Abstract

We propose a novel linear blind deconvolution method for bi-level images. The proposed method seeks an optimal point spread function and two parameters that maximize a high order statistics based objective function. Unlike existing minimum entropy deconvolution and least squares minimization methods, the proposed method requires neither unrealistic assumption that the pixel values of a bi-level image are independently identically distributed samples of a random variable nor tuning of regularization parameters. We demonstrate the effectiveness of the proposed method in simulations and experiments.

© 2010 Optical Society of America

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  1. T. J. Holmes, "Blind deconvolution of quantum-limited incorehent imagery:maximum-likelihood approach," J. Opt. Soc. Am. A 9, 1052-1061 (1992).
    [CrossRef] [PubMed]
  2. D. A. Fish, A. M. Brinicombe, E. R. Pike and G. Walker, "Blind deconvolution by means of the Richardson-Lucy algorithm," J. Opt. Soc. Am. A 12, 58-65 (1995).
    [CrossRef]
  3. S. Esedoglu, "Blind deconvolution of bar code signals," Inverse Probl. 20, 121-135 (2004).
    [CrossRef]
  4. E. Y. Lam, "Blind bi-level image restoration with iterated quadratic programming," IEEE Trans. Circ. Syst. Part 2 52, 52-56 (2007).
    [CrossRef]
  5. J. Kim and H. Lee, "Joint nonuniform illumination estimation and deblurring for bar code signals," Opt. Express 17, 14817-14837 (2007).
    [CrossRef]
  6. D. Kundur and D. Hatzinakos, "A novel blind deconvolution scheme for image restoration using recursive filtering," IEEE Trans. Signal Process. 45, 375-390 (1998).
    [CrossRef]
  7. G. R. Ayers, and J. C. Dainty, ‘Iterative blind deconvolution method and its application," Opt. Lett. 13, 547-549 (1998).
    [CrossRef]
  8. T. Li and K. Lii, "A joint estimation approach for two-tone image deblurring by blind deconvolution," IEEE Trans. Image Process. 11, 847-858 (2002).
    [CrossRef]
  9. H. Wu, "Minimum entropy deconvolution for restoration of blurred two-tone images," Electronics Letters 26, 1183-1184 (1990).
    [CrossRef]
  10. N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
    [CrossRef]
  11. D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Trans. Image Process. 2, 223-235 (1993).
  12. J. A. Cadzow, "Blind deconvolution via cumulant extrema," IEEE Signal Processing Magazine., 24-41 (1996).
    [CrossRef]
  13. P. Campisi and K. Egiazarian, eds., Blind image deconvolution: Theory and applications, (CRC, New York, 2007).
    [CrossRef]
  14. H. Lee and J. Kim, "Retrospective correction of nonuniform illumination on bi-level images," Opt. Express 15, 23880-23893 (2009).
    [CrossRef]
  15. Y. Shen, E. Y. Lam, and N. Wong, "Binary image restoration by positive semidefinite programming," Opt. Lett. 32, 121-123 (2007).
    [CrossRef]
  16. M. D. Sacchi, D. R. Velis, and A. H. Comingues, "Minimum entropy deconvolution with frequency-domain constraints," Geophysics 59, 938-945 (1994).
    [CrossRef]
  17. D. Donoho, "On minimum entropy deconvolution," Applied Time Series Analysis II, D. F. Findley ed., (Academic, New York, 1991).
  18. N. F. Law and R. G. Lane, "Blind deconvolution using least squares minimisation," Opt. Commun. 128, 341-352 (1996).
    [CrossRef]
  19. J. Kim, "Restoration of bi-level images via iterative semi-blind Wiener filtering," Trans. KIEE 57, 1290-1294 (2008).
  20. H. L. Van Trees, Detection, estimation, and modulation theory, Part 1 (Wiley, 1968).
  21. R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB, (Prentice Hall, New York, 2002).
  22. T. Mathworks, Optimization toolbox user’s guide (Mathworks Inc., 2003).
  23. T. Chan and C. K. Wong, "Total variation blind deconvolution," IEEE Trans. Image Process. 7, 370-375 (1998).
    [CrossRef]
  24. E. K. P. Chong and S. H.  Zak, An introduction to optimization, 3rd ed., (Wiley-Interscience, New Jersey, 2008).
  25. W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C++, 2nd ed., (Cambridge, 2005).

