Abstract
A crucial problem in statistical shape analysis is establishing the correspondence of shape features across a population. While many solutions are easy to express using boundary representations, this has been a considerable challenge for medial representations. This paper uses a new 3-D medial model that allows continuous interpolation of the medial manifold and provides a map back and forth between it and the boundary. A measure defined on the medial surface then allows one to write integrals over the boundary and the object interior in medial coordinates, enabling the expression of important object properties in an object-relative coordinate system. We use these integrals to optimize correspondence during model construction, reducing variability due to the model parameterization that could potentially mask true shape change effects. Discrimination and hypothesis testing of populations of shapes are expected to benefit, potentially resulting in improved significance of shape differences between populations even with a smaller sample size.
Preview
Unable to display preview. Download preview PDF.
Similar content being viewed by others
References
Kotcheff, A.C.W., Taylor, C.J.: Automatic construction of eigenshape models by direct optimization. Medical Image Analysis (MEDIA) 2(4), 303–314 (1998)
Davies, R.H., Twining, C.J., Cootes, T.F., Waterton, J.C., Taylor, C.J.: A minimum description length approach to statistical shape modeling. IEEE Transactions on Medical Imaging 21(5), 525–537 (2002)
Heimann, T., Wolf, I., Williams, T., Meinzer, H.P.: 3D active shape models using gradient descent optimization of description length. In: Christensen, G.E., Sonka, M. (eds.) IPMI 2005. LNCS, vol. 3565, pp. 566–577. Springer, Heidelberg (2005)
Cates, J., Meyer, M., Fletcher, P.T., Whitaker, R.T.: Entropy-based particle systems for shape correspondence. In: Proc. of the International Workshop on Mathematical Foundations of Computational Anatomy (MFCA’06), Copenhagen, Denmark, (October 2006) pp. 90–99 (2006)
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)
Näf, M., Kübler, O., Kikinis, R., Shenton, M.E., Székely, G.: Shape characterization and recognition of 3D organ shape in medical image analysis using skeletonization. In: Proc. of the IEEE/SIAM Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA’96), San Francisco, California, (June 1996) pp. 139–150 (1996)
Zhang, J., Siddiqi, K., Macrini, D., Shokoufandeh, A., Dickinson, S.: Retrieving articulated 3-D models using medial surfaces and their graph spectra. In: Proc. of the 5th International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition, St. Augustine, Florida, November 2005 (2005)
Malandain, G., Ayache, N., Bertrand, G.: Topological segmentation of discrete surfaces. In: Proc. of IEEE CVPR’91, Maui, Hawaii, (June 1991), pp. 444–449 (1991)
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. 382–387. Springer, Heidelberg (1999)
Golland, P., Grimson, W.E.L.: Fixed topology skeletons. In: Proc. of IEEE CVPR’00, vol. 1, pp. 10–17. IEEE Computer Society Press, Los Alamitos (2000)
Pizer, S.M., Fritsch, D.S., Yushkevich, P.A., Johnson, V.E., Chaney, E.L.: Segmentation, registration, and measurement of shape variation via image object shape. IEEE Transactions on Medical Imaging 18, 851–865 (1999)
Joshi, S.C., Pizer, S.M., Fletcher, P.T., Thall, A., Tracton, G.S.: Multi-scale 3-D deformable model segmentation based on medial description. In: Insana, M.F., Leahy, R.M. (eds.) IPMI 2001. LNCS, vol. 2082, pp. 64–77. Springer, Heidelberg (2001)
Bouix, S., Pruessner, J.C., Collins, D.L., Siddiqi, K.: Hippocampal shape analysis using medial surfaces. NeuroImage 25(4), 1077–1089 (2005)
Styner, M.A., Gerig, G., Joshi, S.C., Pizer, S.M.: Automatic and robust computation of 3D medial models incorporating object variability. International Journal of Computer Vision 55(2–3), 107–122 (2003)
Giblin, P.J., Kimia, B.B.: A formal classification of 3D medial axis points and their local geometry. IEEE PAMI 26(2), 238–251 (2004)
Damon, J.N.: Smoothness and geometry of boundaries associated to skeletal structures I: Sufficient conditions for smoothness. Annales de l’Institut Fourier 53(6), 1941–1985 (2003)
