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Regularized Reconstruction of Shapes with Statistical a priori Knowledge

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Abstract

The reconstruction of geometry or, in particular, the shape of objects is a common issue in image analysis. Starting from a variational formulation of such a problem on a shape manifold we introduce a regularization technique incorporating statistical shape knowledge. The key idea is to consider a Riemannian metric on the shape manifold which reflects the statistics of a given training set. We investigate the properties of the regularization functional and illustrate our technique by applying it to region-based and edge-based segmentation of image data. In contrast to previous works our framework can be considered on arbitrary (finite-dimensional) shape manifolds and allows the use of Riemannian metrics for regularization of a wide class of variational problems in image processing.

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References

  • Boothby, W. M. (1975). Pure and applied mathematics : Vol. 63. An introduction to differentiable manifolds and Riemannian geometry. New York: Academic Press.

    MATH  Google Scholar 

  • Caselles, V., Catté, F., & Dibos, F. (1993). A geometric model for active contours in image processing. Numerische Mathematik, 66, 1–31.

    Article  MATH  MathSciNet  Google Scholar 

  • Caselles, V., Kimmel, R., & Sapiro, G. (1997). Geodesic active contours. International Journal of Computer Vision, 22(1), 61–79.

    Article  MATH  Google Scholar 

  • Chan, T. F., & Vese, L. A. (2001). Active contours without edges. IEEE Transactions on Image Processing, 10(2), 266–277.

    Article  MATH  Google Scholar 

  • Chen, Y., Thiruvenkadam, S., Tagare, H. D., Huang, F., Wilson, D., & Geiser, E. A. (2001). On the incorporation of shape priors into geometric active contours. In 1st IEEE workshop on variational and level set methods in computer vision, July 2001.

  • Chen, Y., Tagare, H. D., Thiruvenkadam, S., Huang, F., Wilson, D., Gopinath, K. S., Briggs, R. W., & Geiser, E. A. (2002). Using prior shapes in geometric active contours in a variational framework. International Journal of Computer Vision, 50(3), 315–328.

    Article  MATH  Google Scholar 

  • Cremers, D., & Schnörr, C. (2003). Statistical shape knowledge in variational motion segmentation. Image and Vision Computing, 21, 77–68.

    Article  Google Scholar 

  • Cremers, D., Tischhäuser, F., Weickert, J., & Schnörr, C. (2002). Diffusion snakes: introducing statistical shape knowledge into the Mumford–Shah functional. International Journal of Computer Vision, 50(3), 295–313.

    Article  MATH  Google Scholar 

  • Cremers, D., Kohlberger, T., & Schnörr, C. (2003a). Shape statistics in kernel space for variational image segmentation. Pattern Recognition, 36(9), 1929–1943.

    Article  MATH  Google Scholar 

  • Cremers, D., Sochen, N., & Schnörr, C. (2003b). Towards recognition-based variational segmentation using shape priors and dynamic labeling. In L. D. Griffin & M. Lillholm (Eds.), Lecture notes in computer science : Vol. 2695. Scale-space 2003 (pp. 388–400). Berlin: Springer.

    Google Scholar 

  • Engl, H. W., Hanke, M., & Neubauer, A. (2000). In Mathematics and its applications : Vol. 375. Regularization of inverse problems. Dordrecht: Kluwer Academic.

    Google Scholar 

  • Fang, W., & Chan, K. L. (2007). Incorporating shape prior into geodesic active contours for detecting partially occluded object. Pattern Recognition, 40(7), 2163–2172.

    Article  MATH  Google Scholar 

  • Fletcher, P. T., & Joshi, S. (2004). Principal geodesic analysis on symmetric spaces: Statistics of diffusion tensors. In M. Sonka, I. A. Kakadiaris, & J. Kybic (Eds.), Lecture notes in computer science : Vol. 3117. Computer vision and mathematical methods in medical and biomedical image analysis, ECCV 2004 workshops (pp. 87–98). Berlin: Springer.

    Google Scholar 

  • Fletcher, P. T., Lu, C., & Joshi, S. (2003). Statistics of shape via principal geodesic analysis on lie groups. In Proceedings of the IEEE computer society conference on computer vision and pattern recognition (Vol. 1, pp. 95–101).

  • Fletcher, P. T., Lu, C., Joshi, S., & Pizer, S. M. (2004). Principal geodesic analysis for the study of nonlinear statistics of shape. IEEE Transactions of Medical Imaging, 23(8), 995–1005.

    Article  Google Scholar 

  • Fritscher, K. D., & Schubert, R. (2006). 3D image segmentation by using statistical deformation models and level sets. International Journal of Computer Assisted Radiology and Surgery, 1(3), 123–135.

