{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2026,5,6]],"date-time":"2026-05-06T02:36:31Z","timestamp":1778034991316,"version":"3.51.4"},"reference-count":61,"publisher":"Springer Science and Business Media LLC","issue":"6","license":[{"start":{"date-parts":[[2023,4,16]],"date-time":"2023-04-16T00:00:00Z","timestamp":1681603200000},"content-version":"tdm","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"},{"start":{"date-parts":[[2023,4,16]],"date-time":"2023-04-16T00:00:00Z","timestamp":1681603200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0"}],"content-domain":{"domain":["link.springer.com"],"crossmark-restriction":false},"short-container-title":["J Math Imaging Vis"],"published-print":{"date-parts":[[2023,12]]},"abstract":"<jats:title>Abstract<\/jats:title>\n                  <jats:p>Group equivariant convolutional neural networks (G-CNNs) have been successfully applied in geometric deep learning. Typically, G-CNNs have the advantage over CNNs that they do not waste network capacity on training symmetries that should have been hard-coded in the network. The recently introduced framework of PDE-based G-CNNs (PDE-G-CNNs) generalizes G-CNNs. PDE-G-CNNs have the core advantages that they simultaneously (1) reduce network complexity, (2) increase classification performance, and (3) provide geometric interpretability. Their implementations primarily consist of linear and morphological convolutions with kernels. In this paper, we show that the previously suggested approximative morphological kernels do not always accurately approximate the exact kernels accurately. More specifically, depending on the spatial anisotropy of the Riemannian metric, we argue that one must resort to sub-Riemannian approximations. We solve this problem by providing a new approximative kernel that works regardless of the anisotropy. We provide new theorems with better error estimates of the approximative kernels, and prove that they all carry the same reflectional symmetries as the exact ones. We test the effectiveness of multiple approximative kernels within the PDE-G-CNN framework on two datasets, and observe an improvement with the new approximative kernels. We report that the PDE-G-CNNs again allow for a considerable reduction of network complexity while having comparable or better performance than G-CNNs and CNNs on the two datasets. Moreover, PDE-G-CNNs have the advantage of better geometric interpretability over G-CNNs, as the morphological kernels are related to association fields from neurogeometry.<\/jats:p>","DOI":"10.1007\/s10851-023-01147-w","type":"journal-article","created":{"date-parts":[[2023,4,16]],"date-time":"2023-04-16T05:01:51Z","timestamp":1681621311000},"page":"819-843","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":11,"title":["Analysis of (sub-)Riemannian PDE-G-CNNs"],"prefix":"10.1007","volume":"65","author":[{"given":"Gijs","family":"Bellaard","sequence":"first","affiliation":[]},{"given":"Daan L. J.","family":"Bon","sequence":"additional","affiliation":[]},{"given":"Gautam","family":"Pai","sequence":"additional","affiliation":[]},{"given":"Bart M. N.","family":"Smets","sequence":"additional","affiliation":[]},{"given":"Remco","family":"Duits","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2023,4,16]]},"reference":[{"key":"1147_CR1","doi-asserted-by":"crossref","unstructured":"Bekkers, E.J., Lafarge, M.W., Veta, M., Eppenhof, K.A.J., Pluim, J.P.W., Duits, R.: