A Review on Image Reconstruction through MRI k-Space Data

Full Text (PDF, 807KB), PP.42-59

Views: 0 Downloads: 0


Tanuj Kumar Jhamb 1,* Vinith Rejathalal 1 V.K. Govindan 1

1. Department of Computer Science and Engineering, National Institute of Technology Calicut, Kerala-673601, India

* Corresponding author.

DOI: https://doi.org/10.5815/ijigsp.2015.07.06

Received: 19 Feb. 2015 / Revised: 27 Mar. 2015 / Accepted: 24 Apr. 2015 / Published: 8 Jun. 2015

Index Terms

MRI, NMR, Modern MRI, k-Space, Fourier Transform, Phase Error, Phase Correction, SURE, SSSVM


Image reconstruction is the process of generating an image of an object from the signals captured by the scanning machine. Medical imaging is an interdisciplinary field combining physics, biology, mathematics and computational sciences. This paper provides a complete overview of image reconstruction process in MRI (Magnetic Resonance Imaging). It reviews the computational aspect of medical image reconstruction. MRI is one of the commonly used medical imaging techniques. The data collected by MRI scanner for image reconstruction is called the k-space data. For reconstructing an image from k-space data, there are various algorithms such as Homodyne algorithm, Zero Filling method, Dictionary Learning, and Projections onto Convex Set method. All the characteristics of k-space data and MRI data collection technique are reviewed in detail. The algorithms used for image reconstruction discussed in detail along with their pros and cons. Various modern magnetic resonance imaging techniques like functional MRI, diffusion MRI have also been introduced. The concepts of classical techniques like Expectation Maximization, Sensitive Encoding, Level Set Method, and the recent techniques such as Alternating Minimization, Signal Modeling, and Sphere Shaped Support Vector Machine are also reviewed. It is observed that most of these techniques enhance the gradient encoding and reduce the scanning time. Classical algorithms provide undesirable blurring effect when the degree of phase variation is high in partial k-space. Modern reconstructions algorithms such as Dictionary learning works well even with high phase variation as these are iterative procedures.

Cite This Paper

Tanuj Kumar Jhamb, Vinith Rejathalal, V.K. Govindan,"A Review on Image Reconstruction through MRI k-Space Data", IJIGSP, vol.7, no.7, pp.42-59, 2015. DOI: 10.5815/ijigsp.2015.07.06


[1]Hongmei Zhu "Medical Image Processing Overview" University of Calgary, Toronto, Ontario M5T 3J1.

[2]MS Nizam BJJ Abdullah Kwan-Hoong Ng, Azlan C Ahmad "Magnetic resonance imaging: Health effects and safety", Proceedings of the International Conference on NonIonizing Radiation at UNITEN ICNIR2003, (ICNIR2003), 20th October 2003.

[3]Purcell EM, Torrey HC, and Pound RV "Resonance absorption by nuclear magnetic moments in a solid" Phys Rev 69: 37-38, 1946., MIT, Massachusetts, December 24, 1945.

[4]Ming Zhao, Daniel A. Beauregard, Louiza Loizou, Bazbek Davletov & Kevin M. Brindle "Non-invasive detection of apoptosis using magnetic resonance imaging and a targeted contrast agent", MRC Laboratory of Molecular Biology, Hills Road, Cambridge, UK,Nature Medicine 7, 1241 - 1244 (2001), doi:10.1038/nm1101-124.

[5]Abi Berger, science editor "Article on Magnetic Resonance Imaging" BMJ. Jan 5, 2002; 324(7328): 35. PMCID: PMC1121941, US National Library of Medicine National Institutes of Health.

[6]Kay Powell "The Difference Between MRI & CT (CAT Scans)", Article on Medical and Dental Expenses: Topic 502, From the IRS, Holland.

[7]Robert Holgersson and Mikael Svärd "Reconstruction of Echo Planar Images used in Functional MRI", Institute of Technology, Linkopin University, Master's Thesis at the Department of Biomedical Engineering, LiU-IMT-EX-359, February 2004.

[8]Joseph P. Hornak, Book Title "The Basics Of MRI", chapter-6, Director of the Magnetic Resonance Laboratory, Rochester Institute of Technology, Rochester, NY 14623-5604.

[9]H. Bruder, H. Fischer, H.-E. Reinfelder and F. Schmitt"Image reconstruction for echo planar imaging with nonequidistant k-space sampling", Article first published online: 18 NOV 2005 DOI: 10.1002/mrm.1910230211.

