{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,2,27]],"date-time":"2026-02-27T03:47:18Z","timestamp":1772164038383,"version":"3.50.1"},"reference-count":54,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T00:00:00Z","timestamp":1609977600000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"},{"start":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T00:00:00Z","timestamp":1609977600000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springer.com\/tdm"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["Found Comput Math"],"published-print":{"date-parts":[[2021,10]]},"DOI":"10.1007\/s10208-020-09484-7","type":"journal-article","created":{"date-parts":[[2021,1,7]],"date-time":"2021-01-07T19:12:30Z","timestamp":1610046750000},"page":"1317-1361","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":6,"title":["Metrics, Quantization and Registration in Varifold Spaces"],"prefix":"10.1007","volume":"21","author":[{"given":"Hsi-Wei","family":"Hsieh","sequence":"first","affiliation":[]},{"given":"Nicolas","family":"Charon","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,1,7]]},"reference":[{"key":"9484_CR1","doi-asserted-by":"crossref","unstructured":"Allard, W.K.: On the first variation of a varifold. Annals of mathematics pp. 417\u2013491 (1972)","DOI":"10.2307\/1970868"},{"issue":"3","key":"9484_CR2","doi-asserted-by":"publisher","first-page":"435","DOI":"10.1016\/j.acha.2011.08.001","volume":"32","author":"WK Allard","year":"2012","unstructured":"Allard, W.K., Chen, G., Maggioni, M.: Multi-scale geometric methods for data sets ii: Geometric multi-resolution analysis. Applied and Computational Harmonic Analysis 32(3), 435\u2013462 (2012)","journal-title":"Applied and Computational Harmonic Analysis"},{"key":"9484_CR3","unstructured":"Almgren, F.J.: Plateau\u2019s problem: an invitation to varifold geometry, vol.\u00a013. American Mathematical Soc. (1966)"},{"key":"9484_CR4","doi-asserted-by":"crossref","DOI":"10.1093\/oso\/9780198502456.001.0001","volume-title":"Functions of bounded variation and free discontinuity problems","author":"L Ambrosio","year":"2000","unstructured":"Ambrosio, L., Fusco, N., Pallara, D.: Functions of bounded variation and free discontinuity problems. Oxford : Clarendon Press (2000)"},{"key":"9484_CR5","doi-asserted-by":"crossref","unstructured":"Ardekani, S., Jain, A., Jain, S., Abraham, T.P., Abraham, M.R., Zimmerman, S., Winslow, R.L., Miller, M.I., Younes, L.: Matching Sparse Sets of Cardiac Image Cross-Sections Using Large Deformation Diffeomorphic Metric Mapping Algorithm. Statistical Atlases and Computational Models of the Heart (2011)","DOI":"10.1007\/978-3-642-28326-0_24"},{"issue":"3","key":"9484_CR6","doi-asserted-by":"publisher","first-page":"975","DOI":"10.1137\/15M104431X","volume":"9","author":"S Arguill\u00e8re","year":"2016","unstructured":"Arguill\u00e8re, S., Miller, M., Younes, L.: Diffeomorphic Surface Registration with Atrophy Constraints. SIAM Journal on Imaging Sciences 9(3), 975\u20131003 (2016)","journal-title":"SIAM Journal on Imaging Sciences"},{"issue":"1","key":"9484_CR7","doi-asserted-by":"publisher","first-page":"139","DOI":"10.1016\/j.matpur.2015.02.004","volume":"104","author":"S Arguill\u00e8re","year":"2015","unstructured":"Arguill\u00e8re, S., Tr\u00e9lat, E., Trouv\u00e9, A., Younes, L.: Shape deformation analysis from the optimal control viewpoint. Journal de Math\u00e9matiques Pures et Appliqu\u00e9es 104(1), 139\u2013178 (2015)","journal-title":"Journal de Math\u00e9matiques Pures et Appliqu\u00e9es"},{"key":"9484_CR8","doi-asserted-by":"publisher","first-page":"337","DOI":"10.1090\/S0002-9947-1950-0051437-7","volume":"68","author":"N Aronszajn","year":"1950","unstructured":"Aronszajn, N.: Theory of reproducing kernels. Trans. Amer. Math. Soc. 68, 337\u2013404 (1950)","journal-title":"Trans. Amer. Math. Soc."