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Dual norm based iterative methods for image restoration. (English) Zbl 1255.94018

Summary: A convergent iterative regularization procedure based on the square of a dual norm is introduced for image restoration models with general (quadratic or non-quadratic) convex fidelity terms. Iterative regularization methods have been previously employed for image deblurring or denoising in the presence of Gaussian noise, which use \(L^2\) and \(L^1\) data fidelity terms, with rigorous convergence results. Recently, A. N. Iusem and E. Resmerita [Set-Valued Var. Anal. 18, No. 1, 109–120 (2010; Zbl 1189.90196)] proposed a proximal point method using inexact Bregman distance for minimizing a convex function defined on a non-reflexive Banach space (e.g. \(BV(\Omega)\)), which is the dual of a separable Banach space. Based on this method, we investigate several approaches for image restoration such as image deblurring in the presence of noise or image deblurring via (cartoon+texture) decomposition. We show that the resulting proximal point algorithms approximate stably a true image. For image denoising-deblurring we consider Gaussian, Laplace, and Poisson noise models with the corresponding convex fidelity terms as in the Bayesian approach. We test the behavior of proposed algorithms on synthetic and real images in several numerical experiments and compare the results with other state-of-the-art iterative procedures based on the total variation penalization as well as the corresponding existing one-step gradient descent implementations. The numerical experiments indicate that the iterative procedure yields high quality reconstructions and superior results to those obtained by one-step standard gradient descent, with faster computational time.

MSC:

94A08 Image processing (compression, reconstruction, etc.) in information and communication theory
49Q20 Variational problems in a geometric measure-theoretic setting
49N45 Inverse problems in optimal control
90C25 Convex programming

Citations:

Zbl 1189.90196

Software:

SQPlab; PLCP
PDFBibTeX XMLCite
Full Text: DOI Link

References:

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