{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,3,26]],"date-time":"2026-03-26T20:36:37Z","timestamp":1774557397884,"version":"3.50.1"},"reference-count":71,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2024,1,13]],"date-time":"2024-01-13T00:00:00Z","timestamp":1705104000000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2024,1,13]],"date-time":"2024-01-13T00:00:00Z","timestamp":1705104000000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["VI.C 202\u2013031"],"award-info":[{"award-number":["VI.C 202\u2013031"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["VI.C 202\u2013031"],"award-info":[{"award-number":["VI.C 202\u2013031"]}],"id":[{"id":"10.13039\/501100003246","id-type":"DOI","asserted-by":"publisher"}]},{"DOI":"10.13039\/501100003246","name":"Nederlandse Organisatie voor Wetenschappelijk Onderzoek","doi-asserted-by":"publisher","award":["VI.C 202\u2013031"],"award-info":[{"award-number":["VI.C 202\u2013031"]}],"id":[{"id":"10.13039\/501100003246","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":[[2024,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>\n                    We introduce a data-driven version of the plus Cartan connection on the homogeneous space\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathbb {M}}_2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    of 2D positions and orientations. We formulate a theorem that describes all shortest and straight curves (parallel velocity and parallel momentum, respectively) with respect to this new data-driven connection and corresponding Riemannian manifold. Then we use these shortest curves for geodesic tracking of complex vasculature in multi-orientation image representations defined on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathbb {M}}_{2}$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    . The data-driven Cartan connection characterizes the Hamiltonian flow of all geodesics. It also allows for improved adaptation to curvature and misalignment of the (lifted) vessel structure that we track via globally optimal geodesics. We compute these geodesics numerically via steepest descent on distance maps on\n                    <jats:inline-formula>\n                      <jats:alternatives>\n                        <jats:tex-math>$${\\mathbb {M}}_2$$<\/jats:tex-math>\n                        <mml:math xmlns:mml=\"http:\/\/www.w3.org\/1998\/Math\/MathML\">\n                          <mml:msub>\n                            <mml:mi>M<\/mml:mi>\n                            <mml:mn>2<\/mml:mn>\n                          <\/mml:msub>\n                        <\/mml:math>\n                      <\/jats:alternatives>\n                    <\/jats:inline-formula>\n                    that we compute by a new modified anisotropic fast-marching method.Our experiments range from tracking single blood vessels with fixed endpoints to tracking complete vascular trees in retinal images. Single vessel tracking is performed in a single run in the multi-orientation image representation, where we project the resulting geodesics back onto the underlying image. The complete vascular tree tracking requires only two runs and avoids prior segmentation, placement of extra anchor points, and dynamic switching between geodesic models. Altogether we provide a geodesic tracking method using a single, flexible, transparent, data-driven geodesic model providing globally optimal curves which correctly follow highly complex vascular structures in retinal images. All\u00a0experiments\u00a0in this article\u00a0can\u00a0be\u00a0reproduced\u00a0via\u00a0documented\n                    <jats:italic>Mathematica<\/jats:italic>\n                    \u00a0notebooks\u00a0available\u00a0 at van den Berg (Data-driven left-invariant tracking in\n                    <jats:italic>Mathematica<\/jats:italic>\n                    , 2022).