Abstract
Any image filtering operator designed for automatic shape restoration should satisfy robustness (whatever the nature and degree of noise is) as well as non-trivial smooth asymptotic behavior. Moreover, a stopping criterion should be determined by characteristics of the evolved image rather than dependent on the number of iterations. Among the several PDE based techniques, curvature flows appear to be highly reliable for strongly noisy images compared to image diffusion processes.
In the present paper, we introduce a regularized curvature flow (RCF) that admits non-trivial steady states. It is based on a measure of the local curve smoothness that takes into account regularity of the curve curvature and serves as stopping term in the mean curvature flow. We prove that this measure decreases over the orbits of RCF, which endows the method with a natural stop criterion in terms of the magnitude of this measure. Further, in its discrete version it produces steady states consisting of piece-wise regular curves. Numerical experiments made on synthetic shapes corrupted with different kinds of noise show the abilities and limitations of each of the current geometric flows and the benefits of RCF. Finally, we present results on real images that illustrate the usefulness of the present approach in practical applications.
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References
L. Alvarez, F. Guichart, P.L. Lions, and J.M. Morel, "Axioms and fundamental equations of image processing," Arch. Ration. Mech. and Anal., Vol. 123, pp. 199-257, 1993.
G. Aubert, M. Barlaud, L. Blanc-Feraud, and P. Chatbonnier. "Deterministic edge-preserving regularization in computer imaging," IEE Trans. Imag. Process., Vol. 6, No. 2, 1997.
F. Catt, P.-L. Lions, J.-M. Morel, and T. Coll, "Image selective smoothing and edge detection by nonlinear image diffusion," SIAM J. Num. Ana., Vol. 29, pp. 182-193, 1992.
Y.G. Chen, Y. Giga, and S. Goto, "Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations," J. Differential Geometry, Vol. 33, pp. 749-786, 1991.
C.K. Chui, An introduction to Waveletes, Wavelet Analisys and its Applications, Academic Press, INC., 1992.
M.G. Crandall, H. Ishii, and P.L. Lions, "User's guide to viscosity solutions of second order partial differential equations," Bulletin of the American Math. Society, Vol. 27, pp. 1-67, 1992.
L.C. Evans, "Partial differential equations," Berkeley Mathematics Lecture Notes, Vol. 3B, 1993.
M. Gage and R.S. Hamilton, "The heat equation shrinking con-vex plane curves," J. Differential Geometry, Vol. 23, pp. 69-96, 1986.
D. Gil and P. Radeva, "Regularized curvature flow," Comp. Vision Center Tech. Report n 63, 2002.
M.A. Grayson. "The heat equation shrinks embedded plane curves to round points," J. Differential Geometry,Vol. 26, pp. 285-314, 1986.
B. Kimia and K. Siddiqi, "Geometric heat equation and non linear diffusion of shapes and images," in CVPR'94, 1994.
B. Kimia, A. Tanenbaum, and S.W. Zucker, "Toward a computational theory of shape: An overwiew," Lecture Notes in Comp. Sci., Vol. 427, pp. 402-407, Springer-Verlag: New York-Berlin, 1990.
B. Kimia, A. Tanenbaum, and S.W. Zucker, "On the evolution of curves via a function of curvature I: The classical case," J. Math. Analysis and Aplications, Vol. 163, pp. 438-458, 1992.
R. Malladi and J.A. Sethian, "Image processing: Flows under min-max curvature and mean curvature," GMIP, Vol. 58, No. 2, pp. 127-141, 1996.
R. Malladi, J.A. Sethian, and B.C. Vemuri, "Shape modeling with front propagation: A level set approach," PAMI, Vol. 17, No. 2, pp. 158-175, 1995.
Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., 1999.
J. Monteil and Azeddine Beghdadi, "A new interpretation an improvement of the nonlinear anisotropic diffusion for image enhancement," IEEE Trans. IP, Vol. 21, No. 9, 1999.
S.J. Osher and J.A. Sethian "Front propagation with curvature dependent speed: Algorithms based on Hamilton-Jacobi formulations," Journal of Computational Physics, Vol. 79, pp. 12-49, 1988.
P. Perona and J. Malik, "Scale space and edge detection us-ing anisotropic diffusion," in Proc. IEEE Comp. Soc. Workshop on Comp. Vision," IEEE Computer Society Press, pp. 16-22, 1987.
G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press: Cambridge, U.K., 2001.
J.A. Sethian, Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Sciences, Cambridge Universirty Press: Cambridge, U.K., 1996.
M. Spivak, A Comprehensive Introduction to Differential Geometry, Houston: Publish or Perish, Cop. 1979.
J. Weickert, "Anisotropic diffusion in image processing," PhD Thesis, Jan. 1996.
J. Weickert, "A review of nonlinear diffusion filtering," in Scale-Space Theory in Computer Vision, B. Haar Romery, l. Florack, J. Koenderink, and M. Viergever <nt>(Eds.)</nt>, Lecture Notes in Computer Science, Vol. 1252, Springer-Verlag: Berlin, 1997, pp. 3-28.
J. Weickert and B. Benhamouda, "A semidiscrete nonlinear scale-space theory and its relation to the Perona-Malik para-dox," in TFCV '96, F. Solina, W.G. Kropatsch, and R. Klette, R. Bajcsy <nt>(Eds.)</nt>, Springer-Verlag, Wien, 1997.
J. Weickert, B.M. ter Haar Romeny, and M.A. Viergever, "Efficient and reliable schemes for nonlinear diffusion filtering," IEEE Trans. IP, 1998.
Y.-L. You and M. Ka veh, "Formation of step images during anisotropic diffusion," in ICIP '97, 1997.
Y.-L. You, W. Xu, A. Tannenbaum, and M. Ka veh, "Behavioral analysis of anisotropic diffusion in image processing," IEEE Trans. IP, 1996.
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Gil, D., Radeva, P. Shape Restoration via a Regularized Curvature Flow. Journal of Mathematical Imaging and Vision 21, 205–223 (2004). https://doi.org/10.1023/B:JMIV.0000043737.96390.dc
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DOI: https://doi.org/10.1023/B:JMIV.0000043737.96390.dc