Spatio-spectral analysis on the unit sphere

This thesis is focussed on the development of new signal processing techniques to analyse signals defined on the sphere. Analysis and processing of signals defined on the sphere find applications in various fields of science and engineering, such as cosmology, geophysics and medical imaging. The objective to develop new signal processing methods is served by formulating, extending and tailoring existing Euclidean domain signal processing theories in ways that they become suitable for analysis of signals defined on the sphere. The first part of this thesis develops a new type of convolution between two signals on the sphere. This is the first type of convolution on the sphere which is commutative. Two other advantages, in comparison with existing definitions in the literature, are that the new convolution admits anisotropic filters and signals and the domain of the output remains on the sphere. The spectral analysis of the convolution is provided and a fast algorithm for efficient computation of convolution output is developed. The second part of the thesis is focused on the development of signal processing techniques to analyse signals on the sphere in joint spatio-spectral~(spatial-spectral) domain. A transform analogous to short-time Fourier transform(STFT) in time-frequency analysis is formulated for signals defined on the sphere, in order to devise a spatio-spectral representation of a signal. The proposed transform is referred as the spatially localized spherical harmonic transform~(SLSHT) and is defined as windowed spherical harmonic transform, resulting in the SLSHT distribution. The properties of the SLSHT distribution and its analysis in the spherical harmonic domain are also provided. Furthermore, examples are provided to demonstrate the capability of SLSHT to reveal spatially localized spectral contents in a signal that were not obtainable from traditional spherical harmonics analysis. With the consideration that data-sets on the sphere can be of considerable size and the 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. A general framework for spatially-varying spectral filtering of signals defined on the unit sphere is also developed, as an analogy to joint time-frequency filtering. For spatio-spectral filtering, the spherical signals are first mapped from the spatial domain into a joint spatio-spectral domain using SLSHT, where a spatio-spectral signal transformation or modification is introduced. Next, a suitable scheme to transform the modified signal from the spatio-spectral domain back to an admissible signal in the spatial domain using the least squares approach is proposed. It is shown that the overall action of the SLSHT and spatio-spectral signal modification can be described through a single transformation matrix, which is useful in practice. Finally, two specific and useful instances of spatially-varying spectral filtering are presented, defined through multiplicative and convolutive modification of the SLSHT distribution. The proposed framework enables filtering or modification in the spatio-spectral domain which cannot be carried out in either the spatial or spectral domain

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

Dynamic Decomposition of Spatiotemporal Neural Signals

Neural signals are characterized by rich temporal and spatiotemporal dynamics that reflect the organization of cortical networks. Theoretical research has shown how neural networks can operate at different dynamic ranges that correspond to specific types of information processing. Here we present a data analysis framework that uses a linearized model of these dynamic states in order to decompose the measured neural signal into a series of components that capture both rhythmic and non-rhythmic neural activity. The method is based on stochastic differential equations and Gaussian process regression. Through computer simulations and analysis of magnetoencephalographic data, we demonstrate the efficacy of the method in identifying meaningful modulations of oscillatory signals corrupted by structured temporal and spatiotemporal noise. These results suggest that the method is particularly suitable for the analysis and interpretation of complex temporal and spatiotemporal neural signals

BIGRE: a low cross-talk integral field unit tailored for extrasolar planets imaging spectroscopy

Integral field spectroscopy (IFS) represents a powerful technique for the detection and characterization of extrasolar planets through high contrast imaging, since it allows to obtain simultaneously a large number of monochromatic images. These can be used to calibrate and then to reduce the impact of speckles, once their chromatic dependence is taken into account. The main concern in designing integral field spectrographs for high contrast imaging is the impact of the diffraction effects and the non-common path aberrations together with an efficient use of the detector pixels. We focus our attention on integral field spectrographs based on lenslet-arrays, discussing the main features of these designs: the conditions of appropriate spatial and spectral sampling of the resulting spectrograph's slit functions and their related cross-talk terms when the system works at the diffraction limit. We present a new scheme for the integral field unit (IFU) based on a dual-lenslet device (BIGRE), that solves some of the problems related to the classical TIGER design when used for such applications. We show that BIGRE provides much lower cross-talk signals than TIGER, allowing a more efficient use of the detector pixels and a considerable saving of the overall cost of a lenslet-based integral field spectrograph.Comment: 17 pages, 18 figures, accepted for publication in Ap

Differential fast fixed-point algorithms for underdetermined instantaneous and convolutive partial blind source separation