\n                  <\/jats:p>","DOI":"10.1007\/s10851-023-01170-x","type":"journal-article","created":{"date-parts":[[2024,1,13]],"date-time":"2024-01-13T06:02:16Z","timestamp":1705125736000},"page":"198-230","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":10,"title":["Geodesic Tracking via New Data-Driven Connections of Cartan Type for Vascular Tree Tracking"],"prefix":"10.1007","volume":"66","author":[{"given":"Nicky J.","family":"van den Berg","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bart M. N.","family":"Smets","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gautam","family":"Pai","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Marie","family":"Mirebeau","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Remco","family":"Duits","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2024,1,13]]},"reference":[{"key":"1170_CR1","unstructured":"Bekkers, E.: Retinal Image Analysis Using Sub-Riemannian Geometry in SE(2). PhD thesis, TU Eindhoven (2017)"},{"issue":"3","key":"1170_CR2","doi-asserted-by":"crossref","first-page":"384","DOI":"10.1097\/OPX.0000000000000484","volume":"92","author":"DL Weiler","year":"2015","unstructured":"Weiler, D.L., Engelke, C.B., Moore, A.L., Harrison, W.W.: Arteriole tortuosity associated with diabetic retinopathy and cholesterol. Optom. Vis. Sci. 92(3), 384\u2013391 (2015)","journal-title":"Optom. Vis. Sci."},{"issue":"4","key":"1170_CR3","first-page":"551","volume":"41","author":"MB Sasongko","year":"2016","unstructured":"Sasongko, M.B., Wong, T.Y., Nguyen, T.T., Cheung, C.Y., Shaw, J.E., Kawasaki, R., Lamoureux, E.L., Wang, J.J.: Retinal vessel tortuosity and its relation to traditional and novel vascular risk markers in persons with diabetes. Curr. Eye Res. 41(4), 551\u2013557 (2016)","journal-title":"Curr. Eye Res."},{"key":"1170_CR4","doi-asserted-by":"crossref","unstructured":"Colligris, P., Perez\u00a0de Lara, M.J., Colligris, B., Pintor, J.: Ocular manifestations of Alzheimer\u2019s and other neurodegenerative diseases: the prospect of the eye as a tool for the early diagnosis of Alzheimer\u2019s disease. J. Ophthalmol. (2018)","DOI":"10.1155\/2018\/8538573"},{"key":"1170_CR5","doi-asserted-by":"crossref","unstructured":"Bekkers, E.J., Zhang, J., Duits, R., ter Haar\u00a0Romeny, B.M.: Curvature based biomarkers for diabetic retinopathy via exponential curve fits in SE(2). In: Proceedings of the Ophthalmic Medical Image Analysis Second International Workshop, OMIA, pp. 113\u2013120 (2015)","DOI":"10.17077\/omia.1034"},{"key":"1170_CR6","doi-asserted-by":"crossref","first-page":"40","DOI":"10.1016\/j.exer.2012.10.015","volume":"106","author":"AA Kalitzeos","year":"2013","unstructured":"Kalitzeos, A.A., Lip, G.Y.H., Heitmar, R.: Retinal vessel tortuosity measures and their applications. Exp. Eye Res. 106, 40\u201346 (2013)","journal-title":"Exp. Eye Res."},{"issue":"3","key":"1170_CR7","doi-asserted-by":"crossref","first-page":"595","DOI":"10.1177\/193229681200600315","volume":"6","author":"C Cheung","year":"2012","unstructured":"Cheung, C., Lamoureux, E., Ikram, M., Sasongko, M., Ding, J., Zheng, Y., Mitchell, P., Wang, J., Wong, T.: Retinal vascular geometry in Asian persons with diabetes and retinopathy. J. Diabetes Sci. Technol. 6(3), 595\u2013605 (2012)","journal-title":"J. Diabetes Sci. Technol."