{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,25]],"date-time":"2025-10-25T12:40:37Z","timestamp":1761396037784,"version":"3.37.3"},"reference-count":40,"publisher":"Springer Science and Business Media LLC","issue":"4","license":[{"start":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T00:00:00Z","timestamp":1622073600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T00:00:00Z","timestamp":1622073600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100004731","name":"Natural Science Foundation of Zhejiang Province","doi-asserted-by":"publisher","award":["LY20A010001"],"award-info":[{"award-number":["LY20A010001"]}],"id":[{"id":"10.13039\/501100004731","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Machine Vision and Applications"],"published-print":{"date-parts":[[2021,7]]},"DOI":"10.1007\/s00138-021-01214-5","type":"journal-article","created":{"date-parts":[[2021,5,27]],"date-time":"2021-05-27T12:05:09Z","timestamp":1622117109000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":4,"title":["Multiplicative noise removal and blind inpainting of ultrasound images based on a new variational framework"],"prefix":"10.1007","volume":"32","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-2914-3227","authenticated-orcid":false,"given":"Fangfang","family":"Dong","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Nannan","family":"Li","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2021,5,27]]},"reference":[{"key":"1214_CR1","doi-asserted-by":"publisher","unstructured":"Lee, J.: Digital image enhancement and noise filtering by use of local statistics. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-2(2), 165\u2013168 (1980). https:\/\/doi.org\/10.1109\/TPAMI.1980.4766994","DOI":"10.1109\/TPAMI.1980.4766994"},{"key":"1214_CR2","doi-asserted-by":"publisher","unstructured":"Frost, V., Stiles, J., Shanmugan, K., Holtzman, J.: A model for radar images and its application to adaptive digital filtering of multiplicative noise. IEEE Trans. Pattern Anal. Mach. Intell. PAMI-4, 157\u2013166 (1982). https:\/\/doi.org\/10.1109\/TPAMI.1982.4767223","DOI":"10.1109\/TPAMI.1982.4767223"},{"key":"1214_CR3","doi-asserted-by":"publisher","first-page":"165","DOI":"10.1109\/TPAMI.1985.4767641","volume":"7","author":"D Kuan","year":"1985","unstructured":"Kuan, D., Sawchuk, A., Strand, T., Chavel, P.: Adaptive noise smoothing filter for images with signal-dependent noise. IEEE Trans. Pattern Anal. Mach. Intell. 7, 165\u2013177 (1985). https:\/\/doi.org\/10.1109\/TPAMI.1985.4767641","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"1214_CR4","doi-asserted-by":"publisher","first-page":"129","DOI":"10.1109\/31.16577","volume":"36","author":"T Loupas","year":"1989","unstructured":"Loupas, T., McDicken, W., Allan, P.: An adaptive weighted median filter for speckle suppression in medical ultrasonic images. IEEE Trans. Circuits Syst. 36, 129\u2013135 (1989). https:\/\/doi.org\/10.1109\/31.16577","journal-title":"IEEE Trans. Circuits Syst."},{"key":"1214_CR5","doi-asserted-by":"publisher","first-page":"2221","DOI":"10.1109\/TIP.2009.2024064","volume":"18","author":"P Coup\u00e9","year":"2009","unstructured":"Coup\u00e9, P., Hellier, P., Kervrann, C., Barillot, C.: Nonlocal means-based speckle filtering for ultrasound images. IEEE Trans. Image Process. 18, 2221\u20132229 (2009). https:\/\/doi.org\/10.1109\/TIP.2009.2024064","journal-title":"IEEE Trans. Image Process."