{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T05:12:58Z","timestamp":1778735578357,"version":"3.51.4"},"reference-count":70,"publisher":"Springer Science and Business Media LLC","issue":"2","license":[{"start":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T00:00:00Z","timestamp":1769990400000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T00:00:00Z","timestamp":1769990400000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"funder":[{"DOI":"10.13039\/501100006368","name":"Universidad de las Palmas de Gran Canaria","doi-asserted-by":"crossref","id":[{"id":"10.13039\/501100006368","id-type":"DOI","asserted-by":"crossref"}]}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2026,4]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>In this paper, we propose an active contour model for silhouette vectorization using cubic B\u00e9zier curves. Among the end points of the B\u00e9zier curves, we distinguish between corner and regular points where the orientation of the tangent vector is prescribed. By minimizing the distance of the B\u00e9zier curves to the silhouette boundary, the active contour model optimizes the location of the B\u00e9zier curve end points, the orientation of the tangent vectors in the regular points and the estimation of the B\u00e9zier curve parameters. This active contour model can use the silhouette vectorization obtained by any method as an initial guess. The proposed method significantly reduces the average distance between the silhouette boundary and its vectorization obtained by the world-class graphic software Inkscape, Adobe Illustrator and a curvature-based vectorization method, which we introduce for comparison. Our method also allows us to impose additional regularity on the B\u00e9zier curves by reducing their lengths.<\/jats:p>","DOI":"10.1007\/s10851-026-01282-0","type":"journal-article","created":{"date-parts":[[2026,2,2]],"date-time":"2026-02-02T14:57:29Z","timestamp":1770044249000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["An Active Contour Model for Silhouette Vectorization using B\u00e9zier Curves"],"prefix":"10.1007","volume":"68","author":[{"given":"Luis","family":"Alvarez","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jean-Michel","family":"Morel","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2026,2,2]]},"reference":[{"key":"1282_CR1","doi-asserted-by":"crossref","unstructured":"Bachmann, T.: Perception of Pixelated Images. Academic Press (2016)","DOI":"10.1016\/B978-0-12-809311-5.00003-9"},{"issue":"4","key":"1282_CR2","doi-asserted-by":"publisher","first-page":"221","DOI":"10.1037\/h0043921","volume":"53","author":"F Attneave","year":"1957","unstructured":"Attneave, F.: Physical determinants of the judged complexity of shapes. J. Exp. Psychol. 53(4), 221 (1957)","journal-title":"J. Exp. Psychol."},{"issue":"2\u20133","key":"1282_CR3","doi-asserted-by":"publisher","first-page":"97","DOI":"10.1023\/A:1008183703117","volume":"34","author":"JH Elder","year":"1999","unstructured":"Elder, J.H.: Are edges incomplete? Int. J. Comput. Vision 34(2\u20133), 97\u2013122 (1999)","journal-title":"Int. J. Comput. Vision"},{"issue":"1","key":"1282_CR4","doi-asserted-by":"publisher","first-page":"123","DOI":"10.1017\/S0952523808080930","volume":"26","author":"A Torralba","year":"2009","unstructured":"Torralba, A.: How many pixels make an image? Vis. Neurosci. 26(1), 123\u2013131 (2009)","journal-title":"Vis. Neurosci."