Metrics of Curves in Shape Optimization and Analysis

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2090)

Abstract

In these lecture notes we will explore the mathematics of the space of immersed curves, as is nowadays used in applications in computer vision. In this field, the space of curves is employed as a “shape space”; for this reason, we will also define and study the space of geometric curves, which are immersed curves up to reparameterizations. To develop the usages of this space, we will consider the space of curves as an infinite dimensional differentiable manifold; we will then deploy an effective form of calculus and analysis, comprising tools such as a Riemannian metric, so as to be able to perform standard operations such as minimizing a goal functional by gradient descent, or computing the distance between two curves. Along this path of mathematics, we will also present some current literature results (and a few examples of different “shape spaces”, including more than only curves).

References

  1. 1.
    L. Ambrosio, G. Da Prato, A.C.G. Mennucci, An introduction to measure theory and probability, 2007. http//dida.sns.it/dida2/cl/07-08/folde2/pdf0
  2. 2.
    T.M. Apostol, Mathematical Analysis (Addison Wesley, Reading, 1974)MATHGoogle Scholar
  3. 3.
    C.J. Atkin, The Hopf-Rinow theorem is false in infinite dimensions. Bull. Lond. Math. Soc. 7(3), 261–266 (1975) doi: 10.1112/blms/7.3.261MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    H. Brezis, Analisi Funzionale (Liguori Editore, Napoli, 1986). Italian translation of Analyse fonctionelle (Masson, Paris, 1983)Google Scholar
  5. 5.
    J. Canny, A computational approach to edge detection. IEEE Trans. Pattern Anal. Mach. Intell. 8(6), 679–698 (1986) ISSN 0162-8828.Google Scholar
  6. 6.
    V. Caselles, F. Catte, T. Coll, F. Dibos, A geometric model for edge detection. Num. Mathematik 66, 1–31 (1993)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    V. Caselles, R. Kimmel, G. Sapiro, Geodesic active contours, in Proceedings of the IEEE International Conference on Computer Vision, Cambridge, MA, June 1995, pp. 694–699Google Scholar
  8. 8.
    T. Chan, L. Vese, Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)CrossRefMATHGoogle Scholar
  9. 9.
    G. Charpiat, O. Faugeras, R. Keriven, Approximations of shape metrics and application to shape warping and empirical shape statistics. Found. Comput. Math. (2004) doi: 10.1007/s10208-003-0094-xgg819. INRIA report 4820 (2003)Google Scholar
  10. 10.
    G. Charpiat, R. Keriven, J.P. Pons, O. Faugeras, Designing spatially coherent minimizing flows for variational problems based on active contours, in ICCV (2005). doi: 10.1109/ICCV.2005.69Google Scholar
  11. 11.
    G. Charpiat, P. Maurel, J.-P. Pons, R. Keriven, O. Faugeras, Generalized gradients: Priors on minimization flows. Int. J. Comp. Vis. (2007). doi: 10.1007/s11263-006-9966-2Google Scholar
  12. 12.
    Y. Chen, H. Tagare, S. Thiruvenkadam, F. Huang, D. Wilson, K. Gopinath, R. Briggs, E. Geiser, Using prior shapes in geometric active contours in a variational framework. Int. J. Comp. Vis. 50(3), 315–328 (2002)CrossRefMATHGoogle Scholar
  13. 13.
    D. Cremers, S. Soatto, A pseuso distance for shape priors in level set segmentation, in 2nd IEEE Workshop on Variational, Geometric and Level Set Methods in Computer Vision, Nice, ed. by N. Paragios, Oct 2003, pp. 169–176Google Scholar
  14. 14.
    M.P. do Carmo, Riemannian Geometry (Birkhäuser, Boston, 1992)Google Scholar
  15. 15.
    A. Duci, A.C.G. Mennucci, Banach-like metrics and metrics of compact sets. arXiv preprint arXiv:0707.1174 (2007)Google Scholar
  16. 16.
