Orthogonal discrete periodic Radon transform: part I: Theory and realization

This paper presents a new fast method to map between images and their digital projections based on the Number Theoretic Transform (NTT) and the Finite Radon Transform (FRT). The FRT is a Discrete Radon Transform (DRT) defined on the same finite geometry as the Finite or Discrete Fourier Transform (DFT). Consequently, it may be inverted directly and exactly via the Fast Fourier Transform (FFT) without any interpolation or filtering [1]. As with the FFT, the FRT can be adapted to square images of arbitrary sizes such as dyadic images, prime-adic images and arbitrary-sized images. However, its simplest form is that of prime-sized images [2]. The FRT also preserves the discrete versions of both the Fourier Slice Theorem (FST) and Convolution Property of the Radon Transform (RT). The NTT is also defined on the same geometry as the DFT and preserves the Circular Convolution Property (CCP) of the DFT [3, 4]. This paper shows that the Slice Theorem is also valid within the NTT and that it can be utilized as a new exact, integer-only and fast inversion scheme for the FRT, with the same computational complexity as the FFT. Digital convolutions and exact digital filtering of projections can also be performed using this Number Theoretic FRT (NFRT).

A Fast Number Theoretic Finite Radon Transform

Proposes a new approach for computing multidimensional DFTs that reduces interprocessor communications and is therefore suitable for efficient implementation on a variety of multiprocessor platforms including MIMD supercomputers and clusters of workstations. Group theoretic concepts are used to formulate a flexible computational strategy that hybrids the reduced transform algorithm (RTA) with the Good-Thomas factorization and can deal efficiently with non-power-of-two sizes without resorting to zero-padding. The RTA algorithm is employed not as a data processing but rather as a bookkeeping tool in order to decompose the problem into many smaller size subproblems (lines) that can be solved independently by the processors. Implementation issues on an Intel iPSC/i860 hypercube are discussed and timing results for large 2D and 3D DFTs with index sets in Z/MP/spl times/Z/KP and Z/N/spl times/Z/MP/spl times/Z/KP respectively are provided, where N, M, K are powers-of-two and P is a small prime number such as 3, 5, or 7. The nonoptimized realizations of the new hybrid RTA approach are shown to outperform by as much as 70% the optimized assembly coded realizations of the traditional row-column method on the iPSC/860. >

A new approach for computing multidimensional DFT's on parallel machines and its implementation on the iPSC/860 hypercube

The authors generalize the weighted redundancy transform (WRT) algorithm for computing the multidimensional discrete Fourier transform (DFT) in the case in which the sample size (blocklength) is not the same on every axis. The proposed algorithm, like the WRT algorithm, is based on the one-dimensional fast Fourier transform (FFT) and, compared to the traditional ways of computing the multidimensional DFT, offers substantial savings in the number of one-dimensional FFT procedure calls. While the algorithm is applicable to transforms of any dimensions, only the two-dimensional case is explored in detail. >

Computing discrete Fourier transform on a rectangular data array

Consider a finite abelian group A. A mapping 

$$ \alpha :A \to A $$

(7.1)

is called a homomorphism of A if it satisfies 

$$ \alpha ({\text{a}}{\mkern 1mu} {\text{ + }}{\mkern 1mu} {\text{b}}) = \alpha ({\text{a}}) + \alpha ({\text{b}}),{\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\mkern 1mu} {\text{a,b}} \in A. $$

(7.2)

Duality of Lines and Planes

An algorithm is introduced for computing the multidimensional finite Fourier transform. The algorithm can be applied to data samples of any size. In most cases, it offers a substantial reduction in the computational complexity. >

M. Vulis

Papers

The weighted redundancy transform