Abstract

This paper discusses the problem of reconstructing distorted (smeared, defocused, noisy) grey and colored images. The smearing and defocusing of the images is eliminated by solving integral equations by the method of Tikhonov regularization or parametric Wiener filtering, while the noise is eliminated by the method of adaptive Wiener filtering or median filtering. A generalized technique of image truncation is proposed to replace the so-called boundary conditions, and a generalized technique of blurring the edges of the image is proposed to reduce the Gibbs effect. An image-rotation technique is proposed to model the smearing of an image at an arbitrary angle. The methods are implemented in the form of <tt>m</tt>-files in the <span style="font-variant: small-caps">MatLab</span> system. Model and actual images are processed.

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  1. V. S. Sizikov, M. V. Rimskikh, and R. K. Mirdzhamolov, "Reconstructing blurred noisy images without using boundary conditions," Opt. Zh. 76, (5), 38 (2009) [J. Opt. Technol. 76, 279 (2009)].
  2. V. S. Sizikov and I. A. Belov, "Reconstruction of smeared and out-of-focus images by regularization," Opt. Zh. 67, (4), 60 (2000) [J. Opt. Technol. 67, 351 (2000)].
  3. V. S. Sizikov, Mathematical Methods for Processing Measurement Results, Politekhnika, St. Petersburg, 2001.
  4. Yu. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications, VSP, Leiden–Boston, 2005.
  5. V. S. Sizikov, "Using an integral-equation apparatus to reconstruct distorted images," Abstracts of the Conf. Integral Equations–2009, 2009, Izd. IPMÉ, Kiev, pp. 128‒130.
  6. V. S. Sizikov, "Integral equations in the new truncation – blurring – rotation approach," Proc. Int. Conf. Integral Equations—2010, 2010, PAIS, Lviv, pp. 138‒142.
  7. A. N. Tikhonov, A. V. Goncharskiĭ, and V. V. Stepanov, A. N. Tikhonov and A. V. Goncharskiĭ, ed., "Inverse problems of processing photographic images," Some Problems of Natural Science, Izd. MGU, Moscow, 1987, pp. 185‒195.
  8. A. B. Bakushinskiĭ and A. V. Goncharskiĭ, Ill-Posed Problems. Numerical Methods and Applications, Izd. MGU, Moscow, 1989.
  9. R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall, Upper Saddle River, N.J., 2002, Tekhnosfera, Moscow, 2005.
  10. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MatLab, Prentice Hall, Upper Saddle River, N.J., 2004, Tekhnosfera, Moscow, 2006.
  11. M. V. Aref’eva and A. F. Sysoev, "Rapid adjustment algorithms for digital reconstruction of images," Vychislit. Met. Progr. 39, 40 (1983).
  12. A. V. Gorshkov, "Improving the image resolution when processing the data of a physical experiment and finding the unknown spread function from the programs of the Reimage software package," Prib. Tekhnika Éksp. (2), 68 (1995).
  13. I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.
  14. Yu. E. Voskoboĭnikov and V. A. Litasov, "A stable image-reconstruction algorithm for the ill-posed problem of the spread function," Avtometriya 42, (6), 3 (2006).
  15. A. G. Yagola and N. A. Koshev, "The reconstruction of smeared and defocused color images," Vychislit. Met. Progr. 9, 207 (2008).
  16. M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
    [CrossRef]
  17. P. Wendykier and J. G. Nagy, "Image processing on modern CPUs and GPUs," Technical Report TR-2008-023, http://www.mathcs.emory.edu/technical-reports/techrep-00148.pdf.
  18. K. Palmer, J. Nagy, and L. Perrone, Iterative methods for image restoration: MatLab object-oriented approach, 2002, http://citeseer.ist.psu.edu/lee02iterative.html
  19. M. Christiansen and M. Hanke, Deblurring methods using antireflective boundary conditions, 2006, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.9837&rep=rep1&tipe=pdf
  20. A. Arico, M. Donatelli, J. Nagy, and S. Serra-Capizzano, "The anti-reflective transform and regularization by filtering," Technical Report TR-2007-006-A, ftp://ftp.mathcs.emory.edu/pub/techreport/TR-2007-006-A.pdf.
  21. V. D’yakonov and I. Abramenkova, "MatLab," Processing Signals and Images, Piter, St. Petersburg, 2002.
  22. Philip Andrews, Digital Photography Manual, Carlton Books, 2004, Rosmen-Izdat, Moscow, 2005.

