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| {{confusing|date=June 2010}}
| | I'm Deandre (25) from Hurth, Austria. <br>I'm learning German literature at a local university and I'm just about to graduate.<br>I have a part time job in a college.<br><br>Here is my blog - [http://www.dochim.com/xe/?document_srl=440869 backup plugin] |
| [[image:Algebraic Reconstruction Technique - animated.gif|frame|right|Animated sequence of reconstruction steps, one iteration.]]
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| The '''Algebraic Reconstruction Technique (ART)''' is a class of iterative algorithms used in [[computed tomography]]. These reconstruct an image from a series of angular projections (a [[Computed axial tomography|sinogram]]). Gordon, Bender and Herman first showed its use in image reconstruction;<ref name="ref3">{{cite journal|last=Gordon|first=R|coauthors=Bender, R; 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}}</ref> whereas the method is known as [[Kaczmarz method]] in numerical linear algebra.<ref name="ref1">{{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 ed.}}</ref><ref>{{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}}</ref>
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| 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 <math> x </math>, and the image process is described by a matrix <math> A </math>. The measured angular projects are collected in a vector <math> b </math>. Given a real or complex
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| <math> m \times n </math>
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| matrix
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| <math> A </math> | |
| and a real or complex vector
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| <math> b </math>,
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| respectively, the method computes an approximation of the solution of the linear
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| systems of equations as in the following formula,
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| : <math>
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| x^{k+1}
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| =
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| x^{k}
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| +
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| \lambda_k
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| \frac{b_{i} - \langle a_{i}, x^{k} \rangle}{\lVert a_{i} \rVert^2} a_{i}
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| </math>
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| where
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| <math> i = k \, \bmod \, m + 1 </math>,
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| <math> a_i </math>
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| is the ''i''-th row of the matrix | |
| <math> A </math>,
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| <math> b_i </math>
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| is the ''i''-th component of the vector
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| <math> b </math>,
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| and
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| <math> \lambda_k </math>
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| is a relaxation parameter. The above formulae gives a simple iteration routine.
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| 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.
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| For further details see [[Kaczmarz method]].
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| ==References==
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| <references/>
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| [[Category:Medical imaging]]
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| [[Category:Radiography]]
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| {{comp-sci-stub}}
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I'm Deandre (25) from Hurth, Austria.
I'm learning German literature at a local university and I'm just about to graduate.
I have a part time job in a college.
Here is my blog - backup plugin