Digital Projections in Prime and Composite Arrays

Abstract

The Discrete Radon Transform (DRT) provides a 1:1 mapping between any discrete array, for example a 3D micro-crystal or a 2D digital photograph, and its digital projections. The DRT is both conceptually simple and computationally efficient, yet has an engaging richness of descriptive power in characterising discrete systems. So far the DRT has been applied to the reconstruction and analysis of digital images based on digital projections, but it would appear that the DRT has much more to offer. The DRT has roots in fundamental number theory through pseudo-random sequences as well as the distribution of the number of primes and hence with the zeros of the Riemann zeta function. The linkage of these number-theoretic aspects to the properties of physical systems means the DRT may find application in diverse fields. It may be possible to model the chaotic behaviour of the energy spectra of discrete systems using a set of basis functions derived from the DRT. This paper describes the intrinsic properties of discrete arrays as defined by the sets of digital projections generated by the 2D DRT. In particular, it presents details of the angle and sampling distance interval distributions for digital projections, for square and hexagonal arrays of prime and composite size. Arrays of composite size are of interest in representing the spectral properties of crystals containing granular domains of variable size.