    Article  Google Scholar 

  • Fuchs, M., Jüttler, B., Scherzer, O., & Yang, H. (2007). Combined evolution of level sets and B -spline curves for imaging (Technical Report 41, FSP 092). Joint Research Program of Industrial Geometry, January 2007.

  • Gastaud, M., Barlaut, M., & Aubert, G. (2004). Combining shape prior and statistical features for active contour segmentation. IEEE Transactions on Circuits and Systems for Video Technology, 14(5), 726–734.

    Article  Google Scholar 

  • Helgason, S. (1978). In Pure and applied mathematics : Vol. 80. Differential geometry, Lie groups, and symmetric spaces. New York: Academic Press.

    MATH  Google Scholar 

  • Huckemann, S., & Ziezold, H. (2006). Principal component analysis for Riemannian manifolds, with an application to triangular shape spaces. Advances in Applied Probability, 38(2), 299–319.

    Article  MATH  MathSciNet  Google Scholar 

  • Joshi, S., Pizer, S., Fletcher, P. T., Yushkevich, P., Thall, A., & Marron, J. S. (2002). Multiscale deformable model segmentation and statistical shape analysis using medial descriptions. IEEE Transactions on Medical Imaging, 21(5), 538–550.

    Article  Google Scholar 

  • Kass, M., Witkin, A., & Terzopoulos, D. (1988). Snakes active contour models. International Journal of Computer Vision, 1(4), 321–331.

    Article  Google Scholar 

  • Le, H., & Kendall, D. G. (1993). The Riemannian structure of Euclidean shape spaces: a novel environment for statistics. The Annals of Statistics, 21(3), 1225–1271.

    Article  MATH  MathSciNet  Google Scholar 

  • Leventon, M. E., Grimson, W. E. L., & Faugeras, O. (2001). Statistical shape influence in geodesic active contours. In IEEE conference on computer vision and pattern recognition (Vol. 1, pp. 316–323), June 2001.

  • Michor, P. W., & Mumford, D. B. (2006). Riemannian geometries on spaces of plane curves. Journal of the European Mathematical Society, 8(1), 1–48.

    Article  MATH  MathSciNet  Google Scholar 

  • Michor, P. W., & Mumford, D. (2007). An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Applied and Computational Harmonic Analysis, 23(1), 74–113.

    Article  MATH  MathSciNet  Google Scholar 

  • Miller, M. I., Trouvé, A., & Younes, L. (2006). Geodesic shooting for computational anatomy. Journal of Mathematical Imaging and Vision, 24(2), 209–228.

    Article  MathSciNet  Google Scholar 

  • Mumford, D., & Shah, J. (1985). Boundary detection by minimizing functionals. In Proceedings of the IEEE conference on computer vision pattern recognition (pp. 22–26).

  • Mumford, D., & Shah, J. (1989). Optimal approximations by piecewise smooth functions and associated variational problems. Communications on Pure and Applied Mathematics, 42(4), 577–684.

    Article  MATH  MathSciNet  Google Scholar 

  • Peterson, P. (1998). In Graduate texts in mathematics : Vol. 171. Riemannian geometry. Berlin: Springer.

    Google Scholar 

  • Rousson, M., & Paragios, N. (2002). Shape priors for level set representations. In A. Heyden, G. Sparr, M. Nielsen, & P. Johansen (Eds.), Lecture notes in computer science : Vol. 2351. Proceedings of the computer vision—ECCV 2002: 7th European conference on computer vision (Part II, pp. 78–92), Copenhagen, Denmark, 28–31 May 2002. Berlin: Springer.

    Google Scholar 

  • Tsai, A., Yezzi, A., Tempany, C., Tucker, D., Fan, A., Grimson, W. E. L., & Willsky, A. (2003). A shape-based approach to the segmentation of medical imagery using level sets. IEEE Transactions on Medical Imaging, 22(2), 137–154.

    Article  Google Scholar 

  • Vaillant, M., Miller, M. I., Younes, L., & Trouvé, A. (2004). Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23, 161–169.

    Article  Google Scholar 

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Authors and Affiliations

  1. Institute of Computer Science, University of Innsbruck, Technikerstr. 21a/2, Innsbruck, 6020, Austria

    Matthias Fuchs & Otmar Scherzer

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  1. Matthias Fuchs
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  2. Otmar Scherzer
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Correspondence to Matthias Fuchs.

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Fuchs, M., Scherzer, O. Regularized Reconstruction of Shapes with Statistical a priori Knowledge. Int J Comput Vis 79, 119–135 (2008). https://doi.org/10.1007/s11263-007-0103-7

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  • DOI: https://doi.org/10.1007/s11263-007-0103-7

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