Abstract

Retinal images are frequently corrupted by unwanted variations in intensity that occur due to general imperfections in the image acquisition process. This inhomogeneous illumination across the retina can limit the useful information accessible within the acquired image. Specifically, this can lead to serious difficulties when performing image processing tasks requiring quantitative analysis of features present on the retina. Given that the spatial frequency content of the shading profile often overlaps with that of retinal features, retrospectively correcting for inhomogeneous illumination while maintaining the radiometric fidelity of the real data can be challenging. This paper describes a simple method for obtaining an estimate of the illumination profile in retinal images, with the particular goal of minimizing its influence upon features of interest. This is achieved by making use of Laplace interpolation and a multiplicative image formation model.

© 2012 Optical Society of America

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2010

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
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L. Kubecka, J. Jan, and R. Kolar, “Retrospective illumination correction of retinal images,” Int. J. Biomed. Imaging 2010, 780262 (2010).
[CrossRef]

2007

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

2006

T. Chanwimaluang, G. Fan, and S. R. Fransen, “Hybrid retinal image registration,” IEEE Trans. Inf. Technol. Biomed. 10, 129–142 (2006).
[CrossRef]

2005

M. Foracchia, E. Grisan, and A. Ruggeri, “Luminosity and contrast normalization in retinal images,” Med. Image Anal. 9, 179–190 (2005).
[CrossRef]

2002

D. Tomazˆeviĉ, B. Likar, and F. Pernuŝ, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208, 212–223 (2002).
[CrossRef]

2001

2000

A. Hoover, V. Kouznetsova, and M. Goldbaum, “Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,” IEEE Trans. Med. Imag. 19, 203–210 (2000).
[CrossRef]

1999

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automated localization of the optic disc, fovea, and retinal blood vessels from digital color fundus images,” Br. J. Ophthalmol. 83, 902–910 (1999).
[CrossRef]

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuŝ, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197, 285–295 (1999).

1997

1994

L. Leistritz and D. Schweitzer, “Automated detection and quantification of exudates in retinal images,” Proc. SPIE 2298, 690 (1994).
[CrossRef]

1993

B. Dawant, A. Zijdenbos, and R. Margolin, “Correction of intensity variations in MR images for computer-aided tissue classification,” IEEE Trans. Med. Imag. 12, 770–781 (1993).
[CrossRef]

1992

B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

1989

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

1985

W. R. Tobler and S. Kennedy, “Smooth multidimensional interpolation,” Geogr. Anal. 17, 251–257 (1985).
[CrossRef]

Alm, A.

P. L. Kaufman and A. Alm, Adler’s Physiology of the Eye: Clinical Application (Mosby, 2003).

Ballester, C.

M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, “Image inpainting,” in Proceedings of ACM Siggraph(Addison-Wesley, 2000), pp. 417–424.

Bertalmio, M.

M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, “Image inpainting,” in Proceedings of ACM Siggraph(Addison-Wesley, 2000), pp. 417–424.

J. Verdera, V. Caselles, M. Bertalmio, and G. Sapiro, “Inpainting surface holes,” in Proceedings of International Conference on Image Processing (IEEE, 2003), pp. 903–906.

Bouzas, B. H.

B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

Boyce, J. F.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automated localization of the optic disc, fovea, and retinal blood vessels from digital color fundus images,” Br. J. Ophthalmol. 83, 902–910 (1999).
[CrossRef]

Burns, S. A.

Caselles, V.

M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, “Image inpainting,” in Proceedings of ACM Siggraph(Addison-Wesley, 2000), pp. 417–424.

J. Verdera, V. Caselles, M. Bertalmio, and G. Sapiro, “Inpainting surface holes,” in Proceedings of International Conference on Image Processing (IEEE, 2003), pp. 903–906.

Chanwimaluang, T.

T. Chanwimaluang, G. Fan, and S. R. Fransen, “Hybrid retinal image registration,” IEEE Trans. Inf. Technol. Biomed. 10, 129–142 (2006).
[CrossRef]

Chatterjee, S.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

Chaudhuri, S.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

Cook, H. L.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automated localization of the optic disc, fovea, and retinal blood vessels from digital color fundus images,” Br. J. Ophthalmol. 83, 902–910 (1999).
[CrossRef]

Dawant, B.

B. Dawant, A. Zijdenbos, and R. Margolin, “Correction of intensity variations in MR images for computer-aided tissue classification,” IEEE Trans. Med. Imag. 12, 770–781 (1993).
[CrossRef]

Dawczynski, J.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

Delori, F. C.