Roto-translation covariant convolutional networks for medical image analysis. In: International Conference on Medical Image Computing and Computer-Assisted Intervention, pp. 440\u2013448. Springer (2018). arXiv:1804.03393","DOI":"10.1007\/978-3-030-00928-1_50"},{"issue":"4","key":"1147_CR2","doi-asserted-by":"publisher","first-page":"541","DOI":"10.1162\/neco.1989.1.4.541","volume":"1","author":"Y LeCun","year":"1989","unstructured":"LeCun, Y., Boser, B., Denker, J.S., Henderson, D., Howard, R.E., Hubbard, W., Jackel, L.D.: Backpropagation applied to handwritten zip code recognition. Neural Comput. 1(4), 541\u2013551 (1989)","journal-title":"Neural Comput."},{"key":"1147_CR3","unstructured":"Krizhevsky, A., Sutskever, I., Hinton, G.E.: ImageNet classification with deep convolutional neural networks. In: Pereira, F., Burges, C.J., Bottou, L., Weinberger, K.Q. (eds.) Advances in Neural Information Processing Systems, vol. 25. Curran Associates, Inc., Red Hook, New York (2012). https:\/\/proceedings.neurips.cc\/paper\/2012\/file\/c399862d3b9d6b76c8436e924a68c45b-Paper.pdf"},{"key":"1147_CR4","doi-asserted-by":"publisher","first-page":"60","DOI":"10.1016\/j.media.2017.07.005","volume":"42","author":"G Litjens","year":"2017","unstructured":"Litjens, G., Bejnodri, B.E., Setio, A.A.A., Ciompi, F., Ghafoorian, M., van der Laak, J.A.W.M., van Ginneken, B., S\u00e1nchez, C.I.: A survey on deep learning in medical image analysis. Med. Image Anal. 42, 60\u201388 (2017)","journal-title":"Med. Image Anal."},{"key":"1147_CR5","unstructured":"Cohen, T.S., Welling, M.: Group equivariant convolutional networks. In: Proceedings of the 33rd International Conference on Machine Learning, vol. 48, pp. 1\u201312 (2016)"},{"key":"1147_CR6","unstructured":"Dieleman, S., De\u00a0Fauw, J., Kavukcuoglu, K.: Exploiting cyclic symmetry in convolutional neural networks. arXiv:1602.02660 (2016)"},{"issue":"2","key":"1147_CR7","doi-asserted-by":"publisher","first-page":"1441","DOI":"10.1093\/mnras\/stv632","volume":"450","author":"S Dieleman","year":"2015","unstructured":"Dieleman, S., Willett, K.W., Dambre, J.: Rotation-invariant convolutional neural networks for galaxy morphology prediction. Mon. Not. R. Astron. Soc. 450(2), 1441\u20131459 (2015)","journal-title":"Mon. Not. R. Astron. Soc."},{"key":"1147_CR8","unstructured":"Winkels, M., Cohen, T.S.: 3D G-CNNs for pulmonary nodule detection. MIDL, 1\u201311 (2018)"},{"key":"1147_CR9","first-page":"585","volume":"2018","author":"D Worrall","year":"2018","unstructured":"Worrall, D., Brostow, G.: CubeNet: equivariance to 3D rotation and translation. ECCV 2018, 585\u2013602 (2018)","journal-title":"ECCV"},{"key":"1147_CR10","doi-asserted-by":"crossref","unstructured":"Oyallon, E., Mallat, S.: Deep roto-translation scattering for object classification. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 2865\u20132873 (2015)","DOI":"10.1109\/CVPR.2015.7298904"},{"key":"1147_CR11","doi-asserted-by":"crossref","unstructured":"Weiler, M., Hamprecht, F.A., Storath, M.: Learning steerable filters for rotation equivariant CNNs. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 849\u2013858 (2018)","DOI":"10.1109\/CVPR.2018.00095"},{"key":"1147_CR12","unstructured":"Bekkers, E.J.: B-spline CNNs on Lie groups. (2019) arXiv:1909.12057"},{"key":"1147_CR13","unstructured":"Finzi, M., Stanton, S., Izmailov, P., Wilson, A.G.: Generalizing convolutional neural networks for equivariance to Lie groups on arbitrary continuous data. In: III, H.D., Singh, A. (eds.) Proceedings of the 37th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 119, pp. 3165\u20133176. PMLR, Virtual (2020). http:\/\/proceedings.mlr.press\/v119\/finzi20a.html"},{"key":"1147_CR14","unstructured":"Cohen, T.S., Geiger, M., Weiler, M.: A general theory of equivariant CNNs on homogeneous spaces. Adv. Neural Inf. Process. Syst. 32 (2019)"},{"key":"1147_CR15","doi-asserted-by":"crossref","unstructured":"Worrall, D.E., Garbin, S.J., Turmukhambetov, D., Brostow, G.J.: Harmonic networks: Deep translation and rotation equivariance. In: Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition, pp. 5028\u20135037 (2017)","DOI":"10.1109\/CVPR.2017.758"},{"key":"1147_CR16","unstructured":"Kondor, R., Trivedi, S.: On the generalization of equivariance and convolution in neural networks to the action of compact groups. In: Dy, J., Krause, A. (eds.) Proceedings of the 35th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 80, pp. 2747\u20132755. PMLR, Stockholmsm\u00e4ssan, Stockholm Sweden (2018). http:\/\/proceedings.mlr.press\/v80\/kondor18a.html"},{"key":"1147_CR17","doi-asserted-by":"crossref","unstructured":"Esteves, C., Allen-Blanchette, C., Makadia, A., Daniilidis, K.: Learning SO(3) equivariant representations with spherical CNNs. In: Proceedings of the European Conference on Computer Vision (ECCV), pp. 52\u201368 (2018)","DOI":"10.1007\/978-3-030-01261-8_4"},{"key":"1147_CR18","unstructured":"Weiler, M., Cesa, G.: General E(2)-equivariant steerable CNNs. In: Advances in Neural Information Processing Systems, pp. 14334\u201314345 (2019)"},{"key":"1147_CR19","doi-asserted-by":"publisher","first-page":"179575","DOI":"10.1109\/ACCESS.2020.3027776","volume":"8","author":"ME Paoletti","year":"2020","unstructured":"Paoletti, M.E., Haut, J.M., Roy, S.K., Hendrix, E.M.T.: Rotation equivariant convolutional neural networks for hyperspectral image classification. IEEE Access 8, 179575\u2013179591 (2020). https:\/\/doi.org\/10.1109\/ACCESS.2020.3027776","journal-title":"IEEE Access"},{"key":"1147_CR20","unstructured":"Weiler, M., Forr\u00e9, P., Verlinde, E., Welling, M.: Coordinate Independent Convolutional Networks\u2014Isometry and Gauge Equivariant Convolutions on Riemannian Manifolds (2021). arXiv:2106.06020"},{"key":"1147_CR21","unstructured":"Cohen, T.S., Weiler, M., Kicanaoglu, B., Welling, M.: Gauge equivariant convolutional networks and the icosahedral CNN. In: Chaudhuri, K., Salakhutdinov, R. (eds.) Proceedings of the 36th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 97, pp. 1321\u20131330. PMLR, Long Beach, California (2019). https:\/\/proceedings.mlr.press\/v97\/cohen19d.html"},{"key":"1147_CR22","unstructured":"Bogatskiy, A., Anderson, B., Offermann, J.T., Roussi, M., Miller, D.W., Kondor, R.: Lorentz Group Equivariant Neural Network for Particle Physics (2020). arXiv:2006.04780"},{"key":"1147_CR23","doi-asserted-by":"publisher","unstructured":"Sifre, L., Mallat, S.: Rotation, scaling and deformation invariant scattering for texture discrimination. In: 2013 IEEE Conference on Computer Vision and Pattern Recognition, pp. 1233\u20131240 (2013). https:\/\/doi.org\/10.1109\/CVPR.2013.163","DOI":"10.1109\/CVPR.2013.163"},{"issue":"2","key":"1147_CR24","doi-asserted-by":"publisher","first-page":"452","DOI":"10.1109\/TPAMI.2017.2652452","volume":"40","author":"EJ Bekkers","year":"2018","unstructured":"Bekkers, E.J., Loog, M., ter Haar Romeny, B.M., Duits, R.: Template matching via densities on the roto-translation group. IEEE Trans. Pattern Anal. Mach. Intell. 