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.
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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.
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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”.
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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
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DOI: https://doi.org/10.1007/s10589-012-9527-2