{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,1,1]],"date-time":"2026-01-01T10:13:14Z","timestamp":1767262394214,"version":"3.41.0"},"reference-count":42,"publisher":"Springer Science and Business Media LLC","issue":"8","license":[{"start":{"date-parts":[[2018,4,4]],"date-time":"2018-04-04T00:00:00Z","timestamp":1522800000000},"content-version":"tdm","delay-in-days":0,"URL":"http:\/\/www.springer.com\/tdm"}],"funder":[{"DOI":"10.13039\/501100000781","name":"European Research Council","doi-asserted-by":"publisher","award":["ARTIV1 contract number 727283"],"award-info":[{"award-number":["ARTIV1 contract number 727283"]}],"id":[{"id":"10.13039\/501100000781","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001665","name":"Agence Nationale de la Recherche","doi-asserted-by":"publisher","award":["ANR-11-LABX-0056-LMH"],"award-info":[{"award-number":["ANR-11-LABX-0056-LMH"]}],"id":[{"id":"10.13039\/501100001665","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100008530","name":"European Regional Development Fund","doi-asserted-by":"publisher","award":["POCI-01-0145-FEDER-006933\/SYSTEC"],"award-info":[{"award-number":["POCI-01-0145-FEDER-006933\/SYSTEC"]}],"id":[{"id":"10.13039\/501100008530","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100001871","name":"Funda\u00e7\u00e3o para a Ci\u00eancia e a Tecnologia","doi-asserted-by":"publisher","award":["POCI-01-0145-FEDER-006933\/SYSTEC"],"award-info":[{"award-number":["POCI-01-0145-FEDER-006933\/SYSTEC"]}],"id":[{"id":"10.13039\/501100001871","id-type":"DOI","asserted-by":"publisher"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2018,10]]},"DOI":"10.1007\/s10851-018-0810-4","type":"journal-article","created":{"date-parts":[[2018,4,19]],"date-time":"2018-04-19T15:06:31Z","timestamp":1524150391000},"page":"1231-1245","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":21,"title":["Highly Corrupted Image Inpainting Through Hypoelliptic Diffusion"],"prefix":"10.1007","volume":"60","author":[{"given":"Ugo V.","family":"Boscain","sequence":"first","affiliation":[]},{"ORCID":"https:\/\/orcid.org\/0000-0002-5179-4344","authenticated-orcid":false,"given":"Roman","family":"Chertovskih","sequence":"additional","affiliation":[]},{"given":"Jean-Paul","family":"Gauthier","sequence":"additional","affiliation":[]},{"given":"Dario","family":"Prandi","sequence":"additional","affiliation":[]},{"given":"Alexey","family":"Remizov","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2018,4,4]]},"reference":[{"unstructured":"Abas, F.: Analysis of craquelure patterns for content-based retrieval. PhD thesis, University of Southampton (2004)","key":"810_CR1"},{"unstructured":"Agrachev, A., Barilari, D., Boscain, U.: Introduction to Riemannian and Sub-Riemannian Geometry (Lecture Notes). http:\/\/webusers.imj-prg.fr\/~davide.barilari\/notes.php","key":"810_CR2"},{"doi-asserted-by":"crossref","unstructured":"Bella\u00efche, A.: The tangent space in sub-Riemannian geometry. In: Sub-Riemannian Geometry, volume 144 of Progress in Mathematics, pp. 1\u201378. Birkh\u00e4user, Basel (1996)","key":"810_CR3","DOI":"10.1007\/978-3-0348-9210-0_1"},{"doi-asserted-by":"crossref","unstructured":"Bertalmio, M., Sapiro, G., Caselles, V., Ballester, C.: Image inpainting. In: Proceedings of SIGGRAPH 2000, New Orleans, USA, pp. 417\u2013424 (2000)","key":"810_CR4","DOI":"10.1145\/344779.344972"},{"issue":"1","key":"810_CR5","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1007\/s10851-016-0669-1","volume":"57","author":"A Bohi","year":"2017","unstructured":"Bohi, A., Prandi, D., Guis, V., Bouchara, F., Gauthier, J.