Editing Iterative reconstruction (section)

== 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>