[10]G.T. Herman "Image Reconstruction From Projections", Medical Image Processing Group, Department of Radiology, University of Pennsylvania, Philadelphia, PA 19104, USA gabor@mipg.upenn.edu Available online 2 May 2002, Volume 1, Issue 1, April 1995, Pages 3–18, doi:10.1006/rtim.1995.1002.

[11]D. R. Smith, Willie J. Padilla, D. C. Vier, S. C. Nemat-Nasser, and S. Schultz "Composite Medium with Simultaneously Negative Permeability and Permittivity", Phys. Rev. Lett. 84, 4184 – Published 1 May 2000, Department of Physics, University of California, San Diego, 9500 Gilman Drive, La Jolla, California 92093-0319.

[12]Jeffrey G. Jarvik, MD, MPH; and Richard A. Deyo, MD, MPH "Diagnostic Evaluation of Low Back Pain with Emphasis on Imaging", University of Washington, Seattle, Washington, Ann Intern Med. 2002; 137(7):586-597. doi:10.7326/0003-4819-137-7-200210010-00010.

[13]Stephan G. Nekolla, Axel Martinez-Moeller, Antti Saraste "PET and MRI in cardiac imaging: from validation studies to integrated applications" Published online: 23 December 2008 Springer-Verlag 2008 Eur J Nucl Med Mol Imaging (2009) 36 (Suppl 1):S121-S130 DOI 10.1007/s00259-008-0980-1.

[14]Kevin Cleary, Charles Nguyen "State of the art in surgical robotics: Clinical applications and technology challenges", Volume 6, Issue 6, pages 312–328, 2001, Article first published online: 4 APR 2002 DOI: 10.1002/igs.10019, Copyright © 2002 Wiley-Liss.

[15]Uduma Felix U, Fokam Pius G "Magnetic Resonance Imaging (Mri), The Prefered Evaluation Tool In Soft Tissue Sarcoma: Literature Review, Demonstrated With A Case Report", Journal of Asian Scientific Research, 2012, vol. 2, issue 2, pages 87-99.

[16]K. Tadakuma, L.M. DeVita, J.S. Plante, Y. Shaoze and S. Dubowsky, IEEE Fellow "The experimental study of a precision parallel manipulator with binary actuation: With application to MRI cancer treatment" 2008. ICRA 2008. …, 2008 - ieeexplore.ieee.org.

[17]Helene Benvenistea, Steve Blackbandc "MR microscopy and high resolution small animal MRI: applications in neuroscience research", Medical Department, Brookhaven National Laboratory, NY 11793, USA, Received 13 February 2002, Accepted 7 June 2002, Available online 10 September 2002, doi:10.1016/S0301-0082(02)00020-5.

[18]Mark A. Horsfield1, Derek K. Jones"Applications of diffusion-weighted and diffusion tensor MRI to white matter diseases – a review" Article first published online: 5 DEC 2002 DOI: 10.1002/nbm.787 Copyright © 2002 John Wiley & Sons, Ltd.

[19]David Berreby "Scanning Brains for Insights on Racial Perception" Article publish at NewYork Times, Science Section, September 5, 2000, Copyright © 2000 The New York Times Co. Reprinted with permission.

[20]Douglas C. Noll "Notes on Physics of Magnetic Resonance Imaging" Co-Director, Functional MRI Laboratory Department of Biomedical Engineering, University of Michigan, United States of America, Fall-2006, Notes-1.

[21]D Moratal, Vallés-Luch, MSc, L Martí-Bonmatí, ME Brummer "k-Space tutorial: an MRI educational tool for a better understanding of k-space", Biomedical Imaging and Intervention Journal Tutorial, Received 8 January 2008; received in revised form 21 February 2008; accepted 10 March 2008.

[22]A.J. Bohris, U. Goerke, P.J. McDonald, M. Mulheron†, B. Newling, B. Le Page "A broad line NMR and MRI study of water and water transport in portland cement pastes", published on ELSEVIER Available online 29 January 1999, doi:10.1016/S0730-725X(98)00072-1.

[23]Moriel NessAiver "Magnetic Resonance Imaging", Article at Simply Physics, Science Journal of Doctrates, NessAiver, Ph.D., 3406 Labyrinth RdBaltimore, MD 21215.

