{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,10]],"date-time":"2026-05-10T15:52:07Z","timestamp":1778428327381,"version":"3.51.4"},"reference-count":24,"publisher":"MDPI AG","issue":"2","license":[{"start":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T00:00:00Z","timestamp":1611792000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["J. Imaging"],"abstract":"<jats:p>This paper is concerned with the reconstruction of relaxation time distributions in Nuclear Magnetic Resonance (NMR) relaxometry. This is a large-scale and ill-posed inverse problem with many potential applications in biology, medicine, chemistry, and other disciplines. However, the large amount of data and the consequently long inversion times, together with the high sensitivity of the solution to the value of the regularization parameter, still represent a major issue in the applicability of the NMR relaxometry. We present a method for two-dimensional data inversion (2DNMR) which combines Truncated Singular Value Decomposition and Tikhonov regularization in order to accelerate the inversion time and to reduce the sensitivity to the value of the regularization parameter. The Discrete Picard condition is used to jointly select the SVD truncation and Tikhonov regularization parameters. We evaluate the performance of the proposed method on both simulated and real NMR measurements.<\/jats:p>","DOI":"10.3390\/jimaging7020018","type":"journal-article","created":{"date-parts":[[2021,1,28]],"date-time":"2021-01-28T09:03:45Z","timestamp":1611824625000},"page":"18","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["A New Hybrid Inversion Method for 2D Nuclear Magnetic Resonance Combining TSVD and Tikhonov Regularization"],"prefix":"10.3390","volume":"7","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-1250-8218","authenticated-orcid":false,"given":"Germana","family":"Landi","sequence":"first","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-3961-4037","authenticated-orcid":false,"given":"Fabiana","family":"Zama","sequence":"additional","affiliation":[{"name":"Department of Mathematics, University of Bologna, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-9419-1965","authenticated-orcid":false,"given":"Villiam","family":"Bortolotti","sequence":"additional","affiliation":[{"name":"Department of Civil, Chemical, Environmental, and Materials Engineering, University of Bologna, 40126 Bologna, Italy"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2021,1,28]]},"reference":[{"key":"ref_1","unstructured":"Ernst, R., Bodenhausen, G., and Wokaun, A. (1997). Principles of Nuclear Magnetic Resonance in One and Two Dimensions, Oxford University Press. [2nd ed.]."},{"key":"ref_2","unstructured":"Bl\u00fcmich, B. (2005). Essential NMR, Springer."},{"key":"ref_3","doi-asserted-by":"crossref","first-page":"65","DOI":"10.1006\/jmre.1998.1387","article-title":"Uniform-Penalty Inversion of Multiexponential Decay Data","volume":"132","author":"Borgia","year":"1998","journal-title":"J. Magn. Reson."},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"1017","DOI":"10.1109\/78.995059","article-title":"Solving Fredholm integrals of the first kind with tensor product structure in 2 and 2.5 dimensions","volume":"50","author":"Venkataramanan","year":"2002","journal-title":"IEEE Trans. Signal Process."},{"key":"ref_5","first-page":"218","article-title":"Analysis of biological NMR relaxation data with continuous distributions of relaxation times","volume":"69","author":"Kroeker","year":"1986","journal-title":"J. Magn. Reson."},{"key":"ref_6","doi-asserted-by":"crossref","unstructured":"Hansen, P. (1998). Rank-Deficient and Discrete Ill-Posed Problems, Society for Industrial and Applied Mathematics.","DOI":"10.1137\/1.9780898719697"},{"key":"ref_7","doi-asserted-by":"crossref","unstructured":"Engl, H., Hanke, M., and Neubauer, A. (2000). Regularization of Inverse Problems, Springer. Mathematics and Its Applications.","DOI":"10.1007\/978-94-009-1740-8_3"},{"key":"ref_8","first-page":"134","article-title":"Algorithms for the regularization of ill-conditioned least squares problems","volume":"17","year":"1977","journal-title":"Behav. Inf. Technol."