},{"key":"1214_CR6","doi-asserted-by":"publisher","first-page":"2661","DOI":"10.1109\/TIP.2009.2029593","volume":"18","author":"C Deledalle","year":"2010","unstructured":"Deledalle, C., Denis, L., Tupin, F.: Iterative weighted maximum likelihood denoising with probabilistic patch-based weights. IEEE Trans. Image Process. 18, 2661\u20132672 (2010). https:\/\/doi.org\/10.1109\/TIP.2009.2029593","journal-title":"IEEE Trans. Image Process."},{"key":"1214_CR7","doi-asserted-by":"publisher","first-page":"925","DOI":"10.1137\/060671814","volume":"68","author":"G Aubert","year":"2008","unstructured":"Aubert, G., Aujol, J.F.: A variational approach to removing multiplicative noise. SIAM J. Appl. Math. 68, 925\u2013946 (2008). https:\/\/doi.org\/10.1137\/060671814","journal-title":"SIAM J. Appl. Math."},{"key":"1214_CR8","doi-asserted-by":"publisher","first-page":"294","DOI":"10.1137\/070689954","volume":"1","author":"J Shi","year":"2008","unstructured":"Shi, J., Osher, S.: A nonlinear inverse scale space method for a convex multiplicative noise model. SIAM J. Imaging Sci. 1, 294\u2013321 (2008). https:\/\/doi.org\/10.1137\/070689954","journal-title":"SIAM J. Imaging Sci."},{"key":"1214_CR9","doi-asserted-by":"publisher","first-page":"20","DOI":"10.1137\/080712593","volume":"2","author":"YW Wen","year":"2009","unstructured":"Wen, Y.W., Ng, M., Wen, Y.W.: A new total variation method for multiplicative noise removal. SIAM J. Imaging Sci. 2, 20\u201340 (2009). https:\/\/doi.org\/10.1137\/080712593","journal-title":"SIAM J. Imaging Sci."},{"key":"1214_CR10","doi-asserted-by":"publisher","first-page":"103","DOI":"10.1007\/0-387-21810-6_6","volume":"4","author":"L Rudin","year":"2003","unstructured":"Rudin, L., Lions, P., Osher, S.: Multiplicative denoising and deblurring: Theory and algorithms. Geom. Level Set Methods Imaging Vis. Graphics 4, 103\u2013120 (2003). https:\/\/doi.org\/10.1007\/0-387-21810-6_6","journal-title":"Geom. Level Set Methods Imaging Vis. Graphics"},{"key":"1214_CR11","doi-asserted-by":"publisher","first-page":"168","DOI":"10.1007\/s10851-009-0179-5","volume":"36","author":"G Steidl","year":"2010","unstructured":"Steidl, G., Teuber, T.: Removing multiplicative noise by Douglas\u2013Rachford splitting methods. J. Math. Imaging Vis. 36, 168\u2013184 (2010). https:\/\/doi.org\/10.1007\/s10851-009-0179-5","journal-title":"J. Math. Imaging Vis."},{"key":"1214_CR12","doi-asserted-by":"publisher","unstructured":"Shen, C., Pi, L., Peng, Y., Li, Z.: Variational-based speckle noise removal of SAR imagery. In: 2007 IEEE International Geoscience and Remote Sensing Symposium, pp. 532\u2013535 (2007). https:\/\/doi.org\/10.1109\/IGARSS.2007.4422848","DOI":"10.1109\/IGARSS.2007.4422848"},{"key":"1214_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/090748421","volume":"3","author":"F Li","year":"2010","unstructured":"Li, F., Ng, M., Shen, C.: Multiplicative noise removal with spatially varying regularization parameters. SIAM J. Imaging Sci. 3, 1\u201320 (2010). https:\/\/doi.org\/10.1137\/090748421","journal-title":"SIAM J. Imaging Sci."},{"key":"1214_CR14","doi-asserted-by":"publisher","unstructured":"Seabra, J., Xavier, J.a., Sanches, J.a.: Convex ultrasound image reconstruction with Log\u2013Euclidean priors. In: 2008 30th Annual International Conference of the IEEE Engineering in Medicine and Biology Society 2008, 435\u2013438 (2008). https:\/\/doi.org\/10.1109\/IEMBS.2008.4649183","DOI":"10.1109\/IEMBS.2008.4649183"},{"key":"1214_CR15","doi-asserted-by":"publisher","first-page":"51","DOI":"10.1016\/j.apm.2018.05.007","volume":"62","author":"J Lu","year":"2018","unstructured":"Lu, J., Yang, Z., Shen, L., Lu, Z., Yang, H., Xu, C.: A framelet algorithm for de-blurring images corrupted by multiplicative noise. Appl. Math. Model. 