Archive for Rational Mechanics and Analysis

, Volume 123, Issue 3, pp 199–257

Axioms and fundamental equations of image processing

  • Luis Alvarez
  • Frédéric Guichard
  • Pierre -Louis Lions
  • Jean -Michel Morel
Article

Abstract

Image-processing transforms must satisfy a list of formal requirements. We discuss these requirements and classify them into three categories: “architectural requirements” like locality, recursivity and causality in the scale space, “stability requirements” like the comparison principle and “morphological requirements”, which correspond to shape-preserving properties (rotation invariance, scale invariance, etc.). A complete classification is given of all image multiscale transforms satisfying these requirements. This classification yields a characterization of all classical models and includes new ones, which all are partial differential equations. The new models we introduce have more invariance properties than all the previously known models and in particular have a projection invariance essential for shape recognition. Numerical experiments are presented and compared. The same method is applied to the multiscale analysis of movies. By introducing a property of Galilean invariance, we find a single multiscale morphological model for movie analysis.

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References

  1. 1.
    M. Allmen & C. R. Dyer. Computing spatiotemporal surface flow. Conference on Computer Vision, IEEE Computer Society Press. 303–309, 1991.Google Scholar
  2. 2.
    L. Alvarez, P. L. Lions & J. M. Morel. Image selective smoothing and edge detection by nonlinear diffusion (II). SIAM J. Num. Anal, 29, 845–866, 1992.Google Scholar
  3. 3.
    H. Asada & M. Brady. The curvature primal sketch. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 2–14, 1986.Google Scholar
  4. 4.
    E. B. Barret, P. M. Payton, N. N. Haag & M. H. Brill. General methods for determining projective invariants in imagery. Computer Vision Graphics Image Proc., 1991.Google Scholar
  5. 5.
    G. Barles & P. E. Souganidis. Convergence of approximation schemes for fully nonlinear second order equation, Asymp. Anal, to appear.Google Scholar
  6. 6.
    H. Brezis. Analyse Fonctionelle, Théorie et Applications. Masson, Paris, 1987.Google Scholar
  7. 7.
    M. Campani & A. Verri. Computing optical flow from an overconstrained system of linear algebraic equations. Conference on Computer Vision 1991. IEEE Computer Society Press.Google Scholar
  8. 8.
    J. Canny. Finding edges and lines in images. Technical Report 720, MIT, Artificial Intelligence Laboratory, 1983.Google Scholar
  9. 9.
    F. Catté, T. Coll, P. L. Lions & J. M. Morel. Image selective smoothing and edge detection by nonlinear diffusion. Preprint, CEREMADE 1990. To appear in SIAM J. Num. Anal. Google Scholar
  10. 10.
    Y -G. Chen, Y. Giga & S. Goto. Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Preprint, Hokkaido University, 1989.Google Scholar
  11. 11.
    Y -G. Chen, Y. Giga, T. Hitaka & M. Honma. Numerical analysis for motion of a surface by its mean curvature. Preprint, Hokkaido University.Google Scholar
  12. 12.
    M. G. Crandall, H. Ishii & P. L. Lions. User's guide to viscosity solution of second order partial differential equation. Preprint, CEREMADE, 1990.Google Scholar
  13. 13.
    J. I. Diaz. A nonlinear parabolic equation arising in image processing. Extracta Matematicae, Universidad de Extremadura 1990.Google Scholar
  14. 14.
    G. Dubek & J. Tsotsos. Recognising planar curves using curvature-tuned smoothing. Third Conference on Computer Vision 1990. IEEE Computer Society Press.Google Scholar
  15. 15.
    G. Dubek & J. Tsotsos. Shape representation and recognition from curvature. Third Conference on Computer Vision 1990. IEEE Computer Society Press.Google Scholar