2009 (1)

H. Lee and J. Kim, "Retrospective correction of nonuniform illumination on bi-level images," Opt. Express 15, 23880-23893 (2009).
[CrossRef]

2008 (1)

J. Kim, "Restoration of bi-level images via iterative semi-blind Wiener filtering," Trans. KIEE 57, 1290-1294 (2008).

2007 (3)

2004 (1)

S. Esedoglu, "Blind deconvolution of bar code signals," Inverse Probl. 20, 121-135 (2004).
[CrossRef]

2002 (1)

T. Li and K. Lii, "A joint estimation approach for two-tone image deblurring by blind deconvolution," IEEE Trans. Image Process. 11, 847-858 (2002).
[CrossRef]

1998 (3)

D. Kundur and D. Hatzinakos, "A novel blind deconvolution scheme for image restoration using recursive filtering," IEEE Trans. Signal Process. 45, 375-390 (1998).
[CrossRef]

G. R. Ayers, and J. C. Dainty, ‘Iterative blind deconvolution method and its application," Opt. Lett. 13, 547-549 (1998).
[CrossRef]

T. Chan and C. K. Wong, "Total variation blind deconvolution," IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

1996 (2)

J. A. Cadzow, "Blind deconvolution via cumulant extrema," IEEE Signal Processing Magazine., 24-41 (1996).
[CrossRef]

N. F. Law and R. G. Lane, "Blind deconvolution using least squares minimisation," Opt. Commun. 128, 341-352 (1996).
[CrossRef]

1995 (1)

1994 (1)

M. D. Sacchi, D. R. Velis, and A. H. Comingues, "Minimum entropy deconvolution with frequency-domain constraints," Geophysics 59, 938-945 (1994).
[CrossRef]

1993 (1)

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Trans. Image Process. 2, 223-235 (1993).

1992 (2)

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

T. J. Holmes, "Blind deconvolution of quantum-limited incorehent imagery:maximum-likelihood approach," J. Opt. Soc. Am. A 9, 1052-1061 (1992).
[CrossRef] [PubMed]

1990 (1)

H. Wu, "Minimum entropy deconvolution for restoration of blurred two-tone images," Electronics Letters 26, 1183-1184 (1990).
[CrossRef]

Ayers, G. R.

Baba, N.

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

Brinicombe, A. M.

Cadzow, J. A.

J. A. Cadzow, "Blind deconvolution via cumulant extrema," IEEE Signal Processing Magazine., 24-41 (1996).
[CrossRef]

Chan, T.

T. Chan and C. K. Wong, "Total variation blind deconvolution," IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

Comingues, A. H.

M. D. Sacchi, D. R. Velis, and A. H. Comingues, "Minimum entropy deconvolution with frequency-domain constraints," Geophysics 59, 938-945 (1994).
[CrossRef]

Dainty, J. C.

Esedoglu, S.

S. Esedoglu, "Blind deconvolution of bar code signals," Inverse Probl. 20, 121-135 (2004).
[CrossRef]

Fish, D. A.

Hatzinakos, D.

D. Kundur and D. Hatzinakos, "A novel blind deconvolution scheme for image restoration using recursive filtering," IEEE Trans. Signal Process. 45, 375-390 (1998).
[CrossRef]

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Trans. Image Process. 2, 223-235 (1993).

Holmes, T. J.

Isobe, S.

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

Kim, J.

H. Lee and J. Kim, "Retrospective correction of nonuniform illumination on bi-level images," Opt. Express 15, 23880-23893 (2009).
[CrossRef]

J. Kim, "Restoration of bi-level images via iterative semi-blind Wiener filtering," Trans. KIEE 57, 1290-1294 (2008).

J. Kim and H. Lee, "Joint nonuniform illumination estimation and deblurring for bar code signals," Opt. Express 17, 14817-14837 (2007).
[CrossRef]

Kundur, D.

D. Kundur and D. Hatzinakos, "A novel blind deconvolution scheme for image restoration using recursive filtering," IEEE Trans. Signal Process. 45, 375-390 (1998).
[CrossRef]

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Trans. Image Process. 2, 223-235 (1993).