Fan, G.

T. Chanwimaluang, G. Fan, and S. R. Fransen, “Hybrid retinal image registration,” IEEE Trans. Inf. Technol. Biomed. 10, 129–142 (2006).
[CrossRef]

Flannery, B.

W. Press, B. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Press Syndicate, University Cambridge, 2007).

Fleming, A. D.

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

Foracchia, M.

M. Foracchia, E. Grisan, and A. Ruggeri, “Luminosity and contrast normalization in retinal images,” Med. Image Anal. 9, 179–190 (2005).
[CrossRef]

Fransen, S. R.

T. Chanwimaluang, G. Fan, and S. R. Fransen, “Hybrid retinal image registration,” IEEE Trans. Inf. Technol. Biomed. 10, 129–142 (2006).
[CrossRef]

Goatman, K. A.

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

Goger, D. G.

Goldbaum, M.

A. Hoover, V. Kouznetsova, and M. Goldbaum, “Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,” IEEE Trans. Med. Imag. 19, 203–210 (2000).
[CrossRef]

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

Grisan, E.

M. Foracchia, E. Grisan, and A. Ruggeri, “Luminosity and contrast normalization in retinal images,” Med. Image Anal. 9, 179–190 (2005).
[CrossRef]

Hammer, M.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

Hammond, B. R.

Hoover, A.

A. Hoover, V. Kouznetsova, and M. Goldbaum, “Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,” IEEE Trans. Med. Imag. 19, 203–210 (2000).
[CrossRef]

Jan, J.

L. Kubecka, J. Jan, and R. Kolar, “Retrospective illumination correction of retinal images,” Int. J. Biomed. Imaging 2010, 780262 (2010).
[CrossRef]

L. Kubecka, R. Kolar, J. Jan, and R. Jirik, “Retrospective illumination correction of retinal images,” presented at Eurasip Biosignal, Brno, Czech Republic, 28–30 June 2006.

Jentsch, S.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

Jirik, R.

L. Kubecka, R. Kolar, J. Jan, and R. Jirik, “Retrospective illumination correction of retinal images,” presented at Eurasip Biosignal, Brno, Czech Republic, 28–30 June 2006.

Katz, N.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

Kaufman, P. L.

P. L. Kaufman and A. Alm, Adler’s Physiology of the Eye: Clinical Application (Mosby, 2003).

Kennedy, S.

W. R. Tobler and S. Kennedy, “Smooth multidimensional interpolation,” Geogr. Anal. 17, 251–257 (1985).
[CrossRef]

Kolar, R.

L. Kubecka, J. Jan, and R. Kolar, “Retrospective illumination correction of retinal images,” Int. J. Biomed. Imaging 2010, 780262 (2010).
[CrossRef]

L. Kubecka, R. Kolar, J. Jan, and R. Jirik, “Retrospective illumination correction of retinal images,” presented at Eurasip Biosignal, Brno, Czech Republic, 28–30 June 2006.

Kouznetsova, V.

A. Hoover, V. Kouznetsova, and M. Goldbaum, “Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,” IEEE Trans. Med. Imag. 19, 203–210 (2000).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics (Wiley, 1993).

Kubecka, L.

L. Kubecka, J. Jan, and R. Kolar, “Retrospective illumination correction of retinal images,” Int. J. Biomed. Imaging 2010, 780262 (2010).
[CrossRef]

L. Kubecka, R. Kolar, J. Jan, and R. Jirik, “Retrospective illumination correction of retinal images,” presented at Eurasip Biosignal, Brno, Czech Republic, 28–30 June 2006.

Leistritz, L.

L. Leistritz and D. Schweitzer, “Automated detection and quantification of exudates in retinal images,” Proc. SPIE 2298, 690 (1994).
[CrossRef]

Likar, B.

D. Tomazˆeviĉ, B. Likar, and F. Pernuŝ, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208, 212–223 (2002).
[CrossRef]

B. Likar, M. A. Viergever, and F. Pernuŝ, “Retrospective correction of MR intensity inhomogeneity by information minimization,” IEEE Trans. Med. Imag. 20, 1398–1410(2001).
[CrossRef]

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuŝ, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197, 285–295 (1999).

Maintz, J. B. A.

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuŝ, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197, 285–295 (1999).

Margolin, R.

B. Dawant, A. Zijdenbos, and R. Margolin, “Correction of intensity variations in MR images for computer-aided tissue classification,” IEEE Trans. Med. Imag. 12, 770–781 (1993).
[CrossRef]

Nelson, M.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

Nill, B.