},{"issue":"9","key":"1170_CR8","doi-asserted-by":"crossref","first-page":"2409","DOI":"10.1007\/s00125-011-2200-y","volume":"54","author":"M Sasongko","year":"2011","unstructured":"Sasongko, M., Wong, T., Nguyen, T., Cheung, C., Shaw, J., Wang, J.: Retinal vascular tortuosity in persons with diabetes and diabetic retinopathy. Diabetologia 54(9), 2409\u20132416 (2011)","journal-title":"Diabetologia"},{"key":"1170_CR9","doi-asserted-by":"crossref","unstructured":"Chen, D., Cohen, L.D.: A new dynamic minimal path model for tubular structure centerline delineation. In: 2018 24th International Conference on Pattern Recognition (ICPR), pp. 3001\u20133006 (2018)","DOI":"10.1109\/ICPR.2018.8545534"},{"key":"1170_CR10","doi-asserted-by":"crossref","unstructured":"Liu, L., Chen, D., Shu, M., Shu, H., Cohen, L.D.: A new tubular structure tracking algorithm based on curvature-penalized perceptual grouping. In: ICASSP 2021\u20132021 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), pp. 2195\u20132199 (2021)","DOI":"10.1109\/ICASSP39728.2021.9414114"},{"key":"1170_CR11","doi-asserted-by":"crossref","unstructured":"Liu, L., Chen, D., Cohen, L.D., Shu, H., P\u00e2ques, M.: Vessel extraction using crossing-adaptive minimal path model with anisotropic enhancement and curvature constraint. In: 2019 IEEE 16th International Symposium on Biomedical Imaging (ISBI 2019), pp. 20\u201323 (2019)","DOI":"10.1109\/ISBI.2019.8759435"},{"issue":"4","key":"1170_CR12","doi-asserted-by":"crossref","first-page":"2740","DOI":"10.1137\/15M1018460","volume":"8","author":"EJ Bekkers","year":"2015","unstructured":"Bekkers, E.J., Duits, R., Mashtakov, A., Sanguinetti, G.R.: A PDE approach to data-driven sub-Riemannian geodesics in SE(2). SIAM J. Imag. Sci. 8(4), 2740\u20132770 (2015)","journal-title":"SIAM J. Imag. Sci."},{"issue":"1","key":"1170_CR13","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1006\/cviu.1995.1004","volume":"61","author":"TF Cootes","year":"1995","unstructured":"Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models-their training and application. Comput. Vis. Image Underst. 61(1), 38\u201359 (1995)","journal-title":"Comput. Vis. Image Underst."},{"key":"1170_CR14","doi-asserted-by":"crossref","first-page":"61","DOI":"10.1023\/A:1007979827043","volume":"22","author":"V Caselles","year":"1997","unstructured":"Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vision 22, 61\u201379 (1997)","journal-title":"Int. J. Comput. Vision"},{"issue":"3","key":"1170_CR15","doi-asserted-by":"crossref","first-page":"225","DOI":"10.1023\/A:1011281928379","volume":"14","author":"LD Cohen","year":"2001","unstructured":"Cohen, L.D.: Multiple contour finding and perceptual grouping using minimal paths. J. Math. Imaging Vis. 14(3), 225\u2013236 (2001)","journal-title":"J. Math. Imaging Vis."},{"key":"1170_CR16","volume-title":"Geometric Partial Differential Equations and Image Analysis","author":"G Sapiro","year":"2006","unstructured":"Sapiro, G.: Geometric Partial Differential Equations and Image Analysis. Cambridge University Press, Cambridge (2006)"},{"issue":"3\u20134","key":"1170_CR17","first-page":"197","volume":"5","author":"G Peyr\u00e9","year":"2010","unstructured":"Peyr\u00e9, G., P\u00e9chaud, M., Keriven, R., Cohen, L.D., et al.: Geodesic methods in computer vision and graphics. Found. Trends Comput. Graph. Vis. 5(3\u20134), 197\u2013397 (2010)","journal-title":"Found. Trends Comput. Graph. Vis."},{"key":"1170_CR18","doi-asserted-by":"crossref","first-page":"5056","DOI":"10.1109\/TIP.2021.3078102","volume":"30","author":"D Chen","year":"2021","unstructured":"Chen, D., Spencer, J., Mirebeau, J.M., Chen, K., Shu, M., Cohen, L.D.: A generalized asymmetric dual-front model for active contours and image segmentation. IEEE Trans. Image Process. 30, 5056\u20135071 (2021)","journal-title":"IEEE Trans. Image Process."},{"key":"1170_CR19","doi-asserted-by":"crossref","unstructured":"P\u00e9chaud, M., Descoteaux, M., Keriven, R.: Brain connectivity using geodesics in HARDI. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 482\u2013489. Springer, Berlin (2009)","DOI":"10.1007\/978-3-642-04271-3_59"},{"key":"1170_CR20","doi-asserted-by":"crossref","first-page":"309","DOI":"10.1007\/978-3-030-05831-9_24","volume-title":"Computational Diffusion MRI","author":"J Portegies","year":"2019","unstructured":"Portegies, J., Meesters, S., Ossenblok, P., Fuster, A., Florack, L., Duits, R.: Brain connectivity measures via direct sub-Finslerian front propagation on the 5D sphere bundle of positions and directions. In: Bonet-Carne, E., Grussu, F., Ning, L., Sepehrband, F., Tax, C.M.W. (eds.) Computational Diffusion MRI, pp. 309\u2013321. Springer, Cham (2019)"},{"key":"1170_CR21","unstructured":"Chen, D.: New Minimal Paths Models for Tubular Structure Extraction and Image Segmentation. Ph.D. thesis, University Paris-Dauphine (2016)"},{"issue":"2","key":"1170_CR22","doi-asserted-by":"crossref","first-page":"192","DOI":"10.1007\/s11263-010-0331-0","volume":"92","author":"F Benmansour","year":"2011","unstructured":"Benmansour, F., Cohen, L.D.: Tubular structure segmentation based on minimal path method and anisotropic enhancement. Int. J. Comput. Vis. 92(2), 192\u2013210 (2011)","journal-title":"Int. J. Comput. Vis."},{"issue":"3","key":"1170_CR23","doi-asserted-by":"crossref","first-page":"583","DOI":"10.1007\/s10851-013-0488-6","volume":"49","author":"E Bekkers","year":"2014","unstructured":"Bekkers, E., Duits, R., Berendschot, T., ter Haar Romeny, B.: A multi-orientation analysis approach to retinal vessel tracking. J. Math. Imaging Vis. 49(3), 583\u2013610 (2014)","journal-title":"J. Math. Imaging Vis."},{"issue":"3","key":"1170_CR24","doi-asserted-by":"crossref","first-page":"253","DOI":"10.1007\/s11263-009-0213-5","volume":"85","author":"E Franken","year":"2009","unstructured":"Franken, E., Duits, R.: Crossing\u2013preserving coherence-enhancing diffusion on invertible orientation scores. Int. J. Comput. Vis. 85(3), 253 (2009)","journal-title":"Int. J. Comput. Vis."},{"key":"1170_CR25","doi-asserted-by":"crossref","first-page":"43","DOI":"10.1134\/S0081543810030041","volume":"270","author":"Ugo Boscain","year":"2010","unstructured":"Boscain, Ugo, Charlot, Gr\u00e9goire., Rossi, Francesco: 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":"1170_CR26","doi-asserted-by":"crossref","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."},{"key":"1170_CR27","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4612-1268-3","volume-title":"An Introduction to Riemann\u2013Finsler Geometry","author":"D Bao","year":"2000","unstructured":"Bao, D., Chern, S.S., Shen, Z.: An Introduction to Riemann\u2013Finsler Geometry. Springer, New York (2000)"},{"key":"1170_CR28","doi-asserted-by":"crossref","first-page":"816","DOI":"10.1007\/s10851-018-0795-z","volume":"60","author":"R Duits","year":"2018","unstructured":"Duits, R., Meesters, S.P., Mirebeau, J.M., Portegies, J.M.: Optimal paths for variants of the 2D and 3D Reeds\u2013Shepp car with applications in image analysis. J. Math. Imaging Vis. 60, 816\u2013848 (2018)","journal-title":"J. Math. Imaging Vis."},{"issue":"2","key":"1170_CR29","doi-asserted-by":"crossref","first-page":"367","DOI":"10.2140\/pjm.1990.145.367","volume":"145","author":"J Reeds","year":"1990","unstructured":"Reeds, J., Shepp, L.: Optimal paths for a car that goes both forwards and backwards. Pac. J. Math. 145(2), 367\u2013393 (1990)","journal-title":"Pac. J. Math."