Editing Iterative reconstruction

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[[File:FBP Iter single.jpg|thumb|Example showing differences between filtered backprojection (right half) and iterative reconstruction method (left half)]]
'''Iterative reconstruction''' refers to [[Iteration|iterative]] [[algorithms]] used to reconstruct 2D and [[3D reconstruction|3D images]] in certain [[Digital imaging|imaging]] techniques.
For example, in [[computed tomography]] an image must be reconstructed from projections of an object. Here, iterative reconstruction techniques are usually a
better, but computationally more expensive alternative to the common [[filtered back projection]] (FBP) method, which directly calculates the image in
a single reconstruction step.<ref name="ref1">Herman, G. T., [https://books.google.com/books?id=BhtGTkEjkOQC Fundamentals of computerized tomography: Image reconstruction from projection], 2nd edition, Springer, 2009</ref> In recent research works, scientists have shown that extremely fast computations and massive parallelism is possible for iterative reconstruction, which makes iterative reconstruction practical for commercialization.<ref>{{Cite book|last1=Wang|first1=Xiao|last2=Sabne|first2=Amit|last3=Kisner|first3=Sherman|last4=Raghunathan|first4=Anand|last5=Bouman|first5=Charles|last6=Midkiff|first6=Samuel|title=Proceedings of the 21st ACM SIGPLAN Symposium on Principles and Practice of Parallel Programming |chapter=High performance model based image reconstruction |date=2016-01-01|series=PPoPP '16|location=New York, NY, USA|publisher=ACM|pages=2:1–2:12|doi=10.1145/2851141.2851163|isbn=9781450340922|s2cid=16569156|url=https://zenodo.org/record/895136}}</ref>

== 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>

== Advantages ==
[[File:Heart-direct-vs-iterative-reconstruction.png|frame|A single frame from a [[real-time MRI]] (rt-MRI) movie of a [[human heart]]. a) direct reconstruction b) iterative (nonlinear inverse) reconstruction<ref name=uecker2010 />]]

The advantages of the iterative approach include improved insensitivity to [[signal noise|noise]] and capability of reconstructing an [[Optimization (mathematics)|optimal]] image in the case of incomplete data. The method has been applied in emission tomography modalities like [[SPECT]] and [[Positron emission tomography|PET]], where there is significant attenuation along ray paths and [[noise statistics]] are relatively poor.

'''Statistical, likelihood-based approaches''': Statistical, likelihood-based iterative [[expectation-maximization algorithm]]s<ref>{{cite journal|last=Carson|first=Lange|author2=Richard Carson |title=EM reconstruction algorithm for emission and transmission tomography|journal=Journal of Computer Assisted Tomography|date=1984|volume=8|issue=2|pages=306–316|pmid=6608535}}</ref><ref>{{cite journal|last=Vardi|first=Y.|author2=L. A. Shepp |author3=L. Kaufman |title=A statistical model for positron emission tomography|journal=Journal of the American Statistical Association|date=1985|volume=80|issue=389|pages=8–37|doi=10.1080/01621459.1985.10477119}}</ref>
are now the preferred method of reconstruction.  Such algorithms compute estimates of the likely distribution of annihilation events that led to the measured data, based on statistical principle, often providing better noise profiles and resistance to the streak artifacts common with FBP. Since the density of radioactive tracer is a function in a function space, therefore of extremely high-dimensions, methods which regularize the maximum-likelihood solution turning it towards penalized or maximum a-posteriori methods can have significant advantages for low counts. Examples such as [[Ulf Grenander]]'s [[Sieve estimator]]<ref>{{Cite journal|title = On the Use of the Method of Sieves for Positron Emission Tomography |journal = IEEE Transactions on Medical Imaging|date = 1985 |pages =3864–3872|volume = NS-32(5) |issue = 5|doi= 10.1109/TNS.1985.4334521|first2 = Michael I.|last2 = Miller|first1 = Donald L.|last1 = Snyder |bibcode = 1985ITNS...32.3864S|s2cid = 2112617}}</ref><ref>{{cite journal|last1=Snyder|first1=D.L.|last2=Miller|first2=M.I.|last3=Thomas|first3=L.J.|last4=Politte|first4=D.G.|title=Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography|journal=IEEE Transactions on Medical Imaging|date=1987|volume=6|issue=3|pages=228–238|doi=10.1109/tmi.1987.4307831|pmid=18244025|s2cid=30033603}}</ref>
or Bayes penalty methods,<ref>{{Cite journal|title = Bayesian image analysis: An application to single photon emission tomography |journal = Proceedings Amererican Statistical Computing|date = 1985 |pages =12–18|first2 = Donald E. |last2 = McClure|first1 =Stuart|last1 = Geman|url=http://www.dam.brown.edu/people/geman/Homepage/Image%20processing,%20image%20analysis,%20Markov%20random%20fields,%20and%20MCMC/1985GemanMcClureASA.pdf}}
</ref><ref>{{Cite journal|title = Bayesian Reconstructions for Emission Tomography Data Using a Modified EM Algorithm |journal = IEEE Transactions on Medical Imaging|date = 1990 |pmid = 18222753|pages =84–93|volume = 9 | issue = 1 |doi= 10.1109/42.52985 |first = Peter J. |last = Green |citeseerx = 10.1.1.144.8671}}</ref> or via [[I.J. Good]]'s roughness method<ref>{{Cite journal|title = The role of likelihood and entropy in incomplete data problems: Applications to estimating point-process intensites and toeplitz constrained covariance estimates | journal = Proceedings of the IEEE|date = 1987 |pages =3223–3227|volume = 5 | issue = 7  |doi = 10.1109/PROC.1987.13825|first1 = Michael I.|last1 = Miller|first2 = Donald L. |last2 = Snyder| s2cid = 23733140}}</ref><ref>{{Cite journal|title = Bayesian image reconstruction for emission tomography incorporating Good's roughness prior on massively parallel processors | journal = Proceedings of the National Academy of Sciences of the United States of America|date = April 1991 |pmid = 2014243|pages =3223–3227|volume = 88 | issue = 8|doi= 10.1073/pnas.88.8.3223|first1 = Michael I.|last1 = Miller|first2 = Badrinath |last2 = Roysam |pmc=51418| bibcode = 1991PNAS...88.3223M| doi-access = free}}</ref> may yield superior performance to expectation-maximization-based methods which involve a Poisson likelihood function only.