This paper concerns underdetermined linear instantaneous and convolutive blind source separation (BSS), i.e., the case when the number of observed mixed signals is lower than the number of sources.We propose partial BSS methods, which separate supposedly nonstationary sources of interest (while keeping residual components for the other, supposedly stationary, "noise" sources). These methods are based on the general differential BSS concept that we introduced before. In the instantaneous case, the approach proposed in this paper consists of a differential extension of the FastICA method (which does not apply to underdetermined mixtures). In the convolutive case, we extend our recent time-domain fast fixed-point C-FICA algorithm to underdetermined mixtures. Both proposed approaches thus keep the attractive features of the FastICA and C-FICA methods. Our approaches are based on differential sphering processes, followed by the optimization of the differential nonnormalized kurtosis that we introduce in this paper. Experimental tests show that these differential algorithms are much more robust to noise sources than the standard FastICA and C-FICA algorithms.Comment: this paper describes our differential FastICA-like algorithms for linear instantaneous and convolutive underdetermined mixture

Detecting Sunyaev-Zel'dovich clusters with PLANCK: III. Properties of the expected SZ-cluster sample

The PLANCK-mission is the most sensitive all-sky submillimetric mission currently being planned and prepared. Special emphasis is given to the observation of clusters of galaxies by their thermal Sunyaev-Zel'dovich (SZ) effect. In this work, the results of a simulation are presented that combines all-sky maps of the thermal and kinetic SZ-effect with cosmic microwave background (CMB) fluctuations, Galactic foregrounds (synchrotron emission, thermal emission from dust, free-free emission and rotational transitions of carbon monoxide molecules) and sub-millimetric emission from planets and asteroids of the Solar System. Observational issues, such as PLANCKs beam shapes, frequency response and spatially non-uniform instrumental noise have been incorporated. Matched and scale-adaptive multi-frequency filtering schemes have been extended to spherical coordinates and are now applied to the data sets in order to isolate and amplify the weak thermal SZ-signal. The properties of the resulting SZ-cluster sample are characterised in detail: Apart from the number of clusters as a function of cluster parameters such as redshift z and total mass M, the distribution n(sigma)d sigma of the detection significance sigma, the number of detectable clusters in relation to the model cluster parameters entering the filter construction, the position accuracy of an SZ-detection and the cluster number density as a function of ecliptic latitude beta is examined.Comment: 14 pages, 16 figures, 13 tables, submitted to MNRAS, 16.Feb.200

Multichannel interference mitigation methods in radio astronomy

Radio-astronomical observations are increasingly corrupted by RF interference, and online detection and filtering algorithms are becoming essential. To facilitate the introduction of such techniques into radio astronomy, we formulate the astronomical problem in an array signal processing language, and give an introduction to some elementary algorithms from that field. We consider two topics in detail: interference detection by rank estimation of short-term covariance matrices, and spatial filtering by subspace estimation and projection. We discuss experimental data collected at the Westerbork radio telescope, and illustrate the effectiveness of the space-time detection and blanking process on the recovery of a 3C48 absorption line in the presence of GSM mobile telephony interference.Comment: 39 pages, 18 figures.Enhanced figures can be downloaded from http://cas.et.tudelft.nl/~leshem/postscripts/leshem/figs34567.ps.gz To appear in Astrophysical Journal Supplements serie

Radio astronomical imaging in the presence of strong radio interference

Radio-astronomical observations are increasingly contaminated by interference, and suppression techniques become essential. A powerful candidate for interference mitigation is adaptive spatial filtering. We study the effect of spatial filtering techniques on radio astronomical imaging. Current deconvolution procedures such as CLEAN are shown to be unsuitable to spatially filtered data, and the necessary corrections are derived. To that end, we reformulate the imaging (deconvolution/calibration) process as a sequential estimation of the locations of astronomical sources. This not only leads to an extended CLEAN algorithm, the formulation also allows to insert other array signal processing techniques for direction finding, and gives estimates of the expected image quality and the amount of interference suppression that can be achieved. Finally, a maximum likelihood procedure for the imaging is derived, and an approximate ML image formation technique is proposed to overcome the computational burden involved. Some of the effects of the new algorithms are shown in simulated images. Keywords: Radio astronomy, synthesis imaging, parametric imaging, interference mitigation, spatial filtering, maximum likelihood, minimum variance, CLEAN.Comment: 27 pages, 7 figures. Paper with higher resolution color figures at http://cobalt.et.tudelft.nl/~leshem/postscripts/leshem/imaging.ps.g