Generalised finite radon transform for N×N images

This paper extends the domain of the finite radon transform (FRT) to apply to square arrays of arbitrary size. The FRT is a discrete formalism of the Radon transform that assumes the image is periodic over the finite array Z"N^2 and requires only arithmetic operations for both forward and exact inverse transformation. The FRT is useful in image processing applications such as tomographic reconstruction [I. Svalbe, D. van der Spek, Reconstruction of tomographic images using analog projections and the digital Radon transform, Linear Algebra and Its Applications 339 (15) (2001) 125-145.], image representation [M. Do, M. Vetterli, Finite ridgelet transform for image representation, IEEE Transactions on Image Processing 12(1).] image convolution [D. Lun, T. Chan, T. Hsung, D. Feng, Y. Chan, Efficient blind image restoration using discrete periodic Radon transform, IEEE Transactions on Image Processing 13(2) (2004) 188-200.], image watermarking and encryption [A. Kingston, I. Svalbe, Projective transforms on periodic discrete image arrays, to appear in P. Hawkes (Ed), Advances in Imaging and Electron Physics (2006).], and robust data transmission [A. Kingston, I. Svalbe, Geometric shape effects in redundant keys used to encrypt data transformed by finite discrete Radon projections, In: Proc. 8th Int. Conf. on Digital Image Computing: Techniques and Applications (2005).]. The original definition by Matus and Flusser in 1993 [F. Matus, J. Flusser, Image representation via a finite Radon transform, IEEE Transactions on Pattern Analysis and Machine Intelligence 15(10) (1993) 996-1006.] was restricted to apply only to square arrays of prime size, pxp. Hsung, Lun and Siu developed an FRT that also applied to dyadic square arrays, 2^nx2^n, called the discrete periodic radon transform (DPRT) [T. Hsung, D. Lun, W. Siu, The discrete periodic Radon transform, IEEE Transactions on Signal Processing 44(10) (1996) 2651-2657.]. Kingston further extended this to define an FRT that applies to prime-adic arrays, p^nxp^n. This paper defines a generalised FRT that applies to square arrays of arbitrary size, NxN for N@?N The Fourier slice theorem and convolution property (two important properties of the classical Radon transform) are established for this FRT. The original image can be reconstructed exactly from the FRT projections using Fourier inversion and back-projection. New methods are identified to correct for the over-representation of pixels due to the compositeness of N. A remarkable result is established that enables 2D sampling patterns to be corrected by an angle invariant 1D filter prior to back-projection.