  16. 16.
    L. C. Evans & J. Spruck. Motion of level sets by mean curvature, I. Preprint.Google Scholar
  17. 17.
    S. V. Fogel. A nonlinear approach to the motion correspondance problem. Second Conference on Computer Vision 1988, IEEE Computer Society Press. 619–628.Google Scholar
  18. 18.
    D. Forsyth, J. L. Mundy, A. Zisserman, C. Coelho, A. Heller & C. Rothwell. Invariant descriptors for 3-D object recognition and pose. IEEE Transaction of Pattern Analysis and Machine Intelligence, 13, 971–991, 1991.Google Scholar
  19. 19.
    Y. Giga & S. Goto. Motion of hypersurfaces and geometric equations. J. Math. Soc. Japan, 44, 1992.Google Scholar
  20. 20.
    Y. Giga, S. Goto, H. Ishii & M. -H. Sato. Comparison principle and convexity preserving properties for singular degenerate parabolic equation on unbounded domains. Preprint, Hokkaido University, 1990.Google Scholar
  21. 21.
    R. E. Graham. Snow removal: A noise-stripping process for TV signals, IRE Trans. Information Theory IT-9, 129–144, 1962.Google Scholar
  22. 22.
    G. H. Granlukd, H. Knutsson & R. Wilson. Anisotropic non-stationary image and its applications. Picture Processing Laboratory, Linkoeping University, lS-581 83 Linkoeping, Sweden.Google Scholar
  23. 23.
    N. M. Grzywacz & A. L. Yuille. The motion coherence theory. Second Conference on Computer Vision 1988, IEEE Computer Society Press. 344–353.Google Scholar
  24. 24.
    R. Hartshorne. Foundations of Projective Geometry. Benjamin, 1967.Google Scholar
  25. 25.
    D. J. Heeger. Optical flow from spatiotemporal filters. Int. J. Computer Vision, 1, 279–302, 1988.Google Scholar
  26. 26.
    E. Hildreth. The measurement of visual motion. Cambridge, MIT Press 1984.Google Scholar
  27. 27.
    K. Hollig & J. A. Nohel. A diffusion equation with a nonmonotone constitutive function. System of Nonlinear Partial Differential Equations, J. Ball, ed., Reidel, 409–422, 1983.Google Scholar
  28. 28.
    M. Kass, A. Witkin & D. Terzopoulos. Snakes: active contour models. First Conference on Computer Vision, 1987. IEEE Computer Society Press. 259–268.Google Scholar
  29. 29.
    G. Kanizsa. Grammatica del vedere. Tl Mulino, Bologna 1980.Google Scholar
  30. 30.
    J. J. Koenderink. The structure of images, Biol. Cybern. 50, 363–370, 1984.Google Scholar
  31. 31.
    O. A. Ladyženskaja, V. A. Solonnikov & N. N. Ural'tseva. Linear and Quasilinear Equations of Parabolic Type. Amer. Math. Soc., Providence, 1968.Google Scholar
  32. 32.
    Y. Lamdan, J. T. Schwartz & H. J. Wolfson. Object recognition by affine invariant matching. In Proc. Computer Version and Pattern Recognition 88, 1988.Google Scholar
  33. 33.
    D. Lee, A. Papageorgiou & G. W. Wasilkowski. Computational aspects of determining optical flow. Second Conference on Computer Vision 1988, IEEE Computer Society Press, 612–618.Google Scholar
  34. 34.
    P. L. Lions. Generalized Solutions of Hamilton-]acobi Equations. Research Notes in Mathematics, 69, Pitman, Boston, 1982.Google Scholar
  35. 35.
    H-C. Liu & M. D. Srinath. Partial shape classification using contour matching in distance transformation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 1072–1079, 1990.Google Scholar
  36. 36.
    A. Mackworth & F. Mokhtarian. Scale-Based description and recognition of planar curves and two-dimensional shapes. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 34–43, 1986.Google Scholar
  37. 37.
    S. Mallat & S. Zhong. Complete signal representation with multiscale edges. Technical report n∘ 483, Robotics Report n∘ 219, Courant Institute, Computer Science Division.Google Scholar
  38. 38.
    P. Maragos. Tutorial on advances in morphological image processing and analysis. Optical Engineering, 26, 623–632, 1987.Google Scholar
