algebraic reconstruction technique

{{Short description|Technique in computed tomography}}

{{Use dmy dates|date=September 2017}}

image:Algebraic Reconstruction Technique - animated.gif

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;{{cite journal|last=Gordon|first=R|author2=Bender, R|author3= Herman, GT|title=Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography|journal=Journal of Theoretical Biology|date=December 1970|volume=29|issue=3|pages=471–81|pmid=5492997|doi=10.1016/0022-5193(70)90109-8|bibcode=1970JThBi..29..471G}} whereas the method is known as Kaczmarz method in numerical linear algebra.{{cite book|last=Herman|first=Gabor T.|title=Fundamentals of computerized tomography : image reconstruction from projections|year=2009|publisher=Springer|location=Dordrecht|isbn=978-1-85233-617-2|edition=2nd}}{{cite book|last=Natterer|first=F.|title=The mathematics of computerized tomography|year=1986|publisher=B.G. Teubner|location=Stuttgart|isbn=0-471-90959-9}}

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 $A\; x\; =\; b$, where:

: $A$ is a sparse $m\; \backslash times\; n$ matrix whose values represent the relative contribution of each output pixel to different points in the sinogram ($m$ being the number of individual values in the sinogram, and $n$ being the number of output pixels);

: $x$ represents the pixels in the generated (output) image, arranged as a vector, and:

: $b$ is a vector representing the sinogram. Each projection (row) in the sinogram is made up of a number of discrete values, arranged along the transverse axis. $b$ is made up of all of these values, from each of the individual projections.{{cite book|last1=Kak|first1=Avinash|last2=Slaney|first2=Malcolm|title=Principles of Computerized Tomographic Imaging|url=https://archive.org/details/principlescomput00akak|url-access=limited|year=1999|publisher=IEEE Press|location=New York|isbn=978-0898714944|pages=[https://archive.org/details/principlescomput00akak/page/n280 276]–277, 284}}

Given a real or complex 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}{\|a_i\|^2} a_i^T

where $i\; =\; k\; \backslash bmod\; m\; +\; 1$, $a\_i$ is the i-th row of the matrix $A$, $b\_i$ is the i-th component of the vector $b$.

$\backslash lambda\_k$ is an optional relaxation parameter, of the range . 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 $\backslash lambda\_k$ is reduced with each successive iteration.

A further development of the ART algorithm is the simultaneous algebraic reconstruction technique (SART) algorithm.