40(2), 452\u2013466 (2018). https:\/\/doi.org\/10.1109\/TPAMI.2017.2652452","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"1147_CR25","unstructured":"Worrall, D., Welling, M.: Deep scale-spaces: Equivariance over scale. In: Wallach, H., Larochelle, H., Beygelzimer, A., d\u2019 Alch\u00e9-Buc, F., Fox, E., Garnett, R. (eds.) Advances in Neural Information Processing Systems, vol. 32. Curran Associates, Inc., Red Hook, New York (2019). https:\/\/proceedings.neurips.cc\/paper\/2019\/file\/f04cd7399b2b0128970efb6d20b5c551-Paper.pdf"},{"key":"1147_CR26","unstructured":"Satorras, V.G., Hoogeboom, E., Welling, M.: E(n) equivariant graph neural networks. In: Meila, M., Zhang, T. (eds.) Proceedings of the 38th International Conference on Machine Learning. Proceedings of Machine Learning Research, vol. 139, pp. 9323\u20139332. PMLR, Virtual (2021). https:\/\/proceedings.mlr.press\/v139\/satorras21a.html"},{"key":"1147_CR27","unstructured":"Bronstein, M.M., Bruna, J., Cohen, T.S., Veli\u010dkovi\u0107, P.: Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges (2021). arXiv:2104.13478"},{"key":"1147_CR28","doi-asserted-by":"publisher","DOI":"10.1007\/s10851-022-01114-x","author":"BMN Smets","year":"2022","unstructured":"Smets, B.M.N., Portegies, J.W., Bekkers, E.J., Duits, R.: PDE-based group equivariant convolutional neural networks. J. Math. Imaging Vis. (2022). https:\/\/doi.org\/10.1007\/s10851-022-01114-x","journal-title":"J. Math. Imaging Vis."},{"key":"1147_CR29","first-page":"255","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 $${SE}(2)$$. QAM-AMS 68, 255\u2013292 (2010)","journal-title":"QAM-AMS"},{"key":"1147_CR30","unstructured":"Duits, R.: Perceptual organization in image analysis: a mathematical approach based on scale, orientation and curvature. PhD thesis, Eindhoven University of Technology (2005)"},{"issue":"3","key":"1147_CR31","doi-asserted-by":"publisher","first-page":"326","DOI":"10.1007\/s10851-012-0387-2","volume":"46","author":"R Duits","year":"2013","unstructured":"Duits, R., Dela Haije, T.C.J., Creusen, E., Ghosh, A.: Morphological and linear scale spaces for fiber enhancement in DW-MRI. J. Math. Imaging Vis. 46(3), 326\u2013368 (2013)","journal-title":"J. Math. Imaging Vis."},{"key":"1147_CR32","doi-asserted-by":"crossref","unstructured":"Duits, R., Burgeth, B.: Scale spaces on Lie groups. In: International Conference on Scale Space and Variational Methods in Computer Vision, pp. 300\u2013312 (2007). Springer","DOI":"10.1007\/978-3-540-72823-8_26"},{"key":"1147_CR33","unstructured":"Franken, E.M.: Enhancement of crossing elongated structures in images. PhD thesis, Eindhoven University of Technology (2008)"},{"key":"1147_CR34","unstructured":"Bekkers, E.J.: Retinal image analysis using sub-Riemannian geometry in SE(2). PhD thesis, Eindhoven University of Technology (2017)"},{"key":"1147_CR35","doi-asserted-by":"publisher","first-page":"574","DOI":"10.1113\/jphysiol.1959.sp006308","volume":"148","author":"DH Hubel","year":"1959","unstructured":"Hubel, D.H., Wiesel, T.N.: Receptive fields of single neurons in the cat\u2019s striate cortex. J. Physiol. 148, 574\u2013591 (1959)","journal-title":"J. Physiol."