Abstract
We seek a form of object model that exactly and completely captures the interior of most non-branching anatomic objects and simultaneously is well suited for probabilistic analysis on populations of such objects. We show that certain nearly medial, skeletal models satisfy these requirements. These models are first mathematically defined in continuous three-space, and then discrete representations formed by a tuple of spoke vectors are derived. We describe means of fitting these skeletal models into manual or automatic segmentations of objects in a way stable enough to support statistical analysis, and we sketch means of modifying these fits to provide good correspondences of spoke vectors across a training population of objects. Understanding will be developed that these discrete skeletal models live in an abstract space made of a Cartesian product of a Euclidean space and a collection of spherical spaces. Based on this understanding and the way objects change under various rigid and nonrigid transformations, a method analogous to principal component analysis called composite principal nested spheres will be seen to apply to learning a more efficient collection of modes of object variation about a new and more representative mean object than those provided by other representations and other statistical analysis methods. The methods are illustrated by application to hippocampi.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Arsigny, V., Commowick, O., Pennec, X., Ayache, N.: A log-Euclidean framework for statistics on diffeomorphisms. In: Medical Image Computing and Computer-Assisted Intervention, vol. 9, pp. 924–931. Springer, Berlin/Heidelberg (2006)
Blum, H.: A transformation for extracting new descriptors of shape. In: Wathen-Dunn, W. (ed.) Models for the Perception of Speech and Visual Form. MIT, Cambridge, MA (1967)
Cates, J., Fletcher, P.T., Styner, M.E., Shenton, R.T.W.: Shape modeling and analysis with entropy-based particle systems. Inf. Process. Med. Imaging 20, 333–345 (2007)
Cootes, T.F., Taylor, C., Cooper, D., Graham, J.: Training models of shape from sets of examples. In: Hogg, D., Boyle, R. (eds.) Proceedings of British Machine Vision Conference, pp. 9–18. Springer, Berlin (1992)
Cootes, T.F., Twining, C.J., Petrović, V.S., Babalola, K.O., Taylor, C.J.: Computing accurate correspondences across groups of images. IEEE Trans. Pattern Anal. Mach. Intell. 32, 1994–2005 (2010)
Damon, J.: Smoothness and geometry of boundaries associated to skeletal structures I: sufficient conditions for smoothness. Ann. Inst. Fourier 53, 1001–1045 (2003)
Damon, J.: Swept regions and surfaces: modeling and volumetric properties. Conf. Computational Alg. Geom. 2006, in honor of Andre Galligo. Spl. Issue Theor. Comp. Sci. 392, 66–91 (2008)
Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. Wiley, Chichester (1998)
Fletcher, P.T., Lu, C., Pizer, S.M., Joshi, S.: Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Trans. Med. Imaging 23, 995–1005 (2004)
Fletcher, P.T., Venkatasubramanian, S., Joshi, S.: The geometric median on Riemmanian manifolds with application to robust atlas estimation. NeuroImage 45(1), S143–S152 (2009)
Han, Q., Pizer, S.M., Damon, J.N.: Interpolation in discrete single figure medial objects. In: Computer Vision and Pattern Recognition (CVPR) – Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA). IEEE Press (2006)
Huckemann, S., Ziezold, H.: Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Adv. Appl. Probab. 38(2), 299–319 (2006)
Huckemann, S., Hotz, T., Munk, A.: Intrinsic shape analysis: geodesic PCA for Riemannian manifolds modulo isometric lie group actions. Stat. Sin. 20(1), 1–58 (2010)
Jung, S., Liu, X., Marron, J.S., Pizer, S.M.: Generalized PCA via the backward stepwise approach in image analysis. In: Angeles, J. (ed.) Brain, Body and Machine: Proceedings of an International Symposium on the 25th Anniversary of McGill University Centre for Intelligent Machines. Advances in Intelligent and Soft Computing, vol. 83, pp. 111–123. Springer, Berlin/Heidelberg (2010)