-P.: Fourier descriptors based on the structure of the human primary visual cortex with applications to object recognition. J. Math. Imaging Vis. 57(1), 117\u2013133 (2017)","journal-title":"J. Math. Imaging Vis."},{"key":"810_CR6","doi-asserted-by":"publisher","first-page":"43","DOI":"10.1134\/S0081543810030041","volume":"270","author":"U Boscain","year":"2010","unstructured":"Boscain, U., Charlot, G., Rossi, F.: Existence of planar curves minimizing length and curvature. Proc. Steklov Inst. Math. 270, 43\u201356 (2010)","journal-title":"Proc. Steklov Inst. Math."},{"issue":"2","key":"810_CR7","doi-asserted-by":"publisher","first-page":"669","DOI":"10.1137\/130924731","volume":"7","author":"U Boscain","year":"2014","unstructured":"Boscain, U., Chertovskih, R.A., Gauthier, J.-P., Remizov, A.O.: Hypoelliptic diffusion and human vision: a semidiscrete new twist. SIAM J. Imaging Sci. 7(2), 669\u2013695 (2014)","journal-title":"SIAM J. Imaging Sci."},{"issue":"3","key":"810_CR8","doi-asserted-by":"publisher","first-page":"748","DOI":"10.1051\/cocv\/2013082","volume":"20","author":"U Boscain","year":"2014","unstructured":"Boscain, U., Duits, R., Rossi, F., Sachkov, Yu.: Curve cuspless reconstruction via sub-Riemannian geometry. ESAIM Control Optim. Calc. Var. 20(3), 748\u2013770 (2014)","journal-title":"ESAIM Control Optim. Calc. Var."},{"issue":"3","key":"810_CR9","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1137\/11082405X","volume":"50","author":"U Boscain","year":"2012","unstructured":"Boscain, U., Duplaix, J., Gauthier, J.-P., Rossi, F.: Anthropomorphic image reconstruction via hypoelliptic diffusion. SIAM J. Control Optim. 50(3), 1\u201325 (2012)","journal-title":"SIAM J. Control Optim."},{"doi-asserted-by":"crossref","unstructured":"Boscain, U., Gauthier, J.-P., Prandi, D., Remizov, A.: Image reconstruction via non-isotropic diffusion in Dubins\/Reed-Shepp-like control systems. In: 53rd IEEE Conference on Decision and Control, pp. 4278\u20134283 (2014)","key":"810_CR10","DOI":"10.1109\/CDC.2014.7040056"},{"issue":"10","key":"810_CR11","doi-asserted-by":"publisher","first-page":"2634","DOI":"10.1109\/TIP.2010.2049240","volume":"19","author":"A Bugeau","year":"2010","unstructured":"Bugeau, A., Bertalm\u00edo, M., Caselles, V., Sapiro, G.: A comprehensive framework for image inpainting. IEEE Trans. Image Process. 19(10), 2634\u20132645 (2010)","journal-title":"IEEE Trans. Image Process."},{"issue":"2","key":"810_CR12","first-page":"564","volume":"63","author":"TF Chan","year":"2002","unstructured":"Chan, T.F., Kang, S.H., Shen, J.: Euler\u2019s elastica and curvature-based inpainting. SIAM J. Appl. Math. 63(2), 564\u2013592 (2002)","journal-title":"SIAM J. Appl. Math."},{"issue":"1","key":"810_CR13","doi-asserted-by":"publisher","first-page":"212","DOI":"10.1137\/15M1013572","volume":"9","author":"G Citti","year":"2016","unstructured":"Citti, G., Franceschiello, B., Sanguinetti, G., Sarti, A.: Sub-Riemannian mean curvature flow for image processing. SIAM J. Imaging Sci. 9(1), 212\u2013237 (2016)","journal-title":"SIAM J. Imaging Sci."},{"issue":"3","key":"810_CR14","doi-asserted-by":"publisher","first-page":"307","DOI":"10.1007\/s10851-005-3630-2","volume":"24","author":"G Citti","year":"2006","unstructured":"Citti, G., Sarti, A.: A cortical based model of perceptual completion in the roto-translation space. J. Math. Imaging Vis. 24(3), 307\u2013326 (2006)","journal-title":"J. Math. Imaging Vis."