62, 51\u201361 (2018). https:\/\/doi.org\/10.1016\/j.apm.2018.05.007","journal-title":"Appl. Math. Model."},{"key":"1214_CR16","doi-asserted-by":"publisher","first-page":"112684","DOI":"10.1016\/j.cam.2019.112684","volume":"370","author":"C Li","year":"2019","unstructured":"Li, C., Ren, Z., Tang, L.: Multiplicative noise removal via using nonconvex regularizers based on total variation and wavelet frame. J. Comput. Appl. Math. 370, 112684 (2019). https:\/\/doi.org\/10.1016\/j.cam.2019.112684","journal-title":"J. Comput. Appl. Math."},{"key":"1214_CR17","doi-asserted-by":"publisher","first-page":"939","DOI":"10.1016\/j.mcm.2011.09.021","volume":"55","author":"F Dong","year":"2012","unstructured":"Dong, F., Zhang, H., Kong, D.X.: Nonlocal total variation models for multiplicative noise removal using split Bregman iteration. Math. Comput. Model. 55, 939\u2013954 (2012). https:\/\/doi.org\/10.1016\/j.mcm.2011.09.021","journal-title":"Math. Comput. Model."},{"issue":"5","key":"1214_CR18","doi-asserted-by":"publisher","first-page":"862","DOI":"10.1049\/iet-ipr.2018.5930","volume":"14","author":"P Liu","year":"2020","unstructured":"Liu, P.: Hybrid higher-order total variation model for multiplicative noise removal. IET Image Process. 14(5), 862\u2013873 (2020). https:\/\/doi.org\/10.1049\/iet-ipr.2018.5930","journal-title":"IET Image Process."},{"key":"1214_CR19","doi-asserted-by":"publisher","first-page":"109","DOI":"10.1016\/j.amc.2015.12.005","volume":"276","author":"M Shama","year":"2016","unstructured":"Shama, M., Huang, T., Liu, J., Wang, S.: A convex total generalized variation regularized model for multiplicative noise and blur removal. Appl. Math. Comput. 276, 109\u2013121 (2016). https:\/\/doi.org\/10.1016\/j.amc.2015.12.005","journal-title":"Appl. Math. Comput."},{"key":"1214_CR20","first-page":"747","volume":"32","author":"D Tian","year":"2016","unstructured":"Tian, D., Du, Y., Chen, D.: An adaptive fractional-order variation method for multiplicative noise removal. J. Inf. Sci. Eng. 32, 747\u2013762 (2016)","journal-title":"J. Inf. Sci. Eng."},{"key":"1214_CR21","first-page":"483","volume":"58","author":"G Chen","year":"2020","unstructured":"Chen, G., Li, G., Liu, Y., Zhang, X.P., Zhang, L.: Sar image despeckling based on combination of fractional-order total variation and nonlocal low rank regularization. IEEE Trans. Geosci. Remote Sens. 58, 483\u2013491 (2020)","journal-title":"IEEE Trans. Geosci. Remote Sens."},{"key":"1214_CR22","doi-asserted-by":"publisher","first-page":"703","DOI":"10.1515\/jiip-2016-0051","volume":"26","author":"Y Gao","year":"2018","unstructured":"Gao, Y., Yang, X.: Tgv-based multiplicative noise removal approach: Models and algorithms. J. Inverse Ill-posed Probl. 26, 703\u2013727 (2018). https:\/\/doi.org\/10.1515\/jiip-2016-0051","journal-title":"J. Inverse Ill-posed Probl."},{"key":"1214_CR23","doi-asserted-by":"publisher","first-page":"492","DOI":"10.1137\/090769521","volume":"3","author":"K Bredies","year":"2010","unstructured":"Bredies, K., Kunisch, K., Pock, T.: Total generalized variation. SIAM J. Imaging Sci. 3, 492\u2013526 (2010). https:\/\/doi.org\/10.1137\/090769521","journal-title":"SIAM J. Imaging Sci."