{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T02:07:37Z","timestamp":1761962857098,"version":"build-2065373602"},"reference-count":38,"publisher":"Springer Science and Business Media LLC","issue":"5","license":[{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T00:00:00Z","timestamp":1758844800000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2025,10]]},"DOI":"10.1007\/s10851-025-01269-3","type":"journal-article","created":{"date-parts":[[2025,9,26]],"date-time":"2025-09-26T09:14:23Z","timestamp":1758878063000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["Vertex Characterization Via Second-Order Topological Derivatives"],"prefix":"10.1007","volume":"67","author":[{"given":"Peter","family":"Gangl","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Bochra","family":"Mejri","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Otmar","family":"Scherzer","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,9,26]]},"reference":[{"key":"1269_CR1","volume-title":"Polarization and Moment Tensors: With applications to inverse problems and effective medium theory","author":"H Ammari","year":"2007","unstructured":"Ammari, H., Kang, H.: Polarization and Moment Tensors: With applications to inverse problems and effective medium theory. Springer, New York, NY (2007)"},{"key":"1269_CR2","doi-asserted-by":"crossref","unstructured":"Amstutz, S.: Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49, 87 (2006)","DOI":"10.3233\/ASY-2006-778"},{"key":"1269_CR3","doi-asserted-by":"publisher","unstructured":"Amstutz, S., Fehrenbach, J.: Edge detection using topological gradients: a scale-space approach. J. Math. Imag. Vis. 52, 49\u2013266 (2015). https:\/\/doi.org\/10.1007\/s10851-015-0558-z","DOI":"10.1007\/s10851-015-0558-z"},{"key":"1269_CR4","doi-asserted-by":"publisher","first-page":"169","DOI":"10.1553\/etna_vol51s169","volume":"51","author":"S Amstutz","year":"2019","unstructured":"Amstutz, S., Gangl, P.: Topological derivative for the nonlinear magnetostatic problem. Electron. Trans. Numer. Anal. 51, 169\u2013218 (2019). https:\/\/doi.org\/10.1553\/etna_vol51s169","journal-title":"Electron. Trans. Numer. Anal."},{"key":"1269_CR5","unstructured":"Amstutz, S., Horchani, I., Masmoudi, M.: Crack detection by the topological gradient method. Control Cybern. 34, 81\u2013101 (2005)"},{"key":"1269_CR6","doi-asserted-by":"publisher","DOI":"10.1155\/2011\/592924","volume":"20111","author":"D Auroux","year":"2011","unstructured":"Auroux, D., Cohen, L.D., Masmoudi, M.: Contour detection and completion for inpainting and segmentation based on topological gradient and fast marching algorithms. Int. J. Biomed. Imag. 20111, 592924 (2011). https:\/\/doi.org\/10.1155\/2011\/592924","journal-title":"Int. J. Biomed. Imag."},{"key":"1269_CR7","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1108\/EC-07-2021-0407","volume":"39","author":"P Baumann","year":"2022","unstructured":"Baumann, P., Sturm, K.: Adjoint-based methods to compute higher-order topological derivatives with an application to elasticity. Eng. Comput. 39, 60\u2013114 (2022)","journal-title":"Eng. Comput."},{"key":"1269_CR8","doi-asserted-by":"publisher","first-page":"313","DOI":"10.1016\/j.crma.2005.12.009","volume":"342","author":"LJ Belaid","year":"2006","unstructured":"Belaid, L.J., et al.: Image restoration and edge detection by topological asymptotic expansion. Compt. Rendus Math. 342, 313\u2013318 (2006)","journal-title":"Compt. Rendus Math."