},{"key":"1170_CR30","volume-title":"Optimization Algorithms on Matrix Manifolds","author":"PA Absil","year":"2009","unstructured":"Absil, P.A., Mahony, R., Sepulchre, R.: Optimization Algorithms on Matrix Manifolds. Princeton University Press, Princeton (2009)"},{"key":"1170_CR31","doi-asserted-by":"crossref","unstructured":"Bekkers, E.J., Chen, D., Portegies, J.M.: Nilpotent approximations of sub-Riemannian distances for fast perceptual grouping of blood vessels in 2D and 3D. J. Math. Imaging Vis. 60, 1\u201318 (2018)","DOI":"10.1007\/s10851-018-0787-z"},{"issue":"4","key":"1170_CR32","doi-asserted-by":"crossref","first-page":"224","DOI":"10.1177\/1748301816656289","volume":"10","author":"D Chen","year":"2016","unstructured":"Chen, D., Mirebeau, J.M., Cohen, L.D.: Vessel tree extraction using radius-lifted keypoints searching scheme and anisotropic fast marching method. J. Algorithms Comput. Technol. 10(4), 224\u2013234 (2016)","journal-title":"J. Algorithms Comput. Technol."},{"issue":"3","key":"1170_CR33","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s00422-017-0718-x","volume":"111","author":"S Abbasi-Sureshjani","year":"2017","unstructured":"Abbasi-Sureshjani, S., Zhang, J., Duits, R., ter Haar Romeny, B.: Retrieving challenging vessel connections in retinal images by line co-occurrence statistics. Biol. Cybern. 111(3), 237\u2013247 (2017)","journal-title":"Biol. Cybern."},{"issue":"6","key":"1170_CR34","doi-asserted-by":"crossref","first-page":"784","DOI":"10.1007\/s10851-017-0778-5","volume":"60","author":"JM Mirebeau","year":"2018","unstructured":"Mirebeau, J.M.: Fast-marching methods for curvature penalized shortest paths. J. Math. Imaging Vis. 60(6), 784\u2013815 (2018)","journal-title":"J. Math. Imaging Vis."},{"key":"1170_CR35","doi-asserted-by":"crossref","first-page":"73","DOI":"10.1007\/978-3-030-80209-7_9","volume-title":"Geometric Science of Information","author":"A Mashtakov","year":"2021","unstructured":"Mashtakov, A.: Extremal controls for the duits car. In: Nielsen, F., Barbaresco, F. (eds.) Geometric Science of Information, pp. 73\u201381. Springer, Cham (2021)"},{"key":"1170_CR36","doi-asserted-by":"crossref","first-page":"47","DOI":"10.5201\/ipol.2019.227","volume":"9","author":"JM Mirebeau","year":"2019","unstructured":"Mirebeau, J.M., Portegies, J.: Hamiltonian fast marching: a numerical solver for anisotropic and non-holonomic Eikonal PDEs. Image Process. Line 9, 47\u201393 (2019)","journal-title":"Image Process. Line"},{"issue":"12","key":"1170_CR37","doi-asserted-by":"crossref","first-page":"2631","DOI":"10.1109\/TMI.2016.2587062","volume":"35","author":"J Zhang","year":"2016","unstructured":"Zhang, J., Dashtbozorg, B., Bekkers, E.J., Pluim, J.P., Duits, R., HaarRomeny, BMt.: Robust retinal vessel segmentation via locally adaptive derivative frames in orientation scores. IEEE Trans. Med. Imaging 35(12), 2631\u20132644 (2016)","journal-title":"IEEE Trans. Med. Imaging"},{"key":"1170_CR38","doi-asserted-by":"crossref","first-page":"325","DOI":"10.1007\/978-3-319-20801-5_35","volume-title":"Image Analysis and Recognition","author":"S Abbasi-Sureshjani","year":"2015","unstructured":"Abbasi-Sureshjani, S., Smit-Ockeloen, I., Zhang, J., Ter Haar Romeny, B.: Biologically-inspired supervised vasculature segmentation in SLO retinal fundus images. In: Kamel, M., Campilho, A. (eds.) Image Analysis and Recognition, pp. 325\u2013334. Springer, Cham (2015)"},{"key":"1170_CR39","unstructured":"Duits, R.: Perceptual Organization in Image Analysis: A Mathematical Approach Based on Scale, Orientation and Curvature. Ph.D. thesis, Eindhoven University of Technology (2005)"},{"issue":"1","key":"1170_CR40","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1134\/S1054661807010063","volume":"17","author":"R Duits","year":"2007","unstructured":"Duits, R., Duits, M., van Almsick, M., ter Haar Romeny, B.: Invertible orientation scores as an application of generalized wavelet theory. Pattern Recognit Image Anal. 17(1), 42\u201375 (2007)","journal-title":"Pattern Recognit Image Anal."