Abstract
In this paper tube methods for reconstructing discontinuous data from noisy and blurred observation data are considered. It is shown that discrete bounded variation (BV)-regularization (commonly used in inverse problems and image processing) and the taut-string algorithm (commonly used in statistics) select reconstructions in a tube. A version of the taut-string algorithm applicable for higher dimensional data is proposed. This formulation results in a bilateral contact problem which can be solved very efficiently using an active set strategy. As a by-product it is shown that the Lagrange multiplier of the active set strategy is an efficient parameter for edge detection.
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References
R. Acar and C.R. Vogel, “Analysis of bounded variation penalty methods for ill-posed problems,” Inverse Probl., Vol. 10, pp. 1217-1229, 1994.
L. Ambrosio, N. Fusco, and D. Pallara, Functions of Bounded Variation and Free Discontinuity Problems, The Clarendon Press, Oxford University Press: New York, 2000.
L. Ambrosio and V.M. Tortorelli, “On the approximation of free discontinuity problems,” Boll. Un. Mat. Ital. B (7), Vol. 6, pp. 105-123, 1992.
A. Chambolle and P.L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math., Vol. 76, pp. 167-188, 1997.
T. Chan, G. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput., Vol. 20, No. 6, pp. 1964-1977, 1999.
P.L. Davies and A. Kovac, “Local extremes, runs, strings and multiresolution,” Ann. Statist., Vol. 29, pp. 1-65, 2001. With discussion and rejoinder by the authors.
D.C. Dobson and C.R. Vogel, “Convergence of an iterative method for total variation denoising,” SIAM J. Num. Anal., Vol. 34, pp. 1779-1791, 1997.
L.C. Evans and R.F. Gariepy, Measure Theory and Fine Properties of Functions, CRC-Press: Boca Raton, 1992.
D. Geman and C. Yang, “Nonlinear image recovery with half quadratic regularization,” IEEE Transactions on Image Processing, Vol. 4, pp. 932-945, 1995.
C. Großmann and H.-G. Roos, Numerik Partieller Differentialgleichungen, Teubner, Stuttgart, 1992.
M. Hintermüller, K. Ito, and K. Kunisch, “The primal-dual active set strategy as a semi-smooth Newton method,” SIAM J. Optimization, to appear.
K. Ito and K. Kunisch, “Lagrangian formulation of nonsmooth convex optimization in Hilbert spaces,” in Control of Partial Differential Equations and Applications, Lect. Notes Pure Appl. Math. 174, E. Casas (Ed.), New York, 1996, pp. 107-117, Proceedings of the IFIP TC7.
K. Ito and K. Kunisch, “An active set strategy based on the augmented Lagrangian formulation for image restoration,” M2AN Math. Model. Numer. Anal., Vol. 33, pp. 1-21, 1999.
K. Ito and K. Kunisch, “Restoration of edge-flat-grey scales,” Inverse Problems, Vol. 65, pp. 273-298, 2000.
S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed., Academic Press, San Diego, CA, 1999.
E. Mammen and S. van de Geer, “Locally adaptive regression splines,” Ann. Statist., Vol. 25, pp. 387-413, 1997.
D. Mumford and J. Shah, “Optimal approximations by piecewise smooth functions and associated variational problems,” Commun. Pure Appl. Math., Vol. 42, pp. 577-685, 1989.
M.Z. Nashed and O. Scherzer, “Least squares and bounded variation regularization with nondifferentiable functional,” Num. Funct. Anal. and Optimiz., Vol. 19, pp. 873-901, 1998.
M. Nikolova, “Local strong homogeneity of a regularized estimator,” SIAM J. Appl. Math., Vol. 61, pp. 633-658, 2000.
O. Scherzer and J. Weickert, “Relations between regularization and diffusion filtering,” J. Math. Imag.Vision, Vol. 12, pp. 43-63, 2000.
C. Vogel, “A multigrid method for total variation-based image denoising,” in Computation and Control IV, K. Bowers and J. Lund (Eds.), Vol. 20. Birkhäuser, 1995. Progess in Systems and Control Theory.
C.R.Vogel and M.E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput., Vol. 17, pp. 227-238, 1996.
J. Weickert, Anistropic Diffusion in Image Processing, Teubner, Stuttgart, 1998.
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Hinterberger, W., Hintermüller, M., Kunisch, K. et al. Tube Methods for BV Regularization. Journal of Mathematical Imaging and Vision 19, 219–235 (2003). https://doi.org/10.1023/A:1026276804745
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DOI: https://doi.org/10.1023/A:1026276804745