Lam, E. Y.

Y. Shen, E. Y. Lam, and N. Wong, "Binary image restoration by positive semidefinite programming," Opt. Lett. 32, 121-123 (2007).
[CrossRef]

E. Y. Lam, "Blind bi-level image restoration with iterated quadratic programming," IEEE Trans. Circ. Syst. Part 2 52, 52-56 (2007).
[CrossRef]

Lane, R. G.

N. F. Law and R. G. Lane, "Blind deconvolution using least squares minimisation," Opt. Commun. 128, 341-352 (1996).
[CrossRef]

Law, N. F.

N. F. Law and R. G. Lane, "Blind deconvolution using least squares minimisation," Opt. Commun. 128, 341-352 (1996).
[CrossRef]

Lee, H.

H. Lee and J. Kim, "Retrospective correction of nonuniform illumination on bi-level images," Opt. Express 15, 23880-23893 (2009).
[CrossRef]

J. Kim and H. Lee, "Joint nonuniform illumination estimation and deblurring for bar code signals," Opt. Express 17, 14817-14837 (2007).
[CrossRef]

Li, T.

T. Li and K. Lii, "A joint estimation approach for two-tone image deblurring by blind deconvolution," IEEE Trans. Image Process. 11, 847-858 (2002).
[CrossRef]

Lii, K.

T. Li and K. Lii, "A joint estimation approach for two-tone image deblurring by blind deconvolution," IEEE Trans. Image Process. 11, 847-858 (2002).
[CrossRef]

Miura, N.

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

Noguchi, M.

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

Norimoto, Y.

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

Pike, E. R.

Sacchi, M. D.

M. D. Sacchi, D. R. Velis, and A. H. Comingues, "Minimum entropy deconvolution with frequency-domain constraints," Geophysics 59, 938-945 (1994).
[CrossRef]

Shen, Y.

Velis, D. R.

M. D. Sacchi, D. R. Velis, and A. H. Comingues, "Minimum entropy deconvolution with frequency-domain constraints," Geophysics 59, 938-945 (1994).
[CrossRef]

Walker, G.

Wong, C. K.

T. Chan and C. K. Wong, "Total variation blind deconvolution," IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

Wong, N.

Wu, H.

H. Wu, "Minimum entropy deconvolution for restoration of blurred two-tone images," Electronics Letters 26, 1183-1184 (1990).
[CrossRef]

Electronics Letters (1)

H. Wu, "Minimum entropy deconvolution for restoration of blurred two-tone images," Electronics Letters 26, 1183-1184 (1990).
[CrossRef]

Geophysics (1)

M. D. Sacchi, D. R. Velis, and A. H. Comingues, "Minimum entropy deconvolution with frequency-domain constraints," Geophysics 59, 938-945 (1994).
[CrossRef]

IEEE Signal Processing Magazine. (1)

J. A. Cadzow, "Blind deconvolution via cumulant extrema," IEEE Signal Processing Magazine., 24-41 (1996).
[CrossRef]

IEEE Trans. Circ. Syst (1)

E. Y. Lam, "Blind bi-level image restoration with iterated quadratic programming," IEEE Trans. Circ. Syst. Part 2 52, 52-56 (2007).
[CrossRef]

IEEE Trans. Image Process. (3)

T. Li and K. Lii, "A joint estimation approach for two-tone image deblurring by blind deconvolution," IEEE Trans. Image Process. 11, 847-858 (2002).
[CrossRef]

T. Chan and C. K. Wong, "Total variation blind deconvolution," IEEE Trans. Image Process. 7, 370-375 (1998).
[CrossRef]

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Trans. Image Process. 2, 223-235 (1993).

