TY - JOUR
T1 - Constrained iterations for blind deconvolution and convexity issues
AU - Spaletta, Giulia
AU - Caucci, Luca
N1 - Funding Information:
We are grateful to Prof. Reichel, for his suggestions and improvements to this paper, and to Dr. Sofroniou, for his help with the latest version of the package and with this research. We further wish to thank Proff. Sergeyev and Sordoni for helpful discussions. This work has been partially supported by the GNCS Research Project 2002 ‘Analysis and Development of Parallel Computational Kernels for Problem Solution’.
PY - 2006/12/1
Y1 - 2006/12/1
N2 - The need for image restoration arises in many applications of various scientific disciplines, such as medicine and astronomy and, in general, whenever an unknown image must be recovered from blurred and noisy data [M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Philadelphia, PA, USA, 1998]. The algorithm studied in this work restores the image without the knowledge of the blur, using little a priori information and a blind inverse filter iteration. It represents a variation of the methods proposed in Kundur and Hatzinakos [A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process. 46(2) (1998) 375-390] and Ng et al. [Regularization of RIF blind image deconvolution, IEEE Trans. Image Process. 9(6) (2000) 1130-1134]. The problem of interest here is an inverse one, that cannot be solved by simple filtering since it is ill-posed. The imaging system is assumed to be linear and space-invariant: this allows a simplified relationship between unknown and observed images, described by a point spread function modeling the distortion. The blurring, though, makes the restoration ill-conditioned: regularization is therefore also needed, obtained by adding constraints to the formulation. The restoration is modeled as a constrained minimization: particular attention is given here to the analysis of the objective function and on establishing whether or not it is a convex function, whose minima can be located by classic optimization techniques and descent methods. Numerical examples are applied to simulated data and to real data derived from various applications. Comparison with the behavior of methods [D. Kundur, D. Hatzinakos, A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process. 46(2) (1998) 375-390] and [M. Ng, R.J. Plemmons, S. Qiao, Regularization of RIF Blind Image Deconvolution, IEEE Trans. Image Process. 9(6) (2000) 1130-1134] show the effectiveness of our variant.
AB - The need for image restoration arises in many applications of various scientific disciplines, such as medicine and astronomy and, in general, whenever an unknown image must be recovered from blurred and noisy data [M. Bertero, P. Boccacci, Introduction to Inverse Problems in Imaging, Institute of Physics Publishing, Philadelphia, PA, USA, 1998]. The algorithm studied in this work restores the image without the knowledge of the blur, using little a priori information and a blind inverse filter iteration. It represents a variation of the methods proposed in Kundur and Hatzinakos [A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process. 46(2) (1998) 375-390] and Ng et al. [Regularization of RIF blind image deconvolution, IEEE Trans. Image Process. 9(6) (2000) 1130-1134]. The problem of interest here is an inverse one, that cannot be solved by simple filtering since it is ill-posed. The imaging system is assumed to be linear and space-invariant: this allows a simplified relationship between unknown and observed images, described by a point spread function modeling the distortion. The blurring, though, makes the restoration ill-conditioned: regularization is therefore also needed, obtained by adding constraints to the formulation. The restoration is modeled as a constrained minimization: particular attention is given here to the analysis of the objective function and on establishing whether or not it is a convex function, whose minima can be located by classic optimization techniques and descent methods. Numerical examples are applied to simulated data and to real data derived from various applications. Comparison with the behavior of methods [D. Kundur, D. Hatzinakos, A novel blind deconvolution scheme for image restoration using recursive filtering, IEEE Trans. Signal Process. 46(2) (1998) 375-390] and [M. Ng, R.J. Plemmons, S. Qiao, Regularization of RIF Blind Image Deconvolution, IEEE Trans. Image Process. 9(6) (2000) 1130-1134] show the effectiveness of our variant.
KW - Blind restoration
KW - Constrained minimization
KW - Fourier transform
KW - Inverse problems
KW - Regularization
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U2 - 10.1016/j.cam.2005.10.020
DO - 10.1016/j.cam.2005.10.020
M3 - Article
AN - SCOPUS:33746798593
SN - 0377-0427
VL - 197
SP - 29
EP - 43
JO - Journal of Computational and Applied Mathematics
JF - Journal of Computational and Applied Mathematics
IS - 1
ER -