Abstract
Geodesic regression generalizes linear regression to general Riemannian manifolds. Applied to images, it allows for a compact approximation of an image time-series through an initial image and an initial momentum. Geodesic regression requires the definition of a squared residual (squared distance) between the regression geodesic and the measurement images. In principle, this squared distance should also be defined through a geodesic connecting an image on the regression geodesic to its respective measurement. However, in practice only standard registration distances (such as sum of squared distances) are used, to reduce computation time. This paper describes a simplified geodesic regression method which approximates the registration-based distances with respect to a fixed initial image. This results in dramatically simplified computations. In particular, the method becomes straightforward to implement using readily available large displacement diffeomorphic metric mapping (LDDMM) shooting algorithms and decouples the problem into pairwise image registrations allowing parallel computations. We evaluate the approach using 2D synthetic images and real 3D brain images.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Beg, M.F., Miller, M.I., Trouvé, A., Younes, L.: Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision 61(2), 139–157 (2005)
Durrleman, S., Pennec, X., Trouvé, A., Gerig, G., Ayache, N.: Spatiotemporal Atlas Estimation for Developmental Delay Detection in Longitudinal Datasets. In: Yang, G.-Z., Hawkes, D., Rueckert, D., Noble, A., Taylor, C. (eds.) MICCAI 2009, Part I. LNCS, vol. 5761, pp. 297–304. Springer, Heidelberg (2009)
Niethammer, M., Hart, G., Zach, C.: An optimal control approach for the registration of image time series. In: Conference on Decision and Control (CDC), pp. 2427–2434. IEEE (2009)
Trouvé, A., Vialard, F.X.: Shape splines and stochastic shape evolutions: A second order point of view. Arxiv preprint arXiv:1003.3895 (2010)
Fishbaugh, J., Durrleman, S., Gerig, G.: Estimation of Smooth Growth Trajectories with Controlled Acceleration from Time Series Shape Data. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 401–408. Springer, Heidelberg (2011)
Davis, B.C., Fletcher, P.T., Bullitt, E., Joshi, S.: Population Shape Regression from Random Design Data. In: 11th IEEE ICCV (2007)
Niethammer, M., Huang, Y., Vialard, F.-X.: Geodesic Regression for Image Time-Series. In: Fichtinger, G., Martel, A., Peters, T. (eds.) MICCAI 2011, Part II. LNCS, vol. 6892, pp. 655–662. Springer, Heidelberg (2011)
Fletcher, P.T.: Geodesic regression on Riemannian manifolds. In: Proceedings of International Workshop on Mathematical Foundations of Computational Anatomy, MFCA (2011)
Yang, X., Goh, A., Qiu, A.: Approximations of the Diffeomorphic Metric and Their Applications in Shape Learning. In: Székely, G., Hahn, H.K. (eds.) IPMI 2011. LNCS, vol. 6801, pp. 257–270. Springer, Heidelberg (2011)
Trouvé, A., Younes, L.: Metamorphoses through Lie group action. Foundations of Computational Mathematics 5(2), 173–198 (2005)
Vialard, F.X., Risser, L., Rueckert, D., Cotter, C.J.: Diffeomorphic 3D image registration via geodesic shooting using an efficient adjoint calculation. International Journal of Computer Vision, 1–13 (2011)
Risser, L., Vialard, F.-X., Wolz, R., Murgasova, M., Holm, D.D., Rueckert, D.: Simultaneous Multi-Scale Registration Using Large Deformation Diffeomorphic Metric Mapping. IEEE Transcations on Medical Imaging 30(10), 1746–1759 (2011)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2012 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Hong, Y., Shi, Y., Styner, M., Sanchez, M., Niethammer, M. (2012). Simple Geodesic Regression for Image Time-Series. In: Dawant, B.M., Christensen, G.E., Fitzpatrick, J.M., Rueckert, D. (eds) Biomedical Image Registration. WBIR 2012. Lecture Notes in Computer Science, vol 7359. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-31340-0_2
Download citation
DOI: https://doi.org/10.1007/978-3-642-31340-0_2
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-31339-4
Online ISBN: 978-3-642-31340-0
eBook Packages: Computer ScienceComputer Science (R0)