Sampling and Interpolation in Two Dimensions

Sampling and interpolation in two dimensions is much richer than in one dimension. Not only are there polar coordinates and other coordinate systems in addition to cartesian, but sampling can be done along lines as well as at points. The distinction between point and line sampling will be discussed first. Since sampling at regular intervals plays a major role, we pass on to regular point sampling as expressed in one dimension by the shah function III(x), an entity that was introduced earlier in connection with delta functions. In two dimensions we find the direct generalization 2III(x, y), which occupies a square lattice of points, together with a range of other manifestations of the shah function. Since the Fourier transform of a sampling pattern is a transfer function, this treatment brings us to the brink of digital filtering. Before developing this subject in the next chapter, however, we discuss the sampling theorem and interpolation.