Abstract
We propose the minimization of a nonquadratic functional or, equivalently, a nonlinear diffusion model to smooth noisy image functionsg:Ω ⊂R n →R while preserving significant transitions of the data. The model is chosen such that important properties of the conventional quadratic-functional approach still hold: (1) existence of a unique solution continuously depending on the datag and (2) stability of approximations using the standard finite-element method. Relations with other global approaches for the segmentation of image data are discussed. Numerical experiments with real data illustrate this approach.
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L. Ambrosio and V.M. Tortorelli, “Approximation of functionals depending on jumps by elliptic functionals viaΓ-convergence,”Comm. Pure Appl. Math., vol. 43, pp. 999–1036, 1990.
A. Blake, A. Zisserman,Visual Reconstruction, MIT Press: Cambridge, MA, 1987.
F.E. Browder, “Existence theorems for nonlinear partial differential equations, part I,” inGlobal Analysis, Proceedings of Symposia in Pure Mathematics, vol. XVI, American Mathematical Society: Providence, RI, 1970, pp. 1–60.
G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,”IEEE Trans. Acoust. Speech, Signal Process., vol. ASSP-37, pp. 2024–2036, 1989.
D. Geiger and A. Yuille, “A common framework for image segmentation,”Int. J. Comput. Vis., vol. 6, pp. 227–243, 1991.
S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,”IEEE Trans. Patt. Anal. Mach. Intell. vol. PAMI-6, pp. 721–741, 1984.
W. Hackbusch,Multi-Grid Methods and Applications Springer-Verlag: Berlin, 1985.
B.K.P. Horn and B.G. Schunck, “Determining optical flow,”Artif. Intell., vol. 17, pp. 185–203, 1981.
J. Hutchinson, C. Koch, J. Luo, and C. Mead, “Computing motion using analog and binary resistive networks,”IEEE Comput. Mag., March 1988, pp. 52–63.
K. Ikeuchi and B.K.P. Horn, “Numerical shape from shading and occluding boundaries,”Artif. Intell. vol. 17, pp. 141–185, 1981.
M.A. Krasnoselskii,Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press: Oxford, England, 1964.
S.Z. Li, “Invariant surface segmentation through energy minimization with discontinuities,”Int. J. Comput. Vis., vol. 5, pp. 161–194, 1990.
P.L. Lions, L. Alvarez, and J.M. Morel, “Image selective smoothing and edge detection by non-linear diffusion (II),”SIAM J. Numer Anal. vol. 29, pp. 845–866, 1992.
R. March, “Visual reconstruction with discontinuities using variational methods,”Image Vis. Comput., vol. 10, pp. 30–38, 1992.
J. Marroquin, S. Mitter, and T. Poggio, “Probabilistic solution of ill-posed problems in computational vision,”J. Amer. Statist. Assoc., vol. 82, pp. 76–89, 1987.
D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,”Comm. Pure Appl. Math., vol. 42, pp. 577–685, 1989.
D.W. Murray and B.F. Buxton, “Scene segmentation from visual motion using global optimization,”IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-9, pp. 220–228, 1987.
N. Nordström, “Biased anisotropic diffusion—a unified regularization and diffusion approach to edge detection,” inECCV'90, Lecture Notes in Computer Science, vol. 427, Springer-Verlag: Berlin, 1990, pp. 18–27; also inImage Vis. Comput., vol. 8, pp. 318–327, 1990.
J.M. Ortega, W.C. Rheinboldt,Iterative Solution of Nonlinear Equations in Several Variables Academic Press: New York, 1970.
P. Perona and J. Malik, “Scale-space and edge detection using anisotropic diffusion,” inProc. IEEE Workshop on Computer Vision, Miami, FL, 1987, pp. 16–27; also inIEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-12, pp. 629–639, 1990.
T. Poggio, V. Torre, and C. Koch,“Computational vision and regularization theory,”Nature, vol. 317, pp. 314–319, 1985.
T. Poggio and C. Koch, “Ill-posed problems in early vision: from computational theory to analogue networks,”Proc. Roy. Soc. London B, vol. 226, pp. 303–323, 1985.
C. Schnörr and B. Neumann, “Ein Ansatz zur effizienten und eindeutigen Rekonstruktion stückweise glatter Funktionen,” in14. DAGM-Symposium Mustererkennung 1992, Springer-Verlag: Berlin, pp. 411–416, 1992.
D. Terzopoulos, “Multilevel computational processes for visual surface reconstruction,”Comp. Vis., Graph., Image Process vol. 24, pp. 52–96, 1983.
D. Terzopoulos, “Regularization of inverse visual problems involving discontinuities,”IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-8, pp. 413–424, 1986.
D. Terzopoulos, “The computation of visible-surface representations,”IEEE Trans. Patt. Anal. Mach. Intell., vol. PAMI-10, pp. 417–438, 1988.
A.N. Tikhonov and V.Y. Arsenin,Solutions of Ill-Posed Problems, Wiley: New York, 1977.
M. Vainberg,Variational Method and Method of Monotone Operators in the Theory of Nonlinear Equations, Wiley: New York, 1973.
E. Zeidler,Nonlinear Functional Analysis and its Applications II, Springer-Verlag: Berlin, 1990.
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This work was supported by the ESPRIT project SUBSYM.
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Schnörr, C. Unique reconstruction of piecewise-smooth images by minimizing strictly convex nonquadratic functionals. J Math Imaging Vis 4, 189–198 (1994). https://doi.org/10.1007/BF01249896
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DOI: https://doi.org/10.1007/BF01249896