},{"key":"1269_CR9","first-page":"389","volume":"8","author":"E Beretta","year":"2014","unstructured":"Beretta, E., et al.: A variational algorithm for the detection of line segments. Inv. Prob. Imag. 8, 389\u2013408 (2014)","journal-title":"Inv. Prob. Imag."},{"key":"1269_CR10","doi-asserted-by":"publisher","first-page":"2275","DOI":"10.1016\/j.ijsolstr.2009.01.021","volume":"46","author":"M Bonnet","year":"2009","unstructured":"Bonnet, M.: Higher-order topological sensitivity for 2-D potential problems. Application to fast identification of inclusions. Int. J. Solids Struct. 46, 2275\u20132292 (2009)","journal-title":"Int. J. Solids Struct."},{"key":"1269_CR11","doi-asserted-by":"publisher","unstructured":"Bonnet, M., Cornaggia, R.: \u201cHigher order topological derivatives for three-dimensional anisotropic elasticity\u201d. ESAIM: Mathematical Modelling and Numerical Analysis 51.6 (2017), pp.\u00a02069\u20132092. doi: https:\/\/doi.org\/10.1051\/m2an\/2017015","DOI":"10.1051\/m2an\/2017015"},{"key":"1269_CR12","doi-asserted-by":"publisher","unstructured":"Canelas,A., Novotny,A.A., Roche, J.R.: \u201cTopology design of inductors in electromagnetic casting using level-sets and second order topological derivatives\u201d. Structural and Multidisciplinary Optimization 50.6 (May 2014), pp.\u00a01151\u20131163. ISSN: 1615-1488. doi: https:\/\/doi.org\/10.1007\/s00158-014-1103-1","DOI":"10.1007\/s00158-014-1103-1"},{"key":"1269_CR13","doi-asserted-by":"publisher","first-page":"171","DOI":"10.1016\/S0734-189X(87)80001-4","volume":"37","author":"P Cavanagh","year":"1987","unstructured":"Cavanagh, P.: Reconstructing the third dimension: interactions between color, texture, motion, binocular disparity, and shape. Comput. Vis. Graphics Image Proc. 37, 171\u2013195 (1987)","journal-title":"Comput. Vis. Graphics Image Proc."},{"key":"1269_CR14","doi-asserted-by":"crossref","unstructured":"Clowes, M.B.: \u201cOn seeing things\u201d. In: Artificial Intelligence 2.1 (1971), pp.\u00a079\u2013116. ISSN: 0004-3702. doi: https:\/\/doi.org\/10.1016\/0004-3702(71)90005-1","DOI":"10.1016\/0004-3702(71)90005-1"},{"key":"1269_CR15","doi-asserted-by":"crossref","unstructured":"Deny, J., Lions, J.L.: \u201cLes espaces du type de Beppo Levi\u201d. Ann. Inst. Fourier, Grenoble 5 (1954), 305\u2013370 (1955). ISSN: 0373-0956","DOI":"10.5802\/aif.55"},{"key":"1269_CR16","doi-asserted-by":"crossref","unstructured":"Dong, G., et al.: \u201cScale and edge detection with topological derivatives\u201d. In: Scale Space and Variational Methods in Computer Vision. Ed. by Arjan Kuijper et al. Berlin, Heidelberg: Springer Berlin Heidelberg, 2013, pp.\u00a0404\u2013415. ISBN: 978-3-642-38267-3","DOI":"10.1007\/978-3-642-38267-3_34"},{"key":"1269_CR17","doi-asserted-by":"publisher","unstructured":"Gangl, P., Mejri, B., Scherzer, O.: Supplementary material to \"Vertex characterization via second-order topological derivatives\". Version\u00a0NETGEN-6.2.2204-8-g43fbd57a. 2024. doi: https:\/\/doi.org\/10.5281\/zenodo.13384684","DOI":"10.5281\/zenodo.13384684"},{"key":"1269_CR18","doi-asserted-by":"crossref","unstructured":"Gangl, P., Nees, N., Stingl, M.: Efficient and accurate separable models for discretized material optimization: a continuous perspective based on topological derivatives. J. Geomet. Ana. 34, 206 (2024)","DOI":"10.1007\/s12220-024-01663-0"},{"key":"1269_CR19","doi-asserted-by":"crossref","unstructured":"Gangl, P., Sturm, K.: \u201cA simplified derivation technique of topological derivatives for quasi-linear transmission problems\u201d. ESAIM: COCV 26 (2020), p.