Zero-based ensemble deconvolution and EEG spectral topography. (1994)
Type of ContentTheses / Dissertations
Thesis DisciplineElectrical Engineering
Degree NameDoctor of Philosophy
PublisherUniversity of Canterbury
AuthorsSatherley, Brenda Leeshow all
This thesis presents theoretical and practical aspects of two distinctly different topics - zero based two-dimensional ensemble blind deconvolution and spectral topography of the electroencephalogram.
Reconstructing an estimate of some true image from a series of differently blurred and contaminated versions of that same true image constitutes the ensemble blind deconvolution problem, which most commonly arises in astronomical speckle imaging. Speckle imaging and a series of algorithms for recovering high spatial resolution estimates of astronomical objects are comprehensively reviewed. Many of these algorithms depend on prior knowledge of an estimate of the blurring characteristics of the atmosphere.
A new algorithm, capable of two-dimensional ensemble blind deconvolution and demonstrating a potential for application to astronomical speckle images, is introduced. This algorithm, referred to as zero track-based two-dimensional zero-and-add, is based on a principle that has been previously applied to ensembles of one-dimensional speckle images.
Fundamental to the new algorithm is the unique representation of any compact image by the points at which its analytically continued Fourier transform is equal to zero; these points form a zero sheet, a two-dimensional analytic surface in a four-dimensional space. The zero sheet of the spectrum of a convolution can be separated into two sheets which each describe one of the component images of the convolution. Given sufficient knowledge of a zero sheet, it is possible to reconstruct, to within a complex scaling factor, the image from which the zero sheet was derived. Thus, partitioning the zero sheet representing a convolution into separate analytic surfaces is equivalent to effecting deconvolution.
Because the true image is constant throughout an ensemble of speckle images, its zero sheet is a component of the zero sheets representing each member of the ensemble. Thus, the new algorithm attempts to recognise the common zero sheet representing the true image and thereby to obtain an estimate of that image. Extensive results are presented which demonstrate that the new algorithm can deconvolve a faithful estimate from a modestly sized ensemble of small images degraded by low levels of contamination.
Reconstruction of an image from its zero sheet has previously been implemented in two quite different ways, both placing certain restrictions upon zero sheet-based deconvolution schemes. A generalisation of one of these methods is formulated and demonstrated. This generalisation has greatly enhanced the practicality of the zero track-based two-dimensional zero-and-add algorithm.
In the presence of additive contamination, the zero sheets representing the components of a convolution merge together. Consequently, partitioning the zero sheet of a convolution is no longer possible without breaking the zero sheet in the regions of component sheet merging. This effect of contamination presents a challenging obstacle which zero sheet-based deconvolution algorithms must overcome in order to be of practical use. The merging of the component zero sheets is demonstrated with pseudo three-dimensional visualisations.
The importance of spectral analysis of the rhythmic activity of the electroencephalogram (EEG) is outlined. Traditionally, this analysis has been conducted visually by an electroencephalographer, however computer-based analysis of the background EEG is rapidly gaining popularity. An EEG spectral topography system computes the frequency components of the EEG waveforms and displays the results on topographic maps to provide an indication of the spatial distribution of the activity. A comprehensive review of EEG spectral topography systems is presented and the development of a new PC-based system is described.