Fast Image Reconstruction Based on a Radon Inversion Formula Appropriate for Rapidly Collected Data

Source abstract

An arrangement of a single x-ray source and a circular array (centered at the source) of detectors allows rapid collection of a number of x-ray projections of a cross section of an object. A class of appropriately fast algorithms for reconstructing the internal structure of the cross section from such data is rigorously derived starting from the classical Radon inversion formula. A new Radon inversion formula for divergent beams is given as a singlular integral. It is shown that this singular integral can be regularized a number of ways, each one of which leads to an efficient numerical evaluation. The structure, and hence the speed, of the resulting algorithms is similar to the so-called “convolution reconstruction techniques” which have been previously derived for data slowly collected along parallel rays and have been found both fast and accurate. It is demonstrated that, provided the same number of x-ray photons is used, the method of this paper gives as accurate reconstructions for diverging x-rays as previously used algorithms gave for parallel x-rays. Medical applicability of the method is demonstrated by the reconstruction of the total content of the thorax of an intact dog from diverging x-ray projections, as well as realistic mathematically described phantoms representing the human head and thorax from computer generated projections along diverging beams.