[24]Seon Yeong Lee and Dympna Gallagher "Assessment methods in human body composition", Clinical nutrition and metabolic care 2008; 11(5):566-572. doi:10.1097/MCO.0b013e32830b5f23.

[25]Turner, RE, Snider, RF, Fleming, DG "SPIN RELAXATION OF HYDROGEN-ATOM ISOTOPES VIA ELECTRON-SPIN EXCHANGE WITH PARAMAGNETIC GASES", Journal Physical Review A, Volume 41 Pagination 1505-1516 Date Published Feb 1990 ISBN Number 1050-2947.

[26]John Scampini, "Introduction to Magnetic Resonance Imaging (MRI)", TUTORIAL 4681, Executive Director in the Industrial Communications and Ultrasound Business Unit @ May 10, 2010, Maxim Integrated Products, Inc.

[27]B.J. Casey, Matthew Davidson and Bruce Rosen "Functional magnetic resonance imaging: Basic principles of and application to developmental science", Blackwell Publishers Ltd. 200 Developmental Science 5:3 (2002), pp 301–309.

[28]Thulborn KR, Waterton JC, Matthews PM, Radda GK "Oxygenation dependence of the transverse relaxation time of water protons in whole blood at high field". Biochim. Biophys. Acta 714 (2): 265–70. doi:10.1016/0304-4165(82)90333-6. PMID 6275909.

[29]John C. Gore "Principles and practice of functional MRI of the human brain", J Clin Invest. Jul 1, 2003; 112(1): 4–9. doi: 10.1172/JCI200319010.

[30]SUSUMU MORI AND PETER B. BARKER "Diffusion Magnetic Resonance Imaging:Its Principle and Applications", Department of Radiology Anat Rec (New Anat) 257:102–109, 1999. r 1999 Wiley-Liss, Inc.

[31]Aaron G. Filler "The History, Development and Impact of Computed Imaging in Neurological Diagnosis and Neurosurgery: CT, MRI, and DTI", Received 12 July 2009 06:43 UTC; Posted 13 July 2009 Nature Precedings. doi:10.1038/npre.2009.3267.5.

[32]S. J. Holdsworth, S. Skare, and R. Bammer "On the application of phase correction and use of k-space entropy in partial Fourier diffusion-weighted EPI" Lucas MRS/I Center, Stanford University, Stanford, CA, United States.

[33]Martin Uecker, Shuo Zhang, Dirk Voit, Alexander Karaus, Klaus-Dietmar Merboldt and Jens Frahm "Real-time MRI at a resolution of 20 ms", NMR in Biomedicine, Volume 23, Issue 8, pages 986–994, October 2010Article first published online: 27 AUG 2010 OI: 10.1002/nbm.1585.

[34]A Haase, J Frahm, D Matthaei, W Hänicke, KD Merboldt "FLASH imaging: rapid NMR imaging using low flip angle pulses",Max-Planck-Institut für biophysikalische Chemie, Germany, Volume 67, Issue 2, Journal of Magnetic Resonance 67 (2): 258–266. doi:10.1016/0022-2364(86)90433-6.

[35]Schnall MD, Bolinger L, Renshaw PF, Haselgrove JC, Subramanian VH, Eleff SM, Barlow C, Leigh JS Jr, Chance B."Multinuclear MR imaging: a technique for combined anatomic and physiologic studies", published at ncbi, Radiology. 1987 Mar; 162(3):863-6.

[36]Ann Scott "Interventional and Intraoperative Magnetic Resonance Imaging", Alberta Heritage Foundation for Medical Research,Edmonton, AB T5J 4A7 CANADA, March 2004 IP 17: Information Paper.

[37]E. Mark Haacke "Susceptibility Weighted Imaging" Opening new doors to clinical applications of magnetic resonance imaging 5th Edition (2006) Department of Radiology, Wayne State University 3990 John R Detroit, MI 48202.

[38]Mansur G. Abdullah,Michael C. Anderson "Magnetic susceptibility" Article Written by: The Editors of Encyclopædia Britannica, www.britannica.com.

[39]Gujar, Sachin K; Maheshwari, Björkman-Burtscher, Isabella MD, Sundgren, Pia C MD, "Magnetic Resonance Spectroscopy", Journal of Neuro-Ophthalmology, United State of AmericaSeptember 2005 - Volume 25 - Issue 3 - pp 217-226.

