Nonuniform generalized sampling
University of Cambridge
Department of Applied Mathematics and Theoretical Physics
Doctor of Philosophy (PhD)
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Gatarić, M. (2016). Nonuniform generalized sampling (Doctoral thesis). https://doi.org/10.17863/CAM.8433
In this thesis we study a novel approach to stable recovery of unknown compactly supported L2 functions from finitely many nonuniform samples of their Fourier transform, so-called Nonuniform Generalized Sampling (NUGS). This framework is based on a recently introduced idea of generalized sampling for stable sampling and reconstruction in abstract Hilbert spaces, which allows one to tailor the reconstruction space to suit the function to be approximated and thereby obtain rapidly-convergent approximations. While preserving this important hallmark, NUGS describes sampling by the use of weighted Fourier frames, thus allowing for highly nonuniform sampling schemes with the points taken arbitrarily close. The particular setting of NUGS directly corresponds to various image recovery models ubiquitous in applications such as magnetic resonance imaging, computed tomography and electron microscopy, where Fourier samples are often taken not necessarily on a Cartesian grid, but rather along spiral trajectories or radial lines. Specifically, NUGS provides stable recovery in a desired reconstruction space subject to sufficient sampling density and sufficient sampling bandwidth, where the latter depends solely on the particular reconstruction space. For univariate compactly supported wavelets, we show that only a linear scaling between the number of wavelets and the sampling bandwidth is both sufficient and necessary for stable recovery. Furthermore, in the wavelet case, we provide an efficient implementation of NUGS for recovery of wavelet coefficients from Fourier data. Additionally, the sufficient relation between the dimension of the reconstruction space and the bandwidth of the nonuniform samples is analysed for the reconstruction spaces of piecewise polynomials or splines with a nonequidistant sequence of knots, and it is shown that this relation is also linear for splines and piecewise polynomials of fixed degree, but quadratic for piecewise polynomials of varying degree. In order to derive explicit guarantees for stable recovery from nonuniform samples in terms of the sampling density, we also study conditions sufficient to ensure existence of a particular frame. Firstly, we establish the sharp and dimensionless sampling density that is sufficient to guarantee a weighted Fourier frame for the space of multivariate compactly supported L2 functions. Furthermore, subject to non-sharp densities, we improve existing estimates of the corresponding frame bounds. Secondly, we provide sampling densities sufficient to ensure a frame, as well as, estimates of the corresponding frame bounds, when a multivariate bandlimited function and its derivatives are sampled at nonuniformly spaced points.
This record's DOI: https://doi.org/10.17863/CAM.8433