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On Shape of Plane Elastic Curves

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Abstract

We study shapes of planar arcs and closed contours modeled on elastic curves obtained by bending, stretching or compressing line segments non-uniformly along their extensions. Shapes are represented as elements of a quotient space of curves obtained by identifying those that differ by shape-preserving transformations. The elastic properties of the curves are encoded in Riemannian metrics on these spaces. Geodesics in shape spaces are used to quantify shape divergence and to develop morphing techniques. The shape spaces and metrics constructed are novel and offer an environment for the study of shape statistics. Elasticity leads to shape correspondences and deformations that are more natural and intuitive than those obtained in several existing models. Applications of shape geodesics to the definition and calculation of mean shapes and to the development of shape clustering techniques are also investigated.

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References

  • Beg, M.F., Miller, M.I., Trouve, A., and Younes, L. 2005. Computing large deformation metric mappings via geodesic flows of diffeomorphisms. International Journal of Computer Vision, 61:139–157.

    Article  Google Scholar 

  • Bookstein, F.L. 1986. Size and shape spaces for landmark data in two dimensions. Statistical Science, 1:181–242.

    Google Scholar 

  • Cohen, I., Ayache, N., and Sulger, P. 1992. Tracking points on deformable objects using curvature information. In Lecture Notes in Computer Science, vol. 588.

  • Cootes, T.F., Taylor, C.J., Cooper, D.H., and Graham, J. 1995. Active shape models: Their training and applications. Computer Vision and Image Understanding, 61:38–59.

    Article  Google Scholar 

  • do Carmo, Manfredo, 1994. Riemannian Geometry. Birkhauser.

  • Dryden, I.L. and Mardia, K.V. 1998. Statistical Shape Analysis. John Wiley & Son.

  • Geiger, D., Gupta, A., Costa, L.A., and Vlontzos, J. 1995. Dynamic programming for detecting, tracking and matching elastic contours. IEEE Trans. on Pattern Analysis and Machine Intelligence, 17(3):294–302.

    Article  Google Scholar 

  • Grenander, U. 1993. General Pattern Theory. Oxford University Press.

  • Karcher, H. 1977. Riemann center of mass and mollifier smoothing. Comm. Pure Appl. Math., 30:509–541. v

    Google Scholar 

  • Kendall, D.G. 1984. Shape manifolds, Procrustean metrics and complex projective spaces. Bulletin of London Mathematical Society, 16:81–121.

    MathSciNet  Google Scholar 

  • Klassen, E., Srivastava, A., Mio, W., and Joshi, S. 2004. Analysis of planar shapes using geodesic paths on shape manifolds. IEEE Trans. on Pattern Analysis and Machine Intelligence, 26:372–383.

    Article  Google Scholar 

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

    MathSciNet  Google Scholar 

  • Mennucci, A. and Yezzi, A. 2004. Metrics in the space of curves. Technical report.

  • Michor, P. and Mumford, D. Riemannian geometries on spaces of plane curves. J. Eur. Math. Soc. in press.

  • Sebastian, T.B., Klein, P.N., and Kimia, B.B. 2003. On aligning curves. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(1):116–125.

    Article  Google Scholar 

  • Sharon, E. and Mumford, D. 2004. 2D-Shape analysis using conformal mappings. In Proceedings of IEEE Conference on Computer Vision, pp. 350–357.

  • Srivastava, A., Joshi, S., Mio, W., and Liu, X. 2005. Statistical shape analysis: Clustering, learning and testing. IEEE Trans. on Pattern Analysis and Machine Intelligence, 27:590–602.

    Article  Google Scholar 

  • Tagare, H.D. 1999. Shape-based non-rigid correspondence with applications to heart motion analysis. IEEE Trans. on Medical Imaging, 8(7):570–579.

    Article  MathSciNet  Google Scholar 

  • Tagare, H.D., O’Shea, D., and Groisser, D. 2002. Non-rigid shape comparison of plane curves in images. Journal of Mathematical Imaging and Vision, 16:57–68.

    Article  MathSciNet  Google Scholar 

  • Thompson, D.W. 1992. On Growth and Form: The Complete Revised Edition. Dover.

  • Vailliant, M., Miller, M.I., and Younes, L. 2004. Statistics on diffeomorphisms via tangent space representations. NeuroImage, 23:161–169.

    Article  Google Scholar 

  • Younes, L. 1998. Computable elastic distance between shapes. SIAM Journal of Applied Mathematics, 58:565–586.

    Article  MathSciNet  Google Scholar 

  • Younes, L. 1999. Optimal matching between shapes via elastic deformations. Journal of Image and Vision Computing, 17(5/6):381–389.

    Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, Florida State University, Tallahassee, FL, 32306-4510

    Washington Mio

  2. Department of Statistics, Florida State University, Tallahassee, FL, 32306-4330

    Anuj Srivastava

  3. Department of Electrical and Computer Engineering, Florida State University, Tallahassee, FL, 32310-6046

    Shantanu Joshi

Authors
  1. Washington Mio
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  2. Anuj Srivastava
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  3. Shantanu Joshi
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Mio, W., Srivastava, A. & Joshi, S. On Shape of Plane Elastic Curves. Int J Comput Vision 73, 307–324 (2007). https://doi.org/10.1007/s11263-006-9968-0

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  • DOI: https://doi.org/10.1007/s11263-006-9968-0

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