Algebraic reconstruction technique

The algebraic reconstruction technique (ART) is an iterative reconstruction technique used in computed tomography. It reconstructs an image from a series of angular projections (a sinogram). Gordon, Bender and Herman first showed its use in image reconstruction; whereas the method is known as Kaczmarz method in numerical linear algebra.
An advantage of ART over other reconstruction methods (such as filtered backprojection) is that it is relatively easy to incorporate prior knowledge into the reconstruction process.
ART can be considered as an iterative solver of a system of linear equations formula_1, where:
Given a real or complex matrix formula_2 and a real or complex vector formula_7, respectively, the method computes an approximation of the solution of the linear systems of equations as in the following formula,
where formula_12, formula_13 is the "i"-th row of the matrix formula_2, formula_15 is the "i"-th component of the vector formula_7.
formula_17 is an optional relaxation parameter, of the range formula_18. The relaxation parameter is used to slow the convergence of the system. This increases computation time, but can improve the signal-to-noise ratio of the output. In some implementations, the value of formula_17 is reduced with each successive iteration.