\u00a0106","DOI":"10.1051\/cocv\/2020035"},{"key":"1269_CR20","doi-asserted-by":"publisher","unstructured":"Gangl, P., Sturm, K.: \u201cAsymptotic analysis and topological derivative for 3D quasi-linear magnetostatics\u201d. ESAIM: M2AN 55 (2021), S853\u2013S875. doi: https:\/\/doi.org\/10.1051\/m2an\/2020060","DOI":"10.1051\/m2an\/"},{"key":"1269_CR21","doi-asserted-by":"publisher","DOI":"10.1016\/j.cma.2022.115288","volume":"398","author":"P Gangl","year":"2022","unstructured":"Gangl, P., Sturm, K.: Automated computation of topological derivatives with application to nonlinear elasticity and reaction-diffusion problems. Comput. Method. Appl. Mech. Eng. 398, 115288 (2022)","journal-title":"Comput. Method. Appl. Mech. Eng."},{"key":"1269_CR22","doi-asserted-by":"publisher","first-page":"1756","DOI":"10.1137\/S0363012900369538","volume":"39","author":"S Garreau","year":"2001","unstructured":"Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for PDE systems: the elasticity case. SIAM J. Control Optim. 39, 1756\u20131778 (2001)","journal-title":"SIAM J. Control Optim."},{"key":"1269_CR23","doi-asserted-by":"publisher","first-page":"141","DOI":"10.4171\/ifb\/298","volume":"15","author":"M Grasmair","year":"2013","unstructured":"Grasmair, M., Muszkieta, M., Scherzer, O.: An approach to the minimization of the Mumford-Shah functional using $$\\Gamma $$-convergence and topological asymptotic expansion. Interf. Free Bound. 15, 141\u2013166 (2013)","journal-title":"Interf. Free Bound."},{"key":"1269_CR24","doi-asserted-by":"publisher","unstructured":"Guzm\u00e1n, A.: \u201cDecomposition of a visual scene into three-dimensional bodies\u201d. In: Proceedings of the December 9-11, 1968, Fall Joint Computer Conference, Part I. AFIPS \u201968 (Fall, part I). San Francisco, California: Association for Computing Machinery, 1968, pp.\u00a0291\u2013304. ISBN: 9781450378994. doi: https:\/\/doi.org\/10.1145\/1476589.1476631","DOI":"10.1145\/1476589.1476631"},{"key":"1269_CR25","doi-asserted-by":"crossref","unstructured":"Kanizsa, G.: \u201cSubjective contours\u201d. Scientific American 234.4 (1976), pp.\u00a048\u201353. ISSN: 00368733, 19467087. (Visited on 02\/29\/2024)","DOI":"10.1038\/scientificamerican0476-48"},{"key":"1269_CR26","doi-asserted-by":"crossref","unstructured":"Kozlov, V., Maz\u2019ya, V., Movchan, A.: Asymptotic analysis of fields in multi-structures. Oxford University Press, Aug. 1999. ISBN: 9780198514954","DOI":"10.1093\/oso\/9780198514954.001.0001"},{"key":"1269_CR27","doi-asserted-by":"crossref","unstructured":"Larrabide, I., et al. \u201cTopological derivative: A tool for image processing\u201d. In: Computers & Structures 86.13 (2008). Structural Optimization, pp.\u00a01386\u20131403. ISSN: 0045-7949. doi: https:\/\/doi.org\/10.1016\/j.compstruc.2007.05.004","DOI":"10.1016\/j.compstruc.2007.05.004"},{"key":"1269_CR28","doi-asserted-by":"crossref","unstructured":"Mehrotra, R., Nichani, S., Ranganathan, N.: \u201cCorner detection\u201d. In: Pattern Recognition 23.11 (1990), pp.\u00a01223\u20131233. ISSN: 0031-3203. doi: https:\/\/doi.org\/10.1016\/ 0031-3203(90)90118-5","DOI":"10.1016\/0031-3203(90)90118-5"},{"key":"1269_CR29","doi-asserted-by":"publisher","first-page":"577","DOI":"10.1002\/cpa.3160420503","volume":"429","author":"D Mumford","year":"1989","unstructured":"Mumford, D., Shah, J.: Optimal approximations by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 429, 577\u2013685 (1989)","journal-title":"Commun. Pure Appl. Math."},{"key":"1269_CR30","doi-asserted-by":"publisher","unstructured":"Muszkieta, M.: \u201cOptimal edge detection by topological asymptotic analysis\u201d. Mathematical Models and Methods in Applied Sciences 19.11 (2009), pp.