Editing Iterative reconstruction (section)

== Basic concepts ==
[[File:CT scan Iterative reconstruction (left) versus filtered backprojection (right).jpg|thumb|CT scan using iterative reconstruction (left) versus filtered backprojection (right)]]
The reconstruction of an image from the acquired data is an [[inverse problem]]. Often, it is not possible to exactly solve the inverse
problem directly. In this case, a direct algorithm has to approximate the solution, which might cause visible reconstruction [[Digital artifact|artifacts]] in the image. Iterative algorithms approach the correct solution using multiple iteration steps, which allows to obtain a better reconstruction at the cost of a higher computation time.

There are a large variety of algorithms, but each starts with an assumed image, computes projections from the image, compares the original projection data and updates the image based upon the difference between the calculated and the actual projections.

=== Algebraic reconstruction ===
{{Main|Algebraic reconstruction technique}}
The Algebraic Reconstruction Technique (ART) was the first iterative reconstruction technique used for [[computed tomography]] by [[Godfrey Hounsfield|Hounsfield]].

=== Iterative Sparse Asymptotic Minimum Variance ===
{{Main|SAMV (algorithm)}}
The [[SAMV (algorithm)|iterative sparse asymptotic minimum variance]] algorithm is an iterative, parameter-free [[Super-resolution imaging|superresolution]] [[tomographic reconstruction]] method inspired by [[compressed sensing]], with applications in [[synthetic-aperture radar]], [[CT scan|computed tomography scan]], and [[Magnetic resonance imaging|magnetic resonance imaging (MRI)]].

=== Statistical reconstruction ===
There are typically five components to statistical iterative image reconstruction algorithms, e.g.<ref name="pwl">{{cite journal | author = Fessler J A | year = 1994 | title = Penalized weighted least-squares image reconstruction for positron emission tomography | url =https://deepblue.lib.umich.edu/bitstream/2027.42/85851/1/Fessler105.pdf | journal = IEEE Transactions on Medical Imaging | volume = 13 | issue = 2| pages = 290–300 | doi=10.1109/42.293921| pmid = 18218505 | hdl = 2027.42/85851 | hdl-access = free }}</ref>
# An object model that expresses the unknown continuous-space function <math>f(r)</math> that is to be reconstructed in terms of a finite series with unknown coefficients that must be estimated from the data.
# A system model that relates the unknown object to the "ideal" measurements that would be recorded in the absence of measurement noise.  Often this is a linear model of the form <math>\mathbf{A}x+\epsilon</math>, where <math>\epsilon</math> represents the noise.
# A [[statistical model]] that describes how the noisy measurements vary around their ideal values. Often [[Gaussian noise]] or [[Poisson statistics]] are assumed. Because [[Poisson statistics]] are closer to reality, it is more widely used.
# A [[Loss function|cost function]] that is to be minimized to estimate the image coefficient vector. Often this cost function includes some form of [[regularization (mathematics)|regularization]]. Sometimes the regularization is based on [[Markov random fields]].
# An [[algorithm]], usually iterative, for minimizing the cost function, including some initial estimate of the image and some stopping criterion for terminating the iterations.

=== Learned Iterative Reconstruction ===
In learned iterative reconstruction, the updating algorithm is learned from training data using techniques from [[machine learning]] such as [[convolutional neural networks]], while still incorporating the image formation model. This typically gives faster and higher quality reconstructions and has been applied to CT<ref>{{Cite journal|last1=Adler|first1=J.|last2=Öktem|first2=O.|date=2018|title=Learned Primal-dual Reconstruction|journal=IEEE Transactions on Medical Imaging|volume=PP|issue=99|pages=1322–1332|doi=10.1109/tmi.2018.2799231|pmid=29870362|issn=0278-0062|arxiv=1707.06474|s2cid=26897002}}</ref> and MRI reconstruction.<ref>{{Cite journal|last1=Hammernik|first1=Kerstin|last2=Klatzer|first2=Teresa|last3=Kobler|first3=Erich|last4=Recht|first4=Michael P.|last5=Sodickson|first5=Daniel K.|last6=Pock|first6=Thomas|last7=Knoll|first7=Florian|title=Learning a variational network for reconstruction of accelerated MRI data|journal=Magnetic Resonance in Medicine|language=en|volume=79|issue=6|pages=3055–3071|doi=10.1002/mrm.26977|pmid=29115689|pmc=5902683|issn=1522-2594|year=2018|arxiv=1704.00447}}</ref>