Summary
In this paper, we consider error estimation for image restoration problems based on generalized Bregman distances. This error estimation technique has been used to derive convergence rates of variational regularization schemes for linear and nonlinear inverse problems by the authors before (cf. Burger in Inverse Prob 20: 1411–1421, 2004; Resmerita in Inverse Prob 21: 1303–1314, 2005; Inverse Prob 22: 801–814, 2006), but so far it was not applied to image restoration in a systematic way. Due to the flexibility of the Bregman distances, this approach is particularly attractive for imaging tasks, where often singular energies (non-differentiable, not strictly convex) are used to achieve certain tasks such as preservation of edges. Besides the discussion of the variational image restoration schemes, our main goal in this paper is to extend the error estimation approach to iterative regularization schemes (and time-continuous flows) that have emerged recently as multiscale restoration techniques and could improve some shortcomings of the variational schemes. We derive error estimates between the iterates and the exact image both in the case of clean and noisy data, the latter also giving indications on the choice of termination criteria. The error estimates are applied to various image restoration approaches such as denoising and decomposition by total variation and wavelet methods. We shall see that interesting results for various restoration approaches can be deduced from our general results by just exploring the structure of subgradients.
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References
Acar R. and Vogel C.R. (1994). Analysis of bounded variation penalty method for ill-posed problems. Inverse Prob 10: 1217–1229
Ambrosio L., Fusco N. and Pallara D. (2000). Functions of bounded variation and free discontinuity problems. Oxford University Press, Oxford
Andreu F., Ballester C., Caselles V. and Mazon J.M. (2001). Minimizing total variation flow. Diff Int Equ 14: 321–360
Aubert G. and Aujol J. F. (2005). Modeling very oscillating signals, application to image processing. Appl Math Optim 51: 163–182
Bachmayr M. (2007). Iterative total variation methods for nonlinear inverse problems. Master Thesis. Johannes Kepler University, Linz
Bregman L.M. (1967). The relaxation method for finding the common point of convex sets and its application to the solution of problems in convex programming. USSR Comp Math Math Phys 7: 200–217
Burger M. and Osher S. (2004). Convergence rates for convex variational regularization. Inverse Prob 20: 1411–1421
Burger M., Frick K., Osher S. and Scherzer O. (2007). Inverse total variation flow. SIAM Multiscale Mod Simul 6(2): 366–395
Burger M., Gilboa G., Osher S. and Xu J. (2006). Nonlinear inverse scale space methods. Commun Math Sci 4: 179–212
Chambolle A. (2004). An algorithm for total variation regularization and denoising. J Math Imaging Vis 20: 89–97
Chambolle A., DeVore R., Lee N.Y. and Lucier B. (1998). Nonlinear wavelet image processing: variational problems, compression, and noise removal through wavelet shrinkage. IEEE Trans Image Proc. 7: 319–335
Chan T. and Shen J. (2005). Image processing and analysis. SIAM, Philadelphia
Daubechies, I., Teschke, G.: Wavelet-based image decompositions by variational functionals. In: (Truchetet, F., ed) Wavelet applications in industrial processing. Proc SPIE 5266, pp. 94–105 (2004).
Donoho D. and Johnstone I. (1994). Ideal spatial adaptation via wavelet shrinkage. Biometrika 81: 425–455
Engl, H. W., Hanke, M., Neubauer, A.: Regularization of inverse problems. Kluwer, Dordrecht (1996) (Paperback edition 2000)
Ekeland I. and Temam R. (1999). Convex analysis and variational problems. Corrected reprint edition. SIAM, Philadelphia
Feng X. and Prohl A. (2003). Analysis of total variation flow and its finite element approximations. Math Meth M2AN 37: 533–556
Gao H.Y. and Bruce A.G. (1997). WaveShrink with firm shrinkage. Stat Sin 7: 855–874
Gilboa G., Sochen N. and Zeevi Y.Y. (2006). Estimation of optimal PDE-based denoising in the SNR sense. IEEE TIP 15(8): 2269–2280
He, L., Chung, T. C., Osher, S., Fang, T., Speier, P.: MR image reconstruction by using the iterative refinement method and nonlinear inverse scale space methods. UCLA, CAM 06-35
Holmes R.B. (1975). Geometric functional analysis and its applications. Springer, New York
Kindermann S., Osher S. and Xu J. (2006). Denoising by BV-duality. J Sci Comput 28: 411–444
Koenderink J.J. (1988). Scale-time. Biol Cybern 58: 159–162
Lie J. and Nordbotten J.M. (2007). Inverse scale spaces for nonlinear regularization. J Math Imaging Vis 27(1): 41–50
Meyer, Y.: Oscillating patterns in image processing and nonlinear evolution equations. AMS, Providence (2001)
Osher S., Burger M., Goldfarb D., Xu J. and Yin W. (2005). An iterative regularization method for total variation-based image restoration. SIAM Multiscale Model Simul 4: 460–489
Osher S.J., Sole A. and Vese L. (2003). Image decomposition and restoration using total variation minimization and the H −1 norm. SIAM Multiscale Model Simul 1: 349–370
Perona P. and Malik J. (1990). Scale-space and edge detection using anisotropic diffusion. IEEE Trans Pattern Anal Mach Intell 12: 629–639
Resmerita E. (2005). Regularization of ill-posed problems in Banach spaces: convergence rates. Inverse Prob 21: 1303–1314
Resmerita E. and Scherzer O. (2006). Error estimates for non-quadratic regularization and the relation to enhancing. Inverse Prob 22: 801–814
Rudin L.I., Osher S.J. and Fatemi E. (1992). Nonlinear total variation based noise removal algorithms. Physica D 60: 259–268
Scherzer O. and Groetsch C. (2001). Inverse scale space theory for inverse problems. In: Kerckhove, M. (eds) Scale-space and morphology in computer vision. Proc. 3rd Int. Conf. Scale-space, pp 317–325. Springer, Berlin
Schoepfer F., Louis A.K. and Schuster T. (2006). Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Prob 22: 311–329
Tadmor E., Nezzar S. and Vese L. (2004). A multiscale image representation using hierarchical (BV;L2) decompositions. Multiscale Model Simul 2: 554–579
Witkin, A. P.: Scale-space filtering. In: Proc. Int. Joint Conf. on Artificial Intelligence, Karlsruhe 1983, pp. 1019–1023
Xu J. and Osher S. (2007). Iterative regularization and nonlinear inverse scale space applied to wavelet based denoising. IEEE Trans Image Proc 16(2): 534–544
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Burger, M., Resmerita, E. & He, L. Error estimation for Bregman iterations and inverse scale space methods in image restoration. Computing 81, 109–135 (2007). https://doi.org/10.1007/s00607-007-0245-z
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DOI: https://doi.org/10.1007/s00607-007-0245-z