Mathematical topics in imaging: Sampling theory and eigenfunction analysis of imaging systems

We use the Whittaker-Shannon sampling theorem as a guideline to explore several mathematical ideas that originate from problems in imaging. First a modified form of sampling theorem is derived for efficient sampling of carrier-frequency type signals and applied to direct coarse sampling and demodulation of electronic holograms. The concepts such as space-bandwidth product and uncertainty are studied in detail from the point of view of energy concentration and a connection between the sampling theorem and the eigenfunctions of the sinc-kernel (prolate spheroidal wave functions) is established. The prolate spheroids are further used to introduce a fractional version of the finite Fourier transform and the associated inverse problems are studied. The sampling formulae for carrier-frequency signals are employed to construct a new set of orthogonal functions---that we name as "bandpass prolate spheroids"---which is an efficient basis set for representing carrier-frequency type signals on finite intervals. The sampling theorem based treatment of prolate spheroids is then extended to study the eigenvalue problems associated with general bandlimited integral kernels and a non-iterative method for computing eigenwavefronts of linear space-invariant imaging systems (including aberrated ones) is presented. An important result established in this analysis is that the number of significant eigenvalues of an imaging system is of the order of its space-bandwidth product. A definition for the space-bandwidth product or the information carrying capacity of an imaging system is proposed with the help of eigenwavefronts. The application of eigenwavefronts to inverse or de-convolution problems in imaging is also demonstrated.