# Algebraic Reconstruction Algorithms

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Algebraic reconstruction techniques are based on relaxation methods for solving systems of linear equalities or inequalities. Recent developments suggest that with proper tuning, the convergence of these algorithms can be very fast. However, the quality of the reconstructions has not been investigated thoroughly. In this section, a summary overview of algebraic reconstruction techniques is first set out, followed by a description of the algorithmic implementation of the classical algebraic reconstruction technique (ART) and its variants.

The ART algorithm is based on the Kaczmarz method and uses a successive over-relaxation method well-known in numerical linear algebra.4 It was first proposed as a reconstruction method in 1970 by Bender et al.5 and was applied mainly to X-ray photography and electron microscopy. The early developments addressed mainly reconstruction of X-ray CT data but more recently, ART-type algorithms have been proposed specifically for PET.6 The principle of the basic ART algorithm consists of describing every iteration point by point (row-action method), and in correcting all voxels in the image which are found on a projection ray, so as to minimise the difference between the values of the calculated and measured projections at the point under consideration. The process comes to a stop when a certain criterion becomes relatively small. For example the sum of squared differences between the calculated and measured projections can be used as such a criterion.

In fact, ART consists of guessing at a value for all the voxels fj, and then modifying each element along each ray by a factor which compensates for the discrepancy between the measured and the calculated ray sum:

Here fnew and fold refer to the current and previous estimates of the reconstructed object respectively. If the calculated ray sum is the same as the measured value, it implies that the guessed value is correct for a particular projection; however, for another projection there might be a large discrepancy. Thus the pixels of the last views (while lying in the ray for the new view) will be modified according to the discrepancy between the new ray and the measured value. Thus, each ray from each projection is examined and values of f falling within that ray are changed iteratively for all the