\u00a02127\u20132143. doi: https:\/\/doi.org\/10.1142\/S0218202509004066","DOI":"10.1142\/S0218202509004066"},{"key":"1269_CR31","doi-asserted-by":"crossref","unstructured":"Nitzberg, M., Mumford, D.,Shiota, T.: Filtering, segmentation and depth. Vol.\u00a0662. Springer, 1993","DOI":"10.1007\/3-540-56484-5"},{"key":"1269_CR32","doi-asserted-by":"publisher","unstructured":"Novotny, A.A., Sokolowski, J., Zochowski, A.: \u201cTopological derivatives of shape functionals. Part III: Second-order method and applications\u201d. In: Journal of Optimization Theory and Applications 181.1 (2018), pp.\u00a01\u201322. ISSN: 1573-2878. doi: https:\/\/doi.org\/10.1007\/s10957-018-1420-4","DOI":"10.1007\/s10957-"},{"key":"1269_CR33","doi-asserted-by":"crossref","unstructured":"Rocha de Faria, J., Novotny, A.A.: \u201cOn the second order topological asymptotic expansion\u201d. Structural and Multidisciplinary Optimization 39.6 (2009), p.\u00a0547","DOI":"10.1007\/s00158-009-0436-7"},{"key":"1269_CR34","doi-asserted-by":"crossref","unstructured":"Rocha de Faria, J.R., et al.: \u201cSecond order topological sensitivity analysis\u201d. In: International Journal of Solids and Structures 44.14 (2007), pp.\u00a04958\u20134977. ISSN: 0020-7683. doi: https: \/\/doi.org\/10.1016\/j.ijsolstr.2006.12.013","DOI":"10.1016\/j.ijsolstr.2006.12.013"},{"key":"1269_CR35","unstructured":"Sch\u00f6berl, J. C++11 Implementation of finite elements in NGSolve. Tech. rep. 30. Institute for Analysis and Scientific Computing, Vienna University of Technology, 2014"},{"key":"1269_CR36","doi-asserted-by":"crossref","unstructured":"Sokolowski, J., Zochowski, A.: \u201cOn the topological derivative in shape optimization\u201d. SIAM Journal on Control and Optimization 37.4 (1999), pp.\u00a01251\u20131272","DOI":"10.1137\/S0363012997323230"},{"key":"1269_CR37","doi-asserted-by":"publisher","unstructured":"Sturm, K.: \u201cTopological sensitivities via a Lagrangian approach for semilinear problems\u201d. In: Nonlinearity 33.9 (July 2020), pp.\u00a04310\u20134337. ISSN: 1361-6544. https:\/\/doi.org\/10.1088\/1361-6544\/ab86cb","DOI":"10.1088\/1361-"},{"key":"1269_CR38","doi-asserted-by":"crossref","unstructured":"Watanabe, T., Cavanagh, P.: \u201cTransparent surfaces defined by implicit X junctions\u201d. In: Vision Research 33.16 (1993), pp.\u00a02339\u20132346. ISSN: 0042-6989. doi: https: \/\/doi.org\/10.1016\/0042-6989(93)90111-9","DOI":"10.1016\/0042-6989(93)90111-9"}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-025-01269-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10851-025-01269-3\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-025-01269-3.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T02:02:17Z","timestamp":1761962537000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10851-025-01269-3"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2025,9,26]]},"references-count":38,"journal-issue":{"issue":"5","published-print":{"date-parts":[[2025,10]]}},"alternative-id":["1269"],"URL":"https:\/\/doi.org\/10.1007\/s10851-025-01269-3","relation":{},"ISSN":["0924-9907","1573-7683"],"issn-type":[{"type":"print","value":"0924-9907"},{"type":"electronic","value":"1573-7683"}],"subject":[],"published":{"date-parts":[[2025,9,26]]},"assertion":[{"value":"28 August 2024","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"5 September 2025","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"26 September 2025","order":3,"name":"first_online","label":"First Online","group":{"name":"ArticleHistory","label":"Article History"}}],"article-number":"53"}}