{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,6,4]],"date-time":"2026-06-04T20:07:09Z","timestamp":1780603629202,"version":"3.54.1"},"reference-count":45,"publisher":"MDPI AG","issue":"8","license":[{"start":{"date-parts":[[2021,7,31]],"date-time":"2021-07-31T00:00:00Z","timestamp":1627689600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100001691","name":"Japan Society for the Promotion of Science","doi-asserted-by":"publisher","award":["21K04080"],"award-info":[{"award-number":["21K04080"]}],"id":[{"id":"10.13039\/501100001691","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Entropy"],"abstract":"<jats:p>The problem of tomographic image reconstruction can be reduced to an optimization problem of finding unknown pixel values subject to minimizing the difference between the measured and forward projections. Iterative image reconstruction algorithms provide significant improvements over transform methods in computed tomography. In this paper, we present an extended class of power-divergence measures (PDMs), which includes a large set of distance and relative entropy measures, and propose an iterative reconstruction algorithm based on the extended PDM (EPDM) as an objective function for the optimization strategy. For this purpose, we introduce a system of nonlinear differential equations whose Lyapunov function is equivalent to the EPDM. Then, we derive an iterative formula by multiplicative discretization of the continuous-time system. Since the parameterized EPDM family includes the Kullback\u2013Leibler divergence, the resulting iterative algorithm is a natural extension of the maximum-likelihood expectation-maximization (MLEM) method. We conducted image reconstruction experiments using noisy projection data and found that the proposed algorithm outperformed MLEM and could reconstruct high-quality images that were robust to measured noise by properly selecting parameters.<\/jats:p>","DOI":"10.3390\/e23081005","type":"journal-article","created":{"date-parts":[[2021,8,1]],"date-time":"2021-08-01T21:51:07Z","timestamp":1627854667000},"page":"1005","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Noise-Robust Image Reconstruction Based on Minimizing Extended Class of Power-Divergence Measures"],"prefix":"10.3390","volume":"23","author":[{"ORCID":"https:\/\/orcid.org\/0000-0002-4699-1642","authenticated-orcid":false,"given":"Ryosuke","family":"Kasai","sequence":"first","affiliation":[{"name":"Graduate School of Health Sciences, Tokushima University, 3-18-15 Kuramoto, Tokushima 770-8509, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8303-828X","authenticated-orcid":false,"given":"Yusaku","family":"Yamaguchi","sequence":"additional","affiliation":[{"name":"Shikoku Medical Center for Children and Adults, National Hospital Organization, 2-1-1 Senyu, Zentsuji 765-8507, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0001-8612-5362","authenticated-orcid":false,"given":"Takeshi","family":"Kojima","sequence":"additional","affiliation":[{"name":"Institute of Biomedical Sciences, Tokushima University, 3-18-15 Kuramoto, Tokushima 770-8509, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0002-7883-676X","authenticated-orcid":false,"given":"Omar M.","family":"Abou Al-Ola","sequence":"additional","affiliation":[{"name":"Faculty of Science, Tanta University, El-Giesh Street, Tanta 31527, Egypt"}],"role":[{"vocabulary":"crossref","role":"author"}]},{"ORCID":"https:\/\/orcid.org\/0000-0003-4994-0522","authenticated-orcid":false,"given":"Tetsuya","family":"Yoshinaga","sequence":"additional","affiliation":[{"name":"Institute of Biomedical Sciences, Tokushima University, 3-18-15 Kuramoto, Tokushima 770-8509, Japan"}],"role":[{"vocabulary":"crossref","role":"author"}]}],"member":"1968","published-online":{"date-parts":[[2021,7,31]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","first-page":"471","DOI":"10.1016\/0022-5193(70)90109-8","article-title":"Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and X-ray photography","volume":"29","author":"Gordon","year":"1970","journal-title":"J. Theor. Biol."