},{"key":"1282_CR5","unstructured":"Tian, X., G\u00fcnther, T.: A survey of smooth vector graphics: recent advances in representation, creation, rasterization and image vectorization. IEEE Trans. Visual Comput Graph (2022)"},{"key":"1282_CR6","doi-asserted-by":"crossref","unstructured":"Dziuba, M., Jarsky, I., Efimova, V., Filchenkov, A.: Image vectorization: a review. arXiv preprint arXiv:2306.06441 (2023)","DOI":"10.1007\/s10958-024-07422-4"},{"key":"1282_CR7","doi-asserted-by":"crossref","unstructured":"He, Y., Kang, S.H., Alvarez, L.: Finding the skeleton of 2d shape and contours: Implementation of hamilton-jacobi skeleton. Image Processing On Line (2021)","DOI":"10.5201\/ipol.2021.296"},{"key":"1282_CR8","doi-asserted-by":"crossref","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Silhouette vectorization by affine scale-space. J Math Imaging Vis 1\u201316 (2022)","DOI":"10.1007\/s10851-021-01053-z"},{"issue":"6","key":"1282_CR9","doi-asserted-by":"publisher","first-page":"874","DOI":"10.1007\/s10851-023-01149-8","volume":"65","author":"Y He","year":"2023","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Topology-and perception-aware image vectorization. J Math Imaging Vis 65(6), 874\u2013893 (2023)","journal-title":"J Math Imaging Vis"},{"issue":"3","key":"1282_CR10","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/1360612.1360691","volume":"27","author":"A Orzan","year":"2008","unstructured":"Orzan, A., Bousseau, A., Winnem\u00f6ller, H., Barla, P., Thollot, J., Salesin, D.: Diffusion curves: a vector representation for smooth-shaded images. ACM Trans Graph (TOG) 27(3), 1\u20138 (2008)","journal-title":"ACM Trans Graph (TOG)"},{"issue":"6","key":"1282_CR11","doi-asserted-by":"publisher","first-page":"890","DOI":"10.1109\/TPAMI.2006.127","volume":"28","author":"X Hilaire","year":"2006","unstructured":"Hilaire, X., Tombre, K.: Robust and accurate vectorization of line drawings. IEEE Trans. Pattern Anal. Mach. Intell. 28(6), 890\u2013904 (2006)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"issue":"1","key":"1282_CR12","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/2421636.2421640","volume":"32","author":"G Noris","year":"2013","unstructured":"Noris, G., Hornung, A., Sumner, R.W., Simmons, M., Gross, M.: Topology-driven vectorization of clean line drawings. ACM Trans Graph (TOG) 32(1), 1\u201311 (2013)","journal-title":"ACM Trans Graph (TOG)"},{"issue":"4","key":"1282_CR13","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/2897824.2925946","volume":"35","author":"J-D Favreau","year":"2016","unstructured":"Favreau, J.-D., Lafarge, F., Bousseau, A.: Fidelity vs. simplicity: a global approach to line drawing vectorization. ACM Trans Graph (TOG) 35(4), 1\u201310 (2016)","journal-title":"ACM Trans Graph (TOG)"},{"issue":"4","key":"1282_CR14","doi-asserted-by":"publisher","first-page":"618","DOI":"10.1109\/TVCG.2009.9","volume":"15","author":"S-H Zhang","year":"2009","unstructured":"Zhang, S.-H., Chen, T., Zhang, Y.-F., Hu, S.-M., Martin, R.R.: Vectorizing cartoon animations. IEEE Trans. Visual Comput. Graphics 15(4), 618\u2013629 (2009)","journal-title":"IEEE Trans. Visual Comput. Graphics"},{"issue":"2","key":"1282_CR15","doi-asserted-by":"publisher","first-page":"1063","DOI":"10.1109\/TVCG.2015.2440273","volume":"22","author":"M Yang","year":"2015","unstructured":"Yang, M., Chao, H., Zhang, C., Guo, J., Yuan, L., Sun, J.: Effective clipart image vectorization through direct optimization of bezigons. IEEE Trans. Visual Comput. Graphics 22(2), 1063\u20131075 (2015)","journal-title":"IEEE Trans. Visual Comput. Graphics"},{"issue":"6","key":"1282_CR16","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3130800.3130888","volume":"36","author":"J-D Favreau","year":"2017","unstructured":"Favreau, J.