    A. Duci, A.J. Yezzi, S.K. Mitter, S.Soatto, Shape representation via harmonic embedding, in International Conference on Computer Vision (ICCV03), vol. 1, Washington, DC, pp. 656–662 (IEEE Computer Society, Silver Spring, 2003). ISBN 0-7695-1950-4. doi: 10.1109/ICCV.2003.1238410Google Scholar
  17. 17.
    A. Duci, A.J. Yezzi, S. Soatto, K. Rocha, Harmonic embeddings for linear shape. J. Math Imag. Vis 25, 341–352 (2006). doi: 10.1007/s10851-006-7249-8MathSciNetCrossRefGoogle Scholar
  18. 18.
    J. Eells, K.D. Elworthy, Open embeddings of certain Banach manifolds. Ann. Math. (2) 91, 465–485 (1970)Google Scholar
  19. 19.
    I. Ekeland, The Hopf-Rinow theorem in infinite dimension. J. Differ. Geom. 13(2), 287–301 (1978)MathSciNetMATHGoogle Scholar
  20. 20.
    A.T. Fomenko, The Plateau Problem. Studies in the Development of Modern Mathematics (Gordon and Breach, New York, 1990)Google Scholar
  21. 21.
    M. Gage, R.S. Hamilton, The heat equation shrinking convex plane curves. J. Differ. Geom. 23, 69–96 (1986)MathSciNetMATHGoogle Scholar
  22. 22.
    J. Glaunès, A. Trouvé, L. Younes, Modeling planar shape variation via Hamiltonian flows of curves, in Analysis and Statistics of Shapes, ed. by A. Yezzi, H. Krim. Modeling and Simulation in Science, Engineering and Technology, chapter 14 (Birkhäuser, Basel, 2005)Google Scholar
  23. 23.
    M. Grayson, The heat equation shrinks embedded planes curves to round points. J. Differ. Geom. 26, 285–314 (1987)MathSciNetMATHGoogle Scholar
  24. 24.
    R.S. Hamilton, The inverse function theorem of Nash and Moser. Bull. Am. Math. Soc. (N.S.) 7(1), 65–222 (1982) ISSN 0273-0979.Google Scholar
  25. 25.
    J. Itoh, M. Tanaka, The Lipschitz continuity of the distance function to the cut locus. Trans. A.M.S. 353(1), 21–40 (2000)Google Scholar
  26. 26.
    H. Karcher, Riemannian center of mass and mollifier smoothing. Comm. Pure Appl. Math. 30, 509–541 (1977)MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    M. Kass, A. Witkin, D. Terzopoulos, Snakes: active contour models. Int. J. Comp. Vis. 1, 321–331 (1987)CrossRefGoogle Scholar
  28. 28.
    S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, A. Yezzi, Gradient flows and geometric active contour models, in Proceedings of the IEEE International Conference on Computer Vision, Cambridge, MA (1995), pp. 810–815. doi:10.1109/ICCV.1995.466855Google Scholar
  29. 29.
    E. Klassen, A. Srivastava, W. Mio, S.H. Joshi, Analysis of planar shapes using geodesic paths on shape spaces. IEEE Trans. Pattern Anal. Mach. Intell. 26, 372–383 (2004). ISSN 0162-8828. doi: 10.1109/TPAMI.2004.1262333Google Scholar
  30. 30.
    W. Klingenberg, Riemannian Geometry (W. de Gruyter, Berlin, 1982)MATHGoogle Scholar
  31. 31.
    S. Lang, Fundamentals of Differential Geometry (Springer, Berlin, 1999)CrossRefMATHGoogle Scholar
  32. 32.
    M. Leventon, E. Grimson, O. Faugeras, Statistical shape influence in geodesic active contours, IEEE Conference on Computer Vision and Pattern Recognition, Hilton Head Island, SC, vol. 1 (2000), pp. 316–323. doi:10.1109/CVPR.2000.855835Google Scholar
  33. 33.
    Y. Li, L. Nirenberg, The distance function to the boundary, Finsler geometry and the singular set of viscosity solutions of some Hamilton-Jacobi equations. Comm. Pure Appl. Math. 58, 85–146 (2005)MathSciNetMATHGoogle Scholar
  34. 34.
    R. Malladi, J. Sethian, B. Vemuri, Shape modeling with front propagation: a level set approach. IEEE Trans. Pattern Anal. Mach. Intell. 17, 158–175 (1995)CrossRefGoogle Scholar
  35. 35.
    A.C.G. Mennucci, A. Yezzi, G. Sundaramoorthi, Properties of Sobolev Active Contours. Interf. Free Bound. 10, 423–445 (2008)MathSciNetCrossRefMATHGoogle Scholar
  36. 36.
