# Algebraic Reconstruction Technique

Animated sequence of reconstruction steps, one iteration.

The Algebraic Reconstruction Technique (ART) is an iterative algorithm (in fact a class of iterative algorithms) for the reconstruction of an images from a series of angular projections (a sinogram), used in computed tomography. For image reconstruction it was introduced in,[1] while in numerical linear algebra the method is known as Kaczmarz method[2] .[3]

In fact, ART can be considered as an iterative solver of a system of linear equations (the values of the pixels are considered as variables collected in a vector $x$, the image process is described by a matrix $A$, and the measured angular projects are collected in a vector $b$). Given a real or complex $m \times n$ matrix $A$ and a real or complex vector $b$, respectively, the method computes an approximation of the solution of the linear systems of equations as in the following formula,

$x^{k+1} = x^{k} + \lambda_k \frac{b_{i} - \langle a_{i}, x^{k} \rangle}{\lVert a_{i} \rVert^2} a_{i}$

where $i = k \, \bmod \, m + 1$, $a_i$ is the i-th row of the matrix $A$, $b_i$ is the i-th component of the vector $b$, and $\lambda_k$ is a relaxation parameter. The above formulae gives a simple iteration routine.

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.

For further details see Kaczmarz method.

## References

1. ^ Gordon, R; Bender, R; Herman, GT (1970 Dec). "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography.". Journal of theoretical biology 29 (3): 471–81. PMID 5492997.
2. ^ Herman, Gabor T. (2009). Fundamentals of computerized tomography : image reconstruction from projections (2nd ed. ed.). Dordrecht: Springer. ISBN 978-1-85233-617-2.
3. ^ Natterer, F. (1986). The mathematics of computerized tomography. Stuttgart: B.G. Teubner. ISBN 0-471-90959-9.