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Geometric Analysis of Continuous, Planar Shapes

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Energy Minimization Methods in Computer Vision and Pattern Recognition (EMMCVPR 2003)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 2683))

Abstract

We propose two differential geometric representations of planar shapes using: (i) direction functions and (ii) curvature functions, of their boundaries. Under either representation, planar shapes are treated as elements of infinite-dimensional shape spaces. Pairwise differences between the shapes are quantified using the lengths of geodesics connecting them on the shape spaces. We specify the geometry of the two shape spaces and utilize numerical methods for finding geodesics on them. Some applications of this shape analysis are illustrated including: (i) interpolation between shapes, (ii) clustering of objects according to their shapes, and (iii) computation of intrinsic mean shapes.

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References

  1. Amit, Y., Grenander, U., Piccioni, M.: Structural image restoration through deformable templates. J. American Statistical Association (1991)

    Google Scholar 

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

    Article  MATH  Google Scholar 

  3. Do Carmo, M.P.: Differential Geometry of Curves and Surfaces. Prentice Hall, Inc.,Englewood Cliffs (1976)

    MATH  Google Scholar 

  4. Dryden, I.L., Mardia, K.V.: Statistical Shape Analysis. John Wiley & Son, Chichester (1998)

    MATH  Google Scholar 

  5. Goodall, C.R.: Procrustes methods in the statistical analysis of shapes. Journal of Royal Statistical Society B 53, 285–339 (1991)

    MATH  MathSciNet  Google Scholar 

  6. Grenander, U.: General Pattern Theory. Oxford University Press, Oxford (1993)

    Google Scholar 

  7. Grenander, U., Miller, M.I.: Computational anatomy: An emerging discipline. Quarterly of Applied Mathematics, LVI 4, 617–694 (1998)

    MathSciNet  Google Scholar 

  8. Grenander, U., Miller, M.I., Srivastava, A.: Hilbert-schmidt lower bounds for estimators on matrix lie groups for atr. IEEE Transactions on PAMI 20(8), 790–802 (1998)

    Google Scholar 

  9. Kendall, D.G.: Shape manifolds, procrustean metrics and complex projective spaces. Bulletin of London Mathematical Society 16, 81–121 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  10. Le, H.: Locating frechet means with application to shape spaces. Advances in Applied Probability 33(2), 324–338 (2001)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  12. Miller, M.I., Younes, L.: Group actions, homeomorphisms, and matching: A general framework. International Journal of Computer Vision 41(1/2), 61–84 (2002)

    Article  MATH  Google Scholar 

  13. Mokhtarian, F., Abbasi, S., Kittler, J.: Efficient and robust shape retrieval by shape content through curvature scale space. In: Proceedings of First International Conference on Image Database and MultiSearch, pp. 35–42 (1996)

    Google Scholar 

  14. Mokhtarian, F., Abbasi, S., Kittler, J.: Robust and efficient shape indexing through curvature scale space. In: Proceedings of Sixth British Machine Vision Conference, pp. 53–62 (1996)

    Google Scholar 

  15. Small, C.G.: The Statistical Theory of Shape. Springer, Heidelberg (1996)

    Google Scholar 

  16. Thompson, D.W.: On Growth and Form: The Complete Revised Edition. Dover, New York (1992)

    Google Scholar 

  17. Trouve, A.: Diffemorphisms groups and pattern matching in image analysis. International Journal of Computer Vision 28(3), 213–221 (1998)

    Article  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Statistics, Florida State University, Tallahassee, FL

    Anuj Srivastava

  2. Department of Mathematics, Florida State University, Tallahassee, FL

    Washington Mio & Eric Klassen

  3. Department of Electrical Engineering, Florida State University, Tallahassee, FL

    Shantanu Joshi

Authors
  1. Anuj Srivastava
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  2. Washington Mio
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  3. Eric Klassen
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  4. Shantanu Joshi
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Editor information

Editors and Affiliations

  1. Department of CISE, University of Florida, Gainesville, FL, USA

    Anand Rangarajan

  2. Instituto de Telecomunicações, Instituto Superior Técnico, T.U. Lisbon, Av. Rovisco Pais, 1049-001, Lisboa, Portugal

    Mário Figueiredo

  3. ARIANA Research Group, INRIA/I3S, Sophia Antipolis, France

    Josiane Zerubia

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© 2003 Springer-Verlag Berlin Heidelberg

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Srivastava, A., Mio, W., Klassen, E., Joshi, S. (2003). Geometric Analysis of Continuous, Planar Shapes. In: Rangarajan, A., Figueiredo, M., Zerubia, J. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2003. Lecture Notes in Computer Science, vol 2683. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-45063-4_22

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  • DOI: https://doi.org/10.1007/978-3-540-45063-4_22

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-40498-9

  • Online ISBN: 978-3-540-45063-4

  • eBook Packages: Springer Book Archive

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