},{"key":"9484_CR9","first-page":"72","volume":"25","author":"M Bauer","year":"2019","unstructured":"Bauer, M., Bruveris, M., Charon, N., M\u00f8ller-Andersen, J.: A relaxed approach for curve matching with elastic metrics. ESAIM: Control, Optimisation and Calculus of Variations 25, 72 (2019)","journal-title":"ESAIM: Control, Optimisation and Calculus of Variations"},{"key":"9484_CR10","doi-asserted-by":"crossref","unstructured":"Beg, M.F., Miller, M.I., Trouv\u00e9, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International journal of computer vision 61(139-157) (2005)","DOI":"10.1023\/B:VISI.0000043755.93987.aa"},{"key":"9484_CR11","doi-asserted-by":"crossref","unstructured":"Bogachev, V.I.: Measure theory, vol.\u00a02. Springer Science & Business Media (2007)","DOI":"10.1007\/978-3-540-34514-5"},{"issue":"4","key":"9484_CR12","doi-asserted-by":"publisher","first-page":"1344","DOI":"10.1137\/110846324","volume":"10","author":"M Bruveris","year":"2012","unstructured":"Bruveris, M., Risser, L., Vialard, F.X.: Mixture of Kernels and Iterated Semidirect Product of Diffeomorphisms Groups. Multiscale Modeling and Simulation 10(4), 1344\u20131368 (2012)","journal-title":"Multiscale Modeling and Simulation"},{"issue":"1","key":"9484_CR13","doi-asserted-by":"publisher","first-page":"28","DOI":"10.1515\/geofl-2018-0004","volume":"3","author":"B Buet","year":"2018","unstructured":"Buet, B., Leonardi, G.P., Masnou, S.: Discretization and approximation of surfaces using varifolds. Geometric Flows 3(1), 28\u201356 (2018)","journal-title":"Geometric Flows"},{"issue":"01","key":"9484_CR14","doi-asserted-by":"publisher","first-page":"19","DOI":"10.1142\/S0219530510001503","volume":"8","author":"C Carmeli","year":"2010","unstructured":"Carmeli, C., De\u00a0Vito, E., Toigo, A., Umanita, V.: Vector valued reproducing kernel Hilbert spaces and universality. Analysis and Applications 8(01), 19\u201361 (2010)","journal-title":"Analysis and Applications"},{"issue":"2","key":"9484_CR15","doi-asserted-by":"publisher","first-page":"287","DOI":"10.1007\/s10208-015-9288-2","volume":"17","author":"B Charlier","year":"2017","unstructured":"Charlier, B., Charon, N., Trouv\u00e9, A.: The Fshape Framework for the Variability Analysis of Functional Shapes. Foundations of Computational Mathematics 17(2), 287\u2013357 (2017)","journal-title":"Foundations of Computational Mathematics"},{"key":"9484_CR16","unstructured":"Charlier, B., Feydy, J., Glaun\u00e8s, J.: Keops (software). https:\/\/www.kernel-operations.io\/ (2018)"},{"key":"9484_CR17","doi-asserted-by":"crossref","unstructured":"Charon, N., Charlier, B., Glaun\u00e8s, J., Gori, P., Roussillon, P.: Fidelity metrics between curves and surfaces: currents, varifolds, and normal cycles. In: Riemannian Geometric Statistics in Medical Image Analysis, pp. 441\u2013477 (2020)","DOI":"10.1016\/B978-0-12-814725-2.00021-2"},{"issue":"4","key":"9484_CR18","doi-asserted-by":"publisher","first-page":"2547","DOI":"10.1137\/130918885","volume":"6","author":"N Charon","year":"2013","unstructured":"Charon, N., Trouv\u00e9, A.: The varifold representation of non-oriented shapes for diffeomorphic registration. SIAM journal of Imaging Sciences 6(4), 2547\u20132580 (2013)","journal-title":"SIAM journal of Imaging Sciences"},{"issue":"1","key":"9484_CR19","doi-asserted-by":"publisher","first-page":"83","DOI":"10.1007\/s00365-016-9346-2","volume":"45","author":"N Chauffert","year":"2017","unstructured":"Chauffert, N., Ciuciu, P., Kahn, J., Weiss, P.: A projection method on measures sets. Constructive Approximation 45(1), 83\u2013111 (2017)","journal-title":"Constructive Approximation"},{"issue":"1","key":"9484_CR20","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10208-016-9331-y","volume":"18","author":"L