nach oben

Erschienen in:

2017 | OriginalPaper | Buchkapitel

Stochastic Development Regression on Non-linear Manifolds

verfasst von : Line Kühnel, Stefan Sommer

Erschienen in: Information Processing in Medical Imaging

Verlag: Springer International Publishing

Einloggen

Aktivieren Sie unsere intelligente Suche, um passende Fachinhalte oder Patente zu finden.

search-config

KI-gestützte Suche

Aus

Abstract

We introduce a regression model for data on non-linear manifolds. The model describes the relation between a set of manifold valued observations, such as shapes of anatomical objects, and Euclidean explanatory variables. The approach is based on stochastic development of Euclidean diffusion processes to the manifold. Defining the data distribution as the transition distribution of the mapped stochastic process, parameters of the model, the non-linear analogue of design matrix and intercept, are found via maximum likelihood. The model is intrinsically related to the geometry encoded in the connection of the manifold. We propose an estimation procedure which applies the Laplace approximation of the likelihood function. A simulation study of the performance of the model is performed and the model is applied to a real dataset of Corpus Callosum shapes.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

über 102.000 Bücher
über 537 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Finance + Banking
Management + Führung
Marketing + Vertrieb
Maschinenbau + Werkstoffe
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 390 Zeitschriften

aus folgenden Fachgebieten:

Automobil + Motoren
Bauwesen + Immobilien
Business IT + Informatik
Elektrotechnik + Elektronik
Energie + Nachhaltigkeit
Maschinenbau + Werkstoffe

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

über 67.000 Bücher
über 340 Zeitschriften

aus folgenden Fachgebieten:

Bauwesen + Immobilien
Business IT + Informatik
Finance + Banking
Management + Führung
Marketing + Vertrieb
Versicherung + Risiko

Jetzt Wissensvorsprung sichern!

Jetzt informieren

Vorheriges Kapitel Conditional Local Distance Correlation for Manifold-Valued Data

Nächstes Kapitel Spectral Kernels for Probabilistic Analysis and Clustering of Shapes

Aswani, A., Bickel, P., Tomlin, C.: Regression on manifolds: estimation of the exterior derivative. Ann. Stat. 39(1), 48–81 (2011). arXiv:1103.1457

Banerjee, M., Chakraborty, R., Ofori, E., Okun, M.S., Vaillancourt, D.E., Vemuri, B.C.: A nonlinear regression technique for manifold valued data with applications to medical image analysis. In: 2016 IEEE Conference on CVPR, pp. 4424–4432, June 2016

Banerjee, M., Chakraborty, R., Ofori, E., Vaillancourt, D., Vemuri, B.C.: Nonlinear regression on Riemannian manifolds and its applications to neuro-image analysis. In: Navab, N., Hornegger, J., Wells, W.M., Frangi, A.F. (eds.) MICCAI 2015. LNCS, vol. 9349, pp. 719–727. Springer, Cham (2015). doi:10.1007/978-3-319-24553-9_88 CrossRef

Cheng, M., Wu, H.: Local linear regression on manifolds and its geometric interpretation. J. Am. Stat. Assoc. 108(504), 1421–1434 (2013)MathSciNetCrossRefMATH

Cornea, E., Zhu, H., Kim, P., Ibrahim, J.G., The Alzheimer’s Disease Neuroimaging Initiative: Regression models on Riemannian symmetric spaces. J. Roy. Stat. Soc. B 79, 463–482 (2017)

Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population shape regression from random design data. In: 2007 IEEE 11th International Conference on Computer Vision, pp. 1–7, October 2007

Thomas Fletcher, P.: Geodesic regression and the theory of least squares on riemannian manifolds. Int. J. Comput. Vis. 105(2), 171–185 (2012)MathSciNetCrossRef

Hinkle, J., Muralidharan, P., Fletcher, P.T., Joshi, S.: Polynomial regression on Riemannian manifolds. arXiv:1201.2395, January 2012

Hong, Y., Kwitt, R., Singh, N., Vasconcelos, N., Niethammer, M.: Parametric regression on the Grassmannian. IEEE Trans. Pattern Anal. Mach. Intell. 38(11), 2284–2297 (2016)CrossRef

10.

