Abstract
Regularization methods for inverse problems formulated in Hilbert spaces usually give rise to over-smoothness, which does not allow to obtain a good contrast and localization of the edges in the context of image restoration.
On the other hand, regularization methods recently introduced in Banach spaces allow in general to obtain better localization and restoration of the discontinuities or localized impulsive signals in imaging applications.
We present here an expository survey of the topic focused on the iterative Landweber method. In addition, preconditioning techniques previously proposed for Hilbert spaces are extended to the Banach setting and numerically tested.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Notes
A complementary effect can be observed for the other duality map \(J_{s^{*}}^{X^{*}}=J_{2}^{L^{p^{*}}}\) which acts on the reconstructions. Since p ∗>2 for 1<p<2, the factor \(|x|^{p^{*}-1}\) in (8) tends to emphasize the largest components and to reduce the smallest ones; in other words, the contrast of the reconstructed image is somehow enhanced, avoiding over-smoothness.
References
Asplund, E.: Fréchet differentiability of convex functions. Acta Math. 121, 31–47 (1968)
Beauzamy, B.: Introduction to Banach Spaces and Their Geometry, 2nd revised edn. North-Holland, Amsterdam (1985)
Benvenuto, F., Zanella, R., Zanni, L., Bertero, M.: Nonnegative least-squares image deblurring: improved gradient projection approaches. Inverse Probl. 26, 025004 (2010) (18 pp.)
Bertero, M., Boccacci, P.: Introduction to Inverse Problems in Imaging. Institute of Physics Publ., Bristol (1998)
Bertsekas, D.P.: Nonlinear Programming. Athena Scientific, Belmont (1995)
Bonettini, S., Zanella, R., Zanni, L.: A scaled gradient projection method for constrained image deblurring. Inverse Probl. 25, 015002 (2009) (23 pp.)
Brianzi, P., Di Benedetto, F., Estatico, C.: Improvement of space-invariant image deblurring by preconditioned Landweber iterations. SIAM J. Sci. Comput. 30, 1430–1458 (2008)
Canuto, C., Urban, K.: Adaptive optimization of convex functionals in Banach spaces. SIAM J. Numer. Anal. 42, 2043–2075 (2005)
Censor, Y., Elfving, T., Herman, G.T., Nikazad, T.: On diagonally-relaxed orthogonal projection methods. SIAM J. Sci. Comput. 30, 473–504 (2008)
Chan, T.: An optimal circulant preconditioner for Toeplitz systems. SIAM J. Sci. Comput. 9, 766–771 (1988)
Cimmino, G.: Calcolo approssimato per le soluzioni dei sistemi di equazioni lineari. Ric. Sci., Ser. II, Anno IX XVI, 326–333 (1938)
Daubechies, I., Defrise, M., De Mol, C.: An iterative thresholding algorithm for linear inverse problems with a sparsity constraint. Commun. Pure Appl. Math. 57, 1413–1457 (2004)
Elfving, T., Nikazad, T., Hansen, P.C.: Semi-convergence and relaxation parameters for a class of SIRT algorithm. Electron. Trans. Numer. Anal. 37, 321–336 (2010)
Engl, H.W., Hanke, M., Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic, Dordrecht (1996)
Estatico, C., Pastorino, M., Randazzo, A.: A novel microwave imaging approach based on regularization in L p Banach spaces. IEEE Trans. Antennas Propag. 60, 3373–3381 (2012)
Hanke, M., Nagy, J.: Inverse Toeplitz preconditioners for ill-posed problems. Linear Algebra Appl. 284, 137–156 (1998)
Hanke, M., Nagy, J., Vogel, C.: Quasi-Newton approach to nonnegative image restorations. Linear Algebra Appl. 316, 223–236 (2000)
Hansen, P.C., Nagy, J., O’Leary, D.P.: Deblurring Images: Matrices, Spectra, and Filtering. SIAM, Philadelphia (2006)
Hein, T., Kazimierski, K.S.: Accelerated Landweber iteration in Banach spaces. Inverse Probl. 26, 055002 (2010) (17 pp.)
Hein, T., Kazimierski, K.S.: Modified Landweber iteration in Banach spaces—convergence and convergence rates. Numer. Funct. Anal. Optim. 31, 1158–1189 (2010)
Kaltenbacher, B.: Some Newton-type methods for the regularization of nonlinear ill-posed problems. Inverse Probl. 13, 729–753 (1997)
Kamm, J., Nagy, J.: Kronecker product and SVD approximations in image restoration. Linear Algebra Appl. 284, 177–192 (1998)
Landweber, L.: An iteration formula for Fredholm integral equations of the first kind. Am. J. Math. 73, 615–624 (1951)
Natterer, F.: The Mathematics of Computerized Tomography. John Wiley, New York (1986)
Piana, M., Bertero, M.: Projected Landweber method and preconditioning. Inverse Probl. 13, 441–464 (1997)
Rieder, A.: On the regularization of nonlinear ill-posed problems via inexact Newton iterations. Inverse Probl. 15, 309–327 (1999)
Roggemann, M.C., Welsh, B.: Imaging Through Turbulence. CRC Press, Boca Raton (1996)
Scherzer, M.O., Grasmair, M., Grossauer, H., Haltmeier, M., Lenzen, F.: Variational Methods in Imaging. Springer, Berlin (2008)
Schöpfer, F., Louis, A.K., Schuster, T.: Nonlinear iterative methods for linear ill-posed problems in Banach spaces. Inverse Probl. 22, 311–329 (2006)
Trefethen, L., Bau, D.: Numerical Linear Algebra. SIAM, Philadelphia (1997)
Author information
Authors and Affiliations
Corresponding author
Additional information
This work was partially supported by MIUR grant number 20083KLJEZ, and by GNCS-INDAM projects “Analisi di strutture nella ricostruzione di immagini e monumenti” and “Precondizionamento e metodi Multigrid per il calcolo veloce di soluzioni accurate”.
Rights and permissions
About this article
Cite this article
Brianzi, P., Di Benedetto, F. & Estatico, C. Preconditioned iterative regularization in Banach spaces. Comput Optim Appl 54, 263–282 (2013). https://doi.org/10.1007/s10589-012-9527-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10589-012-9527-2