Reconstruction of Structured Functions From Sparse Fourier Data

In several scientific areas, such as radio astronomy, computed tomography, and magnetic resonance imaging, the reconstruction of structured functions from the knowledge of samples of their Fourier transform is a common problem. For the analysis of the examined object, it is important to reconstruct the underlying original signal as exactly as possible. The dissertation on hand aims to uniquely recover structured functions from a smallest possible set of Fourier data. For this purpose, the Prony method, which is a deterministic method for the recovery of sparse trigonometric functions, is used as key instrument to derive algorithms for unique recovery by means of a smallest possible set of Fourier data. First, in the univariate case, reconstruction methods for B-spline functions with non-uniform knots and linear combinations of non-uniform translates of a known low-pass filter function are proposed. Then these reconstruction results are transferred to the bivariate case in a similar way by considering separable functions such as tensor-products of non-uniform spline functions and non-uniform translates. Further, the case of non-separable functions in two dimensions is studied by examining linear combinations of N non-uniform shifts of a given bivariate function. Here, the unknown shift parameters and corresponding coefficients in the linear combination are recovered from sparse Fourier data. Unique recovery of the parameters is possible by using only 3N+1 Fourier samples on three lines through the origin. For this purpose, two predetermined lines are considered, while the third sampling line is chosen dependently on the results obtained by employing the samples from the first two lines. These results are compared to the case when only predetermined sampling lines are employed, and a generalization to higher dimensions is proposed. Further, the reconstruction of polygonal shapes in the real plane is examined. Here, a convex or non-convex polygonal domain D with N vertices is considered. It is shown that the vertices and their order can be reconstructed by taking 3N samples of the Fourier transform of the characteristic function of the polygonal domain D. Again, two predetermined sampling lines and an appropriately chosen third line are used. In several cases, the reconstruction results are illustrated with numerical experiments