As another example, it is considered superior when one does not have a large set of projections available, when the projections are not distributed uniformly in angle, or when the projections are sparse or missing at certain orientations. These scenarios may occur in [[intraoperative]] CT, in [[cardiac]] CT, or when metal [[artifact (observational)|artifacts]]<ref>{{cite journal| last1=Wang|first1=G.E.| last2=Snyder|first2=D.L.| last3=O'Sullivan|first3=J.A.| last4=Vannier|first4=M.W.| title=Iterative deblurring for CT metal artifact reduction| journal=IEEE Transactions on Medical Imaging| volume=15|issue=5|pages=657–664| doi=10.1109/42.538943|pmid=18215947|year=1996}}
</ref><ref name="mdt">{{cite journal |vauthors=Boas FE, Fleischmann D | year = 2011 | title = Evaluation of two iterative techniques for reducing metal artifacts in computed tomography | url = http://radiology.rsna.org/content/early/2011/02/17/radiol.11101782.abstract | archive-url = https://web.archive.org/web/20111201001343/http://radiology.rsna.org/content/early/2011/02/17/radiol.11101782.abstract | url-status = dead | archive-date = 2011-12-01 | journal = Radiology | volume = 259 | issue =  3| pages = 894–902 | doi = 10.1148/radiol.11101782 | pmid = 21357521 | url-access = subscription }}</ref>
require the exclusion of some portions of the projection data.