},{"key":"1214_CR24","doi-asserted-by":"publisher","first-page":"117","DOI":"10.3934\/ipi.2019007","volume":"13","author":"H Na","year":"2019","unstructured":"Na, H., Kang, M., Jung, M., Kang, M.: Nonconvex tgv regularization model for multiplicative noise removal with spatially varying parameters. Inverse Probl. Imaging 13, 117\u2013147 (2019). https:\/\/doi.org\/10.3934\/ipi.2019007","journal-title":"Inverse Probl. Imaging"},{"key":"1214_CR25","doi-asserted-by":"publisher","unstructured":"Milici, C., Draganescu, G., Tenreiro\u00a0Machado, J.: Fractional Differential Equations, pp. 47\u201386 (2019). https:\/\/doi.org\/10.1007\/978-3-030-00895-6_4","DOI":"10.1007\/978-3-030-00895-6_4"},{"key":"1214_CR26","doi-asserted-by":"publisher","first-page":"61","DOI":"10.1016\/j.imavis.2005.12.008","volume":"25","author":"C Barcelos","year":"2007","unstructured":"Barcelos, C., Batista, M.: Image restoration using digital inpainting and noise removal. Image Vis. Comput. 25, 61\u201369 (2007). https:\/\/doi.org\/10.1016\/j.imavis.2005.12.008","journal-title":"Image Vis. Comput."},{"key":"1214_CR27","doi-asserted-by":"publisher","unstructured":"Rodr\u00edguez, P., Rojas, R., Wohlberg, B.: Mixed gaussian-impulse noise image restoration via total variation. In: 2012 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 1077\u20131080 (2012). https:\/\/doi.org\/10.1109\/ICASSP.2012.6288073","DOI":"10.1109\/ICASSP.2012.6288073"},{"key":"1214_CR28","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1098\/rsos.171176","volume":"5","author":"D Thai","year":"2016","unstructured":"Thai, D., Gottschlich, C.: Simultaneous inpainting and denoising by directional global three-part decomposition: connecting variational and fourier domain based image processing. R. Soc. Open Sci. 5, 171\u2013176 (2016). https:\/\/doi.org\/10.1098\/rsos.171176","journal-title":"R. Soc. Open Sci."},{"key":"1214_CR29","doi-asserted-by":"publisher","unstructured":"Dong, B., Ji, H., Li, J., Shen, Z., Xu, Y.: Wavelet frame based blind image inpainting. Appl. Comput. Harmon. Anal. 32, 268\u2013279 (2012). https:\/\/doi.org\/10.1016\/j.acha.2011.06.001","DOI":"10.1016\/j.acha.2011.06.001"},{"key":"1214_CR30","doi-asserted-by":"publisher","unstructured":"Yang, J.: A tv-based approach to blind image inpainting. In: 2011 4th International Congress on Image and Signal Processing 2, 779\u2013781 (2011). https:\/\/doi.org\/10.1109\/CISP.2011.6100364","DOI":"10.1109\/CISP.2011.6100364"},{"key":"1214_CR31","doi-asserted-by":"publisher","first-page":"1227","DOI":"10.1137\/12087178X","volume":"6","author":"M Yan","year":"2013","unstructured":"Yan, M.: Restoration of images corrupted by impulse noise and mixed gaussian impulse noise using blind inpainting. SIAM J. Imaging Sci. 6, 1227\u20131245 (2013). https:\/\/doi.org\/10.1137\/12087178X","journal-title":"SIAM J. Imaging Sci."},{"key":"1214_CR32","doi-asserted-by":"publisher","first-page":"526","DOI":"10.1137\/110843642","volume":"6","author":"Y Wang","year":"2013","unstructured":"Wang, Y., Szlam, A., Lerman, G.: Robust locally linear analysis with applications to image denoising and blind inpainting. SIAM J. Imaging Sci. 6, 526\u2013562 (2013). https:\/\/doi.org\/10.1137\/110843642","journal-title":"SIAM J. Imaging Sci."},{"key":"1214_CR33","doi-asserted-by":"publisher","first-page":"64","DOI":"10.1007\/s10851-015-0589-5","volume":"54","author":"Y Shen","year":"2015","unstructured":"Shen, Y., Han, B., Braverman, E.: Removal of mixed gaussian and impulse noise using directional tensor product complex tight framelets. J. Math. Imaging Vis. 54, 64\u201377 (2015). https:\/\/doi.org\/10.1007\/s10851-015-0589-5","journal-title":"J. Math. Imaging Vis."