A Discrete Modulo N Projective Radon Transform for N × N Images

Abstract

AbstractThis paper presents a Discrete Radon Transform (DRT) based on congruent mathematics that applies to N × N arrays where \(N \in \mathcal{N}\). This definition incorporates and is a natural extension of the more restricted cases of the finite Radon transform [1] where N must be prime, the discrete periodic Radon transform [2] where N must be a power of 2, and the DRT over p n [3], where N must be a power of a single prime. The DRT exactly and invertibly maps a 2-D image to a set of 1-D projections of length N. Projections are found as the sum of the pixels centred on a parallel set of discrete lines. The image is assumed to be periodic and these lines wrap around the array under modulo N arithmetic. Properties of the continuous Radon transform are preserved in the DRT; a discrete form of the Fourier slice theorem applies, as does the convolution property. A formula is given to find the projection set required to be exactly invertible for arrays with N any composite number, as well as a means to determine the level of redundancy in sampling that is introduced on such composite arrays.KeywordsDiscrete FormInversion ProcessProjection AngleRadon TransformShort VectorThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.