Signal concentration and related concepts in time-frequency and on the unit sphere

Abstract

Unit sphere signal processing is an increasingly active area of research with applications in computer vision, medical imaging, geophysics, cosmology and wireless communications. However, comparing with signal processing in time-frequency domain, characterization and processing of signals defined on the unit sphere is relatively unfamiliar for most of the engineering researchers. In order to better understand and analysis the current issues using the spherical model, such as analysis of brain neural electronic activities in medical imaging and neuroscience, target detection and tracking in radar systems, earthquake occurrence prediction and seismic origin detection in seismology, it is necessary to set up a systematic theory for unit sphere signal processing. How to efficiently analyze and represent functions defined on the unit sphere are central for the unit sphere signal processing, such as filtering, smoothing, detection and estimation in the presence of noise and interference. Slepian-Landau-Pollak time-frequency energy concentration theory and the essential dimensionality of time-frequency signals by the Fourier transform are the fundamental tools for signal processing in the time-frequency domain. Therefore, our research work starts from the analogies of signals between time-frequency and spatial-spectral. In this thesis, we first formulate the k-th moment time-duration weighting measure for a band-limited signal using a general constrained variational method, where a complete, orthonormal set of optimal band-limited functions with the minimum fourth moment time-duration measure is obtained and the prospective applications are discussed. Further, the formulation to an arbitrary signal with second and fourth moment weighting in both time and frequency domain is also developed and the corresponding optimal functions are obtained, which are helpful for practical waveform designs in communication systems. Next, we develop a k-th spatially global moment azimuthal measure (GMZM) and a k-th spatially local moment zenithal measure (LMZM) for real-valued spectral-limited signals. The corresponding sets of optimal functions are solved and compared with the spherical Slepian functions. In addition, a harmonic multiplication operation is developed on the unit sphere. Using this operation, a spectral moment weighting measure to a spatial-limited signal is formulated and the corresponding optimal functions are solved. However, the performance of these sets of functions and their perspective applications in real world, such as efficiently analysis and representation of spherical signals, is still in exploration. Some spherical quadratic functionals by spherical harmonic multiplication operation are formulated in this thesis. Next, a general quadratic variational framework for signal design on the unit sphere is developed. Using this framework and the quadratic functionals, the general concentration problem to an arbitrary signal defined on the unit sphere to simultaneously achieve maximum energy in the finite spatial region and finite spherical spectrum is solved. Finally, a novel spherical convolution by defining a linear operator is proposed, which not only specializes the isotropic convolution, but also has a well defined spherical harmonic characterization. Furthermore, using the harmonic multiplication operation on the unit sphere, a reconstruction strategy without consideration of noise using analysis-synthesis filters under three different sampling methods is discussed.