Chizat","year":"2018","unstructured":"Chizat, L., Peyr\u00e9, G., Schmitzer, B., Vialard, F.X.: An Interpolating Distance Between Optimal Transport and Fisher-Rao Metrics. Foundations of Computational Mathematics 18(1), 1\u201344 (2018)","journal-title":"Foundations of Computational Mathematics"},{"key":"9484_CR21","doi-asserted-by":"crossref","unstructured":"Cohen-Steiner, D., Morvan, J.: Restricted Delaunay triangulation and normal cycle. Comput. Geom pp. 312\u2013321 (2003)","DOI":"10.1145\/777792.777839"},{"issue":"3","key":"9484_CR22","doi-asserted-by":"publisher","first-page":"1073","DOI":"10.1016\/j.neuroimage.2010.11.056","volume":"55","author":"S Durrleman","year":"2010","unstructured":"Durrleman, S., Fillard, P., Pennec, X., Trouv\u00e9, A., Ayache, N.: Registration, atlas estimation and variability analysis of white matter fiber bundles modeled as currents. NeuroImage 55(3), 1073\u20131090 (2010)","journal-title":"NeuroImage"},{"issue":"5","key":"9484_CR23","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1016\/j.media.2009.07.007","volume":"13","author":"S Durrleman","year":"2009","unstructured":"Durrleman, S., Pennec, X., Trouv\u00e9, A., Ayache, N.: Statistical models of sets of curves and surfaces based on currents. Medical image analysis 13(5), 793\u2013808 (2009)","journal-title":"Medical image analysis"},{"key":"9484_CR24","doi-asserted-by":"crossref","unstructured":"Evans, L.: Measure theory and fine properties of functions. Routledge (2018)","DOI":"10.1201\/9780203747940"},{"key":"9484_CR25","unstructured":"Federer, H.: Geometric measure theory. Springer (1969)"},{"key":"9484_CR26","doi-asserted-by":"crossref","unstructured":"Feydy, J., Charlier, B., Vialard, F.X., Peyr\u00e9, G.: Optimal Transport for Diffeomorphic Registration. In: Medical Image Computing and Computer Assisted Intervention, pp. 291\u2013299 (2017)","DOI":"10.1007\/978-3-319-66182-7_34"},{"issue":"3","key":"9484_CR27","doi-asserted-by":"publisher","first-page":"317","DOI":"10.1007\/s11263-008-0141-9","volume":"80","author":"J Glaun\u00e8s","year":"2008","unstructured":"Glaun\u00e8s, J., Qiu, A., Miller, M., Younes, L.: Large deformation diffeomorphic metric curve mapping. Int J Comput Vis 80(3), 317\u2013336 (2008)","journal-title":"Int J Comput Vis"},{"key":"9484_CR28","first-page":"712","volume":"2","author":"J Glaun\u00e8s","year":"2004","unstructured":"Glaun\u00e8s, J., Trouv\u00e9, A., Younes, L.: Diffeomorphic matching of distributions: A new approach for unlabelled point-sets and sub-manifolds matching. Computer Vision and Pattern Recognition (CVPR) 2, 712\u2013718 (2004)","journal-title":"Computer Vision and Pattern Recognition (CVPR)"},{"key":"9484_CR29","doi-asserted-by":"crossref","unstructured":"Glaun\u00e8s, J., Vaillant, M.: Surface matching via currents. Proceedings of Information Processing in Medical Imaging (IPMI), Lecture Notes in Computer Science 3565(381-392) (2006)","DOI":"10.1007\/11505730_32"},{"key":"9484_CR30","doi-asserted-by":"crossref","unstructured":"Gori, P., Colliot, O., Marrakchi-Kacem, L., Worbe, Y., Fallani, F.D.V., Chavez, M., Poupon, C., Hartmann, A., Ayache, N., Durrleman, S.: Parsimonious Approximation of Streamline Trajectories in White Matter Fiber Bundles. IEEE Transactions on Medical Imaging PP(99) (2016)","DOI":"10.1109\/TMI.2016.2591080"},{"key":"9484_CR31","unstructured":"Graf, S., Luschgy, H.: Foundations of quantization for probability distributions. Springer (2007)"},{"key":"9484_CR32","volume-title":"General pattern theory: A mathematical study of regular structures","author":"U Grenander","year":"1993","unstructured":"Grenander, U.: General pattern theory: A mathematical study of regular structures. Clarendon Press Oxford (1993)"},{"key":"9484_CR33","doi-asserted-by":"crossref","unstructured":"Hastie, T., Tibshirani, R., Friedman, J.: The elements of statistical learning. Springer (2001)","DOI":"10.1007\/978-0-387-21606-5"},{"key":"9484_CR34","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1142\/9789811200137_0003","volume":"37","author":"HW Hsieh","year":"2019","unstructured":"Hsieh, H.W., Charon, N.: Diffeomorphic registration of discrete geometric distributions. Mathematics Of Shapes And Applications 37, 45 (2019)","journal-title":"Mathematics Of Shapes And Applications"},{"key":"9484_CR35","unstructured":"Hu, Y., Hudelson, M., Krishnamoorthy, B., Tumurbaatar, A., Vixie, K.R.: Median Shapes. preprint (2018)"},{"key":"9484_CR36","unstructured":"Jacobs, H.: Symmetries in LDDMM with higher-order momentum distributions. Mathematical Foundations of Computational Anatomy (MFCA) Proceedings pp. 20\u201331 (2013)"},{"key":"9484_CR37","doi-asserted-by":"crossref","unstructured":"Kaltenmark, I., Charlier, B., Charon, N.: A general framework for curve and surface comparison and registration with oriented varifolds. Computer Vision and Pattern Recognition (CVPR) (2017)","DOI":"10.1109\/CVPR.2017.487"},{"issue":"2","key":"9484_CR38","first-page":"343","volume":"18","author":"B Kloeckner","year":"2012","unstructured":"Kloeckner, B.: Approximation by finitely supported measures. ESAIM: Control, Optimisation and Calculus of Variations 18(2), 343\u2013359 (2012)","journal-title":"ESAIM: Control, Optimisation and Calculus of Variations"},{"issue":"1","key":"9484_CR39","doi-asserted-by":"publisher","first-page":"57","DOI":"10.4310\/GIC.2014.v1.n1.a2","volume":"1","author":"M Micheli","year":"2014","unstructured":"Micheli, M., Glaun\u00e8s, J.: Matrix-valued kernels for shape deformation analysis. Geometry, Imaging and Computing 1(1), 57\u2013139 (2014)","journal-title":"Geometry, Imaging and Computing"},{"key":"9484_CR40","doi-asserted-by":"publisher","first-page":"16","DOI":"10.1016\/j.neuroimage.2008.10.044","volume":"45","author":"M Miller","year":"2009","unstructured":"Miller, M., Qiu, A.: The emerging discipline of computational functional anatomy. NeuroImage 45, 16\u201339 (2009)","journal-title":"NeuroImage"},{"key":"9484_CR41","doi-asserted-by":"crossref","unstructured":"Miller, M., Ratnanather, J.T., Tward, D.J., Brown, T., Lee, D., Ketcha, M., Mori, K., Wang, M.C., Mori, S., Albert, M., Younes, L., \u00a0, B.R.T.: Network Neurodegeneration in Alzheimer\u2019s Disease via MRI Based Shape Diffeomorphometry and High-Field Atlasing. Frontiers in Bioengineering and Biotechnology 3, 54 (2015)","DOI":"10.3389\/fbioe.2015.00054"},{"issue":"17","key":"9484_CR42","doi-asserted-by":"publisher","first-page":"447","DOI":"10.1146\/annurev-bioeng-071114-040601","volume":"7","author":"M Miller","year":"2015","unstructured":"Miller, M., Trouv\u00e9, A., Younes, L.: Hamiltonian Systems and Optimal Control in Computational Anatomy: 100 Years Since D\u2019Arcy Thompson. Annu Rev Biomed Eng 7(17), 447\u2013509 (2015)","journal-title":"Annu Rev Biomed Eng"},{"key":"9484_CR43","unstructured":"Overton, M.: HANSO: hybrid algorithm for non-smooth optimization 2.2 (2016). https:\/\/cs.nyu.edu\/overton\/software\/hanso\/"},{"issue":"1","key":"9484_CR44","doi-asserted-by":"publisher","first-page":"335","DOI":"10.1007\/s00205-013-0669-x","volume":"211","author":"B Piccoli","year":"2014","unstructured":"Piccoli, B., Rossi, F.: Generalized Wasserstein Distance and its Application to Transport Equations with Source. Archive for Rational Mechanics and Analysis 211(1), 335\u2013358 (2014)","journal-title":"Archive for Rational Mechanics and Analysis"},{"key":"9484_CR45","unstructured":"Pontryagin, L., Boltyanskii, V., Gamkrelidze, R., Mishchenko, E.: The Mathematical Theory of Optimal Processes. John Wiley & Sons (1962)"},{"issue":"4","key":"9484_CR46","doi-asserted-by":"publisher","first-page":"1991","DOI":"10.1137\/16M1070529","volume":"9","author":"P Roussillon","year":"2016","unstructured":"Roussillon, P., Glaun\u00e8s, J.: Kernel Metrics on Normal Cycles and Application to Curve Matching. SIAM Journal on Imaging Sciences 9(4), 1991\u20132038 (2016)","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"9484_CR47","doi-asserted-by":"crossref","unstructured":"Roussillon, P., Glaun\u00e8s, J.: Surface Matching Using Normal Cycles. Geometric Science of Information pp. 73\u201380 (2017)","DOI":"10.1007\/978-3-319-68445-1_9"},{"key":"9484_CR48","unstructured":"Simon, L.: Lecture notes on geometric measure theory. Australian national university (1983)"},{"issue":"1","key":"9484_CR49","doi-asserted-by":"publisher","first-page":"341","DOI":"10.1137\/110859002","volume":"6","author":"S Sommer","year":"2013","unstructured":"Sommer, S., Nielsen, M., Darkner, S., Pennec, X.: Higher-Order Momentum Distributions and Locally Affine LDDMM Registration. SIAM Journal on Imaging Sciences 6(1), 341\u2013367 (2013)","journal-title":"SIAM Journal on Imaging Sciences"},{"key":"9484_CR50","first-page":"2389","volume":"12","author":"BK Sriperumbudur","year":"2011","unstructured":"Sriperumbudur, B.K., Fukumizu, K., Lanckriet, G.R.: Universality, characteristic kernels and RKHS embedding of measures. Journal of Machine Learning Research 12, 2389\u20132410 (2011)","journal-title":"Journal of Machine Learning Research"},{"issue":"2","key":"9484_CR51","doi-asserted-by":"publisher","first-page":"229","DOI":"10.1007\/s11263-011-0481-8","volume":"97","author":"FX Vialard","year":"2012","unstructured":"Vialard, F.X., Risser, L., Rueckert, D., Cotter, C.: Diffeomorphic 3D Image Registration via Geodesic Shooting Using an Efficient Adjoint Calculation. International Journal of Computer Vision 97(2), 229\u2013241 (2012)","journal-title":"International Journal of Computer Vision"},{"key":"9484_CR52","unstructured":"Villani, C.: Optimal transport: old and new, vol. 338. Springer Science & Business Media (2008)"},{"key":"9484_CR53","doi-asserted-by":"crossref","unstructured":"Younes, L.: Shapes and diffeomorphisms. Springer (2019)","DOI":"10.1007\/978-3-662-58496-5"},{"issue":"1","key":"9484_CR54","doi-asserted-by":"publisher","first-page":"84","DOI":"10.2307\/1968882","volume":"43","author":"LC Young","year":"1942","unstructured":"Young, L.C.: Generalized surfaces in the calculus of variations. Annals of mathematics 43(1), 84\u2013103 (1942)","journal-title":"Annals of mathematics"}],"container-title":["Foundations of Computational Mathematics"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09484-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10208-020-09484-7\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10208-020-09484-7.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,8,22]],"date-time":"2024-08-22T01:36:58Z","timestamp":1724290618000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10208-020-09484-7"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2021,1,7]]},"references-count":54,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2021,10]]}},"alternative-id":["9484"],"URL":"https:\/\/doi.org\/10.1007\/s10208-020-09484-7","relation":{},"ISSN":["1615-3375","1615-3383"],"issn-type":[{"value":"1615-3375","type":"print"},{"value":"1615-3383","type":"electronic"}],"subject":[],"published":{"date-parts":[[2021,1,7]]},"assertion":[{"value":"3 July 2019","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"10 November 2020","order":2,"name":"revised","label":"Revised","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"12 November 2020","order":3,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"7 January 2021","order":4,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}]}}