Image Selective Smoothing and Edge Detection by Nonlinear Diffusion. II

A stable algorithm is proposed for image restoration based on the “mean curvature motion” equation. Existence and uniqueness of the “viscosity” solution of the equation are proved, a $L^\infty $ stable algorithm is given, experimental results are shown, and the subjacent vision model is compared with those introduced recently by several vision researchers. The algorithm presented appears to be the sharpest possible among the multiscale image smoothing methods preserving uniqueness and stability.

  • [1]  G. Barles and , P. E. Souganidis, Convergence of approximation schemes for fully nonlinear second order equations, Asymptotic Anal., 4 (1991), 271–283 92d:35137 0729.65077 CrossrefGoogle Scholar

  • [2]  H. Brezis, Analyse Fonctionnelle, Théorie et Applications, Masson, Paris, 1987 Google Scholar

  • [3]  J. Canny, Finding edges and lines in images, Tech. Report, 720, Artificial Intelligence Laboratory, Massachusetts Institute of Technology, Cambridge, MA, 1983 Google Scholar

  • [4]  Francine Catté, Tomeu Coll, P. L. Lions and , J. M. Morel, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182–193 10.1137/0729012 93a:94005 0746.65091 LinkISIGoogle Scholar

  • [5]  Yun-Gang Chen, Yoshikazu Giga and , Shun'ichi Goto, Uniqueness and existence of viscosity solutions of generalized mean curvature flow equations, Proc. Japan Acad. Ser. A Math. Sci., 65 (1989), 207–210 91b:35049 0735.35082 CrossrefGoogle Scholar

  • [6]  Michael G. Crandall, Hitoshi Ishii and , Pierre Louis Lions, User's guide to viscosity solutions of second order partial differential equations, Bull. Amer. Math. Soc. (N.S.), 27 (1992), 1–67 92j:35050 0755.35015 CrossrefISIGoogle Scholar

  • [7]  J. I. Diaz, A nonlinear parabolic equation arising in image processing, Extracta Math., (1990), , Universidad de Extremadura Google Scholar

  • [8]  L. C. Evans and , J. Spruck, Motion of level sets by mean curvature, I, preprint Google Scholar

  • [9]  Y. Giga, S. Goto, H. Ishii and , M.-H. Sato, Comparison principle and convexity preserving properties for singular degenerate parabolic equations on unbounded domains, Indiana Univ. Math. J., 40 (1991), 443–470 92h:35010 0836.35009 CrossrefISIGoogle Scholar

  • [10]  R. E. Graham, Snow removal: A noise-stripping process for TV signals, IRE Trans. Information Theory, IT-9 (1962), 129–144 10.1109/TIT.1962.1057690 CrossrefISIGoogle Scholar

  • [11]  Klaus Höllig and , John A. Nohel, John Ball, A diffusion equation with a nonmonotone constitutive functionSystems of nonlinear partial differential equations (Oxford, 1982), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., Vol. 111, Reidel, Dordrecht, 1983, 409–422 84k:35082 0531.35045 CrossrefGoogle Scholar

  • [12]  M. Kass, A. Witkin and , D. Terzopoulos, Snakes: active contour models, 1987, ICCV 1987, IEEE 777 Google Scholar

  • [13]  Jan J. Koenderink, The structure of images, Biol. Cybernet., 50 (1984), 363–370 758 126 0537.92011 CrossrefISIGoogle Scholar

  • [14]  O. A. Ladyzhenskaya, V. A. Solonnikov and , N. N. Uralt'seva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, Providence, RI, 1968 0174.15403 CrossrefGoogle Scholar

  • [15]  Pierre Louis Lions, Generalized solutions of Hamilton-Jacobi equations, Research Notes in Mathematics, Vol. 69, Pitman (Advanced Publishing Program), Boston, Mass., 1982iv+317 84a:49038 0497.35001 Google Scholar

  • [16]  S. Mallat and , S. Zhong, Complete signal representation with multiscale edges, Tech. Rep., 483, Computer Science Division, Courant Institute, New York, Robotics Report 219 Google Scholar

  • [17]  D. Marr, Vision, W. H. Freeman, San Francisco, CA, 1982 Google Scholar

  • [18]  D. Marr and , E. Hildreth, Theory of edge detection, Proc. Roy. Soc. London Ser. B, 207 (1980), 187–217 CrossrefISIGoogle Scholar

  • [19]  Jean Michel Morel and , Sergio Solimini, Segmentation of images by variational methods: a constructive approach, Rev. Mat. Univ. Complut. Madrid, 1 (1988), 169–182 89k:90183 0679.68205 Google Scholar

  • [20]  D. Mumford and , J. Shah, Boundary detection by minimizing functionals, IEEE Conf. Comput. Vision and Pattern Recognition, San Francisco, CA, 1985 Google Scholar

  • [21]  M. Nitzberg and , T. Shiota, Nonlinear image smoothing with edge and corner enhancement, Tech. Report, 90-2, Division of Applied Sciences, Harvard University, Cambridge, MA, 1990 Google Scholar

  • [22]  K. N. Nordström, Biased anisotropic diffusion—A unified approach to edge detection, 1989, Dept. of Electrical Engineering and Computer Sciences, University of California, Berkeley, CA, preprint Google Scholar

  • [23]  S. Osher and , L. Rudin, Feature-oriented image enhancement using shock filters, SIAM J. Numer. Anal., 27 (1990), 919–940 10.1137/0727053 0714.65096 LinkISIGoogle Scholar

  • [24]  Stanley Osher and , James Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49 10.1016/0021-9991(88)90002-2 89h:80012 0659.65132 CrossrefISIGoogle Scholar

  • [25]  P. Perona and , J. Malik, Scale space and edge detection using anisotropic diffusion, Proc. IEEE Comput. Soc. Workshop on Comput. Vision, 1987 Google Scholar

  • [26]  T. Richardson, Masters Thesis, Ph.D. dissertation, Massachusetts Institute of Technology, Cambridge, MA, 1990 Google Scholar

  • [27]  A. Rosenfeld and , M. Thurston, Edge and curve detection for visual scene analysis, IEEE Trans. on Comput., C-20 (1971), 562–569 CrossrefISIGoogle Scholar

  • [28]  M. Soner, Motion of a set by the curvature of its mean boundary, preprint Google Scholar

  • [29]  A. P. Witkin, Scale-space filtering, Proc. IJCAI, Karlsruhe, 1983, 1019–1021 Google Scholar

  • [30]  A. Yuille and , T. Poggio, Scaling theorems for zero crossings, IEEE Trans. on Pattern Analysis and Machine Intelligence, 8 (1986), 0575.94001 CrossrefISIGoogle Scholar