challenge and gains a lot of attention. Images provided by conventional direct reconstruction methods usually bear ringing, streaking, and other leakage artifacts caused by discontinuous structures. In this paper, we tackle these problems by analyzing the principal point spread function (PSF) of non-Cartesian reconstruction and propose a leakage reduction reconstruction scheme based on discontinuity subtraction. Data fidelity in k-space is enforced during each iteration. Multidimensional nonuniform fast Fourier transform (NUFFT) algorithms are utilized to simulate the k-space samples as well as to reconstruct images. The proposed method is compared to the direct reconstruction method on computer-simulated phantoms and physical scans. Non-Cartesian sampling trajectories including 2D spiral, 2D and 3D radial trajectories are studied. The proposed method is found useful on reducing artifacts due to high image discontinuities. It also improves the quality of images reconstructed from undersampled data. Copyright © 2006 J. Song and Q. H. Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.