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UBC Theses and Dissertations

Restoration of randomly blurred images Guan, Ling


This thesis considers the restoration of images blurred by random point spread functions. The stochastic point spread functions are represented by the sum of a deterministic blur function plus an image-dependent noise term. The stochastic properties of such systems are studied. Several linear filters are derived. These filters are based on the following criteria: the Wiener, the minimum variance unbiased, and the constrained least squares. A filter based on the geometrical mean methodology is also presented. The popular Wiener filter, and the minimum variance unbiased filter have already been applied to this problem for the one-dimensional case. We generalize these filters to the two-dimensional case. These filters are found to give good restoration results when the blur and/or the additive noise effects are not severe. When either or both of the above conditions are not satisfied, numerical instability may occur. The constrained least squares filter does not suffer from such ill conditioning. However, the performance of this method needs to be manually controlled in order to yield good restoration results. The geometrical mean filter is formed by combining the modified Wiener filter and the constrained least squares filter. This filter gives the best restoration amongst the four linear-based filters because it does not have the shortcomings the other filters have. A nonlinear filter, the maximum a posteriori filter is also derived and implemented. Since the random blur effect is a nonlinear stochastic process, we expect the nonlinear filtering scheme to simulate the reverse of the blurring process more closely. The superiority of this filter over the modified Wiener filter is demonstrated by examples. In addition, the restoration of random time-varying blurred images is also discussed. The Wiener criterion is adopted. The solution of this problem involves the processing of a huge amount of multi-image data. To circumvent this problem we present two approaches to realize this Wiener filter. One is based on the Karhunen-Loeve transformation, and the other is a direct Wiener filter implementation. These two approaches give better results than that given by processing any single frame individually. Since image restoration involves computation of very large systems, The computational effort is an important factor to consider in the implementation of all restoration algorithms. By means of the circulant matrix approximation and the fast Fourier transform, our methods are implemented in the frequency domain. The computational efforts are reduced significantly.

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