Damon, J.N.: Smoothness and geometry of boundaries associated to skeletal structures II: Geometry in the Blum case. Compositio Mathematica 140(6), 1657–1674 (2004)
Thall, A.: Deformable Solid Modeling via Medial Sampling and Displacement Subdivision. PhD thesis, University of North Carolina at Chapel Hill (2004)
Han, Q., Pizer, S.M., Damon, J.N.: Interpolation in discrete single figure medial objects. In: Proc. of the IEEE Workshop on Mathematical Methods in Biomedical Image Analysis (MMBIA’06), New York City, New York June 2006 (2006)
Yushkevich, P.A.: Statistical shape characterization using the medial representation. PhD thesis, University of North Carolina at Chapel Hill Department of Computer Science (2003)
Yushkevich, P.A., Zhang, H., Gee, J.C.: 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.) IPMI 2005. LNCS, vol. 3565, pp. 162–173. Springer, Heidelberg (July 2005)
Yushkevich, P.A., Zhang, H., Gee, J.C.: Continuous medial representation for anatomical structures. IEEE TMI 25(12), 1–18 (2006)
Terriberry, T.B., Gerig, G.: A continuous 3-D medial shape model with branching. In: Proc. of the International Workshop on Mathematical Foundations of Computational Anatomy (MFCA’06), Copenhagen, Denmark, (October 2006), pp. 80–89 (2006)
Terriberry, T.B.: Continuous Medial Models in Two-Sample Statistics of Shape. PhD thesis, University of North Carolina at Chapel Hill (2006)
Stam, J.: Exact evaluation of Catmull-Clark subdivision surfaces at arbitrary parameter values. In: Zorin, D., Schröder, P. (eds.): Lecture Notes for the SIGGRAPH’99 Course on Subdivision for Modeling and Animation, pp. 89–110 (1999)
Damon, J.N.: Global geometry of regions and boundaries via skeletal and medial integrals (2005) Preprint: http://www.math.unc.edu/Faculty/jndamon/Skel.Str.IV.r.pdf
Joshi, S.C., Davis, B., Jomier, M., Gerig, G.: Unbiased diffeomorphic atlas construction for computational anatomy. NeuroImage; Supplemental issue on Mathematics in Brain Imaging 23(Supp. 1), S151–S160 (2004)
Škrinjar, O., Tagare, H.: Symmetric, transitive, geometric deformation and intensity variation invariant nonrigid image registration. In: Proc. of IEEE ISBI’04, Arlington, Virginia, April 2004, pp. 920–923. IEEE Computer Society Press, Los Alamitos (2004)
Loop, C.: Second order smoothness over extraordinary vertices. In: Proc. of the 2004 Eurographics/ ACM SIGGRAPH Symposium on Geometry Processing, Nice, France, July 2004, pp. 165–174. ACM Press, New York (2004)
Halstead, M., Kass, M., DeRose, T.: Efficient, fair interpolation using Catmull-Clark surfaces. In: Proc. of the 20th Annual SIGGRAPH Conference on Computer Graphics and Interactive Techniques, Anaheim, California, August 1993, pp. 35–44 (1993)
Han, Q,: Personal communication (2006)
Styner, M.A., Gorczowski, K., Fletcher, P.T., Jeong, J.Y., Pizer, S.M., Gerig, G.: Statistics of pose and shape in multi-object complexes using prinipal geodesic analysis. In: Yang, G.-Z., Jiang, T., Shen, D., Gu, L., Yang, J. (eds.) Medical Imaging and Augmented Reality. LNCS, vol. 4091, pp. 1–8. Springer, Heidelberg (August 2006)
Brechbühler, C., Gerig, G., Kübler, O.: Parameterization of closed surfaces for 3-D shape description. Computer Vision, Graphics, and Image Processing: Image Understanding 61(2), 195–170 (1995)
Möller, T., Trumbore, B.: Fast, minimum storage ray/triangle intersection. Journal of Graphics Tools 2(1), 21–28 (1997)
Gottschalk, S., Lin, M.C., Manocha, D.: OBB-tree: A hierarchical structure for rapid interference detection. In: Proc. of the 23rd Annual SIGGRAPH Conference on Computer Graphics and Interactive Techniques, August 1996, pp. 171–180 (1996)
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 2007 Springer Berlin Heidelberg
About this paper
Cite this paper
Terriberry, T.B., Damon, J.N., Pizer, S.M., Joshi, S.C., Gerig, G. (2007). Population-Based Fitting of Medial Shape Models with Correspondence Optimization. In: Karssemeijer, N., Lelieveldt, B. (eds) Information Processing in Medical Imaging. IPMI 2007. Lecture Notes in Computer Science, vol 4584. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-73273-0_58
Download citation
DOI: https://doi.org/10.1007/978-3-540-73273-0_58
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-73272-3
Online ISBN: 978-3-540-73273-0
eBook Packages: Computer ScienceComputer Science (R0)