-D., Lafarge, F., Bousseau, A.: Photo2clipart: image abstraction and vectorization using layered linear gradients. ACM Trans Graph (TOG) 36(6), 1\u201311 (2017)","journal-title":"ACM Trans Graph (TOG)"},{"key":"1282_CR17","doi-asserted-by":"publisher","first-page":"22","DOI":"10.5201\/ipol.2023.401","volume":"13","author":"Y He","year":"2023","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Binary shape vectorization by affine scale-space. Image Processing On Line 13, 22\u201337 (2023)","journal-title":"Image Processing On Line"},{"key":"1282_CR18","doi-asserted-by":"crossref","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Viva: a variational image vectorization algorithm on dual-primal graph pairs. In: 2023 IEEE Int. Conf. Image Process. (ICIP), pp. 1285\u20131289 (2023). IEEE","DOI":"10.1109\/ICIP49359.2023.10222783"},{"key":"1282_CR19","doi-asserted-by":"crossref","unstructured":"Reddy, P., Gharbi, M., Lukac, M., Mitra, N.J.: Im2vec: Synthesizing vector graphics without vector supervision. In: Proceedings of the IEEE\/CVF conference on computer vision and pattern recognition, pp. 7342\u20137351 (2021)","DOI":"10.1109\/CVPR46437.2021.00726"},{"issue":"6","key":"1282_CR20","first-page":"1","volume":"39","author":"T-M Li","year":"2020","unstructured":"Li, T.-M., Luk\u00e1\u010d, M., Gharbi, M., Ragan-Kelley, J.: Differentiable vector graphics rasterization for editing and learning. ACM Trans. Graph. (TOG) 39(6), 1\u201315 (2020)","journal-title":"ACM Trans. Graph. (TOG)"},{"key":"1282_CR21","unstructured":"Ferraiolo, J., Jun, F., Jackson, D.: Scalable Vector Graphics (SVG) 1.0 Specification. iuniverse Bloomington (2000)"},{"key":"1282_CR22","unstructured":"Adobe: Adobe Illustrator. https:\/\/www.adobe.com\/products\/illustrator.html"},{"issue":"3","key":"1282_CR23","doi-asserted-by":"publisher","first-page":"11","DOI":"10.1145\/1276377.1276391","volume":"26","author":"J Sun","year":"2007","unstructured":"Sun, J., Liang, L., Wen, F., Shum, H.-Y.: Image vectorization using optimized gradient meshes. ACM Trans. Graph. (TOG) 26(3), 11 (2007)","journal-title":"ACM Trans. Graph. (TOG)"},{"issue":"3\u20134","key":"1282_CR24","doi-asserted-by":"publisher","first-page":"197","DOI":"10.1561\/0600000029","volume":"5","author":"G Peyr\u00e9","year":"2010","unstructured":"Peyr\u00e9, G., P\u00e9chaud, M., Keriven, R., Cohen, L.D.: Geodesic methods in computer vision and graphics. Found. Trends Comp. Graphics Vision 5(3\u20134), 197\u2013397 (2010)","journal-title":"Found. Trends Comp. Graphics Vision"},{"key":"1282_CR25","doi-asserted-by":"crossref","unstructured":"Lopes, R.G., Ha, D., Eck, D., Shlens, J.: A learned representation for scalable vector graphics. In: Proceedings of the IEEE\/CVF International Conference on Computer Vision, pp. 7930\u20137939 (2019)","DOI":"10.1109\/ICCV.2019.00802"},{"issue":"4","key":"1282_CR26","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1145\/3528223.3530068","volume":"41","author":"Y Vinker","year":"2022","unstructured":"Vinker, Y., Pajouheshgar, E., Bo, J.Y., Bachmann, R.C., Bermano, A.H., Cohen-Or, D., Zamir, A., Shamir, A.: Clipasso: Semantically-aware object sketching. ACM Transactions on Graphics (TOG) 41(4), 1\u201311 (2022)","journal-title":"ACM Transactions on Graphics (TOG)"},{"key":"1282_CR27","doi-asserted-by":"crossref","unstructured":"Ma, X., Zhou, Y., Xu, X., Sun, B., Filev, V., Orlov, N., Fu, Y., Shi, H.: Towards layer-wise image vectorization. In: Proceedings of the IEEE\/CVF Conference on Computer Vision and Pattern Recognition, pp. 16314\u201316323 (2022)","DOI":"10.1109\/CVPR52688.2022.01583"},{"key":"1282_CR28","unstructured":"Pastva, T.A.: Bezier curve fitting. PhD thesis, Monterey, California. Naval Postgraduate School (1998)"},{"key":"1282_CR29","unstructured":"Selinger, P.: Potrace : a polygon-based tracing algorithm. (2003). https:\/\/api.semanticscholar.org\/CorpusID:1419652"},{"issue":"9","key":"1282_CR30","doi-asserted-by":"publisher","first-page":"821","DOI":"10.1016\/S0167-8655(98)00069-5","volume":"19","author":"L Cinque","year":"1998","unstructured":"Cinque, L., Levialdi, S., Malizia, A.: Shape description using cubic polynomial bezier curves. Pattern Recogn. Lett. 19(9), 821\u2013828 (1998). https:\/\/doi.org\/10.1016\/S0167-8655(98)00069-5","journal-title":"Pattern Recogn. Lett."},{"key":"1282_CR31","doi-asserted-by":"crossref","unstructured":"Plass, M., Stone, M.: Curve-fitting with piecewise parametric cubics. In: 10th Annual Conference on Computer Graphics and Interactive Techniques, pp. 229\u2013239 (1983)","DOI":"10.1145\/800059.801153"},{"key":"1282_CR32","doi-asserted-by":"publisher","first-page":"41","DOI":"10.1007\/s10851-021-01053-z","volume":"64","author":"Y He","year":"2022","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Silhouette vectorization by affine scale-space. J. Math. Imaging Vis. 64, 41\u201356 (2022)","journal-title":"J. Math. Imaging Vis."},{"key":"1282_CR33","first-page":"1","volume":"1","author":"Y He","year":"2023","unstructured":"He, Y., Kang, S.H., Morel, J.M.: Topology-and perception-aware image vectorization. J. Math. Imaging Vis. 1, 1\u201320 (2023)","journal-title":"J. Math. Imaging Vis."},{"issue":"4","key":"1282_CR34","doi-asserted-by":"publisher","first-page":"360","DOI":"10.1016\/0146-664X(72)90021-4","volume":"1","author":"T Pavlidis","year":"1972","unstructured":"Pavlidis, T.: Segmentation of pictures and maps through functional approximation. Comput. Graphics Image Process. 1(4), 360\u2013372 (1972)","journal-title":"Comput. Graphics Image Process."},{"key":"1282_CR35","doi-asserted-by":"crossref","unstructured":"Mumford, D.B., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Communications on pure and applied mathematics (1989)","DOI":"10.1002\/cpa.3160420503"},{"key":"1282_CR36","unstructured":"Morel, J.-M., Solimini, S.: Variational Methods in Image Segmentation: with Seven Image Processing Experiments vol. 14. Springer (2012)"},{"key":"1282_CR37","unstructured":"Kornprobst, G.A.P., Aubert, G.: Mathematical problems in image processing. Appl. Math. Sci.147 (2006)"},{"issue":"1","key":"1282_CR38","doi-asserted-by":"publisher","first-page":"368","DOI":"10.1137\/120867068","volume":"6","author":"X Cai","year":"2013","unstructured":"Cai, X., Chan, R., Zeng, T.: two-stage image seagmentation method using a convex variant of the mumford-shah model and thresholding. SIAM J. Imag. Sci. 6(1), 368\u2013390 (2013)","journal-title":"SIAM J. Imag. Sci."