|Ph.D Thesis||Department of Electrical Engineering|
|Full Thesis text|
Continuous-domain operations have a fundamental role in image processing systems although many applications use digital data. This dichotomous situation in which continuous-domain formulation is applied to digitized data is currently overcome by one of two possible approaches. One approach considers continuous-domain models from sampled data (e.g. bandlimited, polynomial), while the other approach considers numerical approximation of continuous-domain operators (derivatives, integrals). In spite of the rich literature on these two approaches, it seems that within the field of image processing and analysis, the sampling process itself is not taken into account in the analysis or design of systems; the nature of the sampling process does not affect currently available continuous-domain models either. Furthermore, it is not clear as to what extent the sampling process keeps continuous-domain operations intact, where of special interest is the inner product operation that describes linear bounded functional in Hilbert spaces.
The main goal of this work is to investigate the effect of sampling on continuous-domain operations within the context of images. In particular, we suggest an alternative approach to image processing and analysis that utilizes the properties of the sampling process itself such as ideal vs. non-ideal sampling, uniform vs. non-uniform, etc. The approach taken is based on functional analysis of Sobolev spaces, incorporating stochastic properties of discrete-domain data into deterministic modeling of signals. It is shown that both ideal and non-ideal sampling procedures correspond to orthogonal projection operations in these spaces, allowing for a minimax approach to be derived for inner product approximation from sampled data. The underlying kernels are exponential functions and are related to stochastic autoregressive image modeling. A tight l2, rather than an L2, upper bound on the approximation error of continuous-domain operators is also derived, providing a useful error characterization for image processing applications. Unlike currently available polynomial-based models, the proposed approach is not restricted by the partition-of-unity condition or by the scaling property of shift-invariant spaces. Experimental results indicate that the proposed approach with properly-tuned signal-dependent weights outperforms currently available polynomial B-spline models of a comparable order. Furthermore, a unified approach to image interpolation by ideal and non-ideal sampling procedures is derived, suggesting that the proposed exponential kernels may have a significant role in image modeling as well.