},{"key":"ref_2","doi-asserted-by":"crossref","first-page":"1455","DOI":"10.1088\/0031-9155\/49\/8\/006","article-title":"Experiments with the nonlinear and chaotic behaviour of the multiplicative algebraic reconstruction technique (MART) algorithm for computed tomography","volume":"49","author":"Badea","year":"2004","journal-title":"Phys. Med. Biol."},{"key":"ref_3","unstructured":"Kak, A.C., and Slaney, M. (1988). Principles of Computerized Tomographic Imaging, IEEE Press."},{"key":"ref_4","unstructured":"Stark, H. (1987). Image Recovery: Theory and Applications, Academic Press."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1097\/RLI.ob013e3181dzfeec","article-title":"Reducing abdominal CT radiation dose with adaptive statistical iterative reconstruction technique","volume":"45","author":"Prakash","year":"2010","journal-title":"Investig. Radiol."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"565","DOI":"10.1148\/radiol.11101450","article-title":"Adaptive statistical iterative reconstruction technique for radiation dose reduction in chest CT: A pilot study","volume":"259","author":"Singh","year":"2011","journal-title":"Radiology"},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"347","DOI":"10.1097\/RCT.0b013e31824e639e","article-title":"Comparison of hybrid and pure iterative reconstruction techniques with conventional filtered back projection: Dose reduction potential in the abdomen","volume":"36","author":"Singh","year":"2012","journal-title":"J. Comput. Assist. Tomogr."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"94","DOI":"10.1016\/j.ejmp.2012.01.003","article-title":"Iterative reconstruction methods in X-ray CT","volume":"28","author":"Beister","year":"2012","journal-title":"Phys. Med."},{"key":"ref_9","doi-asserted-by":"crossref","unstructured":"Huang, H.M., and Hsiao, I.T. (2016). Accelerating an Ordered-Subset Low-Dose X-Ray Cone Beam Computed Tomography Image Reconstruction with a Power Factor and Total Variation Minimization. PLoS ONE, 11.","DOI":"10.1371\/journal.pone.0153421"},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"113","DOI":"10.1109\/TMI.1982.4307558","article-title":"Maximum likelihood reconstruction for emission tomography","volume":"1","author":"Shepp","year":"1982","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_11","doi-asserted-by":"crossref","first-page":"601","DOI":"10.1109\/42.363108","article-title":"Accelerated image reconstruction using ordered subsets of projection data","volume":"13","author":"Hudson","year":"1994","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"1417","DOI":"10.1109\/83.465106","article-title":"Penalized maximum-likelihood image reconstruction using space-alternating generalized EM algorithms","volume":"4","author":"Fessler","year":"1995","journal-title":"IEEE Trans. Image Process."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"100","DOI":"10.1109\/83.650854","article-title":"Accelerating the EMML algorithm and related iterative algorithms by rescaled block-iterative methods","volume":"7","author":"Byrne","year":"1998","journal-title":"IEEE Trans. Image Process."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1088\/0031-9155\/51\/2\/004","article-title":"Convergence study of an accelerated ML-EM algorithm using bigger step size","volume":"51","author":"Hwang","year":"2006","journal-title":"Phys. Med. Biol."},{"key":"ref_15","doi-asserted-by":"crossref","first-page":"792","DOI":"10.1109\/83.499919","article-title":"Block-iterative methods for image reconstruction from projections","volume":"5","author":"Byrne","year":"1996","journal-title":"IEEE Trans. Image Process."