},{"key":"1170_CR41","doi-asserted-by":"crossref","first-page":"255","DOI":"10.1090\/S0033-569X-10-01172-0","volume":"68","author":"R Duits","year":"2010","unstructured":"Duits, R., Franken, E.M.: Left invariant parabolic evolution equations on $${SE}(2)$$ and contour enhancement via invertible orientation scores, part I: Linear left-invariant diffusion equations on $${SE}(2)$$. Quart. Appl. Math. AMS 68, 255\u2013292 (2010)","journal-title":"Quart. Appl. Math. AMS"},{"key":"1170_CR42","doi-asserted-by":"crossref","first-page":"55","DOI":"10.1016\/j.difgeo.2019.03.007","volume":"65","author":"B Franceschiello","year":"2019","unstructured":"Franceschiello, B., Mashtakov, A., Citti, G., Sarti, A.: Geometrical optical illusion via sub-Riemannian geodesics in the roto-translation group. Differ. Geom. Appl. 65, 55\u201377 (2019)","journal-title":"Differ. Geom. Appl."},{"key":"1170_CR43","doi-asserted-by":"crossref","unstructured":"Duits, R., Smets, B., Wemmenhove, J., Portegies, J., Bekkers, E.: Springer Handbook of Mathematical Imaging 2021, chap. Recent Geometric Flows in Multi-Orientation Image Processing via a Cartan Connection, pp. 1\u201360. Springer, Berlin (2021)","DOI":"10.1007\/978-3-030-03009-4_101-1"},{"key":"1170_CR44","volume-title":"Foundations of Differential Geometry","author":"S Kobayashi","year":"1963","unstructured":"Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. 45. Wiley, New York (1963)"},{"key":"1170_CR45","doi-asserted-by":"crossref","first-page":"39","DOI":"10.1016\/j.hm.2017.09.001","volume":"45","author":"A Cogliati","year":"2017","unstructured":"Cogliati, A., Mastrolia, P.: Cartan, Schouten and the search for connection. Hist. Math. 45, 39\u201374 (2017)","journal-title":"Hist. Math."},{"issue":"12","key":"1170_CR46","doi-asserted-by":"crossref","first-page":"2492","DOI":"10.1109\/TPAMI.2015.2408352","volume":"37","author":"E Piuze","year":"2015","unstructured":"Piuze, E., Sporring, J., Siddiqi, K.: Maurer\u2013Cartan forms for fields on surfaces: application to heart fiber geometry. IEEE Trans. Pattern Anal. Mach. Intell. 37(12), 2492\u20132504 (2015)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"1170_CR47","doi-asserted-by":"crossref","unstructured":"Momayyez, P., Siddiqi, K.: 3D stochastic completion fields for fiber tractography. In: 2009 IEEE Computer Society Conference on Computer Vision and Pattern Recognition Workshops, pp. 178\u2013185 (2009)","DOI":"10.1109\/CVPRW.2009.5204044"},{"key":"1170_CR48","doi-asserted-by":"crossref","unstructured":"Duits, R., Smets, B.M.N., Wemmenhove, A.J., Portegies, J.W., Bekkers, E.J.: Recent geometric flows in multi-orientation image processing via a Cartan connection. In: Handbook of Mathematical Models and Algorithms in Computer Vision and Imaging: Mathematical Imaging and Vision pp. 1\u201360 (2021)","DOI":"10.1007\/978-3-030-03009-4_101-1"},{"key":"1170_CR49","doi-asserted-by":"crossref","unstructured":"Hannink, J., Duits, R., Bekkers, E.: Crossing-preserving multi-scale vesselness. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 603\u2013610. Springer, Berlin (2014)","DOI":"10.1007\/978-3-319-10470-6_75"},{"issue":"6","key":"1170_CR50","doi-asserted-by":"crossref","first-page":"766","DOI":"10.1007\/s10851-017-0776-7","volume":"60","author":"D Chen","year":"2018","unstructured":"Chen, D., Cohen, L.D.: Fast asymmetric fronts propagation for image segmentation. J. Math. Imaging Vis. 60(6), 766\u2013783 (2018)","journal-title":"J. Math. Imaging Vis."},{"issue":"2","key":"1170_CR51","doi-asserted-by":"crossref","first-page":"199","DOI":"10.1137\/S0036144598347059","volume":"41","author":"JA Sethian","year":"1999","unstructured":"Sethian, J.A.: Fast marching methods. SIAM Rev. 41(2), 199\u2013235 (1999)","journal-title":"SIAM Rev."