[40]RP Wagner, HK Mitchell "Genetics and metabolism" [BOOK] - 1955 - Am Genetic Assoc, Lucid survey of biochemical basis of inheritance.

[41]Ian Boomer, "Isotopes: theory, principles and practicalities", Stable Isotope & Luminescence Laboratory, Senior Research Fellow (SILLA) University of Birmingham, Edgbaston BirminghamB15 2TT UK.

[42]Paul Suetens (ed): "Fundamentals of Medical Imaging (2nd edition)" [BOOK] European Journal of Nuclear Medicine and Molecular Imaging, Dec 2010.

[43]Douglas C. Noll and Bradley P. Sutton "Gridding Procedures for Non-Cartesian K-space Trajectories" Dept. of Biomedical Engineering, University of Michigan, Ann Arbor, MI, USA.

[44]Denis Hoa "K-space exploration", article at http://www.imaios.com.

[45]Dr Jeremy Jones, Dr Usman Bashir "Article on k-space" on Radiopaedia.org, Royal Marsden Hospital, London, United Kingdom, http://radiopaedia.org/articles/k-space.

[46]Rahul Tayal "MRI Reconstruction" EN 250: Medical Imaging, Spring 2009, Laboratory for Engineering Man-Machine Systems.

[47]James Meakin "Introduction to MRI Acquisition", FMRIB Physics Group, FSL Course, Bristol, September 2012.

[48]Sarah K. Patch "k-Space Data Preprocessing for Artifact Reduction in MR Imaging", Multidimensional Image Processing, Analysis, and Display: RSNA Categorical Course in Diagnostic Radiology Physics 2005; pp 73–87.

[49]Deans, Stanley R. (1983), "The Radon Transform and Some of Its Applications", New York: John Wiley & Sons.

[50]Anthony J. Devaney "Generalized Projection-Slice Theorem for Fan Beam Diffraction Tomography", doi: 10.1177/016173468500700306 Ultrasonic Imaging July 1985 vol. 7 no. 3 264-275.

[51]Dr. Matthew D. Robson, John C. Gore and R. Todd Constable "Measurement of the point spread function in MRI using constant time imaging", Article first published online: 12 DEC 2005 DOI: 10.1002/mrm.1910380509 Magnetic Resonance in Medicine Volume 38, Issue 5, pages 733–740, November 1997.

[52]Jiang Hsieh "Computed Tomography: Principles, Design, Artifacts, and Recent Advances" Published: 2009, DOI:10.1117/3.817303, eISBN: 9780819480422 | Print ISBN13: 9780819475336.

[53]Lauterbur PC (1973). "Image Formation by Induced Local Interactions: Examples of Employing Nuclear Magnetic Resonance". Nature 242 (5394): 190–1. Bibcode: 1973Natur.242.190L. doi: 10.1038/242190a0.

[54]Yuan Cheng, C.Forbes Dewey "Discrete tomography by bayesian labeling with its efficient algorithm" C.Forbes Dewey Proc. SPIE 4121, Mathematical Modeling, Estimation, and Imaging, doi:10.1117/12.402448, October 2000. 

[55]Klaas P. Pruessmann, Markus Weiger, Markus B. Scheidegger, and Peter Boesiger"SENSE: Sensitivity Encoding for Fast MRI", Wiley-Liss, Inc publish in Magnetic Resonance in Medicine 42:952–962 (1999).

[56]Gylson Thomas and V K Govindan, A sequency domain filtered-backprojection algorithm for fast computation of tomographic images, #rd Int nat. Conf on Computational Intelligence, robotics and autonomous systems, National University of Singapore, Dec 2005.

[57]Megha Borse, S.B. Patil, B.S.Patil "LITERATURE SURVEY FOR 3D RECONSTRUCTION OF BRAIN MRI IMAGES", International Journal of Research in Engineering and Technology, Volume: 02 Issue: 11 | Nov-2013.

[58]Andrei V Bronnikov "A filtering approach to image reconstruction in 3D SPECT", published in Physics in Medicine and Biology, Volume 45, Number 9 2000 Phys. Med. Biol. 45 2639 doi:10.1088/0031-9155/45/9/315.

[59]Che-Yen Wen,Chien-Hsiung Lee "Point spread functions and their applications to forensic image restoration Introduction" ,Forensic Science Journal 2002;1:15-26, Department of Forensic Science, Central Police University, Taiwan ROC.