IEEE Trans. Signal Process. (1)

D. Kundur and D. Hatzinakos, "A novel blind deconvolution scheme for image restoration using recursive filtering," IEEE Trans. Signal Process. 45, 375-390 (1998).
[CrossRef]

Inverse Probl. (1)

S. Esedoglu, "Blind deconvolution of bar code signals," Inverse Probl. 20, 121-135 (2004).
[CrossRef]

J. Mod. Opt. (1)

N. Miura, N. Baba, S. Isobe, M. Noguchi, and Y. Norimoto, "Binary star reconstruction with use of the blind deconvolution method," J. Mod. Opt. 39, 1137-1146 (1992).
[CrossRef]

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

N. F. Law and R. G. Lane, "Blind deconvolution using least squares minimisation," Opt. Commun. 128, 341-352 (1996).
[CrossRef]

Opt. Express (2)

J. Kim and H. Lee, "Joint nonuniform illumination estimation and deblurring for bar code signals," Opt. Express 17, 14817-14837 (2007).
[CrossRef]

H. Lee and J. Kim, "Retrospective correction of nonuniform illumination on bi-level images," Opt. Express 15, 23880-23893 (2009).
[CrossRef]

Opt. Lett. (2)

Trans. KIEE (1)

J. Kim, "Restoration of bi-level images via iterative semi-blind Wiener filtering," Trans. KIEE 57, 1290-1294 (2008).

Other (7)

H. L. Van Trees, Detection, estimation, and modulation theory, Part 1 (Wiley, 1968).

R. C. Gonzalez, R. E. Woods, and S. L. Eddins, Digital image processing using MATLAB, (Prentice Hall, New York, 2002).

T. Mathworks, Optimization toolbox user’s guide (Mathworks Inc., 2003).

E. K. P. Chong and S. H.  Zak, An introduction to optimization, 3rd ed., (Wiley-Interscience, New Jersey, 2008).

W. H. Press, S. A. Teukolsky, W. T. Vetterling and B. P. Flannery, Numerical recipes in C++, 2nd ed., (Cambridge, 2005).

D. Donoho, "On minimum entropy deconvolution," Applied Time Series Analysis II, D. F. Findley ed., (Academic, New York, 1991).

P. Campisi and K. Egiazarian, eds., Blind image deconvolution: Theory and applications, (CRC, New York, 2007).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Images for simulation: (a) true barcode image; (b) true text image.

Fig. 2.
Fig. 2.

Barcode images: (a) noisy blurred barcode image (σ=6 pixels, SNR=25dB, COR=0.786); (b) binarization of Fig. 2(a) (BER=12.63%); (c) deblurred barcode image using the DMED method (COR=0.372); (d) binarization of Fig. 2(c) (BER=39.76%); (e) deblurred barcode image using the LSM method (COR=0.813); (f) binarization of Fig. 2(e) (BER=10.06%); (g) deblurred barcode image using the proposed method (COR=0.901); (h) binarization of Fig. 2(f) (BER=3.86%).

Fig. 3.
Fig. 3.

Text images: (a) noisy blurred text image (σ=5 pixels, SNR=35dB, COR=0.751); (b) binarization of Fig. 3(a) (BER=6.69%); (c) deblurred text image using the DMED method (COR=0.811); (d) binarization of Fig. 3(c) (BER=4.93%); (e) deblurred text image using the LSM method (COR=0.754); (f) binarization of Fig. 3(e) (BER=6.67%); (g) deblurred text image using the proposed method (COR=0.894); (h) binarization of Fig. 3(g) (BER=2.18%).

Fig. 4.
Fig. 4.

Images for AR blur: (a) barcode; (b) text; (c) binarization of Fig. 4(a)(BER 8.48%); (d) binarization of Fig. 4(b)(BER 6.14%); (e) barcode by DMED (BER 41.9%); (f) text by DMED(BER 2.92%); (g) barcode by LSM(BER 6.21%); (h) text by LSM (BER 4.66%); (i) barcode by proposed (BER 1.55%) (j) text by proposed (BER 1.96%).

Fig. 5.
Fig. 5.

Real barcode image and deblurred images: (a) real noisy blurred barcode image; (b) binarization of image shown in Fig. 5(a); (c) deblurred real barcode image using the DMED method; (d) binarization of image shown in Fig. 5(c); (e) deblurred real barcode image using the LSM method; (f) binarization of image shown in Fig. 5(e); (e) deblurred real barcode image using the proposed method; (f) binarization of image shown in Fig. 5(g).

Fig. 6.
Fig. 6.