2009 (1)

V. S. Sizikov, M. V. Rimskikh, and R. K. Mirdzhamolov, "Reconstructing blurred noisy images without using boundary conditions," Opt. Zh. 76, (5), 38 (2009) [J. Opt. Technol. 76, 279 (2009)].

2008 (1)

A. G. Yagola and N. A. Koshev, "The reconstruction of smeared and defocused color images," Vychislit. Met. Progr. 9, 207 (2008).

2006 (2)

M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
[CrossRef]

Yu. E. Voskoboĭnikov and V. A. Litasov, "A stable image-reconstruction algorithm for the ill-posed problem of the spread function," Avtometriya 42, (6), 3 (2006).

2000 (1)

V. S. Sizikov and I. A. Belov, "Reconstruction of smeared and out-of-focus images by regularization," Opt. Zh. 67, (4), 60 (2000) [J. Opt. Technol. 67, 351 (2000)].

1995 (1)

A. V. Gorshkov, "Improving the image resolution when processing the data of a physical experiment and finding the unknown spread function from the programs of the Reimage software package," Prib. Tekhnika Éksp. (2), 68 (1995).

1983 (1)

M. V. Aref’eva and A. F. Sysoev, "Rapid adjustment algorithms for digital reconstruction of images," Vychislit. Met. Progr. 39, 40 (1983).

Abramenkova, I.

V. D’yakonov and I. Abramenkova, "MatLab," Processing Signals and Images, Piter, St. Petersburg, 2002.

Andrews, Philip

Philip Andrews, Digital Photography Manual, Carlton Books, 2004, Rosmen-Izdat, Moscow, 2005.

Aref’eva, M. V.

M. V. Aref’eva and A. F. Sysoev, "Rapid adjustment algorithms for digital reconstruction of images," Vychislit. Met. Progr. 39, 40 (1983).

Arico, A.

A. Arico, M. Donatelli, J. Nagy, and S. Serra-Capizzano, "The anti-reflective transform and regularization by filtering," Technical Report TR-2007-006-A, ftp://ftp.mathcs.emory.edu/pub/techreport/TR-2007-006-A.pdf.

Bakushinskii, A. B.

A. B. Bakushinskiĭ and A. V. Goncharskiĭ, Ill-Posed Problems. Numerical Methods and Applications, Izd. MGU, Moscow, 1989.

Belov, I. A.

V. S. Sizikov and I. A. Belov, "Reconstruction of smeared and out-of-focus images by regularization," Opt. Zh. 67, (4), 60 (2000) [J. Opt. Technol. 67, 351 (2000)].

D’yakonov, V.

V. D’yakonov and I. Abramenkova, "MatLab," Processing Signals and Images, Piter, St. Petersburg, 2002.

Donatelli, M.

M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
[CrossRef]

A. Arico, M. Donatelli, J. Nagy, and S. Serra-Capizzano, "The anti-reflective transform and regularization by filtering," Technical Report TR-2007-006-A, ftp://ftp.mathcs.emory.edu/pub/techreport/TR-2007-006-A.pdf.

Eddins, S. L.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MatLab, Prentice Hall, Upper Saddle River, N.J., 2004, Tekhnosfera, Moscow, 2006.

Estatico, C.

M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
[CrossRef]

Goncharskii, A. V.

A. N. Tikhonov, A. V. Goncharskiĭ, and V. V. Stepanov, A. N. Tikhonov and A. V. Goncharskiĭ, ed., "Inverse problems of processing photographic images," Some Problems of Natural Science, Izd. MGU, Moscow, 1987, pp. 185‒195.

A. B. Bakushinskiĭ and A. V. Goncharskiĭ, Ill-Posed Problems. Numerical Methods and Applications, Izd. MGU, Moscow, 1989.

Gonzalez, R. C.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall, Upper Saddle River, N.J., 2002, Tekhnosfera, Moscow, 2005.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MatLab, Prentice Hall, Upper Saddle River, N.J., 2004, Tekhnosfera, Moscow, 2006.

Gorshkov, A. V.

A. V. Gorshkov, "Improving the image resolution when processing the data of a physical experiment and finding the unknown spread function from the programs of the Reimage software package," Prib. Tekhnika Éksp. (2), 68 (1995).

Gruzman, I. S.

I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.

Kirichuk, V. S.

I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.

Koshev, N. A.

A. G. Yagola and N. A. Koshev, "The reconstruction of smeared and defocused color images," Vychislit. Met. Progr. 9, 207 (2008).

Kosykh, V. P.

I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.

Litasov, V. A.