Hsu, E.P.: Stochastic Analysis on Manifolds. American Mathematical Soc., Providence (2002)CrossRefMATH

11.

Kass, R.E., Steffey, D.: Approximate Bayesian inference in conditionally independent hierarchical models (Parametric Empirical Bayes Models). J. Am. Stat. Assoc. 84(407), 717–726 (1989)MathSciNetCrossRef

12.

Lin, L., St Thomas, B., Zhu, H., Dunson, D.B.: Extrinsic local regression on manifold-valued data. arXiv:1508.02201, August 2015

13.

Loubes, J.-M., Pelletier, B.: A kernel-based classifier on a Riemannian manifold. Stat. Decis. Int. Math. J. Stoch. Methods Models 26(1), 35–51 (2009)MathSciNetMATH

14.

Pennec, X.: Intrinsic statistics on Riemannian manifolds: basic tools for geometric measurements. J. Math. Imaging Vis. 25, 127 (2006)MathSciNetCrossRef

15.

Shi, X., Styner, M., Lieberman, J., Ibrahim, J.G., Lin, W., Zhu, H.: Intrinsic regression models for manifold-valued data. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009. LNCS, vol. 5762, pp. 192–199. Springer, Heidelberg (2009). doi:10.1007/978-3-642-04271-3_24 CrossRef

16.

Singh, N., Hinkle, J., Joshi, S., Fletcher, P.T.: Hierarchical geodesic models in diffeomorphisms. Int. J. Comput. Vis. 117, 70–92 (2016)MathSciNetCrossRef

17.

Singh, N., Vialard, F.-X., Niethammer, M.: Splines for diffeomorphisms. Med. Image Anal. 25(1), 56–71 (2015)CrossRef

18.

Sommer, S.H., Svane, A.M.: Modelling anisotropic covariance using stochastic development and sub-Riemannian frame bundle geometry. J. Geom. Mech. (2016). ISSN: 1941-4889. American Institute of Mathematical Sciences

19.

Sommer, S.: Anisotropic distributions on manifolds: template estimation and most probable paths. In: Ourselin, S., Alexander, D.C., Westin, C.-F., Cardoso, M.J. (eds.) IPMI 2015. LNCS, vol. 9123, pp. 193–204. Springer, Cham (2015). doi:10.1007/978-3-319-19992-4_15 CrossRef

20.

Sommer, S.: Anisotropically weighted and nonholonomically constrained evolutions on manifolds. Entropy 18(12), 425 (2016)MathSciNetCrossRef

21.

Steinke, F., Hein, M.: Non-parametric regression between manifolds. In: Advances in Neural Information Processing Systems 21, pp. 1561–1568. Curran Associates, Inc. (2009)

22.

Theano Development Team. Theano: a Python framework for fast computation of mathematical expressions. arXiv e-prints, abs/1605.02688, May 2016

23.

Younes, L., Arrate, F., Miller, M.I.: Evolutions equations in computational anatomy. NeuroImage 45(1 Suppl.), S40–S50 (2009)CrossRef

24.

Younes, L.: Shapes and Diffeomorphisms. Springer, Heidelberg (2010)CrossRefMATH

25.

Yuan, Y., Zhu, H., Lin, W., Marron, J.S.: Local polynomial regression for symmetric positive definite matrices. J. Roy. Stat. Soc. Ser. B Stat. Methodol. 74(4), 697–719 (2012)MathSciNetCrossRef

Titel: Stochastic Development Regression on Non-linear Manifolds
verfasst von: Line Kühnel
Stefan Sommer
Verlag: Springer International Publishing
Buch: Information Processing in Medical Imaging
Print ISBN: 978-3-319-59049-3

Electronic ISBN: 978-3-319-59050-9

Copyright-Jahr: 2017
DOI: https://doi.org/10.1007/978-3-319-59050-9_5

Springer Professional

Abstract

Bitte loggen Sie sich ein, um Zugang zu Ihrer Lizenz zu erhalten.

Sie haben noch keine Lizenz? Dann Informieren Sie sich jetzt über unsere Produkte:

Springer Professional "Wirtschaft+Technik"

Springer Professional "Technik"

Springer Professional "Wirtschaft"