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

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

Sparse image reconstruction on the sphere: implications of a new sampling theorem

We study the impact of sampling theorems on the fidelity of sparse image reconstruction on the sphere. We discuss how a reduction in the number of samples required to represent all information content of a band-limited signal acts to improve the fidelity of sparse image reconstruction, through both the dimensionality and sparsity of signals. To demonstrate this result we consider a simple inpainting problem on the sphere and consider images sparse in the magnitude of their gradient. We develop a framework for total variation (TV) inpainting on the sphere, including fast methods to render the inpainting problem computationally feasible at high-resolution. Recently a new sampling theorem on the sphere was developed, reducing the required number of samples by a factor of two for equiangular sampling schemes. Through numerical simulations we verify the enhanced fidelity of sparse image reconstruction due to the more efficient sampling of the sphere provided by the new sampling theorem.Comment: 11 pages, 5 figure

Fast directional spatially localized spherical harmonic transform

We propose a transform for signals defined on the sphere that reveals their localized directional content in the spatio-spectral domain when used in conjunction with an asymmetric window function. We call this transform the directional spatially localized spherical harmonic transform (directional SLSHT) which extends the SLSHT from the literature whose usefulness is limited to symmetric windows. We present an inversion relation to synthesize the original signal from its directional-SLSHT distribution for an arbitrary window function. As an example of an asymmetric window, the most concentrated band-limited eigenfunction in an elliptical region on the sphere is proposed for directional spatio-spectral analysis and its effectiveness is illustrated on the synthetic and Mars topographic data-sets. Finally, since such typical data-sets on the sphere are of considerable size and the directional SLSHT is intrinsically computationally demanding depending on the band-limits of the signal and window, a fast algorithm for the efficient computation of the transform is developed. The floating point precision numerical accuracy of the fast algorithm is demonstrated and a full numerical complexity analysis is presented.Comment: 12 pages, 5 figure

S2LET: A code to perform fast wavelet analysis on the sphere

We describe S2LET, a fast and robust implementation of the scale-discretised wavelet transform on the sphere. Wavelets are constructed through a tiling of the harmonic line and can be used to probe spatially localised, scale-depended features of signals on the sphere. The scale-discretised wavelet transform was developed previously and reduces to the needlet transform in the axisymmetric case. The reconstruction of a signal from its wavelets coefficients is made exact here through the use of a sampling theorem on the sphere. Moreover, a multiresolution algorithm is presented to capture all information of each wavelet scale in the minimal number of samples on the sphere. In addition S2LET supports the HEALPix pixelisation scheme, in which case the transform is not exact but nevertheless achieves good numerical accuracy. The core routines of S2LET are written in C and have interfaces in Matlab, IDL and Java. Real signals can be written to and read from FITS files and plotted as Mollweide projections. The S2LET code is made publicly available, is extensively documented, and ships with several examples in the four languages supported. At present the code is restricted to axisymmetric wavelets but will be extended to directional, steerable wavelets in a future release.Comment: 8 pages, 6 figures, version accepted for publication in A&A. Code is publicly available from http://www.s2let.or

On the computation of directional scale-discretized wavelet transforms on the sphere

We review scale-discretized wavelets on the sphere, which are directional and allow one to probe oriented structure in data defined on the sphere. Furthermore, scale-discretized wavelets allow in practice the exact synthesis of a signal from its wavelet coefficients. We present exact and efficient algorithms to compute the scale-discretized wavelet transform of band-limited signals on the sphere. These algorithms are implemented in the publicly available S2DW code. We release a new version of S2DW that is parallelized and contains additional code optimizations. Note that scale-discretized wavelets can be viewed as a directional generalization of needlets. Finally, we outline future improvements to the algorithms presented, which can be achieved by exploiting a new sampling theorem on the sphere developed recently by some of the authors.Comment: 13 pages, 3 figures, Proceedings of Wavelets and Sparsity XV, SPIE Optics and Photonics 2013, Code is publicly available at http://www.s2dw.org

Slepian functions and their use in signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and on the surface of a sphere.Comment: Submitted to the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verla

Scalar and vector Slepian functions, spherical signal estimation and spectral analysis

It is a well-known fact that mathematical functions that are timelimited (or spacelimited) cannot be simultaneously bandlimited (in frequency). Yet the finite precision of measurement and computation unavoidably bandlimits our observation and modeling scientific data, and we often only have access to, or are only interested in, a study area that is temporally or spatially bounded. In the geosciences we may be interested in spectrally modeling a time series defined only on a certain interval, or we may want to characterize a specific geographical area observed using an effectively bandlimited measurement device. It is clear that analyzing and representing scientific data of this kind will be facilitated if a basis of functions can be found that are "spatiospectrally" concentrated, i.e. "localized" in both domains at the same time. Here, we give a theoretical overview of one particular approach to this "concentration" problem, as originally proposed for time series by Slepian and coworkers, in the 1960s. We show how this framework leads to practical algorithms and statistically performant methods for the analysis of signals and their power spectra in one and two dimensions, and, particularly for applications in the geosciences, for scalar and vectorial signals defined on the surface of a unit sphere.Comment: Submitted to the 2nd Edition of the Handbook of Geomathematics, edited by Willi Freeden, Zuhair M. Nashed and Thomas Sonar, and to be published by Springer Verlag. This is a slightly modified but expanded version of the paper arxiv:0909.5368 that appeared in the 1st Edition of the Handbook, when it was called: Slepian functions and their use in signal estimation and spectral analysi