Circulant theory of the Radon transform

Abstract

This thesis presents original work on the construction of a Circulant Theory of the Radon Transform (RT). The RT allows one to determine or reconstruct the internal structure of an object from its projected views. The new Circulant theory provides a computationally practical, yet fully digital and exact approach to the RT. There are two major theoretical outcomes from this work. The first part of the theory contains the discretisation of the RT using circulant matrices (or circulants), which represent generalised cyclic convolutions. Cyclic convolutions provide practical realisations of many integral transforms, such as the Fast Fourier Transform (FFT) that originated from the Fourier Transform. The result is a new, fully digital and exact RT called the Number-Theoretic Radon Transform (NRT), a discrete RT mappable to the Number Theoretic Transform [Chandra, 2010]. This new transform is not only faster than previous FFT based discrete RTs, but its computations are performed using only integers. The NRT also has its own integer-only Slice Theorem called the Number-Theoretic Slice Theorem. This makes the NRT the first purely digital analogue to the RT, where all computations (and any image filtering) can always be performed without any round-off errors, while also requiring significantly less memory. The theory also results in a reformulation of the Discrete Radon Transform (DRT) of Bolker [1987], Matus and Flusser [1993] and others, as first suggested by Fill [1989]. The DRT is a discrete (periodic) RT directly mappable to the FFT. The second part of the theory provides a circulant-based treatment of under-determined systems in these discrete RTs. Under-determined systems are comprised of a finite set of measurements, which are insufficient for an unambiguous reconstruction of the object. An insufficient projection set leaves missing projections in the RT. These ill-posed systems create artefacts, called Ghosts, in the reconstructed image. The Finite Ghost Theory allows the removal of these artefacts exactly using the DRT and NRT reconstructions. Ghost removal is made possible if a sufficiently large known or redundant image region is present within the reconstructed image area. This is equivalent to having projections that are over-determined for a given sub-region within this reconstruction [Chandra et al., 2010b]. This algorithm is used as an inversion mechanism for the Mojette Transform (MT), an experimentally practical (aperiodic) discrete linogram RT. The resulting fusion of the MT and the DRT is a computationally and experimentally practical algorithm for Discrete Tomography, denoted as the Fast Mojette Transform [Chandra et al., 2010a, in preparation]. The Ghost removal methods are constructed using the Cyclic n-gons of Bachmann [1971], which provide a useful and visual interpretation of the algorithm. The circulant theory is the culmination of work done in removing Ghosts by Chandra et al. [2008] and Chandra and Svalbe [2008]. The NRT and the circulant theory was constructed out of the necessity to overcome the technical obstacles found in these works. The most significant of these problems was the numerical overflow encountered when removing (in the order of a hundred) Ghosts. Work on more generalised approaches to removing Ghosts, by Svalbe et al. [2010a,b, in preparation], is also presented.