The Image Reconstruction Problem of xray absorption tomography in words

X-rays pass through the subject body and interact to a certain extent with those features they pass. The detected X-ray on the other side of the subject contains information relating to every obstacle in the path, and not just to one. The reconstruction problem is to decode these muddled mixes of information into spatial information throughout the subject, by combining information from many paths through the subject.

The filter used in Filtered Backprojection. Image artefacts of backprojection the filter corrects.

The artefacts are equivalent to the Point Spread Function. The Fourier transform of this function is used.

The filter reduces low frequency information and enhances high frequency information, sharpening edges, and reducing blur.

Image reconstruction problem re-cast as an algebraic problem

The ray integral equation describes how the detected intensity is related to the attenuation coefficients and the incident intensity along the ray path. This means that the problem can be cast as an algebraic situation in which the attenuation coefficients form a large matrix of unknown values, and each ray a data set.

ART has a number of advantages, the first is the limitation in application of filtered back projection (FBP). Where it is not possible to get data representing the full 180 or 360o, the image from FBP is poor, likewise if the projections are not uniformly distributed. Using ART these problems can be resolved at computational cost.

Another advantage is the ability to correct for noise. It is likely that in a given specimen that only certain tissues are expected, for example soft tissue, connective tissue, bone. These have finite ranges of possible attenuation coefficients, and values outside of these ranges are hence likely to be noise, and can be corrected by moving into the nearest expected range, or discarded from the image. Another important example is negative attenuation coefficients, which are known to be an artefact, but are not removed from the image by the FBP methods. Using more advanced recursive techniques these enhancements can all be used to produce images of superior quality from reduced datasets.