},{"key":"1214_CR34","doi-asserted-by":"publisher","unstructured":"Setzer, S.: Split Bregman algorithm, Douglas\u2013Rachford splitting and frame shrinkage. In: Proceeding of Scale Space and Variational Methods in Computer Vision, Second International Conference, vol. 5567, pp. 464\u2013476 (2009). https:\/\/doi.org\/10.1007\/978-3-642-02256-2_39","DOI":"10.1007\/978-3-642-02256-2_39"},{"issue":"7","key":"1214_CR35","doi-asserted-by":"publisher","first-page":"2239","DOI":"10.1109\/TIP.2015.2417505","volume":"24","author":"MV Afonso","year":"2015","unstructured":"Afonso, M.V., Sanches, J.M.R.: Blind inpainting using $$\\ell \\_{0}$$ and total variation regularization. IEEE Trans. Image Process. 24(7), 2239\u20132253 (2015). https:\/\/doi.org\/10.1109\/TIP.2015.2417505","journal-title":"IEEE Trans. Image Process."},{"key":"1214_CR36","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1007\/BF01581204","volume":"55","author":"J Eckstein","year":"1992","unstructured":"Eckstein, J., Bertsekas, D., Systems, M.: On the Douglas\u2013Rachford splitting method and the proximal point algorithm for maximal monotone operators. Math. Program. 55, 293\u2013318 (1992). https:\/\/doi.org\/10.1007\/BF01581204","journal-title":"Math. Program."},{"key":"1214_CR37","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-010-0251-1","author":"A Chambolle","year":"2011","unstructured":"Chambolle, A., Pock, T.: A first-order primal-dual algorithm for convex problems with applications to imaging. J. Math. Imaging Vis. (2011). https:\/\/doi.org\/10.1007\/s10851-010-0251-1","journal-title":"J. Math. Imaging Vis."},{"key":"1214_CR38","doi-asserted-by":"publisher","first-page":"629","DOI":"10.1007\/s00041-008-9035-z","volume":"14","author":"T Blumensath","year":"2008","unstructured":"Blumensath, T., Davies, M.E.: Iterative thresholding for sparse approximations. J. Fourier Anal. Appl. 14, 629\u2013654 (2008). https:\/\/doi.org\/10.1007\/s00041-008-9035-z","journal-title":"J. Fourier Anal. Appl."},{"key":"1214_CR39","doi-asserted-by":"publisher","unstructured":"Knoll, F., Bredies, K., Pock, T., Stollberger, R.: Second order total generalized variation (tgv) for mri. Magnetic resonance in medicine. Off. J. Soc. Magn. Reson. Med. 65, 480\u201391 (2011). https:\/\/doi.org\/10.1002\/mrm.22595","DOI":"10.1002\/mrm.22595"},{"key":"1214_CR40","doi-asserted-by":"publisher","DOI":"10.1155\/2013\/585310","author":"D Chen","year":"2013","unstructured":"Chen, D., Chen, Y., Xue, D.: Fractional-order total variation image restoration based on primal-dual algorithm. Abstr. Appl. Anal. (2013). https:\/\/doi.org\/10.1155\/2013\/585310","journal-title":"Abstr. Appl. Anal."}],"container-title":["Machine Vision and Applications"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00138-021-01214-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s00138-021-01214-5\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s00138-021-01214-5.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2021,7,24]],"date-time":"2021-07-24T17:06:00Z","timestamp":1627146360000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s00138-021-01214-5"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,5,27]]},"references-count":40,"journal-issue":{"issue":"4","published-print":{"date-parts":[[2021,7]]}},"alternative-id":["1214"],"URL":"https:\/\/doi.org\/10.1007\/s00138-021-01214-5","relation":{},"ISSN":["0932-8092","1432-1769"],"issn-type":[{"type":"print","value":"0932-8092"},{"type":"electronic","value":"1432-1769"}],"subject":[],"published":{"date-parts":[[2021,5,27]]},"assertion":[{"value":"17 September 2020","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 May 2021","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 May 2021","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"27 May 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"86"}}