},{"issue":"3","key":"810_CR15","doi-asserted-by":"publisher","first-page":"605","DOI":"10.1016\/j.sigpro.2012.07.022","volume":"93","author":"B Cornelis","year":"2013","unstructured":"Cornelis, B., Ru\u017ei\u0107, T., Gezels, E., Dooms, A., Pi\u017eurica, A., Plati\u0161a, L., Cornelis, J., Martens, M., De Mey, M., Daubechies, I.: Crack detection and inpainting for virtual restoration of paintings: the case of the ghent altarpiece. Signal Process. 93(3), 605\u2013619 (2013)","journal-title":"Signal Process."},{"issue":"2","key":"810_CR16","doi-asserted-by":"publisher","first-page":"384","DOI":"10.1007\/s10851-013-0475-y","volume":"49","author":"R Duits","year":"2014","unstructured":"Duits, R., Boscain, U., Rossi, F., Sachkov, Y.: Association fields via cuspless sub-Riemannian geodesics in SE(2). J. Math. Imaging Vis. 49(2), 384\u2013417 (2014)","journal-title":"J. Math. Imaging Vis."},{"issue":"2","key":"810_CR17","doi-asserted-by":"publisher","first-page":"255","DOI":"10.1090\/S0033-569X-10-01172-0","volume":"68","author":"R Duits","year":"2010","unstructured":"Duits, R., Franken, E.: Left-invariant parabolic evolutions on $${\\rm SE}(2)$$ SE ( 2 ) and contour enhancement via invertible orientation scores. Part I: linear left-invariant diffusion equations on $${\\rm SE}(2)$$ SE ( 2 ) . Q. Appl. Math. 68(2), 255\u2013292 (2010)","journal-title":"Q. Appl. Math."},{"issue":"2","key":"810_CR18","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1090\/S0033-569X-10-01173-3","volume":"68","author":"R Duits","year":"2010","unstructured":"Duits, R., Franken, E.: Left-invariant parabolic evolutions on $${\\rm SE}(2)$$ SE ( 2 ) and contour enhancement via invertible orientation scores. Part II: nonlinear left-invariant diffusions on invertible orientation scores. Q. Appl. Math. 68(2), 293\u2013331 (2010)","journal-title":"Q. Appl. Math."},{"key":"810_CR19","doi-asserted-by":"publisher","first-page":"27","DOI":"10.1090\/S0033-569X-07-01066-0","volume":"66","author":"R Duits","year":"2008","unstructured":"Duits, R., van Almsick, M.A.: The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D euclidean motion group. Q. Appl. Math. 66, 27\u201367 (2008)","journal-title":"Q. Appl. Math."},{"doi-asserted-by":"crossref","unstructured":"Facciolo, G., Arias, P., Caselles, V., Sapiro, G.: Exemplar-based interpolation of sparsely sampled images. In: Cremers, D., Boykov, Yu., Blake, A., Schmidt, F.R. (eds.) Energy Minimization Methods in Computer Vision and Pattern Recognition: 7th International Conference EMMCVPR 2009 (Bonn, Germany, August 24\u201327, 2009) Proceedings, pp. 331\u2013344. Springer (2009)","key":"810_CR20","DOI":"10.21236\/ADA513256"},{"doi-asserted-by":"crossref","unstructured":"Ferziger, J.H., Peri\u0107, M.: Computational Methods for Fluid Dynamics, 3rd rev. edn. Springer, Berlin (2002)","key":"810_CR21","DOI":"10.1007\/978-3-642-56026-2"},{"doi-asserted-by":"crossref","unstructured":"Gromov, M.: Carnot-Carath\u00e9odory spaces seen from within. In: Sub-Riemannian Geometry, volume 144 of Progress in Mathematics, pp. 79\u2013323. Birkh\u00e4user, Basel (1996)","key":"810_CR22","DOI":"10.1007\/978-3-0348-9210-0_2"},{"issue":"1","key":"810_CR23","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10851-009-0167-9","volume":"36","author":"RK Hladky","year":"2010","unstructured":"Hladky, R.K., Pauls, S.D.: Minimal surfaces in the roto-translation group with applications to a neuro-biological image completion model. J. Math. Imaging Vis. 36(1), 1\u201327 (2010)","journal-title":"J. Math. Imaging Vis."