    A.C.G. Mennucci, On asymmetric distances, 2nd version, preprint, 2004. http://cvgmt.sns.it/papers/and04/
  37. 37.
    P.W. Michor, D. Mumford, Vanishing geodesic distance on spaces of submanifolds and diffeomorphisms. Documenta Math. 10, 217–245 (2005). http://www.univie.ac.at/EMIS/journals/DMJDMV/vol-10/05.pdf Google Scholar
  38. 38.
    P.W. Michor, D. Mumford, Riemannian geometris of space of plane curves. J. Eur. Math. Soc. (JEMS) 8, 1–48 (2006)MathSciNetCrossRefMATHGoogle Scholar
  39. 39.
    P.W. Michor, D. Mumford, An overview of the Riemannian metrics on spaces of curves using the Hamiltonian approach. Appl. Comput. Harmonic Anal. 23, 76–113 (2007). doi: 10.1016/j.acha.2006.07.004. http://www.mat.univie.ac.at/~michor/curves-hamiltonian.pdf
  40. 40.
    W. Mio, A. Srivastava, Elastic-string models for representation and analysis of planar shapes, in Proceedings of the 2004 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2004 (CVPR 2004), vol. 2 (2004), pp. 10–15. doi:10.1109/CVPR.2004.1315138Google Scholar
  41. 41.
    W. Mio, A. Srivastava, Elastic-string models for representation and analysis of planar shapes, in Conference on Computer Vision and Pattern Recognition (CVPR), June 2004. http://stat.fsu.edu/~anuj/pdf/papers/CVPR_Paper_04.pdf
  42. 42.
    D. Mumford, J. Shah, Boundary detection by minimizing functionals, in Proceedings CVPR 85: IEEE Computer Society Conference on Computer Vision and Pattern Recognition, June 19–23, 1985, San Francisco, CA, 1985Google Scholar
  43. 43.
    D. Mumford, J. Shah, Optimal approximations by piecewise smooth functions and associated variational problems. Comm. Pure Appl. Math. 42, 577–685 (1989)MathSciNetCrossRefMATHGoogle Scholar
  44. 44.
    S. Osher, J. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on the Hamilton-Jacobi equations. J. Comp. Phys. 79, 12–49 (1988)MathSciNetCrossRefMATHGoogle Scholar
  45. 45.
    T.R. Raviv, N. Kiryati, N. Sochen, Unlevel-set: geometry and prior-based segmentation, in Proceedings of European Conference on Computer Vision, 2004 (Computer Vision-ECCV 2004), ed. by T. Pajdla (Springer, Berlin, 2004). http://dx.doi.org/10.1007/978-3-540-24673-2_5
  46. 46.
    R. Ronfard, Region based strategies for active contour models. Int. J. Comp. Vis. 13(2), 229–251 (1994). http://perception.inrialpes.fr/Publications/1994/Ron94 Google Scholar
  47. 47.
    M. Rousson, N. Paragios, Shape priors for level set representations, in Proceedings of the European Conference on Computer Vision, vol. 2 (2002), pp. 78–93Google Scholar
  48. 48.
    W. Rudin, Functional Analysis (McGraw-Hill, New York, 1973)MATHGoogle Scholar
  49. 49.
    W. Rudin, Real and Complex Analysis (McGraw-Hill, New York, 1987)MATHGoogle Scholar
  50. 50.
    J. Shah, H 0 type Riemannian metrics on the space of planar curves. Q. Appl. Math. 66, 123–137 (2008)MATHGoogle Scholar
  51. 51.
    S. Soatto, A.J. Yezzi, DEFORMOTION: deforming motion, shape average and the joint registration and segmentation of images. ECCV (3), 32–57 (2002)Google Scholar
  52. 52.
    A. Srivastava, S.H. Joshi, W. Mio, X. Liu, Statistical shape analysis: clustering, learning, and testing. IEEE Trans. Pattern Anal. Mach. Intell. 27, 590–602 (2005). ISSN 0162-8828. doi: 10.1109/TPAMI.2005.86Google Scholar
  53. 53.
    G. Sundaramoorthi, A. Yezzi, A.C.G. Mennucci, Sobolev active contours, in VLSM, ed. by N. Paragios, O.D. Faugeras, T. Chan, C. Schnörr. Lecture Notes in Computer Science, vol. 3752 (Springer, Berlin, 2005), pp. 109–120. ISBN 3-540-29348-5. doi: 10.1007/11567646_10Google Scholar
  54. 54.
    G. Sundaramoorthi, J.D. Jackson, A. Yezzi, A.C.G. Mennucci, Tracking with Sobolev active contours, in Conference on Computer Vision and Pattern Recognition (CVPR06) (IEEE Computer Society, Silver Spring, 2006). ISBN 0-7695-2372-2. doi: 10.1109/CVPR.2006.314Google Scholar
