Sampling and Reconstruction of Spherical Signals for Applications in Cosmology, Acoustics and Beyond

Abstract

The main focus of this thesis is using the existing spherical signal processing techniques to sample and reconstruct data under different application driven scenarios. The thesis consist of three major parts and the problems of sampling and reconstruction are discussed first. Then, the significance of signal processing techniques like spatial filtering are discussed in the field of acoustics. A sampling scheme is proposed for the representation of spin-$s$ band-limited functions on the sphere, which requires optimal number of samples equal to the number of degrees of freedom. In comparison to the existing sampling designs, which require ${\sim}2L^2$ samples for the representation of spin-$s$ functions band-limited at $L$, the proposed scheme requires $L^2-s^2$ samples for the accurate computation of the spin-$s$ spherical harmonic transform~($s$-SHT). A method is developed to compute the $s$-SHT and samples are taken such that matrices involved in the computation of $s$-SHT are well conditioned. In order to improve the accuracy further, a multi-pass $s$-SHT method is also proposed. Geometrical properties like sampling efficiency, minimum geodesic distance, mesh norm and mesh ratio give us an insight of the nature of distribution of the points on the sphere. A comparative analysis with the existing schemes show that the proposed sampling design exhibits superior geometrical properties. Algorithms for signal reconstruction on the sphere are developed and analysed for two different scenarios: i) when the measurements are not taken over a pre-defined grid and ii) when the estimation is done from incomplete measurements. For the first one, the generalized iterative residual fitting~(IRF) for the computation of the spherical harmonic transform~(SHT) of band-limited signals on the sphere is presented. The proposed method is based on the partitioning of the subspace of band-limited signals into orthogonal subspaces. There exist sampling schemes on the sphere which support accurate computation of SHT. The proposed IRF method enables accurate computation of SHTs of signals with randomly distributed sufficient number of samples. In order to improve the accuracy of the computation of the SHT, multi-pass IRF is proposed which adds multiple iterative passes to the IRF. An iterative algorithm for the extrapolation of band-limited signals from incomplete measurements on the sphere is proposed. The proposed algorithm improves the accuracy of the extrapolation of band-limited signals by using the information contained in the out-of-band harmonic coefficients of the signal to update the extrapolated signal at each iteration. The proposed algorithm does not only exploit the band-limited property of the signal at each iteration but also uses the harmonic coefficients outside the harmonic domain to improve the accuracy of signal extrapolation. To demonstrate the improvement in the accuracy, numerical experiments are conducted and a comparison is done with the results of the existing iterative conjugate gradient method. The signal processing technique of spatial filtering is exploited in order to design an anti-aliasing filter for the applications in acoustics. In acoustics, the performance of spherical microphone arrays is typically limited by spatial aliasing which introduces side-lobes in the array beam pattern. In order to reduce the aliasing error, a spatially constrained anti-aliasing filter is proposed which approximates an ideal anti-aliasing filter used in literature as a weighted sum of concentrated eigenfunctions obtained by solving the Slepian concentration problem on the sphere. Three performance parameters namely white noise gain~(WNG), directivity index~(DI) and processing loss are employed to compare the performance of proposed filter with the ideal filter. A parameter-constrained filter design is also proposed by maximizing WNG subject to constraints on the DI and processing loss of the proposed filter. Show more