Abstract
In this paper, we look at the fundamental problem of object matching in computational anatomy. We present a new framework for warping pairs of overlapping and non-overlapping shapes, open curves, and landmarks based on the level set approach. When implemented in 3-D, the same framework could be used to warp 3-D objects with minimal modification. Our approach is to use the level set functions to represent the objects to be matched. Using this representation, the problem becomes an energy minimization problem. Cost functions for warping overlapping, non-overlapping, open curves, and landmarks are proposed. Euler-Lagrange equations are applied and gradient descent is used to solve the corresponding partial differential equations. Moreover, a general framework for linking the level set approach and the infinite dimensional group actions is discussed.
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References
Grenander, U., Miller, M.I.: Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics 56, 617–694 (1998)
Thompson, P., Toga, A.W.: A framework for computational anatomy. Computing and Visualization in Science 5, 13–34 (2002)
Bajcsy, R., Kovacic, S.: Multiresolution Elastic Matching. Computer Vision Graphics and Image Processing 46, 1–21 (1989)
Christensen, G.E., Rabbitt, R.D., Miller, M.I.: Deformable templates using large deformation kinematics. IEEE Transactions on Image Processing 5, 1435–1447 (1996)
Christensen, G.E., Rabbitt, R.D., Miller, M.I.: 3D brain mapping using a deformable neuroanatomy (1994)
Christensen, G.E., Joshi, S.C., Miller, M.I.: Volumetric transformation of brain anatomy. IEEE Transactions on Medical Imaging 16, 864–877 (1997)
Miller, M.I., Trouve, A., Younes, L.: On the metrics and Euler-Lagrange equations of computational anatomy. Annual Review of Biomedical Engineering 4, 375–405 (2002)
Dupuis, P., Grenander, U., Miller, M.I.: Variational problems on flows of diffeomorphisms for image matching. Quarterly of Applied Mathematics 56, 587–600 (1998)
Mumford, D.: Pattern Theory: the Mathematics of Perception. In: ICM 2002, vol. 3 (2002)
Osher, S., Sethian, J.A.: Fronts Propagating with Curvature-Dependent Speed – Algorithms Based on Hamilton-Jacobi Formulations. Journal of Computational Physics 79, 12–49 (1988)
Osher, S., Fedkiw, R.P.: Level set methods: An overview and some recent results. Journal of Computational Physics 169, 463–502 (2001)
Whitaker, R.T.: A level-set approach to image blending. IEEE Transactions on Image Processing 9, 1849–1861 (2000)
Paragios, N., Rousson, M., Ramesh, V.: Matching Distance Functions: A Shape-to-Area Variational Approach for Global-to-Local Registration. In: Heyden, A., Sparr, G., Nielsen, M., Johansen, P. (eds.) ECCV 2002. LNCS, vol. 2351, pp. 775–789. Springer, Heidelberg (2002)
Smereka, P.: Spiral crystal growth. Physica D 138, 282–301 (2000)
Liao, W.H., Jhuu, A., Bergsneider, M., Vese, L.A., Huang, S.C., Osher, S.: From Landmark Matching to Shape and Open Curve Matching: A Level Set Approach. In: UCLA CAM report (2002)
Liao, W.H.: Mathematical techniques in object matching and computational anatomy: a new framework based on the level set method. In: Biomathematics. UCLA, Los Angeles (2003)
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Liao, WH., Vese, L., Huang, SC., Bergsneider, M., Osher, S. (2003). Computational Anatomy and Implicit Object Representation: A Level Set Approach. In: Gee, J.C., Maintz, J.B.A., Vannier, M.W. (eds) Biomedical Image Registration. WBIR 2003. Lecture Notes in Computer Science, vol 2717. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-39701-4_5
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DOI: https://doi.org/10.1007/978-3-540-39701-4_5
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