Abstract
This paper presents a new shape representation for a special class of 3-D objects. In a generative approach to object modeling inspired by m-reps [15], skeletons of objects are explicitly defined as continuous manifolds and boundaries are derived from the skeleton by a process that involves solving a Poisson PDE with a non-linear boundary condition. This formulation helps satisfy the equality constraints that are imposed on the parameters of the representation by rules of medial geometry. One benefit of the new approach is the ability to represent different instances of an anatomical structure using a common parametrization domain, simplifying the problem of computing correspondences between instances. Another benefit is the ability to continuously parameterize the volumetric region enclosed by the representation’s boundary in a one-to-one and onto manner, in a way that preserves two of the three coordinates of the parametrization along vectors normal to the boundary. These two features make the new representation an attractive candidate for statistical analysis of shape and appearance. In this paper, the representation is carefully defined and the results of fitting the hippocampus in a deformable templates framework are presented.
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
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Blum, H., Nagel, R.N.: Shape description using weighted symmetric axis features. Pattern Recognition 10(3), 167–180 (1978)
Chakos, M.H., Schobel, S.A., Gu, H., Gerig, G., Bradford, D., Charles, C., Lieberman, J.A.: Duration of illness and treatment effects on hippocampal volume in male patients with schizophrenia. Br. J. Psychiatry 186(1), 26–31 (2005)
Cootes, T.F., Taylor, C.J., Cooper, D.H., Graham, J.: Active shape models – their training and application. Computer Vision and Image Understanding 61(1), 38–59 (1995)
Cootes, T.F., Edwards, G.J., Taylor, C.J.: Active appearance models. In: European Conference on Computer Vision, Freiburg, Germany, June 1998, vol. 2, pp. 484–498 (1998)
Csernansky, J., Joshi, S., Wang, L., Haller, J., Gado, M., Miller, J., Grenander, U., Miller, M.: Hippocampal morphometry in schizophrenia via high dimensional brain mapping. Proc. National Academy of Sciences 95, 11406–11411 (1998)
Damon, J.: Determining the geometry of boundaries of objects from medial data. International Journal of Computer Vision 63(1), 45–64 (2005) (in print)
Gerig, G., Styner, M., Shenton, M.E., Lieberman, J.: Shape versus size: Improved understanding of the morphology of brain structures. In: Niessen, W., Viergever, M. (eds.) Medical Image Computing and Computer-Assisted Intervention (MICCAI), New York, October 2001, vol. 2208, pp. 24–32. Springer, Heidelberg (2001)
Golland, P., Grimson, W.E.L., Kikinis, R.: Statistical shape analysis using fixed topology skeletons: Corpus callosum study. In: Kuba, A., Sámal, M., Todd-Pokropek, A. (eds.) IPMI 1999. LNCS, vol. 1613, pp. 382–388. Springer, Heidelberg (1999)
Haller, J.W., Banerjee, A., Christensen, G.E., Gado, M., Joshi, S., Miller, M.I., Sheline, Y.I., Vannier, M.W., Csernansky, J.G.: Three-dimensional hippocampal MR morphometry by high-dimensional transformation of a neuroanatomic atlas. Radiology 202, 504–510 (1997)
Ho, S., Gerig, G.: Profile scale-spaces for multiscale image match. In: Barillot, C., Haynor, D.R., Hellier, P. (eds.) MICCAI 2004. LNCS, vol. 3216, pp. 176–183. Springer, Heidelberg (2004)
Joshi, S., Grenander, U., Miller, M.: On the geometry and shape of brain sub-manifolds. IEEE Transactions on Pattern Analysis and Machine Intelligence 11, 1317–1343 (1997)
Joshi, S., Pizer, S., Fletcher, P.T., Yushkevich, P., Thall, A., Marron, J.S.: Multiscale deformable model segmentation and statistical shape analysis using medial descriptions. IEEE Transactions on Medical Imaging 21(5), 538–550 (2002)
Kreyszig, E.: Differential Geometry. University of Toronto Press (1959)
Kimia, B.B., Giblin, P.J.: On the intrinsic reconstruction of shape from its symmetries. IEEE PAMI 25(7), 895–911 (2003)
Pizer, S.M., Fletcher, P.T., Joshi, S., Thall, A., Chen, J.Z., Fridman, Y., Fritsch, D.S., Gash, A.G., Glotzer, J.M., Jiroutek, M.R., Lu, C., Muller, K.E., Tracton, G., Yushkevich, P., Chaney, E.L.: Deformable m-reps for 3D medical image segmentation. International Journal of Computer Vision 55(2), 85–106 (2003)
Pizer, S.M., Siddiqi, K., Székely, G., Damon, J.N., Zucker, S.W.: Multiscale medial loci and their properties. International Journal of Computer Vision 55(2- 3), 155–179 (2003)
Schenk, O., Gärtner, K.: Solving unsymmetric sparse systems of linear equations with PARDISO. Journal of Future Generation Computer Systems 20(3), 475–487 (2004)
Smith, G.D.: Numerical Solution of Partial Differential Equations: Finite Difference Methods. Oxford University Press, Oxford (1985)
Staib, L.H., Duncan, J.S.: Boundary finding with parametrically deformable models. IEEE PAMI 14(11), 1061–1075 (1992)
Styner, M., Gerig, G., Joshi, S., Pizer, S.M.: Automatic and robust computation of 3D medial models incorporating object variability. International Journal of Computer Vision 55(2), 107–122 (2003)
Thall, A.: Deformable Solid Modeling via Medial Sampling and Displacement Subdivision. PhD thesis, Dept. of Comp. Sci. UNC Chapel Hill (2004)
Yushkevich, P., Fletcher, P.T., Joshi, S., Thall, A., Pizer, S.M.: Continuous medial representations for geometric object modeling in 2D and 3D. Image and Vision Computing 21(1), 17–28 (2003)
Yushkevich, P., Pizer, S.M., Joshi, S., Marron, J.S.: Intuitive, localized analysis of shape variability. In: International Conference on Information Processing in Medical Imaging, Berlin, Germany, pp. 402–408. Springer, Heidelberg (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Yushkevich, P.A., Zhang, H., Gee, J.C. (2005). Parametric Medial Shape Representation in 3-D via the Poisson Partial Differential Equation with Non-linear Boundary Conditions. In: Christensen, G.E., Sonka, M. (eds) Information Processing in Medical Imaging. IPMI 2005. Lecture Notes in Computer Science, vol 3565. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11505730_14
Download citation
DOI: https://doi.org/10.1007/11505730_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-26545-0
Online ISBN: 978-3-540-31676-3
eBook Packages: Computer ScienceComputer Science (R0)