},{"issue":"6","key":"1170_CR52","doi-asserted-by":"crossref","first-page":"2608","DOI":"10.1137\/17M1127466","volume":"57","author":"JM Mirebeau","year":"2019","unstructured":"Mirebeau, J.M.: Riemannian fast-marching on Cartesian grids, using Voronoi\u2019s first reduction of quadratic forms. SIAM J. Numer. Anal. 57(6), 2608\u20132655 (2019)","journal-title":"SIAM J. Numer. Anal."},{"key":"1170_CR53","first-page":"143","volume":"77","author":"E Selling","year":"1874","unstructured":"Selling, E.: Ueber die bin\u00e4ren und tern\u00e4ren quadratischen formen. Journal f\u00fcr die reine und angewandte Mathematik 77, 143\u2013229 (1874)","journal-title":"Journal f\u00fcr die reine und angewandte Mathematik"},{"issue":"24","key":"1170_CR54","doi-asserted-by":"crossref","first-page":"9248","DOI":"10.1073\/pnas.1120785109","volume":"109","author":"P Savadjiev","year":"2012","unstructured":"Savadjiev, P., Strijkers, G., Bakermans, A., Piuze, E., Zucker, S., Siddiqi, K.: Heart wall myofibers are arranged in minimal surfaces to optimize organ function. PNAS 109(24), 9248\u20139253 (2012)","journal-title":"PNAS"},{"issue":"3","key":"1170_CR55","doi-asserted-by":"crossref","first-page":"367","DOI":"10.1007\/s10851-016-0641-0","volume":"56","author":"R Duits","year":"2016","unstructured":"Duits, R., Janssen, M.H., Hannink, J., Sanguinetti, G.R.: Locally adaptive frames in the roto-translation group and their applications in medical imaging. J. Math. Imaging Vis. 56(3), 367\u2013402 (2016)","journal-title":"J. Math. Imaging Vis."},{"issue":"2","key":"1170_CR56","doi-asserted-by":"crossref","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 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":"1170_CR57","doi-asserted-by":"crossref","DOI":"10.1007\/978-1-4020-8840-7","volume-title":"Front-End Vision and Multi-Scale Image Analysis","author":"BM Haar Romeny","year":"2003","unstructured":"Haar Romeny, B.M.: Front-End Vision and Multi-Scale Image Analysis. Springer, Berlin (2003)"},{"key":"1170_CR58","volume-title":"Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems","author":"JE Marsden","year":"2013","unstructured":"Marsden, J.E., Ratiu, T.S.: Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, vol. 17. Springer, Berlin (2013)"},{"key":"1170_CR59","volume-title":"A Tour of Subriemannian Geometries, Their Geodesics and Applications","author":"R Montgomery","year":"2002","unstructured":"Montgomery, R.: A Tour of Subriemannian Geometries, Their Geodesics and Applications, vol. 91. American Mathematical Society, Providence (2002)"},{"key":"1170_CR60","doi-asserted-by":"crossref","unstructured":"Agrachev, A.A., Sachkov, Y.L.: Control Theory from the Geometrical Viewpoint, vol. 87. Springer, Berlin (2004)","DOI":"10.1007\/978-3-662-06404-7"},{"key":"1170_CR61","doi-asserted-by":"crossref","unstructured":"Agrachev, A., Barilari, D., Boscain, U.: A Comprehensive Introduction to Sub-Riemannian Geometry. Cambridge Studies in Advanced Mathematics. Cambridge University Press, Cambridge (2019)","DOI":"10.1017\/9781108677325"},{"issue":"2","key":"1170_CR62","first-page":"380","volume":"16","author":"I Moiseev","year":"2010","unstructured":"Moiseev, I., Sachkov, Y.L.: Maxwell strata in sub-Riemannian problem on the group of motions of a plane. ESAIM: Control Optim. Calc. Var. 16(2), 380\u2013399 (2010)","journal-title":"ESAIM: Control Optim. Calc. Var."},{"issue":"4","key":"1170_CR63","doi-asserted-by":"crossref","first-page":"771","DOI":"10.1007\/s10883-016-9329-4","volume":"22","author":"R Duits","year":"2016","unstructured":"Duits, R., Ghosh, A., Dela Haije, T., Mashtakov, A.: On sub-Riemannian geodesics in SE(3) whose spatial projections do not have cusps. J. Dyn. Control Syst. 22(4), 771\u2013805 (2016)","journal-title":"J. Dyn. Control Syst."