Yu. E. Voskoboĭnikov and V. A. Litasov, "A stable image-reconstruction algorithm for the ill-posed problem of the spread function," Avtometriya 42, (6), 3 (2006).

Martinelli, A.

M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
[CrossRef]

Mirdzhamolov, R. K.

V. S. Sizikov, M. V. Rimskikh, and R. K. Mirdzhamolov, "Reconstructing blurred noisy images without using boundary conditions," Opt. Zh. 76, (5), 38 (2009) [J. Opt. Technol. 76, 279 (2009)].

Nagy, J.

A. Arico, M. Donatelli, J. Nagy, and S. Serra-Capizzano, "The anti-reflective transform and regularization by filtering," Technical Report TR-2007-006-A, ftp://ftp.mathcs.emory.edu/pub/techreport/TR-2007-006-A.pdf.

Nagy, J. G.

P. Wendykier and J. G. Nagy, "Image processing on modern CPUs and GPUs," Technical Report TR-2008-023, http://www.mathcs.emory.edu/technical-reports/techrep-00148.pdf.

Peretyagin, G. I.

I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.

Petrov, Yu. P.

Yu. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications, VSP, Leiden–Boston, 2005.

Rimskikh, M. V.

V. S. Sizikov, M. V. Rimskikh, and R. K. Mirdzhamolov, "Reconstructing blurred noisy images without using boundary conditions," Opt. Zh. 76, (5), 38 (2009) [J. Opt. Technol. 76, 279 (2009)].

Serra-Capizzano, S.

M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
[CrossRef]

A. Arico, M. Donatelli, J. Nagy, and S. Serra-Capizzano, "The anti-reflective transform and regularization by filtering," Technical Report TR-2007-006-A, ftp://ftp.mathcs.emory.edu/pub/techreport/TR-2007-006-A.pdf.

Sizikov, V. S.

V. S. Sizikov, M. V. Rimskikh, and R. K. Mirdzhamolov, "Reconstructing blurred noisy images without using boundary conditions," Opt. Zh. 76, (5), 38 (2009) [J. Opt. Technol. 76, 279 (2009)].

V. S. Sizikov and I. A. Belov, "Reconstruction of smeared and out-of-focus images by regularization," Opt. Zh. 67, (4), 60 (2000) [J. Opt. Technol. 67, 351 (2000)].

Yu. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications, VSP, Leiden–Boston, 2005.

V. S. Sizikov, "Using an integral-equation apparatus to reconstruct distorted images," Abstracts of the Conf. Integral Equations–2009, 2009, Izd. IPMÉ, Kiev, pp. 128‒130.

V. S. Sizikov, "Integral equations in the new truncation – blurring – rotation approach," Proc. Int. Conf. Integral Equations—2010, 2010, PAIS, Lviv, pp. 138‒142.

V. S. Sizikov, Mathematical Methods for Processing Measurement Results, Politekhnika, St. Petersburg, 2001.

Spektor, A. A.

I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.

Stepanov, V. V.

A. N. Tikhonov, A. V. Goncharskiĭ, and V. V. Stepanov, A. N. Tikhonov and A. V. Goncharskiĭ, ed., "Inverse problems of processing photographic images," Some Problems of Natural Science, Izd. MGU, Moscow, 1987, pp. 185‒195.

Sysoev, A. F.

M. V. Aref’eva and A. F. Sysoev, "Rapid adjustment algorithms for digital reconstruction of images," Vychislit. Met. Progr. 39, 40 (1983).

Tikhonov, A. N.

A. N. Tikhonov, A. V. Goncharskiĭ, and V. V. Stepanov, A. N. Tikhonov and A. V. Goncharskiĭ, ed., "Inverse problems of processing photographic images," Some Problems of Natural Science, Izd. MGU, Moscow, 1987, pp. 185‒195.

Voskoboinikov, Yu. E.

Yu. E. Voskoboĭnikov and V. A. Litasov, "A stable image-reconstruction algorithm for the ill-posed problem of the spread function," Avtometriya 42, (6), 3 (2006).

Wendykier, P.

P. Wendykier and J. G. Nagy, "Image processing on modern CPUs and GPUs," Technical Report TR-2008-023, http://www.mathcs.emory.edu/technical-reports/techrep-00148.pdf.

Woods, R. E.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MatLab, Prentice Hall, Upper Saddle River, N.J., 2004, Tekhnosfera, Moscow, 2006.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall, Upper Saddle River, N.J., 2002, Tekhnosfera, Moscow, 2005.

Yagola, A. G.