B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

Olson, J. A.

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

Pernus, F.

D. Tomazˆeviĉ, B. Likar, and F. Pernuŝ, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208, 212–223 (2002).
[CrossRef]

B. Likar, M. A. Viergever, and F. Pernuŝ, “Retrospective correction of MR intensity inhomogeneity by information minimization,” IEEE Trans. Med. Imag. 20, 1398–1410(2001).
[CrossRef]

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuŝ, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197, 285–295 (1999).

Philip, S.

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

Press, W.

W. Press, B. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Press Syndicate, University Cambridge, 2007).

Ruggeri, A.

M. Foracchia, E. Grisan, and A. Ruggeri, “Luminosity and contrast normalization in retinal images,” Med. Image Anal. 9, 179–190 (2005).
[CrossRef]

Sadiku, M. N. O.

M. N. O. Sadiku, Elements of Electromagnetics (Sounders College, 2000).

Sapiro, G.

J. Verdera, V. Caselles, M. Bertalmio, and G. Sapiro, “Inpainting surface holes,” in Proceedings of International Conference on Image Processing (IEEE, 2003), pp. 903–906.

M. Bertalmio, G. Sapiro, V. Caselles, and C. Ballester, “Image inpainting,” in Proceedings of ACM Siggraph(Addison-Wesley, 2000), pp. 417–424.

Schweitzer, D.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

L. Leistritz and D. Schweitzer, “Automated detection and quantification of exudates in retinal images,” Proc. SPIE 2298, 690 (1994).
[CrossRef]

Sharp, P. F.

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

Sinthanayothin, C.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automated localization of the optic disc, fovea, and retinal blood vessels from digital color fundus images,” Br. J. Ophthalmol. 83, 902–910 (1999).
[CrossRef]

Snodderly, D. M.

Teukolsky, S.

W. Press, B. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Press Syndicate, University Cambridge, 2007).

Tobler, W. R.

W. R. Tobler and S. Kennedy, “Smooth multidimensional interpolation,” Geogr. Anal. 17, 251–257 (1985).
[CrossRef]

Tomazˆevic, D.

D. Tomazˆeviĉ, B. Likar, and F. Pernuŝ, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208, 212–223 (2002).
[CrossRef]

Verdera, J.

J. Verdera, V. Caselles, M. Bertalmio, and G. Sapiro, “Inpainting surface holes,” in Proceedings of International Conference on Image Processing (IEEE, 2003), pp. 903–906.

Vetterling, W. T.

W. Press, B. Flannery, S. Teukolsky, and W. T. Vetterling, Numerical Recipes: The Art of Scientific Computing (Press Syndicate, University Cambridge, 2007).

Viergever, M. A.

B. Likar, M. A. Viergever, and F. Pernuŝ, “Retrospective correction of MR intensity inhomogeneity by information minimization,” IEEE Trans. Med. Imag. 20, 1398–1410(2001).
[CrossRef]

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuŝ, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197, 285–295 (1999).

Williamson, T. H.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automated localization of the optic disc, fovea, and retinal blood vessels from digital color fundus images,” Br. J. Ophthalmol. 83, 902–910 (1999).
[CrossRef]

Wolf, S.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

Wolf-Schnurrbusch, U. E. K.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

Wooten, B. R.

Zijdenbos, A.

B. Dawant, A. Zijdenbos, and R. Margolin, “Correction of intensity variations in MR images for computer-aided tissue classification,” IEEE Trans. Med. Imag. 12, 770–781 (1993).
[CrossRef]

Br. J. Ophthalmol.

C. Sinthanayothin, J. F. Boyce, H. L. Cook, and T. H. Williamson, “Automated localization of the optic disc, fovea, and retinal blood vessels from digital color fundus images,” Br. J. Ophthalmol. 83, 902–910 (1999).
[CrossRef]

Geogr. Anal.

W. R. Tobler and S. Kennedy, “Smooth multidimensional interpolation,” Geogr. Anal. 17, 251–257 (1985).
[CrossRef]

IEEE Trans. Inf. Technol. Biomed.

T. Chanwimaluang, G. Fan, and S. R. Fransen, “Hybrid retinal image registration,” IEEE Trans. Inf. Technol. Biomed. 10, 129–142 (2006).
[CrossRef]

IEEE Trans. Med. Imag.

