{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,11,1]],"date-time":"2025-11-01T02:51:45Z","timestamp":1761965505194,"version":"3.40.3"},"publisher-location":"Cham","reference-count":18,"publisher":"Springer Nature Switzerland","isbn-type":[{"type":"print","value":"9783031577925"},{"type":"electronic","value":"9783031577932"}],"license":[{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"},{"start":{"date-parts":[[2024,1,1]],"date-time":"2024-01-01T00:00:00Z","timestamp":1704067200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/www.springernature.com\/gp\/researchers\/text-and-data-mining"}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":[],"published-print":{"date-parts":[[2024]]},"DOI":"10.1007\/978-3-031-57793-2_18","type":"book-chapter","created":{"date-parts":[[2024,4,10]],"date-time":"2024-04-10T12:01:55Z","timestamp":1712750515000},"page":"229-241","source":"Crossref","is-referenced-by-count":5,"title":["Training Morphological Neural Networks with\u00a0Gradient Descent: Some Theoretical Insights"],"prefix":"10.1007","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0294-8172","authenticated-orcid":false,"given":"Samy","family":"Blusseau","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","reference":[{"key":"18_CR1","doi-asserted-by":"crossref","unstructured":"Aouad, T., Talbot, H.: Binary morphological neural network. In: 2022 IEEE International Conference on Image Processing (ICIP), pp. 3276\u20133280 (2022)","DOI":"10.1109\/ICIP46576.2022.9897474"},{"issue":"3","key":"18_CR2","doi-asserted-by":"publisher","first-page":"299","DOI":"10.1016\/0165-1684(93)90015-3","volume":"30","author":"GJF Banon","year":"1993","unstructured":"Banon, G.J.F., Barrera, J.: Decomposition of mappings between complete lattices by mathematical morphology, Part I. general lattices. Sig. Process. 30(3), 299\u2013327 (1993)","journal-title":"Sig. Process."},{"key":"18_CR3","series-title":"LNCS","doi-asserted-by":"publisher","first-page":"206","DOI":"10.1007\/978-3-031-19897-7_17","volume-title":"DGMM 2022","author":"S Blusseau","year":"2022","unstructured":"Blusseau, S., Velasco-Forero, S., Angulo, J., Bloch, I.: Morphological adjunctions represented by matrices in max-plus algebra for signal and image processing. In: Baudrier, \u00c9., Naegel, B., Kr\u00e4henb\u00fchl, A., Tajine, M. (eds.) DGMM 2022. LNCS, vol. 13493, pp. 206\u2013218. Springer, Cham (2022). https:\/\/doi.org\/10.1007\/978-3-031-19897-7_17"},{"key":"18_CR4","series-title":"LNCS","doi-asserted-by":"publisher","first-page":"3","DOI":"10.1007\/978-3-319-57240-6_1","volume-title":"ISMM 2017","author":"V Charisopoulos","year":"2017","unstructured":"Charisopoulos, V., Maragos, P.: Morphological perceptrons: geometry and training algorithms. In: Angulo, J., Velasco-Forero, S., Meyer, F. (eds.) ISMM 2017. LNCS, vol. 10225, pp. 3\u201315. Springer, Cham (2017). https:\/\/doi.org\/10.1007\/978-3-319-57240-6_1"},{"key":"18_CR5","doi-asserted-by":"crossref","unstructured":"Davidson, J.L., Ritter, G.X.: Theory of morphological neural networks. In: Digital Optical Computing II, vol.\u00a01215, pp. 378\u2013389 (1990)","DOI":"10.1117\/12.18085"},{"key":"18_CR6","doi-asserted-by":"publisher","first-page":"107246","DOI":"10.1016\/j.patcog.2020.107246","volume":"102","author":"G Franchi","year":"2020","unstructured":"Franchi, G., Fehri, A., Yao, A.: Deep morphological networks. Pattern Recogn. 102, 107246 (2020)","journal-title":"Pattern Recogn."},{"issue":"7","key":"18_CR7","doi-asserted-by":"publisher","first-page":"736","DOI":"10.1007\/s10851-022-01091-1","volume":"64","author":"R Hermary","year":"2022","unstructured":"Hermary, R., Tochon, G., Puybareau, \u00c9., Kirszenberg, A., Angulo, J.: Learning grayscale mathematical morphology with smooth morphological layers. J. Math. Imaging Vis. 64(7), 736\u2013753 (2022)","journal-title":"J. Math. Imaging Vis."