},{"key":"1282_CR39","doi-asserted-by":"crossref","unstructured":"Pock, T., Cremers, D., Bischof, H., Chambolle, A.: An algorithm for minimizing the mumford-shah functional. In: 2009 IEEE 12th International Conference on Computer Vision, pp. 1133\u20131140 (2009). IEEE","DOI":"10.1109\/ICCV.2009.5459348"},{"key":"1282_CR40","doi-asserted-by":"crossref","unstructured":"Chan, T.F., Vese, L.A.: A level set algorithm for minimizing the mumford-shah functional in image processing. In: Proceedings IEEE Workshop on Variational and Level Set Methods in Computer Vision, pp. 161\u2013168 (2001). IEEE","DOI":"10.1109\/VLSM.2001.938895"},{"issue":"8","key":"1282_CR41","doi-asserted-by":"publisher","first-page":"999","DOI":"10.1002\/cpa.3160430805","volume":"43","author":"L Ambrosio","year":"1990","unstructured":"Ambrosio, L., Tortorelli, V.M.: Approximation of functional depending on jumps by elliptic functional via t-convergence. Commun. Pure Appl. Math. 43(8), 999\u20131036 (1990)","journal-title":"Commun. Pure Appl. Math."},{"issue":"4","key":"1282_CR42","doi-asserted-by":"publisher","first-page":"321","DOI":"10.1007\/BF00133570","volume":"1","author":"M Kass","year":"1988","unstructured":"Kass, M., Witkin, A., Terzopoulos, D.: Snakes: Active contour models. Int. J. Comput. Vision 1(4), 321\u2013331 (1988)","journal-title":"Int. J. Comput. Vision"},{"issue":"3","key":"1282_CR43","doi-asserted-by":"publisher","first-page":"359","DOI":"10.1109\/83.661186","volume":"7","author":"C Xu","year":"1998","unstructured":"Xu, C., Prince, J.L.: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Process. 7(3), 359\u2013369 (1998)","journal-title":"IEEE Trans. Image Process."},{"key":"1282_CR44","unstructured":"Amini, A.A., Tehrani, S., Weymouth, T.E.: Using dynamic programming for minimizing the energy of active contours in the presence of hard constraints. In: [1988 Proceedings] Second International Conference on Computer Vision, pp. 95\u201399 (1988). IEEE"},{"key":"1282_CR45","doi-asserted-by":"crossref","unstructured":"Bresson, X., Esedo\u1e21lu, S., Vandergheynst, P., Thiran, J.-P., Osher, S.: Fast global minimization of the active contour\/snake model. J. Math. Imaging vis. 28(2), 151\u2013167 (2007)","DOI":"10.1007\/s10851-007-0002-0"},{"key":"1282_CR46","doi-asserted-by":"publisher","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"},{"key":"1282_CR47","doi-asserted-by":"publisher","DOI":"10.1007\/b10404","volume-title":"Geometric Curve Evolution and Image Processing","author":"F Cao","year":"2003","unstructured":"Cao, F.: Geometric Curve Evolution and Image Processing. Springer, Cham (2003)"},{"key":"1282_CR48","unstructured":"Miranda, M.: Caccioppoli sets. Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 14(3), 173\u2013177 (2003)"},{"issue":"1","key":"1282_CR49","doi-asserted-by":"publisher","first-page":"39","DOI":"10.1007\/pl00011302","volume":"3","author":"L Ambrosio","year":"2001","unstructured":"Ambrosio, L., Morel, J.-M., Masnou, S., Caselles, V.: Connected components of sets of finite perimeter and applications to image processing. J. Eur. Math. Soc. 3(1), 39\u201392 (2001)","journal-title":"J. Eur. Math. Soc."},{"issue":"2","key":"1282_CR50","first-page":"285","volume":"26","author":"MA Grayson","year":"1987","unstructured":"Grayson, M.A.: The heat equation shrinks embedded plane curves to round points. J. Diff. Geometry 26(2), 285\u2013314 (1987)","journal-title":"J. Diff. Geometry"},{"key":"1282_CR51","doi-asserted-by":"crossref","unstructured":"Chou, K.-S., Zhu, X.-P.: The Curve Shortening Problem. Chapman and Hall\/CRC (2001)","DOI":"10.1201\/9781420035704"},{"key":"1282_CR52","doi-asserted-by":"publisher","first-page":"2","DOI":"10.1109\/TPAMI.1986.4767747","volume":"1","author":"H Asada","year":"1986","unstructured":"Asada, H., Brady, M.: The curvature primal sketch. IEEE Trans. Pattern Anal. Mach. Intell. 