},{"issue":"3","key":"1170_CR64","doi-asserted-by":"crossref","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 perceptional completion in the roto-translation space. J. Math. Imaging Vis. 24(3), 307\u2013326 (2006)","journal-title":"J. Math. Imaging Vis."},{"key":"1170_CR65","first-page":"293","volume":"17","author":"Y Sachkov","year":"2011","unstructured":"Sachkov, Y.: Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM: Control Optim Calc. Var. 17, 293\u2013321 (2011)","journal-title":"ESAIM: Control Optim Calc. Var."},{"key":"1170_CR66","volume-title":"Partial Differential Quations","author":"LC Evans","year":"1998","unstructured":"Evans, L.C.: Partial Differential Quations. AMS, Providence (1998)"},{"issue":"2","key":"1170_CR67","doi-asserted-by":"crossref","first-page":"237","DOI":"10.1007\/s10851-020-00991-4","volume":"63","author":"BM Smets","year":"2021","unstructured":"Smets, B.M., Portegies, J., St-Onge, E., Duits, R.: Total variation and mean curvature PDEs on the homogeneous space of positions and orientations. J. Math. Imaging Vis. 63(2), 237\u2013262 (2021)","journal-title":"J. Math. Imaging Vis."},{"key":"1170_CR68","doi-asserted-by":"publisher","unstructured":"Smets, B.M.N., Portegies, J., Bekkers, E.J., Duits , R.: PDE-based group equivariant convolutional neural networks. J. Math. Imaging Vis. 65, 209\u2013239 (2023). https:\/\/doi.org\/10.1007\/s10851-022-01114-x","DOI":"10.1007\/s10851-022-01114-x"},{"key":"1170_CR69","doi-asserted-by":"crossref","first-page":"38","DOI":"10.1007\/978-3-030-22368-7_4","volume-title":"Scale Space and Variational Methods in Computer Vision","author":"M Lichtenstein","year":"2019","unstructured":"Lichtenstein, M., Pai, G., Kimmel, R.: Deep eikonal solvers. In: Lellmann, J., Burger, M., Modersitzki, J. (eds.) Scale Space and Variational Methods in Computer Vision, pp. 38\u201350. Springer, Cham (2019)"},{"key":"1170_CR70","unstructured":"Berg, N.J.v.d.: Data-driven left-invariant tracking in Mathematica. https:\/\/github.com\/NickyvdBerg\/DataDrivenTracking (2022)"},{"key":"1170_CR71","first-page":"130","volume":"1496","author":"R Frangi","year":"2000","unstructured":"Frangi, R., Niessen, W., Vincken, K., Viergever, M.: Multiscale vessel enhancement filtering. Med. Image Comput. Comput. Assist. Interv. 1496, 130\u2013137 (2000)","journal-title":"Med. Image Comput. Comput. Assist. Interv."}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-023-01170-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10851-023-01170-x\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-023-01170-x.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,3,11]],"date-time":"2024-03-11T07:19:45Z","timestamp":1710141585000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10851-023-01170-x"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024,1,13]]},"references-count":71,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2024,4]]}},"alternative-id":["1170"],"URL":"https:\/\/doi.org\/10.1007\/s10851-023-01170-x","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2085777\/v1","asserted-by":"object"}]},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"value":"0924-9907","type":"print"},{"value":"1573-7683","type":"electronic"}],"subject":[],"published":{"date-parts":[[2024,1,13]]},"assertion":[{"value":"20 September 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"20 November 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"13 January 2024","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"R.\u00a0Duits is a member of the editorial board of JMIV.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Competing Interests"}}]}}