},{"issue":"6","key":"1147_CR36","doi-asserted-by":"publisher","first-page":"2112","DOI":"10.1523\/JNEUROSCI.17-06-02112.1997","volume":"17","author":"WH Bosking","year":"1997","unstructured":"Bosking, W.H., Zhang, Y., Schofield, B., Fitzpatrick, D.: Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex. J. Neurosci. 17(6), 2112\u20132127 (1997)","journal-title":"J. Neurosci."},{"key":"1147_CR37","doi-asserted-by":"publisher","first-page":"265","DOI":"10.1016\/j.jphysparis.2003.10.010","volume":"97","author":"J Petitot","year":"2003","unstructured":"Petitot, J.: The neurogeometry of pinwheels as a sub-Riemannian contact structure. J. Physiol. - Paris 97, 265\u2013309 (2003)","journal-title":"J. Physiol. - Paris"},{"issue":"3","key":"1147_CR38","doi-asserted-by":"publisher","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."},{"issue":"2","key":"1147_CR39","doi-asserted-by":"publisher","first-page":"173","DOI":"10.1016\/0042-6989(93)90156-Q","volume":"33","author":"DJ Field","year":"1993","unstructured":"Field, D.J., Hayes, A., Hess, R.F.: Contour integration by the human visual system: evidence for a local \u201cassociation field\u2019\u2019. Vis. Res. 33(2), 173\u2013193 (1993). https:\/\/doi.org\/10.1016\/0042-6989(93)90156-Q","journal-title":"Vis. Res."},{"key":"1147_CR40","doi-asserted-by":"publisher","first-page":"41","DOI":"10.3390\/jimaging7030041","volume":"7","author":"E Baspinar","year":"2021","unstructured":"Baspinar, E., Calatroni, L., Franceschi, V., Prandi, D.: A cortical-inspired sub-Riemannian model for Poggendorff-type visual illusions. Journal of Imaging 7, 41 (2021). https:\/\/doi.org\/10.3390\/jimaging7030041","journal-title":"Journal of Imaging"},{"key":"1147_CR41","doi-asserted-by":"publisher","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. Its Appl. 65, 55\u201377 (2019). https:\/\/doi.org\/10.1016\/j.difgeo.2019.03.007","journal-title":"Differ. Geom. Its Appl."},{"issue":"2","key":"1147_CR42","doi-asserted-by":"publisher","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.L.: Association fields via cuspless sub-Riemannian geodesics in SE(2). J. Math. Imaging Vis. 49(2), 384\u2013417 (2014). https:\/\/doi.org\/10.1007\/s10851-013-0475-y","journal-title":"J. Math. Imaging Vis."},{"key":"1147_CR43","doi-asserted-by":"publisher","first-page":"293","DOI":"10.1051\/cocv\/2010005","volume":"17","author":"YL Sachkov","year":"2011","unstructured":"Sachkov, Y.L.: Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane. ESAIM Control Optim. Calcu. Var. 17, 293\u2013321 (2011)","journal-title":"ESAIM Control Optim. Calcu. Var."},{"issue":"2","key":"1147_CR44","doi-asserted-by":"publisher","first-page":"380","DOI":"10.1051\/cocv\/2009004","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. Calcu. Var. 16(2), 380\u2013399 (2010). https:\/\/doi.org\/10.1051\/cocv\/2009004","journal-title":"ESAIM Control Optim. Calcu. Var."},{"key":"1147_CR45","doi-asserted-by":"publisher","first-page":"816","DOI":"10.1007\/s10851-018-0795-z","volume":"60","author":"R Duits","year":"2018","unstructured":"Duits, R., Meesters, S.P.L., 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."