Comments on “Generalised finite Radon transform for N × N images”

In Kingston and Svalbe [1], a generalized finite Radon transform (FRT) that applied to square arrays of arbitrary size N × N was defined and the Fourier slice theorem was established for the FRT. Kingston and Svalbe asserted that “the original definition by Matúš and Flusser was restricted to apply only to square arrays of prime size,” and “Hsung, Lun and Siu developed an FRT that also applied to dyadic square arrays,” and “Kingston further extended this to define an FRT that applies to prime-adic arrays”. It should be said that the presented generalized FRT together with the above FRT definitions repeated the known concept of tensor representation, or tensor transform of images of size N × N which was published earlier by Artyom Grigoryan in 1984–1991 in the USSR. The above mentioned “Fourier slice theorem” repeated the known tensor transform-based algorithm of 2-D DFT [5–11], which was developed for any order N1 × N2 of the transformation, including the cases of N × N, when N = 2r, (r > 1), and N = Lr, (r ≥ 1), where L is an odd prime. The problem of “over-representation” of the two-dimensional discrete Fourier transform in tensor representation was also solved by means of the paired representation in Grigoryan [6–9].

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► Generalized finite Radon transform was the tensor transform published in 1984–1986.
► “Fourier slice theorem” repeated the known tensor transform-based algorithm.
► Grigoryan split the N × N-point 2-D DFT on orbits by the paired transforms in 1986.
► Image reconstruction from their projections was solved by paired transforms in 1986.
► Orthogonal discrete periodic Radon transform was exactly the paired transform.