Jung, S., Dryden, I.L., Marron, J.S.: Analysis of principal nested spheres. Biometrika. 99(3), 551–568 (2012)
Kendall, D.G., Barden, D., Carne, T.K., Le, H.: Shape and Shape Theory. Wiley, Chichester (1999)
Kurtek, S., Ding, Z., Klassen, E., Srivastava, A.: Parameterization-invariant shape statistics and probabilistic classification of anatomical surfaces. Inf. Process. Med. Imaging 22, 147–158 (2011)
Leventon, M., Faugeras, O., Grimson, E., Kikinis, R., Wells, W.: Knowledge-based segmentation of medical images. In: Osher, S., Paragios, N. (eds.) Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer, New York (2003)
Merck, D., Tracton, G., Saboo, R., Levy, J., Chaney, E., Pizer, S.M., Joshi, S.: Training models of anatomic shape variability. Med. Phys. 35, 3584–3596 (2008)
Oguz, I., Cates, J., Fletcher, T., Whitaker, R., Cool, D., Aylward, S., Styner, M.: Entropy-based particle systems and local features for cortical correspondence optimization. In: Proceedings of the IEEE International Symposium on Biomedical Imaging (ISBI), pp. 1637–1641. IEEE Press (2008)
Pennec, X.: Statistical computing on manifolds: from Riemannian geometry to computational anatomy. Emerg. Trends Vis. Comput. 5416, 347–386 (2008)
Saboo, R., Niethammer, M., North, M., Pizer, S.M.: Anti-aliasing discretely sampled object boundaries using fourth-order Laplacian of curvature flow. http://midag.cs.unc.edu/bibliography.html (2011). Accessed 3 Mar. 2012
Savadjiev, P., Campbell, J.S.W., Descoteaux, M., Deriche, R., Pike, G., Siddiqi, K.: Labeling of ambiguous sub-voxel fibre bundle configurations in high angular resolution diffusion MRI. NeuroImage 41(1), 58–68 (2008)
Schulz, J., Jung, S., Huckemann, S.: A collection of internal reports submitted to UNC-Göttingen study group on s-rep change under rotational transformations. University of North Carolina at Chapel Hill (2011)
Sen, S.K., Foskey, M., Marron, J.S., Styner, M.A.: Support vector machine for data on manifolds: an application to image analysis. In: Proceedings of the IEEE International Symposium on Biomedical Imaging (ISBI), pp. 1195–1198. IEEE Press (2008)
Shi, X., Styner, M., Lieberman, J., Ibrahim, J.G., Lin, W., Zhu, H.: Intrinsic regression models for manifold-valued data. In: Medical Image Computing and Computer-Assisted Intervention, vol. 5762, pp. 192–199. Springer, Berlin/Heidelberg (2009)
Siddiqi, K., Pizer, S.: Medial Representations: Mathematics, Algorithms and Applications. Springer, Dordrecht (2008)
Sorensen, P., Lo, J., Petersen, A., Dirksen, A., de Bruijne, M.: Dissimilarity-based classification of anatomical tree structures. Inf. Process. Med. Imaging 22, 475–485 (2011)
Terriberry, T., Joshi, S., Gerig, G.: Hypothesis testing with nonlinear shape models. Inf. Process. Med. Imaging 19, 15–26 (2005)
Wang, H., Marron, J.S.: Object oriented data analysis: sets of trees. Ann. Stat. 35(5), 1849–1873 (2007)
Yang, J., Staib, L., Duncan, J.: Neighbor-constrained segmentation with 3D deformable models. Inf. Process. Med. Imaging 18, 198–209 (2003)
Acknowledgements
We thank Jörn Schulz for studying the effects of global rotation, folding, and twisting on skeletal spoke vectors; Martin Styner for the hippocampus data; and Anna Snyder for help with the references.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this chapter
Cite this chapter
Pizer, S.M. et al. (2013). Nested Sphere Statistics of Skeletal Models. In: Breuß, M., Bruckstein, A., Maragos, P. (eds) Innovations for Shape Analysis. Mathematics and Visualization. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34141-0_5
Download citation
DOI: https://doi.org/10.1007/978-3-642-34141-0_5
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-34140-3
Online ISBN: 978-3-642-34141-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)