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"427","DOI":"10.1111\/j.1475-3995.2008.00683.x","article-title":"Block-iterative algorithms","volume":"16","author":"Byrne","year":"2009","journal-title":"Int. Trans. Oper. Res."},{"key":"ref_17","first-page":"379","article-title":"Utilization of Non-Negativity Constraints in Reconstruction of Emission Tomograms","volume":"Volume 1","author":"Stephen","year":"1986","journal-title":"Information Processing in Medical Imaging"},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"98","DOI":"10.1109\/TMI.1987.4307809","article-title":"A fast reconstruction algorithm for stationary positron emission tomography based on a modified EM algorithm","volume":"6","author":"Tanaka","year":"1987","journal-title":"IEEE Trans. Med. Imaging"},{"key":"ref_19","doi-asserted-by":"crossref","unstructured":"Zeng, G.L. (November, January 31). The ML-EM algorithm is not optimal for poisson noise. Proceedings of the 2015 IEEE Nuclear Science Symposium and Medical Imaging Conference (NSS\/MIC), San Diego, CA, USA.","DOI":"10.1109\/NSSMIC.2015.7582178"},{"key":"ref_20","first-page":"105","article-title":"Extended Ordered-subsets Expectation-maximization Algorithm with Power Exponent for Noise-robust Image Reconstruction in Computed Tomography","volume":"8","author":"Yamaguchi","year":"2019","journal-title":"Radiat. Environ. Med."},{"key":"ref_21","doi-asserted-by":"crossref","unstructured":"Read, T.R.C., and Cressie, N.A.C. (1988). Goodness-of-Fit Statistics for Discrete Multivariate Data, Springer.","DOI":"10.1007\/978-1-4612-4578-0"},{"key":"ref_22","unstructured":"Pardo, L. (2005). Statistical Inference Based on Divergence Measures, Chapman and Hall\/CRC."},{"key":"ref_23","doi-asserted-by":"crossref","first-page":"4394","DOI":"10.1109\/TIT.2006.881731","article-title":"On Divergences and Informations in Statistics and Information Theory","volume":"52","author":"Liese","year":"2006","journal-title":"IEEE Trans. Inf. Theory"},{"key":"ref_24","doi-asserted-by":"crossref","unstructured":"Pardo, L. (2019). New Developments in Statistical Information Theory Based on Entropy and Divergence Measures. Entropy, 21.","DOI":"10.3390\/e21040391"},{"key":"ref_25","doi-asserted-by":"crossref","first-page":"79","DOI":"10.1214\/aoms\/1177729694","article-title":"On information and sufficiency","volume":"22","author":"Kullback","year":"1951","journal-title":"Ann. Math. Stat."},{"key":"ref_26","doi-asserted-by":"crossref","first-page":"445","DOI":"10.1214\/aos\/1176343842","article-title":"Minimum Hellinger distance estimates for parametric models","volume":"5","author":"Beran","year":"1977","journal-title":"Ann. Stat."},{"key":"ref_27","doi-asserted-by":"crossref","first-page":"683","DOI":"10.1007\/BF00773476","article-title":"Minimum disparity estimation for continuous models: Efficiency, distributions and robustness","volume":"46","author":"Basu","year":"1994","journal-title":"Ann. Inst. Stat. Math."},{"key":"ref_28","doi-asserted-by":"crossref","first-page":"549","DOI":"10.1093\/biomet\/85.3.549","article-title":"Robust and efficient estimation by minimising a density power divergence","volume":"85","author":"Basu","year":"1998","journal-title":"Biometrika"},{"key":"ref_29","unstructured":"Ghosh, A., and Basu, A. (2014). Estimation of Multivariate Location and Covariance using the S-Hellinger Distance. arXiv."},{"key":"ref_30","doi-asserted-by":"crossref","first-page":"321","DOI":"10.1016\/0168-9274(95)00065-3","article-title":"Using dynamical systems methods to solve minimization problems","volume":"18","author":"Schropp","year":"1995","journal-title":"Appl. Numer. Math."},{"key":"ref_31","doi-asserted-by":"crossref","first-page":"463","DOI":"10.1142\/S0218202599000233","article-title":"Continuous analog of Gauss-Newton method","volume":"9","author":"Airapetyan","year":"1999","journal-title":"Math. Models Methods Appl. Sci."