1, 2\u201314 (1986)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"1282_CR53","unstructured":"Do\u00a0Carmo, M.P.: Differential Geometry of Curves and Surfaces: Revised and Updated Second Edition. Courier Dover Publications (2016)"},{"issue":"1","key":"1282_CR54","doi-asserted-by":"publisher","first-page":"12","DOI":"10.1016\/0021-9991(88)90002-2","volume":"79","author":"S Osher","year":"1988","unstructured":"Osher, S., Sethian, J.A.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton\u2013Jacobi formulations. J. Comput. Phys. 79(1), 12\u201349 (1988)","journal-title":"J. Comput. Phys."},{"issue":"1","key":"1282_CR55","doi-asserted-by":"publisher","first-page":"25","DOI":"10.1007\/BF01420591","volume":"11","author":"G Sapiro","year":"1993","unstructured":"Sapiro, G., Tannenbaum, A.: Affine invariant scale-space. Int. J. Comput. Vision 11(1), 25\u201344 (1993)","journal-title":"Int. J. Comput. Vision"},{"issue":"1","key":"1282_CR56","doi-asserted-by":"publisher","first-page":"176","DOI":"10.1137\/S0036139994266311","volume":"57","author":"A Tannenbaum","year":"1997","unstructured":"Tannenbaum, A., Sapiro, G., Olver, P.J.: Invariant geometric evolutions of surfaces and volumetric smoothing. SIAM J. Appl. Math. 57(1), 176\u2013194 (1997)","journal-title":"SIAM J. Appl. Math."},{"key":"1282_CR57","doi-asserted-by":"crossref","unstructured":"Angenent, S., Sapiro, G., Tannenbaum, A.: On the affine heat equation for non-convex curves. J. Am. Math. Soc. 11(3), 601\u2013634 (1998)","DOI":"10.1090\/S0894-0347-98-00262-8"},{"issue":"3","key":"1282_CR58","doi-asserted-by":"publisher","first-page":"411","DOI":"10.1109\/83.661191","volume":"7","author":"L Moisan","year":"1998","unstructured":"Moisan, L.: Affine plane curve evolution: A fully consistent scheme. IEEE Trans. Image Process. 7(3), 411\u2013420 (1998)","journal-title":"IEEE Trans. Image Process."},{"key":"1282_CR59","doi-asserted-by":"crossref","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Accurate silhouette vectorization by affine scale-space. In: 2021 IEEE International Conference on Image Processing (ICIP), pp. 1539\u20131543 (2021). IEEE","DOI":"10.1109\/ICIP42928.2021.9506518"},{"key":"1282_CR60","doi-asserted-by":"crossref","unstructured":"He, Y., Kang, S.H., Morel, J.-M.: Vectorizing images of any size. In: 2022 IEEE International Conference on Image Processing (ICIP), pp. 816\u2013820 (2022). IEEE","DOI":"10.1109\/ICIP46576.2022.9897200"},{"key":"1282_CR61","doi-asserted-by":"publisher","first-page":"95","DOI":"10.1023\/A:1007959616598","volume":"25","author":"L Alvarez","year":"1997","unstructured":"Alvarez, L., Morales, F.: Affine morphological multiscale analysis of corners and multiple junctions. Int. J. Comput. Vision 25, 95\u2013107 (1997)","journal-title":"Int. J. Comput. Vision"},{"key":"1282_CR62","doi-asserted-by":"crossref","unstructured":"Alvarez, L.: Corner detection using the affine morphological scale space. In: International Conference on Scale Space and Variational Methods in Computer Vision, pp. 29\u201340 (2017). Springer","DOI":"10.1007\/978-3-319-58771-4_3"},{"key":"1282_CR63","unstructured":"Magni, A., Mantegazza, C., Novaga, M.: Motion by curvature of planar networks ii. arXiv preprint arXiv:1301.3352 (2013)"},{"key":"1282_CR64","doi-asserted-by":"publisher","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"},{"key":"1282_CR65","doi-asserted-by":"publisher","first-page":"315","DOI":"10.1023\/A:1020878408985","volume":"50","author":"Y