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"381","DOI":"10.1137\/0718025","article-title":"Estimating solutions of the first kind integral equations with nonnegative constraints and optimal smoothing","volume":"18","author":"Butler","year":"1981","journal-title":"SIAM J. Numer. Anal."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"139","DOI":"10.1006\/jmra.1995.1073","article-title":"Imaging multiexponential relaxation in the (y, logT1) plane, with application to clay filtration in rock cores","volume":"113","author":"Fordham","year":"1995","journal-title":"J. Magn. Reson. Ser. A"},{"key":"ref_11","doi-asserted-by":"crossref","unstructured":"Chouzenoux, E., Moussaoui, S., Idier, J., and Mariette, F. (2013, January 26\u201331). Primal-Dual Interior Point Optimization for a Regularized Reconstruction of NMR Relaxation Time Distributions. Proceedings of the 38th IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2013), Vancouver, BC, Canada.","DOI":"10.1109\/ICASSP.2013.6639374"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1775","DOI":"10.1137\/130932168","article-title":"Solving 2D Fredholm Integral from Incomplete Measurements Using Compressive Sensing","volume":"7","author":"Cloninger","year":"2014","journal-title":"SIAM J. Imaging Sci."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"658","DOI":"10.1007\/BF01933214","article-title":"The discrete Picard condition for discrete ill-posed problems","volume":"30","author":"Hansen","year":"1990","journal-title":"BIT Numer. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"015003","DOI":"10.1088\/1361-6420\/33\/1\/015003","article-title":"Uniform Penalty inversion of two-dimensional NMR relaxation data","volume":"33","author":"Bortolotti","year":"2016","journal-title":"Inverse Probl."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"195","DOI":"10.1016\/j.micromeso.2017.04.038","article-title":"I2DUPEN: Improved 2DUPEN algorithm for inversion of two-dimensional NMR data","volume":"269","author":"Bortolotti","year":"2017","journal-title":"Microporous Mesoporous Mater."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"085008","DOI":"10.1088\/0266-5611\/29\/8\/085008","article-title":"Regularization with randomized SVD for large-scale discrete inverse problems","volume":"29","author":"Xiang","year":"2013","journal-title":"Inverse Probl."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"085008","DOI":"10.1088\/0266-5611\/31\/8\/085008","article-title":"Randomized algorithms for large-scale inverse problems with general Tikhonov regularizations","volume":"31","author":"Xiang","year":"2015","journal-title":"Inverse Probl."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"221","DOI":"10.1137\/0320018","article-title":"Projected Newton method for optimization with simple constraints","volume":"20","author":"Bertsekas","year":"1982","journal-title":"SIAM J. Control Optim."},{"key":"ref_19","doi-asserted-by":"crossref","first-page":"9","DOI":"10.1080\/00207160.2012.716513","article-title":"Scaling techniques for gradient projection-type methods in astronomical image deblurring","volume":"90","author":"Bonettini","year":"2013","journal-title":"Int. J. Comput. Math."},{"key":"ref_20","doi-asserted-by":"crossref","unstructured":"Bortolotti, V., Landi, G., and Zama, F. (2020). 2DNMR data inversion using locally adapted multi-penalty regularization. arXiv.","DOI":"10.1007\/s10596-021-10049-y"},{"key":"ref_21","doi-asserted-by":"crossref","first-page":"A2741","DOI":"10.1137\/17M1123195","article-title":"Estimation of the Regularization Parameter in Linear Discrete Ill-Posed Problems Using the Picard Parameter","volume":"39","author":"Levin","year":"2017","journal-title":"SIAM J. Sci. Comput."},{"key":"ref_22","doi-asserted-by":"crossref","first-page":"53","DOI":"10.1016\/j.apnum.2016.03.006","article-title":"A stopping criterion for iterative regularization methods","volume":"106","author":"Landi","year":"2016","journal-title":"Appl. Numer. Math."},{"key":"ref_23","unstructured":"Bertsekas, D. (1999). Nonlinear Programming, Athena Scientific. [2nd ed.]."},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Vogel, C.R. (2002). 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