Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-26T06:21:02.859Z Has data issue: false hasContentIssue false
  • English
  • Français

Deblurring and denoising of images with minimization of variation and negative norms

Published online by Cambridge University Press:  17 February 2009

A. Cherid ,
M. A. El-Gebeily ,
Donal O′Regan  and
Ravi Agarwal
A. Cherid
Affiliation:
department of Mathematical Sciences King Fahd University of Petroleum and Minerals, Dhahran 31261 KSA.
M. A. El-Gebeily
Affiliation:
department of Mathematical Sciences King Fahd University of Petroleum and Minerals, Dhahran 31261 KSA.
Donal O′Regan
Affiliation:
department of Mathematics National University of Ireland, Galway Ireland.
Ravi Agarwal
Affiliation:
department of Mathematical Sciences Florida Institute of Technology, 150 West University Blvd Melbourne FL 32901-6975 USA; email: [email protected].
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

A method based on the minimization of variation is presented for the identification of a completely unknown blur operator. We assume the knowledge of a blurred image and its original version. The class of blurring operators is identified in the class of compact operators. A variational method with negative norms is then used for the restoration of a blurred and noised image. The restoration method works for a wide class of blurring operators and we do not assume that the blur operator commutes with the Laplacian.

Keywords

image deblurringimage restorationvariation minimizationnegative norms.
Type
Articles
Information
The ANZIAM Journal , Volume 49 , Issue 2 , October 2007 , pp. 171 - 185
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Aubert, G. and Komprobst, P., Mathematical Problems in Image Processing (Springer, New York, 2002).CrossRefGoogle Scholar
[2]Chambolle, A. and Lions, P., “Image recovery via total variation minimization and related problems”, Numer Math. 76 (1997) 167–188.CrossRefGoogle Scholar
[3]Chatelin, F., Spectral approximation of linear operators (Academic Press, New York, 1983).Google Scholar
[4]Cheema, T. A., Qureshi, I. M., Jalil, A. and Naveed, A., “Blur and image restoration of nonlinear degraded images using neural networks based on modified ARMA model”, in Proceedings of IN MIC 2004, 8th International Multitopic Conference, (2004) 102–107.Google Scholar
[5]Chen, T. F. and Shen, J., Image processing and analysis: variational, PDE, wavelet, and stochastic methods (SIAM, Philadelphia, 2005).CrossRefGoogle Scholar
[6]Ekland, I. and Temam, R., Convex analysis and variational problems (North Holland, Amsterdam, 1979).Google Scholar
[7]Gaal, S. A., Linear analysis and representation theory (Springer-Verlag, Berlin, Heidelberg, New York, 1973).CrossRefGoogle Scholar
[8]Guan, L. and Ward, R. K., “Deblurring random time-varying blur”, J Opt Soc Amer A 6 (1989) 1727–1737.CrossRefGoogle ScholarPubMed
[9]Katsaggelos, A. K., Biemond, J., Schaffer, R. W. and Mersereau, R. M., “A regularized iterative image restoration algorithm”, IEEE Trans Acoust Speech Signal Processing 39 (1991) 914–929.CrossRefGoogle Scholar
[10]Katsaggelos, A. K. and Lay, K. T., “Maximum likelihood blur identification and image restoration using EM algorithm”, IEEE Trans Acoust Speech Signal Processing 39 (1991) 729–733.CrossRefGoogle Scholar
[11]Kelley, C. T., Iterative methods for linear and nonlinear equations Frontiers in Applied Mathematics, 16 (SIAM, Philadelphia, 1995).CrossRefGoogle Scholar
[12]Lay, K. T. and Katsaggelos, A. K., “Image identification and restoration based on the expectationmaximization algorithm”, Opt Eng 29 (1990) 436–445.Google Scholar
[13]Meyer, Y., Oscillating patterns in image processing and nonlinear evolution equations, Volume 22 of Univ. Led. Ser. (AMS, Providence, RI, 2001).Google Scholar
[14]Osher, S., Sole, A. and Vese, L., “Image decomposition and restoration using total variation minimization and the H-l norm”, Multiscale Model Simul. 1 (2003) 349–370.CrossRefGoogle Scholar
[15]Reeves, S. R. and Mersereau, R. M., “Blur identification by the method of generalized crossvalidation”, IEEE Trans Image Processing 1 (1992) 301–311.CrossRefGoogle ScholarPubMed
[16]Rudin, L., Osher, S. and Fatemi, E., “Nonlinear total variation based noise removal algorithms”, Phys D 60 (1992) 259–268.CrossRefGoogle Scholar
[17]Savakis, A. E. and Trussell, H. J., “On the accuracy of PSF representation in image restoration”, IEEE Trans Image Processing 2 (1993) 252–259.CrossRefGoogle ScholarPubMed
[18]Sezan, M. I. and Tekalp, A. M., “Adaptive image restoration with artifact suppression using the theory of convex projection”, IEEE Trans Acoust Speech Signal Processing 38 (1990) 181–185.CrossRefGoogle Scholar
[19]Trussell, H. J. and Fogel, S., “Identification and restoration of spatially variant motion blurs in sequential images”, IEEE Trans Image Processing 1 (1992) 123–126.CrossRefGoogle ScholarPubMed
[20]Yang, Y., Galatsanos, N. P. and Stark, H., “Projection-based blind deconvolution”, J Opt Soc Amer A. 11 (1994) 2104–2109.CrossRefGoogle Scholar
[21]Zeidler, E., Nonlinear functional analysis and its applications. II/B Nonlinear monotone operators (Springer-Verlag, New York, 1990).Google Scholar
[22]Ziemer, W., Weakly differentiable functions. Volume 120 of Graduate Texts in Mathematics (Springer-Verlag, New York, 1989).CrossRefGoogle Scholar