Blind deconvolution and phase retrieval
Thesis DisciplineElectrical Engineering
Degree GrantorUniversity of Canterbury
Degree NameDoctor of Philosophy
Theoretical and practical aspects of identifying and deconvolving a convolution in more than one-dimension are presented. In contrast to conventional techniques which require knowledge of the blurring function, this thesis describes techniques for "blind" deconvolution. The techniques introduced differ from previous work in the field of blind deconvolution because they do not require an ensemble of similarly blurred images, i.e. they can be effectively employed upon a single convolution.
The first method for blind deconvolution introduced relies on the analytic properties of the Fourier spectrum of a compact image. Rather than deal with continuous images, a discrete approximation is employed. It is argued, however, that approximation of the Fourier spectrum by a finite order polynomial model is a logical response to the practical constraints posed by limited a.mounts of noisy data.
Since the convolution of two images is equivalent to a multiplication of their Fourier spectra, deconvolution is consequently equivalent to factorisation of their Fourier spectra. In one dimension it is always possible to factorise a polynomial, even when it is of infinite order. These factors correspond to isolated points, in the complex plane into which the Fourier spectra are analytically continued, where the spectra are zero. Since these points are distinct there are a large number of factors and hence there is usually a large number of ways of deconvolving a one-dimensional image.
By contrast the analytically continued Fourier spectrum of a two-dimensional image exists in a four-dimensional space and is zero on a two-dimensional analytic surface, here called a zero-sheet. Because of the analytic nature of the zero-sheet it is not possible, in general, to factorise a two-dimensional spectrum or equivalently partition its zerosheet into separate analytic surfaces. The major exception is when the true image is a convolution in which case the zero-sheet is, in fact, the union of the zero-sheets of the components of the convolution. As a result the zero-sheet of a convolution can be partitioned into two zero-sheets which can be used to recover, to within a complex constant, the components of the convolution. The addition of noise is shown to link the zero-sheets of the components of the convolution. Consequently it is no longer possible to partition the zero-sheet without isolating and correcting these "bridges" between the zero-sheets of the components.
The Fourier phase problem forms a special subclass of the blind deconvolution problem, one in which the true image and the blurring function are conjugate mirror images of each other. The data in the Fourier phase problem comprises the oversampled magnitude of the Fourier transform of the true image. Consequently, it is necessary to reconstruct the Fourier phase before an estimate of the true image can be formed. It is shown that a solution exists and the accuracy of the solution can be empirically related to the amount of noise present in the Fourier magnitude data.
It is shown that a unique solution to the Fourier phase problem in more than one dimension exists except when the spectrum is the Fourier transform of a convolution. In this case, the number of solutions to the Fourier phase problem is related to the number of component images which have been convolved to produce the convolution. The second technique for deconvolution introduced in this thesis uses these multiple solutions to the Fourier phase problem to recover information about the phase of the spectra of the components of the convolution. The Fourier phase is, however, only recovered modulo π. The problems encountered in the modified magnitude problem, as it is called in this thesis, are analysed and techniques for overcoming these difficulties are described. A final result presented herein is an extension to an existing technique for blind deconvolution of ensembles of two-dimensional speckle images. It is shown that comparing the zero-sheets of the speckle spectra leads to a useful new approach to speckle imaging.