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;[1] whereas the method is known as Kaczmarz method in numerical linear algebra.[2][3]

Animated sequence of reconstruction steps, one iteration.

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 ${\displaystyle Ax=b}$, where:

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

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

${\displaystyle b}$ 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. ${\displaystyle b}$ is made up of all of these values, from each of the individual projections.[4]

Given a real or complex matrix ${\displaystyle A}$ and a real or complex vector ${\displaystyle b}$, respectively, the method computes an approximation of the solution of the linear systems of equations as in the following formula,

${\displaystyle x^{k+1}=x^{k}+\lambda _{k}{\frac {b_{i}-\langle a_{i},x^{k}\rangle }{\|a_{i}\|^{2}}}a_{i}^{T}}$

where ${\displaystyle i=k{\bmod {m}}+1}$, ${\displaystyle a_{i}}$ is the i-th row of the matrix ${\displaystyle A}$, ${\displaystyle b_{i}}$ is the i-th component of the vector ${\displaystyle b}$.

${\displaystyle \lambda _{k}}$ is an optional relaxation parameter, of the range ${\displaystyle 0<\lambda _{k}\leq 1}$. 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 ${\displaystyle \lambda _{k}}$ is reduced with each successive iteration.[4]

References

1. Gordon, R; Bender, R; Herman, GT (December 1970). "Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography". Journal of Theoretical Biology. 29 (3): 471–81. doi:10.1016/0022-5193(70)90109-8. PMID 5492997.
2. Herman, Gabor T. (2009). Fundamentals of computerized tomography : image reconstruction from projections (2nd 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.
4. Kak, Avinash; Slaney, Malcolm (1999). Principles of Computerized Tomographic Imaging. New York: IEEE Press. pp. 276–277, 284. ISBN 978-0898714944.