[61]Alejandro Ribés and Francis Schmitt "Linear Inverse Problems in Imaging" IEEE SIGNAL PROCESSING MAGAZINE JULY 2008 Digital Object Identifier 10.1109/MSP.2008.923099, IEEE Xplore. Downloaded on October 20, 2008 at 15:41.

[62]Magdeleine Dinguirard and Philip N. Slater "Calibration of Space-Multispectral Imaging Sensors: A Review", Optical Sciences Center, University of Arizona, Tucson, AZUSA Received 7 July 1998, Revised 24 November 1998, Available online 21 June 1999 Volume 68, Issue 3, June 1999, Pages 194–205.

[63]Gylson Thomas and V. K. Govindan, Computationally efficient filtered-back projection algorithm for topographic image reconstruction using Walsh transform, Journal of Visual communication and image representation, Vol 17, Issue 3, June 2006.

[64]Jong Chul Ye, Yoram Bresler, Pierre Moulin "A Self-Referencing Level-Set Method for Image Reconstruction from Sparse Fourier Samples" Punblished in International Journal of Computer Vision December 2002, Volume 50, Issue 3, pp 253-270.

[65]Richard Ttsai and Sstanley Oosher "level set methods and their applications in image science", University of California, Berkley, Cambridge University Press.

[66]Laura B. Montefusco, Damiana Lazzaro, "Nonlinear Filtering for Compressed Sensing Sparse Image Reconstruction" Department of Mathematics University of Bologna, Italy Serena Papi CIRI-ICT University of Bologna, Italy.

[67]Michael Lustig, David L. Donoho Member, Juan M. Santos Member and John M. Pauly "Compressed Sensing MRI", Packard 210, Published in Department of Electrical Engineering, 350 Serra Mall, Stanford University Stanford CA, 94305.

[68]Nick Kingsbury, Julian Magarey "Wavelet Transforms in Image Processing", Signal Analysis and Prediction Applied and Numerical Harmonic Analysis 1998, pp 27-46, Springer Copyright 1998 DOI 10.1007/978-1-4612-1768-8_2.

[69]Martin S. Banks, Benjamin R. Stephens, E.Eugenie Hartmann "The development of basic mechanisms of pattern vision: Spatial frequency channels ☆", Journal of Experimental Child Psychology Volume 40, Issue 3, December 1985, Pages 501–527.

[70]M. W. K. Kan, L. H. T. Leung, W. Wong, and N. Lam, "Radiation dose from cone beam computed tomography for image-guided radiation therapy," International Journal of Radiation Oncology Biology Physics, vol. 70, no. 1, pp. 272-279, 2008. 

[71]W. De Vos, J. Casselman, G.R.J. Swennen1, "Cone-beam computerized tomography (CBCT) imaging of the oral and maxillofacial region: A systematic review of the literature" Published at ELSEVIER, Accepted 27 February 2009, Volume 38, Pages 609–625.

[72]Karla Miller "Partial k-space Reconstruction", Lecture Notes, FMRIB Centre, University of Oxford, Nuffield Department of Clinical Neurosciences, University of Oxford, UK.

[73]Bin Rong Wu, Satoshi Ito, Yoshitsugu Kamimura and Yoshifumi Yamada "Zero-filling Technique in Fresnel Transform Image Reconstruction for MR Image Denoising", Progress In Electromagnetics Research Symposium 2005, Hangzhou, China, August 22-26.

[74]Christopher K. Anand, Paul M. Margosian, Francis H. Bearden"Two-dimensional phase-conjugate symmetry reconstruction for 3d spin-warp, echo-planar and echo-volume magnetic resonance imaging", Philips Medical Systems (Cleveland), Inc., Publication type: Grant, Publication date 25 Jun 2002.

[75]Douglas C. Noll, Dwight G. Nishimura, Member, IEEE, and Albert Macovski, Fellow, IEEE "Homodyne Detection in Magnetic Resonance Imaging", IEEE TRANSACTIONS ON MEDICAL IMAGING, VOL. IO, NO. 2, JUNE 1991.

[76]Kyle Herrity, Raviv Raich and AlfredO.HeroIII. "Blind reconstruction of sparse images with unknown point spread function" In HeroIII Department of EECS, University of Michigan, Ann Arbor, MI 48109-2122 School of EECS, Oregon State University, Corvallis, OR 97331-5501, pages 128–137, May 2008.