},{"key":"1147_CR46","doi-asserted-by":"publisher","DOI":"10.1007\/978-3-319-65591-8","volume-title":"Elements of Neurogeometry. Lecture Notes in Morphogenesis","author":"J Petitot","year":"2017","unstructured":"Petitot, J.: Elements of Neurogeometry. Lecture Notes in Morphogenesis. Springer, London (2017). https:\/\/doi.org\/10.1007\/978-3-319-65591-8"},{"issue":"4","key":"1147_CR47","doi-asserted-by":"publisher","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. Imaging Sci. 8(4), 2740\u20132770 (2015)","journal-title":"SIAM J. Imaging Sci."},{"issue":"6","key":"1147_CR48","doi-asserted-by":"publisher","first-page":"882","DOI":"10.1007\/s10851-018-0787-z","volume":"60","author":"EJ Bekkers","year":"2018","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(6), 882\u2013899 (2018). https:\/\/doi.org\/10.1007\/s10851-018-0787-z","journal-title":"J. Math. Imaging Vis."},{"key":"1147_CR49","doi-asserted-by":"publisher","DOI":"10.1201\/9781420041767","volume-title":"Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups","author":"GS Chirikjian","year":"2000","unstructured":"Chirikjian, G.S., Kyatkin, A.B.: Engineering Applications of Noncommutative Harmonic Analysis: With Emphasis on Rotation and Motion Groups. CRC Press, Boca Raton (2000)"},{"issue":"2","key":"1147_CR50","doi-asserted-by":"publisher","first-page":"352","DOI":"10.1007\/s10851-016-0646-8","volume":"56","author":"M Schmidt","year":"2016","unstructured":"Schmidt, M., Weickert, J.: Morphological counterparts of linear shift-invariant scale-spaces. J. Math. Imaging Vis. 56(2), 352\u2013366 (2016)","journal-title":"J. Math. Imaging Vis."},{"issue":"11","key":"1147_CR51","doi-asserted-by":"publisher","first-page":"1101","DOI":"10.1109\/34.334389","volume":"16","author":"R van den Boomgaard","year":"1994","unstructured":"van den Boomgaard, R., Smeulders, A.: The morphological structure of images: the differential equations of morphological scale-space. IEEE Trans. Pattern Anal. Mach. Intell. 16(11), 1101\u20131113 (1994). https:\/\/doi.org\/10.1109\/34.334389","journal-title":"IEEE Trans. Pattern Anal. Mach. Intell."},{"key":"1147_CR52","volume-title":"Partial Differential Equations","author":"LC Evans","year":"2010","unstructured":"Evans, L.C.: Partial Differential Equations, vol. 19. American Mathematical Society, Providence (2010)"},{"key":"1147_CR53","doi-asserted-by":"crossref","unstructured":"Diop, E.H.S., Mbengue, A., Manga, B., Seck, D.: Extension of mathematical morphology in Riemannian spaces. In: Scale Space and Variational Methods in Computer Vision, pp. 100\u2013111. Springer, Cham (2021)","DOI":"10.1007\/978-3-030-75549-2_9"},{"issue":"1\u20132","key":"1147_CR54","doi-asserted-by":"publisher","first-page":"1","DOI":"10.1007\/s00030-007-2047-6","volume":"14","author":"A Fathi","year":"2007","unstructured":"Fathi, A., Maderna, E.: Weak KAM theorem on non compact manifolds. Nonlinear Differ. Equ. Appl. NoDEA 14(1\u20132), 1\u201327 (2007). https:\/\/doi.org\/10.1007\/s00030-007-2047-6","journal-title":"Nonlinear Differ. Equ. Appl. NoDEA"},{"issue":"2","key":"1147_CR55","doi-asserted-by":"publisher","first-page":"304","DOI":"10.1016\/j.jfa.2004.10.008","volume":"220","author":"D Azagra","year":"2005","unstructured":"Azagra, D., Ferrera, J., L\u00f3pez-Mesas, F.: Nonsmooth analysis and Hamilton\u2013Jacobi equations on Riemannian manifolds. J. Funct. Anal. 220(2), 304\u2013361 (2005)","journal-title":"J. Funct. Anal."