A. G. Yagola and N. A. Koshev, "The reconstruction of smeared and defocused color images," Vychislit. Met. Progr. 9, 207 (2008).

Avtometriya (1)

Yu. E. Voskoboĭnikov and V. A. Litasov, "A stable image-reconstruction algorithm for the ill-posed problem of the spread function," Avtometriya 42, (6), 3 (2006).

Inverse Probl. (1)

M. Donatelli, C. Estatico, A. Martinelli, and S. Serra-Capizzano, "Improved image deblurring with anti-reflective boundary conditions and re-blurring," Inverse Probl. 22, 2035 (2006).
[CrossRef]

Opt. Zh. (2)

V. S. Sizikov, M. V. Rimskikh, and R. K. Mirdzhamolov, "Reconstructing blurred noisy images without using boundary conditions," Opt. Zh. 76, (5), 38 (2009) [J. Opt. Technol. 76, 279 (2009)].

V. S. Sizikov and I. A. Belov, "Reconstruction of smeared and out-of-focus images by regularization," Opt. Zh. 67, (4), 60 (2000) [J. Opt. Technol. 67, 351 (2000)].

Prib. Tekhnika Éksp. (1)

A. V. Gorshkov, "Improving the image resolution when processing the data of a physical experiment and finding the unknown spread function from the programs of the Reimage software package," Prib. Tekhnika Éksp. (2), 68 (1995).

Vychislit. Met. Progr. (2)

A. G. Yagola and N. A. Koshev, "The reconstruction of smeared and defocused color images," Vychislit. Met. Progr. 9, 207 (2008).

M. V. Aref’eva and A. F. Sysoev, "Rapid adjustment algorithms for digital reconstruction of images," Vychislit. Met. Progr. 39, 40 (1983).

Other (15)

I. S. Gruzman, V. S. Kirichuk, V. P. Kosykh, G. I. Peretyagin, and A. A. Spektor, Digital Image Processing in Information Systems, Izd. NGTU, Novosibirsk, 2000.

P. Wendykier and J. G. Nagy, "Image processing on modern CPUs and GPUs," Technical Report TR-2008-023, http://www.mathcs.emory.edu/technical-reports/techrep-00148.pdf.

K. Palmer, J. Nagy, and L. Perrone, Iterative methods for image restoration: MatLab object-oriented approach, 2002, http://citeseer.ist.psu.edu/lee02iterative.html

M. Christiansen and M. Hanke, Deblurring methods using antireflective boundary conditions, 2006, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.70.9837&rep=rep1&tipe=pdf

A. Arico, M. Donatelli, J. Nagy, and S. Serra-Capizzano, "The anti-reflective transform and regularization by filtering," Technical Report TR-2007-006-A, ftp://ftp.mathcs.emory.edu/pub/techreport/TR-2007-006-A.pdf.

V. D’yakonov and I. Abramenkova, "MatLab," Processing Signals and Images, Piter, St. Petersburg, 2002.

Philip Andrews, Digital Photography Manual, Carlton Books, 2004, Rosmen-Izdat, Moscow, 2005.

V. S. Sizikov, Mathematical Methods for Processing Measurement Results, Politekhnika, St. Petersburg, 2001.

Yu. P. Petrov and V. S. Sizikov, Well-Posed, Ill-Posed, and Intermediate Problems with Applications, VSP, Leiden–Boston, 2005.

V. S. Sizikov, "Using an integral-equation apparatus to reconstruct distorted images," Abstracts of the Conf. Integral Equations–2009, 2009, Izd. IPMÉ, Kiev, pp. 128‒130.

V. S. Sizikov, "Integral equations in the new truncation – blurring – rotation approach," Proc. Int. Conf. Integral Equations—2010, 2010, PAIS, Lviv, pp. 138‒142.

A. N. Tikhonov, A. V. Goncharskiĭ, and V. V. Stepanov, A. N. Tikhonov and A. V. Goncharskiĭ, ed., "Inverse problems of processing photographic images," Some Problems of Natural Science, Izd. MGU, Moscow, 1987, pp. 185‒195.

A. B. Bakushinskiĭ and A. V. Goncharskiĭ, Ill-Posed Problems. Numerical Methods and Applications, Izd. MGU, Moscow, 1989.

R. C. Gonzalez and R. E. Woods, Digital Image Processing, Prentice Hall, Upper Saddle River, N.J., 2002, Tekhnosfera, Moscow, 2005.

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital Image Processing Using MatLab, Prentice Hall, Upper Saddle River, N.J., 2004, Tekhnosfera, Moscow, 2006.

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