Image Recovery from Sparse Samples, Discrete Sampling Theorem, and Sharply Bounded Band-Limited Discrete Signals

Image sampling is a special case of image discretization, the very first step in digital image processing, storage, and transmission. Generally, discretization is the representation of continuous images by sets of numbers. Discretization basis functions and their reciprocal reconstruction basis functions are physically implemented as point spread functions of discretization and reconstruction devices. Apart from the general desire to reduce the number of image samples required for image storage and transmission, there are many real applications, where, contrary to the common practice of uniform sampling, sampled data are collected in an irregular fashion. Because display devices, sound synthesizers, and other devices for reconstruction of continuous signals from their samples, as well as computer software for processing sampling data, assume the customary use of a regular uniform sampling grid, in all these cases it is necessary to convert irregularly sampled images to regularly sampled ones.