  55. 55.
    G. Sundaramoorthi, A. Yezzi, A.C.G. Mennucci, G. Sapiro, New possibilities with Sobolev active contours, in Scale Space Variational Methods 07 (2007). http://ssvm07.ciram.unibo.it/ssvm07_public/index.html. “Best Numerical Paper-Project Award”; also [58]
  56. 56.
    G. Sundaramoorthi, A. Yezzi, A.C.G. Mennucci, Sobolev active contours. Int. J. Comp. Vis. (2007). doi: 10.1007/s11263-006-0635-2Google Scholar
  57. 57.
    G. Sundaramoorthi, A. Yezzi, A.C.G. Mennucci, Coarse-to-fine segmentation and tracking using Sobolev Active Contours. IEEE Trans. Pattern Anal. Mach. Intell. (TPAMI) (2008). doi: 10.1109/TPAMI.2007.70751Google Scholar
  58. 58.
    G. Sundaramoorthi, A. Yezzi, A.C.G. Mennucci, G. Sapiro, New possibilities with Sobolev active contours. Int. J. Comp. Vis. (2008). doi: 10.1007/s11263-008-0133-9Google Scholar
  59. 59.
    A. Trouvé, L. Younes, Local geometry of deformable templates. SIAM J. Math. Anal. 37(1), 17–59 (electronic) (2005). ISSN 0036-1410Google Scholar
  60. 60.
    A. Tsai, A. Yezzi, W. Wells, C. Tempany, D. Tucker, A. Fan, E. Grimson, A. Willsky, Model-based curve evolution technique for image segmentation, in Proceedings of the 2001 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, 2001 (CVPR 2001), vol. 1, Dec 2001, pp. I-463, I-468. doi: 10.1109/CVPR.2001.990511Google Scholar
  61. 61.
    A. Tsai, A. Yezzi, A.S. Willsky, Curve evolution implementation of the mumford-shah functional for image segmentation, denoising, interpolation, and magnification. IEEE Trans. Image Process. 10(8), 1169–1186 (2001)CrossRefMATHGoogle Scholar
  62. 62.
    L.A. Vese, T.F. Chan, A multiphase level set framework for image segmentation using the mumford and shah model. Int. J. Comp. Vis. 50(3), 271–293 (2002)CrossRefMATHGoogle Scholar
  63. 63.
    A. Yezzi, A.C.G. Mennucci, Geodesic homotopies, in EUSIPCO04 (2004). http://www.eurasip.org/content/Eusipco/2004/defevent/papers/cr1925.pdf
  64. 64.
    A. Yezzi, A.C.G. Mennucci, Metrics in the space of curves. arXiv (2004)Google Scholar
  65. 65.
    A. Yezzi, A.C.G. Mennucci, Conformal metrics and true “gradient flows” for curves, in International Conference on Computer Vision (ICCV05) (2005), pp. 913–919. doi: 10.1109/ICCV.2005.60. URL http://research.microsoft.com/iccv2005/
  66. 66.
    A. Yezzi, A. Tsai, A. Willsky, A statistical approach to snakes for bimodal and trimodal imagery, in The Proceedings of the Seventh IEEE International Conference on Computer Vision, 1999, vol. 2, October 1999, pp. 898, 903. doi:10.1109/ICCV.1999.790317Google Scholar
  67. 67.
    L. Younes, Computable elastic distances between shapes. SIAM J. Appl. Math. 58(2), 565–586 (1998). doi: 10.1137/S0036139995287685MathSciNetCrossRefMATHGoogle Scholar
  68. 68.
    L. Younes, P.W. Michor, J. Shah, D. Mumford, A metric on shape space with explicit geodesics. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl. 19(1), 25–57 (2008). ISSN 1120-6330. doi: 10.4171/RLM/506Google Scholar
  69. 69.
    C.T. Zahn, R.Z. Roskies, Fourier descriptors for plane closed curves. IEEE Trans. Comput. 21(3), 269–281 (1972). ISSN 0018-9340. doi: 10.1109/TC.1972.5008949Google Scholar
  70. 70.
    S.C. Zhu, T.S. Lee, A.L. Yuille, Region competition: Unifying snakes, region growing, energy/bayes/MDL for multi-band image segmentation, in ICCV (1995), p. 416. citeseer.ist.psu.edu/zhu95region.html

Copyright information

© Springer International Publishing Switzerland 2013

Authors and Affiliations

  1. 1.Scuola Normale SuperiorePisaItaly

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