  39. 39.
    D. Marr. Vision. Freeman, 1982.Google Scholar
  40. 40.
    D. Marr & E. Hildreth. Theory of edge detection. Proc. Ray. Soc. Land., B207, 187–217, 1980.Google Scholar
  41. 41.
    J. M. Morel & S. Solimini. Segmentation of images by variational methods: a constructive approach. Revista Matematica de la Universidad Complutense de Madrid, 1, 169–182, 1988.Google Scholar
  42. 42.
    D. Mumford & J. Shah. Boundary detection by minimizing functionals, IEEE Conference on Computer Vision and Pattern Recognition, San Francisco, 1985.Google Scholar
  43. 43.
    M. Nitzberg & Takahiro Shiota. Nonlinear Image Smoothing with Edge and Corner Enhancement. Technical Report n∘ 90-2, Division of Applied Sciences, Harvard University, Cambridge, 1990.Google Scholar
  44. 44.
    K. N. Nordstrom. Biased anisotropic diffusion — A unified Approach to Edge Detection. Preprint. Dept. of Electrical Engineering and Computer Sciences, University of California, Berkeley.Google Scholar
  45. 45.
    S. Osher & L. Rudin. Feature-oriented image enhancement using shock filters. SIAM J. on Numerical Analysis, 27, 919–940, 1990.Google Scholar
  46. 46.
    S. Osher & J. Sethian. Fronts propagating with curvature dependent speed: algorithms based on the Hamilton-Jacobi formulation. J. Comp. Physics, 79, 12–49, 1988.Google Scholar
  47. 47.
    P. Perona & J. Malik. Scale space and edge detection using anisotropic diffusion. Proc. IEEE Computer Soc. Workshop on Computer Vision, 1987.Google Scholar
  48. 48.
    A. Rattarangsi & R. T. Chin. Scale-Based Detection of Corners of Planer Curves. Third Conference on Computer Vision 1990. IEEE Computer Society Press.Google Scholar
  49. 49.
    T.Richardson. Ph.D.Dissertation, MIT 1990.Google Scholar
  50. 50.
    A. Rosenfeld & M. Thurston. Edge and curve detection for visual scene analysis. IEEE Trans. on Computers, C-20, 562–569, May 1971.Google Scholar
  51. 51.
    L. Rudin & S. Osher. Total variation based restoration of noisy, blurred images, submitted to SIAM J. Num. Analysis, 1992.Google Scholar
  52. 52.
    L. Rudin, S. Osher & E. Fatemi. Nonlinear total variation based noise removal algorithms, Physica D., Proceedings of 11th Conf. on Experimental Mathematics, 1992.Google Scholar
  53. 53.
    P. Saint-Marc, J.-S. Chen & G. Medioni. Adaptative smoothing: A general tool for early vision. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 514–529, 1991.Google Scholar
  54. 54.
    J. Serra. Image Analysis and mathematical Morphology, Vol1. Academic Press, 1982.Google Scholar
  55. 55.
    A. Singh. An estimation-theoretic framework for image-flow computation. Third Conference on Computer Vision 1990. IEEE Computer Society Press.Google Scholar
  56. 56.
    D. Sinha & C. R. Giardina. Discrete black and white object Recognition via morphological functions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12, 275–293, 1990.Google Scholar
  57. 57.
    M. A. Snyder. On the mathematical foundations of smoothness constrains for the determination of the optical flow and for surface reconstruction. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13, 1105–1114, 1991.Google Scholar
  58. 58.
    M. Soner. Motion of a set by the curvature of its mean boundary. Preprint.Google Scholar
  59. 59.
    A. Verri & T. Poggio. Against quantitative optical flow. Conference on Computer Vision 1987. IEEE Computer Society Press, 171–180.Google Scholar
  60. 60.
    I. Weiss. Projective invariants of shapes. In Proc. DARPA IU Workshop, 1125–1134, 1989.Google Scholar
  61. 61.
    A. P. Witkin. Scale-space filtering. Proc. of IJCAI, Karlsruhe, 1019–1021, 1983.Google Scholar
  62. 62.