[77]Ludwig Ritschl1, Frank Bergner1, Christof Fleischmann2 and Marc Kachelrieß1 "Improved total variation-based CT image reconstruction applied to clinical data", Phys. Med. Biol. 56 1545. doi:10.1088/0031-9155/56/6/003, Received 24 August 2010, in final form 12 January 2011. Published 16 February 2011. Institute of Physics and Engineering in Medicine.

[78]Herman, Gabor T., "Fundamentals of computerized tomography: Image reconstruction from projection", published by Academic Press, 1980, Advances in Computer Vision and Pattern Recognition, 2nd edition, Springer, 2009.

[79]Alfred M., David L., Donoho Bruckstein and Michael Elad. "From sparse solutions of systems of equations to sparse modeling of signals and images related databases" Publisher: Society for Industrial and Applied Mathematics, DOI: 10.1137/060657704(3):471–81, July 2007.

[80]Candes Emmanuel J. and B. Wakin Michael. "An introduction to compressive sampling" IEEE Signal Processing Magazine, March 2008, Pages-10, ISBN 1053-5888/08/$25.002008IEE.

[81]Raviv Raich, Alfred Michael Ting "Sparse image reconstruction using sparse priors" Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109-2122, USA.

[82]D. P. Wipf and B. D. Rao, "Sparse Bayesian learning for basis selection," IEEE Trans. Signal Processing, vol. 52, no. 8, pp. 2153–2164, 2004.

[83]Jeffrey A. Fessler "MODEL-BASED IMAGE RECONSTRUCTION FOR MRI" Systems Lab of the ECE Division within the Electrical Engineering and Computer Science, GE Healthcare, University of Michigan, 4431 EECS Bldg., 1301 Beal Ave.

[84]Johnes Jose and Govindan V.K.; Image Reconstruction from Projections using Triangular Linear System of Equations; International Journal of Engineering Research & Technology (IJERT); Vol. 3 Issue 1, January – 2014; ISSN: 2278-0181. 

[85]Guo L1, Wang L, Wu Y, Yan W, Shen X. "Research on 3D modeling for head MRI image based on immune sphere-shaped support vector machine.", Conf Proc IEEE Eng Med Biol Soc. 2007;2007:1082-5, Engineering in Medicine and Biology Society, 2007. EMBS 2007. 29th Annual International Conference of the IEEE 

[86]Patrik Brynolfsson "Using Radial k-space Sampling and Temporal Filters in MRI to Improve Temporal Resolution", Medical Physics Conference, Department of Radiation Sciences Umea University, Umea, Sweden 2010.

[87]Dr Yuranga Weerakkody and Dr Usman Bashir "Temporal resolution" Article at Radiopedia, MBBS MMed FRCR Department of Radiology, Royal Marsden Hospital, London, United Kingdom.



[90]Jiaming Chen, Lu Zhang, Yuemin Zhu, Jianhua Luo "MRI Reconstruction From 2D Partial k-Space Using POCS Algorithm",College of Life Science & Technology, Shanghai Jiao Tong University, 200240, Shanghai, P.R.China, 978-1-4244-2902-8/09/$25.00 ©2009 IEEE.

[91]Fadi Yaacoub 1,2, Antoine Abche 1, Elie Karam 1 and Yskandar Hamam "MRI Reconstruction Using SVD in the Least Square Sense", 21st IEEE International Symposium on Computer-Based Medical Systems, 1063-7125/08 $25.00 © 2008 IEEEDOI 10.1109/CBMS.2008.662.3.

[92]Ron Rubinstein, Alfred M. Bruckstein and Michael Elad "Dictionaries for Sparse Representation Modeling" Senior Member, IEEE, IEEE PROCEEDINGS, VOL. 5 Israel Institute of Technology, Haifa 32000.

[93]Ying Song, Zhen Zhu, Yang Lu1, Qiegen Liu1 and Jun Zhao1, "Reconstruction of magnetic resonance imaging by three-dimensional dual-dictionary learning", Published at Wiley, Volume 71, Issue 3, pages 1285–1298, March 2014.

[94]Guoqiang Yu, Liang Li, Jianwei Gu, and Li Zhang "Total Variation Based Iterative Image Reconstruction", Department of Application Science, Nuctech Company Limited, Beijing, P.R. China 100084 CV B I A 2005, LNCS 3765, pp. 526–534, 2005.c Springer-Verlag Berlin Heidelberg 2005.