},{"key":"18_CR8","series-title":"LNCS","doi-asserted-by":"publisher","first-page":"470","DOI":"10.1007\/978-3-030-76657-3_34","volume-title":"DGMM 2021","author":"A Kirszenberg","year":"2021","unstructured":"Kirszenberg, A., Tochon, G., Puybareau, \u00c9., Angulo, J.: Going beyond p-convolutions to learn grayscale morphological operators. In: Lindblad, J., Malmberg, F., Sladoje, N. (eds.) DGMM 2021. LNCS, vol. 12708, pp. 470\u2013482. Springer, Cham (2021). https:\/\/doi.org\/10.1007\/978-3-030-76657-3_34"},{"issue":"6","key":"18_CR9","doi-asserted-by":"publisher","first-page":"586","DOI":"10.1109\/34.24793","volume":"11","author":"P Maragos","year":"1989","unstructured":"Maragos, P.: A representation theory for morphological image and signal processing. IEEE Trans. Pattern Anal. Mach. Intell. 11(6), 586\u2013599 (1989)","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"18_CR10","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1016\/B978-0-12-407702-7.00002-4","volume":"177","author":"P Maragos","year":"2013","unstructured":"Maragos, P.: Chapter two - representations for morphological image operators and analogies with linear operators. Adv. Imaging Electron Phys. 177, 45\u2013187 (2013)","journal-title":"Adv. Imaging Electron Phys."},{"key":"18_CR11","volume-title":"Random Sets and Integral Geometry","author":"G Matheron","year":"1975","unstructured":"Matheron, G.: Random Sets and Integral Geometry. Wiley, New York (1975)"},{"issue":"1","key":"18_CR12","first-page":"87","volume":"4","author":"R Mondal","year":"2020","unstructured":"Mondal, R., Dey, M.S., Chanda, B.: Image restoration by learning morphological opening-closing network. Mathem. Morphol. Theory Appl. 4(1), 87\u2013107 (2020)","journal-title":"Mathem. Morphol. Theory Appl."},{"key":"18_CR13","doi-asserted-by":"publisher","first-page":"45","DOI":"10.1007\/bfb0121154","volume-title":"Nonlinear Analysis and Optimization. Mathematical Programming Studies Part III","author":"SM Robinson","year":"1987","unstructured":"Robinson, S.M.: Local structure of feasible sets in nonlinear programming, stability and sensitivity. In: Cornet, B., Nguyen, V.H., Vial, J.P. (eds.) Nonlinear Analysis and Optimization. Mathematical Programming Studies Part III, vol. 30, pp. 45\u201366. Springer, Heidelberg (1987). https:\/\/doi.org\/10.1007\/bfb0121154"},{"key":"18_CR14","doi-asserted-by":"publisher","DOI":"10.1007\/978-1-4614-4340-7","volume-title":"Introduction to Piecewise Differentiable Equations","author":"S Scholtes","year":"2012","unstructured":"Scholtes, S.: Introduction to Piecewise Differentiable Equations. Springer, New York (2012). https:\/\/doi.org\/10.1007\/978-1-4614-4340-7"},{"key":"18_CR15","unstructured":"Velasco-Forero, S.: Morpholayers (2020)"},{"key":"18_CR16","unstructured":"Velasco-Forero, S., Rhim, A., Angulo, J.: Fixed point layers for geodesic morphological operations. In: BMVC. London, United Kingdom (2022)"},{"key":"18_CR17","doi-asserted-by":"crossref","unstructured":"Wilson, S.S.: Morphological networks. In: Visual Communications and Image Processing IV, vol.\u00a01199, pp. 483\u2013496 (1989)","DOI":"10.1117\/12.970058"},{"key":"18_CR18","series-title":"LNCS","doi-asserted-by":"publisher","first-page":"310","DOI":"10.1007\/978-3-030-20867-7_24","volume-title":"ISMM 2019","author":"Y Zhang","year":"2019","unstructured":"Zhang, Y., Blusseau, S., Velasco-Forero, S., Bloch, I., Angulo, J.: Max-plus operators applied to filter selection and model pruning in neural networks. In: Burgeth, B., Kleefeld, A., Naegel, B., Passat, N., Perret, B. (eds.) ISMM 2019. LNCS, vol. 11564, pp. 310\u2013322. Springer, Cham (2019). https:\/\/doi.org\/10.1007\/978-3-030-20867-7_24"}],"container-title":["Lecture Notes in Computer Science","Discrete Geometry and Mathematical Morphology"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/978-3-031-57793-2_18","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2024,11,16]],"date-time":"2024-11-16T00:51:29Z","timestamp":1731718289000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/978-3-031-57793-2_18"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2024]]},"ISBN":["9783031577925","9783031577932"],"references-count":18,"URL":"https:\/\/doi.org\/10.1007\/978-3-031-57793-2_18","relation":{},"ISSN":["0302-9743","1611-3349"],"issn-type":[{"type":"print","value":"0302-9743"},{"type":"electronic","value":"1611-3349"}],"subject":[],"published":{"date-parts":[[2024]]}}}