    A. Yuille & T. Poggio. Scaling theorems for zero crossings. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8, 15–25, 1986.Google Scholar
  63. [AGLM1]
    L. Alvarez, F. Guichard, P.-L. Lions & J.-M. Morel. Axiomatisation et nouveaux opérateurs de la morphologie mathématique. C. R. Acad. Sci. Paris, 315, Sér. I, 265–268, 1992.Google Scholar
  64. [AGLM2]
    L. Alvarez, F. Guichard, P.-L. Lions & J.-M. Morel. Axiomes et équations fundamentales du traitement d'images (Analyse multiéchelle et E.D.P.). C. R. Acad. Sci. Paris, 315, Sér. I, 135–138, 1992.Google Scholar
  65. [AGLM3]
    L. Alvarez, F. Guichard, P.-L. Lions & J.-M. Morel. Analyse multiéchelle des films. C. R. Acad. Sci. Paris, 315, Sér. I, 1145–1148, 1992.Google Scholar
  66. [Ang]
    S. Angenant. Parabolic equations for curves on surfaces I, II. University of Wisconsin-Madison, Technical Summary Reports, 89-19, 89-24, 1989.Google Scholar
  67. [Bar]
    G. Barles. Remarks on a flame propagation model. Technical Report 464, INRIA Rapports de Recherche, December 1985.Google Scholar
  68. [BaGe]
    G. Barles & C. Georgelin. A simple proof of convergence for an approximation scheme for computing motions by mean curvature. Preprint 1992.Google Scholar
  69. [GaHa]
    M. Gage & R. S. Hamilton. The heat equation shrinking convex plane curves. J. Diff. Geom., 23, 69–96, 1986.Google Scholar
  70. [Gray]
    M. Grayson. The heat equation shrinks embedded plane curves to round points. J. Diff. Geom., 26, 285–314, 1987.Google Scholar
  71. [Hum]
    R. Hummel. Representations based on zero-crossings in scale-space. Proc. IEEE Computer Vision and Pattern Recognition Conf., 204–209, 1986.Google Scholar
  72. [Kim]
    B. B. Kimia. Toward a computational theory of shape. Ph. D. Dissertation, Department of Electrical Engineering, McGill University, Montreal, Canada, August 1990.Google Scholar
  73. [KoDo]
    J. J. Koenderink & A. J. Van Doorn. Dynamic shape. Biol. Cyber., 53, 383–396, 1986.Google Scholar
  74. [KTZ]
    B. B. Kimia, A. Tannenbaum & S. W. Zucker. On the evolution of curves via a function of curvature, 1: the classical case. To appear in J. Math. Anal. Appl. Google Scholar
  75. [LoMo]
    C. Lopez & J. M. Morel. Axiomatisation of shape analysis and application to texture hyperdiscrimination. Proceedings of the Trento Conference on Motion by Mean Curvature and Related Models. Springer, 1992.Google Scholar
  76. [MaMo]
    A. Mackworth & F. Mokhtarian. A theory of multiscale, curvaturebased shape representation for planar curves. IEEE Trans. Pattern Anal. Machine Intell. 14, 789–805, 1992.Google Scholar
  77. [Mara2]
    P. Maragos. Pattern spectrum and multiscale shape representation. IEEE Trans. Pattern Anal. Machine Intell., 11, 701–716, 1989.Google Scholar
  78. [MBO]
    B. Merriman, J. Bence & S. Osher. Diffusion generated motion by mean curvature. April 1992. CAM Report 92-18. Dept. Mathematics. University of California, Los Angeles.Google Scholar
  79. [SaTa1]
    G. Sapiro & A. Tannenbaum. On affine plane curve evolution. Dept. Electrical Engineering. Technion, Israel Institute of Technology, Haifa, Israel. Preprint, February 1992.Google Scholar
  80. [SaTa2]
    G. Sapiro & A. Tannenbaum. Affine shortening of non-convex plane curves. Dept. of Electrical Engineering. Technion, Israel Institute of Technology, Haifa, Israel. Preprint, February 1992.Google Scholar

Copyright information

© Springer-Verlag 1993

Authors and Affiliations

  • Luis Alvarez
    • 1
    • 2
  • Frédéric Guichard
    • 1
    • 2
  • Pierre -Louis Lions
    • 1
    • 2
  • Jean -Michel Morel
    • 1
    • 2
  1. 1.Departamento de Informatica y SistemasUniversidad de Las PalmasLas PalmasSpain
  2. 2.Ceremade, Université Paris-DauphineParis Cedex 16

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