},{"key":"ref_32","first-page":"491","article-title":"Dynamical systems and discrete methods for solving nonlinear ill-posed problems","volume":"Volume 1","author":"Anastassiou","year":"2000","journal-title":"Applied Mathematics Reviews"},{"key":"ref_33","first-page":"111","article-title":"Continuous methods for solving nonlinear ill-posed problems","volume":"Volume 25","author":"Ramm","year":"2000","journal-title":"Operator Theory and Its Applications"},{"key":"ref_34","doi-asserted-by":"crossref","first-page":"383","DOI":"10.1016\/S1007-5704(03)00006-6","article-title":"Dynamical systems method for solving operator equations","volume":"9","author":"Ramm","year":"2004","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_35","doi-asserted-by":"crossref","first-page":"399","DOI":"10.1016\/j.amc.2007.07.070","article-title":"A dynamical system method for solving nonlinear ill-posed problems","volume":"197","author":"Li","year":"2008","journal-title":"J. Comput. Appl. Math."},{"key":"ref_36","doi-asserted-by":"crossref","first-page":"1648","DOI":"10.1016\/j.cnsns.2009.06.025","article-title":"Continuous-time image reconstruction using differential equations for computed tomography","volume":"15","author":"Fujimoto","year":"2010","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_37","first-page":"723509:1","article-title":"Common Lyapunov function based on Kullback\u2013Leibler divergence for a switched nonlinear system","volume":"2011","author":"Fujimoto","year":"2011","journal-title":"Math. Probl. Eng."},{"key":"ref_38","doi-asserted-by":"crossref","first-page":"2081","DOI":"10.1016\/j.cnsns.2013.01.001","article-title":"Continuous-time image reconstruction for binary tomography","volume":"18","author":"Yamaguchi","year":"2013","journal-title":"Commun. Nonlinear Sci. Numer. Simul."},{"key":"ref_39","doi-asserted-by":"crossref","first-page":"1564123:1","DOI":"10.1155\/2017\/1564123","article-title":"Continuous Analog of Accelerated OS-EM Algorithm for Computed Tomography","volume":"2017","author":"Tateishi","year":"2017","journal-title":"Math. Probl. Eng."},{"key":"ref_40","unstructured":"Lyapunov, A.M. (1966). Stability of Motion, Academic Press."},{"key":"ref_41","doi-asserted-by":"crossref","first-page":"265","DOI":"10.1007\/s11071-006-9156-3","article-title":"Multiplicative Runge\u2013Kutta methods","volume":"50","author":"Aniszewska","year":"2007","journal-title":"Nonlinear Dyn."},{"key":"ref_42","doi-asserted-by":"crossref","first-page":"36","DOI":"10.1016\/j.jmaa.2007.03.081","article-title":"Multiplicative calculus and its applications","volume":"337","author":"Bashirov","year":"2008","journal-title":"J. Math. Anal. Appl."},{"key":"ref_43","doi-asserted-by":"crossref","first-page":"21","DOI":"10.1109\/TNS.1974.6499235","article-title":"The Fourier reconstruction of a head section","volume":"21","author":"Shepp","year":"1974","journal-title":"IEEE Trans. Nucl. Sci."},{"key":"ref_44","doi-asserted-by":"crossref","first-page":"600","DOI":"10.1109\/TIP.2003.819861","article-title":"Image quality assessment: From error visibility to structural similarity","volume":"13","author":"Wang","year":"2004","journal-title":"IEEE Trans. Image Process."},{"key":"ref_45","unstructured":"Kyoto Kagaku (2021, June 01). CT Whole Body Phantom PBU-60. Available online: https:\/\/kyotokagaku.com\/en\/products_introduction\/ph-2b\/."}],"container-title":["Entropy"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1005\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T06:38:26Z","timestamp":1760164706000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/1099-4300\/23\/8\/1005"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,7,31]]},"references-count":45,"journal-issue":{"issue":"8","published-online":{"date-parts":[[2021,8]]}},"alternative-id":["e23081005"],"URL":"https:\/\/doi.org\/10.3390\/e23081005","relation":{},"ISSN":["1099-4300"],"issn-type":[{"value":"1099-4300","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,7,31]]}}}