},{"key":"810_CR24","doi-asserted-by":"publisher","first-page":"147","DOI":"10.1007\/BF02392081","volume":"119","author":"L H\u00f6rmander","year":"1967","unstructured":"H\u00f6rmander, L.: Hypoelliptic second order differential equations. Acta Math. 119, 147\u2013171 (1967)","journal-title":"Acta Math."},{"key":"810_CR25","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1113\/jphysiol.1959.sp006308","volume":"148","author":"DH Hubel","year":"1959","unstructured":"Hubel, D.H., Wiesel, T.N.: Receptive fields of single neurones in the cat\u2019s striate cortex. J. Physiol. 148, 574\u2013591 (1959)","journal-title":"J. Physiol."},{"key":"810_CR26","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4613-8150-1","volume-title":"Methods of Numerical Mathematics","author":"GI Marchuk","year":"1982","unstructured":"Marchuk, G.I.: Methods of Numerical Mathematics. Springer, Berlin (1982)"},{"issue":"1167","key":"810_CR27","doi-asserted-by":"publisher","first-page":"187","DOI":"10.1098\/rspb.1980.0020","volume":"207","author":"D Marr","year":"1980","unstructured":"Marr, D., Hildreth, E.: Theory of edge detection. Proc. R. Soc. Lond. B Biol. Sci. 207(1167), 187\u2013217 (1980)","journal-title":"Proc. R. Soc. Lond. B Biol. Sci."},{"issue":"2","key":"810_CR28","doi-asserted-by":"publisher","first-page":"68","DOI":"10.1109\/83.982815","volume":"11","author":"S Masnou","year":"2002","unstructured":"Masnou, S.: Disocclusion: a variational approach using level lines. IEEE Trans. Image Process. 11(2), 68\u201376 (2002)","journal-title":"IEEE Trans. Image Process."},{"doi-asserted-by":"crossref","unstructured":"Masnou, S., Morel, J.-M.: Level lines based disocclusion. In: Proceedings of 5th IEEE International Confernce on Image Processing, vol. 3, pp. 259\u2013263 (1998)","key":"810_CR29","DOI":"10.1109\/ICIP.1998.999016"},{"unstructured":"Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and applications, volume\u00a091 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI (2002)","key":"810_CR30"},{"key":"810_CR31","doi-asserted-by":"publisher","first-page":"117","DOI":"10.1113\/jphysiol.1979.sp012803","volume":"291","author":"L Peichl","year":"1979","unstructured":"Peichl, L., W\u00e4ssle, H.: Size, scatter and coverage of ganglion cell receptive field centres in the cat retina. J. Physiol. 291, 117\u2013141 (1979)","journal-title":"J. Physiol."},{"issue":"2\u20133","key":"810_CR32","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1016\/j.jphysparis.2003.10.010","volume":"97","author":"J Petitot","year":"2003","unstructured":"Petitot, J.: The neurogeometry of pinwheels as a sub-Riemannian contact structure. J. Physiol. Paris 97(2\u20133), 265\u2013309 (2003)","journal-title":"J. Physiol. Paris"},{"unstructured":"Petitot, J.: Neurog\u00e9om\u00e9trie de la vision - Mod\u00e8les math\u00e9matiques et physiques des architectures fonctionnelles. Les \u00c9ditions de l\u2019\u00c9cole Polytechnique (2008)","key":"810_CR33"},{"doi-asserted-by":"crossref","unstructured":"Ponomarenko, N., Jin, L., Lukin, V., Egiazarian, K.: Self-similarity measure for assessment of image visual quality. In: Proceedings of the 13th International Conference on Advanced Concepts for Intelligent Vision Systems, ACIVS\u201911, pp. 459\u2013470, Berlin, Heidelberg (2011)","key":"810_CR34","DOI":"10.1007\/978-3-642-23687-7_42"},{"doi-asserted-by":"crossref","unstructured":"Prandi, D., Boscain, U., Gauthier, J.