In [[Magnetic Resonance Imaging]] it can be used to reconstruct images from data acquired with multiple receive coils and with sampling patterns different from the conventional Cartesian grid<ref>{{cite journal | author = Pruessmann K. P., Weiger M., Börnert P., Boesiger P. | year = 2001 | title = Advances in sensitivity encoding with arbitrary k-space trajectories | journal = Magnetic Resonance in Medicine | volume = 46 | issue = 4| pages = 638–651 | doi = 10.1002/mrm.1241 | pmid = 11590639 | doi-access = free }}</ref> and allows the use of improved regularization techniques (e.g. [[total variation]])<ref>{{cite journal | author = Block K. T., Uecker M., Frahm J. | year = 2007 | title = Undersampled radial MRI with multiple coils. Iterative image reconstruction using a total variation constraint | journal = Magnetic Resonance in Medicine | volume = 57 | issue = 6| pages = 1086–1098 | doi = 10.1002/mrm.21236 | pmid = 17534903 | hdl = 11858/00-001M-0000-0012-E2A3-7 | s2cid = 16396739 | hdl-access = free }}</ref> or an extended modeling of physical processes<ref>{{cite journal | author = Fessler J | year = 2010 | title = Model-based Image Reconstruction for MRI | journal = IEEE Signal Processing Magazine| volume = 27 | issue = 4| pages = 81–89 | doi=10.1109/msp.2010.936726| pmid = 21135916 | pmc = 2996730 | bibcode = 2010ISPM...27...81F }}</ref> to improve the reconstruction. For example, with iterative algorithms it is possible to 
reconstruct images from data acquired in a very short time as required for [[real-time MRI]] (rt-MRI).<ref name=uecker2010>{{cite journal |vauthors=Uecker M, Zhang S, Voit D, Karaus A, Merboldt KD, Frahm J | year = 2010a | title = Real-time MRI at a resolution of 20 ms | url = http://pubman.mpdl.mpg.de/pubman/item/escidoc:587599/component/escidoc:2166988/587599.pdf | journal = NMR Biomed | volume = 23 | issue = 8| pages = 986–994 | doi = 10.1002/nbm.1585 | pmid=20799371| hdl = 11858/00-001M-0000-0012-D4F9-7 | s2cid = 8268489 | hdl-access = free }}</ref>

In [[Cryo-electron tomography|Cryo Electron Tomography]], where the limited number of projections are acquired due to the hardware limitations and to avoid the biological specimen damage, it can be used along with [[compressive sensing]] techniques or regularization functions (e.g. [[Huber loss|Huber function]]) to improve the reconstruction for better interpretation.<ref>{{Cite book|publisher = Springer International Publishing|date = 2015-01-01|isbn = 978-3-319-18430-2|pages = 43–51|series = Lecture Notes in Computational Vision and Biomechanics|doi = 10.1007/978-3-319-18431-9_5|language = en|first1 = Shadi|last1 = Albarqouni|first2 = Tobias|last2 = Lasser|first3 = Weaam|last3 = Alkhaldi|first4 = Ashraf|last4 = Al-Amoudi|first5 = Nassir|last5 = Navab| title=Computational Methods for Molecular Imaging | chapter=Gradient Projection for Regularized Cryo-Electron Tomographic Reconstruction | volume=22 |editor-first = Fei|editor-last = Gao|editor-first2 = Kuangyu|editor-last2 = Shi|editor-first3 = Shuo|editor-last3 = Li}}</ref>

Here is an example that illustrates the benefits of iterative image reconstruction for cardiac MRI.<ref name="ilyas Uyanik 2013">I Uyanik, P Lindner, D Shah, N Tsekos I Pavlidis (2013) Applying a Level Set Method for Resolving Physiologic Motions in Free-Breathing and Non-gated Cardiac MRI. FIMH, 2013, {{cite web |url=http://www.cpl.uh.edu/files/publications/conf_paper_videos/c66.pdf  |archive-url=https://web.archive.org/web/20180722095944/http://www.cpl.uh.edu/files/publications/conf_paper_videos/c66.pdf  |url-status=dead  |archive-date=2018-07-22  |access-date=2013-10-01 |title=Computational Physiology Lab  }}</ref>

== See also ==
*[[Tomographic reconstruction]]
*[[Positron Emission Tomography]]
*[[Tomogram]]
*[[Computed Tomography]]
*[[Magnetic Resonance Imaging]]
*[[Inverse problem]]
*[[Osem (mathematics)|Osem]]
*[[Deconvolution]]
*[[Inpainting]]
*[[Algebraic Reconstruction Technique]]
*[[SAMV (algorithm)|iterative Sparse Asymptotic Minimum Variance]]

== References ==
{{reflist}}
<ref>{{cite journal | author = Bruyant P.P. | year = 2002 | title = Analytic and iterative reconstruction algorithms in SPECT | url = http://jnm.snmjournals.org/content/43/10/1343.long | journal = Journal of Nuclear Medicine | volume = 43 | issue = 10| pages = 1343–1358 | pmid = 12368373 }}</ref>
<ref>{{cite journal | author = Grishentcev A. Jr | year = 2012 | title = Effective compression of images on the basis of differential analysis | url = http://jre.cplire.ru/iso/nov12/1/text.pdf | journal = Journal of Radio Electronics | volume = 11 | pages = 1–42 }}</ref>

{{commons category|Iterative reconstruction}}

{{DEFAULTSORT:Iterative Reconstruction}}
[[Category:Medical imaging]]
[[Category:Image processing]]