Kaczmarz method

The Kaczmarz method, also called Kaczmarz's algorithm, is an iterative algorithm for solving a system of linear equations written Ax = b. It was first discovered by the Polish mathematician Stefan Kaczmarz, and it was rediscovered for image reconstruction from projections by Richard Gordon, Robert Bender and Gabor Herman in 1970, where it is known as the Algebraic Reconstruction Technique, or ART. ART adds a positivity constraint, which makes it nonlinear. The method can be applied to any linear system, but its computational advantage over other methods depends on the system being sparse. In some biomedical imaging applications it has been shown to be superior to filtered backprojection. Its applications range from computed tomography to signal processing, and it can also be obtained by applying the method of successive projections onto convex sets to the hyperplanes described by the linear system. This entry covers it as a numerical algorithm; its mathematical analysis belongs to the mathematics atlas.