S. Chaudhuri, S. Chatterjee, N. Katz, M. Nelson, and M. Goldbaum, “Detection of blood vessels in retinal images using two-dimensional matched filters,” IEEE Trans. Med. Imag. 8, 263–269 (1989).
[CrossRef]

A. Hoover, V. Kouznetsova, and M. Goldbaum, “Locating blood vessels in retinal images by piecewise threshold probing of a matched filter response,” IEEE Trans. Med. Imag. 19, 203–210 (2000).
[CrossRef]

B. Dawant, A. Zijdenbos, and R. Margolin, “Correction of intensity variations in MR images for computer-aided tissue classification,” IEEE Trans. Med. Imag. 12, 770–781 (1993).
[CrossRef]

B. Likar, M. A. Viergever, and F. Pernuŝ, “Retrospective correction of MR intensity inhomogeneity by information minimization,” IEEE Trans. Med. Imag. 20, 1398–1410(2001).
[CrossRef]

Int. J. Biomed. Imaging

L. Kubecka, J. Jan, and R. Kolar, “Retrospective illumination correction of retinal images,” Int. J. Biomed. Imaging 2010, 780262 (2010).
[CrossRef]

J. Biomed. Opt.

D. Schweitzer, S. Jentsch, J. Dawczynski, M. Hammer, U. E. K. Wolf-Schnurrbusch, and S. Wolf, “Simple and objective method for routine detection of the macular pigment xanthophyll,” J. Biomed. Opt. 15, 061714 (2010).
[CrossRef]

J. Microsc.

B. Likar, J. B. A. Maintz, M. A. Viergever, and F. Pernuŝ, “Retrospective shading correction based on entropy minimization,” J. Microsc. 197, 285–295 (1999).

D. Tomazˆeviĉ, B. Likar, and F. Pernuŝ, “Comparative evaluation of retrospective shading correction methods,” J. Microsc. 208, 212–223 (2002).
[CrossRef]

J. Opt. Soc. Am. A

Med. Image Anal.

M. Foracchia, E. Grisan, and A. Ruggeri, “Luminosity and contrast normalization in retinal images,” Med. Image Anal. 9, 179–190 (2005).
[CrossRef]

Opt. Eng.

B. Nill and B. H. Bouzas, “Objective image quality measure derived from digital image power spectra,” Opt. Eng. 31, 813–825 (1992).
[CrossRef]

Phys. Med. Biol.

A. D. Fleming, K. A. Goatman, S. Philip, J. A. Olson, and P. F. Sharp, “Automatic detection of retinal anatomy to assist diabetic retinopathy screening,” Phys. Med. Biol. 52, 331–345 (2007).
[CrossRef]

Proc. SPIE

L. Leistritz and D. Schweitzer, “Automated detection and quantification of exudates in retinal images,” Proc. SPIE 2298, 690 (1994).
[CrossRef]

Other

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Figures (5)

Fig. 1.
Fig. 1.

Illustration of Laplace interpolation as given by Eq. (7).

Fig. 2.
Fig. 2.

(a) Original acquired image, (b) image with vascular detail masked, and (c) image with both vascular detail and the foveal region masked.

Fig. 3.
Fig. 3.

(a) Shading function estimate U^(x,y), after applying the Laplace interpolation procedure and additional smoothing. (b) Estimate of the shading-free image obtained after applying illumination correction.

Fig. 4.
Fig. 4.

3D projection of: (a) the original acquired image, (b) the estimate of the shading function, and (c) the estimate of the shading-free image. The z dimension corresponds to grayscale pixel value. The figures are displayed with 15° horizontal rotation about the z-axis and 60° vertical elevation of the viewpoint.

Fig. 5.
Fig. 5.

Shading correction results for three retinal images: (a) Green illumination, acquired image σμ=0.084, corrected image σμ=0.071 (15.5% reduction). (b) Blue illumination, acquired image σμ=0.128, corrected image σμ=0.111 (13.3% reduction). (c) Green illumination, acquired image σμ=0.082, corrected image σμ=0.071 (13.4% reduction).

Equations (12)

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I(x,y)=U(x,y)It(x,y).
2U=0,
2Ux2+2Uy2=0.
Ω|U(x,y)|2dxdy,
2Ux2|x=x0U(x0+Δx,y0)2U(x0,y0)+U(x0Δx,y0)(Δx)2,
2Uy2|y=y0U(x0,y0+Δy)2U(x0,y0)+U(x0,y0Δy)(Δy)2,
U^0=14(Uu+Ud+Ul+Ur),
Ax=b,
x=A1b.
It^(x,y)=αI(x,y)U^(x,y)+β,
σμ=1Nk(μkμ0)2,
μ0=1Nkμk.

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