-P.: Image processing in the semidiscrete group of rototranslations. In: Geometric Science of Information, volume 9389 of Lecture Notes in Computer Science, pp. 627\u2013634. Springer (2015)","key":"810_CR35","DOI":"10.1007\/978-3-319-25040-3_67"},{"unstructured":"Prandi, D., Gauthier, J.-P.: A semidiscrete version of the Petitot model as a plausible model for anthropomorphic image reconstruction and pattern recognition. Springer Briefs in Mathematics. Springer. arXiv:1704.03069","key":"810_CR36"},{"doi-asserted-by":"crossref","unstructured":"Sanguinetti, G., Citti, G., Sarti, A.: Image completion using a diffusion driven mean curvature flow in a sub-Riemannian space. In: Proceedings of the 3rd International Conference on Computer Vision Theory and Applications (VISAPP 2008), volume\u00a02, pp. 46\u201353 (2008)","key":"810_CR37","DOI":"10.5220\/0001075800460053"},{"issue":"2","key":"810_CR38","doi-asserted-by":"publisher","first-page":"221","DOI":"10.4310\/jdg\/1214440436","volume":"24","author":"RS Strichartz","year":"1986","unstructured":"Strichartz, R.S.: Sub-Riemannian geometry. J. Differ. Geom. 24(2), 221\u2013263 (1986)","journal-title":"J. Differ. Geom."},{"doi-asserted-by":"crossref","unstructured":"Strichartz, R.S.: Corrections to: Sub-Riemannian geometry [J. Differential Geom. 24(2), 221\u2013263 (1986); MR0862049 (88b:53055)]. J. Differ. Geom. 30(2), 595\u2013596 (1989)","key":"810_CR39","DOI":"10.4310\/jdg\/1214440436"},{"unstructured":"Voronin, V.V., Frantc, V.A., Marchuk, V.I., Sherstobitov, A.I., Egiazarian, K.: No-reference visual quality assessment for image inpainting. In: Proceedings of SPIE 9399, Image Processing: Algorithms and Systems XIII (March 16, 2015), p. 93990U (2015)","key":"810_CR40"},{"issue":"7","key":"810_CR41","doi-asserted-by":"publisher","first-page":"753","DOI":"10.1016\/j.image.2013.03.002","volume":"28","author":"M Wang","year":"2013","unstructured":"Wang, M., Yan, B., Ngan, K.N.: An efficient framework for image\/video inpainting. Signal Process. Image Commun. 28(7), 753\u2013762 (2013)","journal-title":"Signal Process. Image Commun."},{"doi-asserted-by":"crossref","unstructured":"Zhang, F., Li, S., Ma, L., Ngan, K.N.: Limitation and challenges of image quality measurement. In: Proceedings of SPIE 7744, Visual Communications and Image Processing 2010 (July 13, 2010), pp. 774402\u2013774402\u20138 (2010)","key":"810_CR42","DOI":"10.1117\/12.863083"}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"http:\/\/link.springer.com\/article\/10.1007\/s10851-018-0810-4\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-018-0810-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"http:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-018-0810-4.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,7,3]],"date-time":"2025-07-03T19:48:25Z","timestamp":1751572105000},"score":1,"resource":{"primary":{"URL":"http:\/\/link.springer.com\/10.1007\/s10851-018-0810-4"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2018,4,4]]},"references-count":42,"journal-issue":{"issue":"8","published-print":{"date-parts":[[2018,10]]}},"alternative-id":["810"],"URL":"https:\/\/doi.org\/10.1007\/s10851-018-0810-4","relation":{},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"type":"print","value":"0924-9907"},{"type":"electronic","value":"1573-7683"}],"subject":[],"published":{"date-parts":[[2018,4,4]]},"assertion":[{"value":"8 December 2017","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 March 2018","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 April 2018","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}