The circular harmonic Radon transform

Abstract

The circular harmonic decomposition method for evaluating the inverse Radon transform is investigated. A discrete, finite set of projection data may be aliased and its interpretation is inevitably non-unique. When the inverse Radon transform is approximated by a summation, the filtered back projection, it is shown that as well as being non-unique, the reconstruction is inconsistent with the data. By contrast, the circular harmonic decomposition produces a consistent image. The stable form of the method is used to develop a simple and efficient numerical algorithm. This is illustrated with various simple examples and head phantoms.

Recent Developments on Compton Scatter Tomography: Theory and Numerical Simulations

“…As we have shown, it is sufficient develop a single reconstruction algorithm for the Radon transform on circles passing through a fixed point, since all other cases can be reduced to that one by a change of variable and a change of functions. Since we use the formalism of angular Fourier components, we shall adapt a procedure set up long ago by C. H. Chapman & P. W. Carey [12] for the classical Radon transform.…”

The integral equations of Compton Scatter Tomography

“…Here ı IRU PRGDOLW\ 7KDQNV WR WKH NH\ IRUPXOD > @ (9) the inversion of (8) can be achieved by a clever application of (9). The resulting explicit inversion formula reads (10) However (10), as such is still improper for setting up a computational algorithm.…”

“…In practice a computing algorithm can be set up from (11) following the approach of Chapman and Carey [9]. It FRQVLVWV LQ FXWWLQJ WKH IJ-integration range into a finite number of integrals and then making the proper change of variables so that each term is represented by an exactly calculated indefinite integral.…”

Novel numerical inversions of two circular-arc radon transforms in Compton scattering tomography

“…Then he derived a consistency condition for the data which permits to regularize the inverse formulas. From the regularized formulas, Chapman and Cary [11] discussed an alternative inversion algorithm to the 'Filtered Back-Projection' (FBP) algorithm for the standard Radon transform (RT) and showed that the fulfilment of the consistency criterion of the data reduces the number of artefacts. This is not so in the well-known FBP algorithm.…”

“…Until now, the CHD approach has not been widely used for image reconstruction. However, this approach offers many advantages, in particular in the case of Radon data [9][10][11], for example consistency between the image reconstruction formula and the data, stable algorithms, reduced artefacts, less complexity in algorithms and less computing time.…”