Chen","year":"2002","unstructured":"Chen, Y., Tagare, H.D., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K.S., Briggs, R.W., Geiser, E.A.: Using prior shapes in geometric active contours in a variational framework. Int. J. Comput. Vision 50, 315\u2013328 (2002)","journal-title":"Int. J. Comput. Vision"},{"key":"1282_CR66","doi-asserted-by":"crossref","unstructured":"Liu, H., Chen, Y., Ho, H.P., Shi, P.: Geodesic active contours with adaptive neighboring influence. In: Gerig, G. (eds.) Medical Image Computing and Computer-Assisted Intervention \u2013 MICCAI 2005, pp. 741\u2013748. Springer, Berlin, Heidelberg (2005)","DOI":"10.1007\/11566489_91"},{"key":"1282_CR67","doi-asserted-by":"crossref","unstructured":"Alvarez, L.: 3D curve regularization. Rev. Real Acad. Cienc. Exactas Fis. Nat. Ser. A-Mat 116 (106), 1\u201310 (2022)","DOI":"10.1007\/s13398-022-01242-4"},{"key":"1282_CR68","doi-asserted-by":"publisher","first-page":"793","DOI":"10.1007\/s11554-015-0531-5","volume":"14","author":"C Cuenca","year":"2018","unstructured":"Cuenca, C., Gonz\u00e1lez, E., Trujillo, A., Esclarin, J., Mazorra, L., Alvarez, L., Martinez, J.A., Tahoces, P.G., Carreira, J.M.: Fast and accurate circle tracking using active contour models. J. Real-Time Image Proc. 14, 793\u2013802 (2018)","journal-title":"J. Real-Time Image Proc."},{"issue":"7","key":"1282_CR69","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s10851-018-0798-9","volume":"60","author":"L Alvarez","year":"2018","unstructured":"Alvarez, L., Gonz\u00e1lez, E., Cuenca, C., Trujillo, A., Tahoces, P.G., Carreira, J.M.: Ellipse motion estimation using parametric snakes. J. Math. Imaging Vision 60(7), 1\u201310 (2018)","journal-title":"J. Math. Imaging Vision"},{"key":"1282_CR70","doi-asserted-by":"publisher","first-page":"18","DOI":"10.5201\/ipol.2021.296","volume":"11","author":"Y He","year":"2021","unstructured":"He, Y., Kang, S.H., Alvarez, L.: Finding the skeleton of 2D shape and contours: implementation of Hamilton\u2013Jacobi skeleton. IPOL J. Image Process. Line 11, 18\u201336 (2021)","journal-title":"IPOL J. Image Process. Line"}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-026-01282-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10851-026-01282-0","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-026-01282-0.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2026,5,14]],"date-time":"2026-05-14T04:47:27Z","timestamp":1778734047000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10851-026-01282-0"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2026,2,2]]},"references-count":70,"journal-issue":{"issue":"2","published-print":{"date-parts":[[2026,4]]}},"alternative-id":["1282"],"URL":"https:\/\/doi.org\/10.1007\/s10851-026-01282-0","relation":{},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"value":"0924-9907","type":"print"},{"value":"1573-7683","type":"electronic"}],"subject":[],"published":{"date-parts":[[2026,2,2]]},"assertion":[{"value":"8 May 2025","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"4 January 2026","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"2 February 2026","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}},{"order":1,"name":"Ethics","group":{"name":"EthicsHeading","label":"Declarations"}},{"value":"The authors declare no Conflict of interest.","order":2,"name":"Ethics","group":{"name":"EthicsHeading","label":"Conflict of interest"}}],"article-number":"6"}}