Real text image and deblurred images: (a) real noisy blurred text image; (b) binarization of image shown in Fig. 6(a); (c) deblurred real text image using the DMED method; (d) binarization of image shown in Fig. 6(c); (e) deblurred real text image using the LSM method; (f) binarization of image shown in Fig. 6(e); (g) deblurred real text image using the proposed method; (h) binarization of image shown in Fig. 6(g).

Tables (6)

Tables Icon

Table 1. COR values of blurred images and deblurred images by the three methods for the barcode image with different amounts of blur and SNR values (unit for SNR is dB.)

Tables Icon

Table 2. BER values of blurred images and deblurred images by the three methods for the barcode image with different amounts of blur and SNR values (unit for BER is %.)

Tables Icon

Table 3. COR values of blurred images and deblurred images by the three methods for the text image with different amounts of blur and SNR values (unit for SNR is dB.)

Tables Icon

Table 4. BER values of blurred images and deblurred images by the three methods for the text image with different amounts of blur and SNR values (unit for BER is %.)

Tables Icon

Table 5. BER values of blurred images and deblurred images by the three methods for the barcode image blurred by AR filter with different amounts of blur and SNR values (unit for BER is %.)

Tables Icon

Table 6. BER values of blurred images and deblurred images by the three methods for the text image blurred by AR filter with different amounts of blur and SNR values (unit for BER is %.)

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

y ( m , n ) = Σ j Σ k h ( n j , m k ) x ( j , k ) + w ( m , n ) , m = 1 , . . . M , n = 1 , . . . , N ,
( x ̂ , h ̂ ) = argmin x { β 1 , β 2 } , h L ( x , h ; y ) + λ 1 R 1 ( x ) + λ 2 R 2 ( h ) ,
( x ̂ , h ̂ ) = argmin x , h Σ n , m y ( n , m ) h ( n , m ) * x ( n , m ) 2 + λ 1 R 1 ( x ) + λ 2 R 2 ( h ) ,
R 1 ( x ) = Σ n , m ( x ( n + 1 , m ) 2 x ( n , m ) + x ( n 1 , m ) ) 2 + ( x ( n , m + 1 ) 2 x ( n , m ) + x ( n 1 , m 1 ) ) 2
R 2 ( h ) = Σ n , m ( h ( n + 1 , m ) 2 h ( n , m ) + h ( n 1 , m ) ) 2 + ( h ( n , m + 1 ) 2 h ( n , m ) + h ( n 1 , m 1 ) ) 2
f ̂ = argmax f Φ z ( f ; y ) ,
z ( m , n ; f ) = Σ j Σ k f ( m j , n k ) y ( j , k ) , m = 1 , . . . M , n = 1 , . . . , N ,
f ̂ = argmax f 1 MN Σ m , n ( z ( m , n ; f ) ) 4 ( 1 MN Σ m , n ( z ( m , n ; f ) ) 2 ) 2 .
f ̂ = argmax f Σ n Σ m d x 4 ( m , n ; f ) ( Σ m d x 2 ( m , n ; f ) ) 2 + Σ m Σ n d y 4 ( m , n ; f ) ( Σ n d y 2 ( m , n ; f ) ) 2 ,
z ( n , m ; f ~ ) α ~ = k , ( m , n ) ,
Φ ( f , α ) = 1 MN Σ m , n ( z ( m , n ; f ) α ) 4 ( 1 MN Σ m , n ( z ( m , n ; f ) α ) 2 ) 2 .
z ( m , n ; h , η ) = 1 { H * ( ω x , ω y ) H ( ω x , ω y ) 2 + η Y ( ω x , ω y ) } ,
( h ̂ , α ̂ , η ̂ ) = argmin h , α 1 α α 2 , η 1 η η 2 Φ z ( h , α , η ) ,
Φ z ( h , α , η ) = 1 MN Σ m , n ( z ( m , n ; h , η ) α ) 4 ( 1 MN Σ m , n ( z ( m , n ; h , η ) α ) 2 ) 2 .
Y ( ω x , ω y ) = ( 1 ρ 4 ) ( 1 ρ e j ω x ) ( 1 ρ e j ω x ) ( 1 ρ e j ω y ) ( 1 ρ e j ω y ) X ( ω x , ω y ) + W ( ω x , ω y ) ,

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