},{"key":"1147_CR56","unstructured":"Lupi, G.: Kernel approximations in lie groups and application to group-invariant CNN. Master thesis, University of Bologna (2021)"},{"issue":"1","key":"1147_CR57","doi-asserted-by":"publisher","first-page":"88","DOI":"10.1006\/jfan.1998.3259","volume":"157","author":"AFM ter Elst","year":"1998","unstructured":"ter Elst, A.F.M., Robinson, D.W.: Weighted subcoercive operators on Lie groups. J. Funct. Anal. 157(1), 88\u2013163 (1998). https:\/\/doi.org\/10.1006\/jfan.1998.3259","journal-title":"J. Funct. Anal."},{"issue":"1","key":"1147_CR58","doi-asserted-by":"publisher","first-page":"1","DOI":"10.2200\/S00049ED1V01Y200609IVM008","volume":"2","author":"P Mordohai","year":"2006","unstructured":"Mordohai, P., Medioni, G.: Tensor voting: a perceptual organization approach to computer vision and machine learning. Synth. Lect. Image Video Multimed. Process. 2(1), 1\u2013136 (2006). https:\/\/doi.org\/10.2200\/S00049ED1V01Y200609IVM008","journal-title":"Synth. Lect. Image Video Multimed. Process."},{"key":"1147_CR59","doi-asserted-by":"publisher","DOI":"10.3390\/app9245507","author":"F Cervantes-Sanchez","year":"2019","unstructured":"Cervantes-Sanchez, F., Cruz-Aceves, I., Hernandez-Aguirre, A., Hernandez-Gonzalez, M.A., Solorio-Meza, S.E.: Automatic segmentation of coronary arteries in X-ray angiograms using multiscale analysis and artificial neural networks. Appl. Sci. (2019). https:\/\/doi.org\/10.3390\/app9245507","journal-title":"Appl. Sci."},{"key":"1147_CR60","first-page":"27","volume":"12679","author":"R Duits","year":"2021","unstructured":"Duits, R., Smets, B.M.N., Bekkers, E.J., Portegies, J.W.: Equivariant deep learning via morphological and linear scale space PDEs on the space of positions and orientations. LNCS 12679, 27\u201339 (2021)","journal-title":"LNCS"},{"issue":"6","key":"1147_CR61","doi-asserted-by":"publisher","first-page":"900","DOI":"10.1007\/s10851-018-0803-3","volume":"60","author":"E Baspinar","year":"2018","unstructured":"Baspinar, E., Citti, G., Sarti, A.: A geometric model of multi-scale orientation preference maps via Gabor functions. J. Math. Imaging Vis. 60(6), 900\u2013912 (2018)","journal-title":"J. Math. Imaging Vis."}],"container-title":["Journal of Mathematical Imaging and Vision"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-023-01147-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/article\/10.1007\/s10851-023-01147-w\/fulltext.html","content-type":"text\/html","content-version":"vor","intended-application":"text-mining"},{"URL":"https:\/\/link.springer.com\/content\/pdf\/10.1007\/s10851-023-01147-w.pdf","content-type":"application\/pdf","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2023,10,20]],"date-time":"2023-10-20T07:13:32Z","timestamp":1697786012000},"score":1,"resource":{"primary":{"URL":"https:\/\/link.springer.com\/10.1007\/s10851-023-01147-w"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2023,4,16]]},"references-count":61,"journal-issue":{"issue":"6","published-print":{"date-parts":[[2023,12]]}},"alternative-id":["1147"],"URL":"https:\/\/doi.org\/10.1007\/s10851-023-01147-w","relation":{"has-preprint":[{"id-type":"doi","id":"10.21203\/rs.3.rs-2191367\/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":[[2023,4,16]]},"assertion":[{"value":"21 October 2022","order":1,"name":"received","label":"Received","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"18 March 2023","order":2,"name":"accepted","label":"Accepted","group":{"name":"ArticleHistory","label":"Article History"}},{"value":"16 April 2023","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"}}]}}