Abstract

We propose a practical sensor deblurring filtering method for images that are contaminated with noise. A sensor blurring function is usually modeled via a Gaussian-like function having a bell shape. The straightforward inverse function results in the magnification of noise at high frequencies. To address this issue, we apply a special spectral window to the inverse blurring function. This special window is called the power window, which is a Fourier-based smoothing window that preserves most of the spatial frequency components in the passband and attenuates quickly at the transition band. The power window is differentiable at the transition point, which gives a desired smooth property and limits the ripple effect. Utilizing the properties of the power window, we design the deblurring filter adaptively by estimating the energy of the signal and the noise of the image to determine the passband and the transition band of the filter. The deblurring filter design criteria are (a) the filter magnitude is less than 1 at the frequencies where the noise is stronger than the desired signal (the transition band), and (b) the filter magnitude is greater than 1 at the other frequencies (the passband). Therefore the adaptively designed deblurring filter is able to deblur the image by a desired amount based on the estimated or known blurring function while suppressing the noise in the output image. The deblurring filter performance is demonstrated by a human perception experiment in which 10 observers are to identify 12 military targets with 12 aspect angles. The results of comparing target identification probabilities with blurred and deblurred images and adding two levels of noise to blurred and deblurred noisy images are reported.

© 2007 Optical Society of America

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References

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  1. H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).
  2. B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Processing, T.S.Huang, ed. (Springer-Verlag, 1975), pp. 177-248.
  3. W. E. Blass and G. W. Halsey, Deconvolution of Absorption Spectra (Academic, 1981).
  4. G. Demoment, "Image reconstruction and restoration: overview of common estimation structures and problems," IEEE Trans. Acoust. Speech , Signal Process. 37, 2024-2036 (1989).
    [CrossRef]
  5. D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13(3), 43-64 (1996).
    [CrossRef]
  6. R. L. Lagendijk, A. Tekal, and J. Biemond, "Maximum likelihood image and blur identification: a unifying approach," Opt. Eng. 29, 422-435 (1990).
    [CrossRef]
  7. M. I. Sezan and A. M. Tekalp, "Tutorial review of recent developments in digital image restoration," in Visual Communications and Image Processing 1990, M. Kunt, ed., Proc. SPIE 1360, 1346-1359 (1990).
    [CrossRef]
  8. D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).
  9. J. Mateos, R. Molina, and A. K. Katsaggelos, "Approximation of posterior distributions in blind deconvolution using variational methods," in Proceedings of IEEE ICIP (IEEE, 2005), pp. II-770-II-773.
  10. A. Stern and N. S. Kopeika, "General restoration filter for vibrated-image restoration," Appl. Opt. 37, 7596-7603 (1998).
    [CrossRef]
  11. E. Lam, "Digital restoration of defocused images in the wavelet domain," Appl. Opt. 41, 4806-4811 (2002).
    [CrossRef] [PubMed]
  12. S. S. Young, "Alias-free image subsampling using Fourier-based windowing methods," Opt. Eng. 43, 843-855 (2004).
  13. M. Soumekh, Fourier Array Imaging (Prentice-Hall, 1994).

2005

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).

2002

1998

1996

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13(3), 43-64 (1996).
[CrossRef]

1990

R. L. Lagendijk, A. Tekal, and J. Biemond, "Maximum likelihood image and blur identification: a unifying approach," Opt. Eng. 29, 422-435 (1990).
[CrossRef]

M. I. Sezan and A. M. Tekalp, "Tutorial review of recent developments in digital image restoration," in Visual Communications and Image Processing 1990, M. Kunt, ed., Proc. SPIE 1360, 1346-1359 (1990).
[CrossRef]

1989

G. Demoment, "Image reconstruction and restoration: overview of common estimation structures and problems," IEEE Trans. Acoust. Speech , Signal Process. 37, 2024-2036 (1989).
[CrossRef]

Andrews, H. C.

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Biemond, J.

R. L. Lagendijk, A. Tekal, and J. Biemond, "Maximum likelihood image and blur identification: a unifying approach," Opt. Eng. 29, 422-435 (1990).
[CrossRef]

Blass, W. E.

W. E. Blass and G. W. Halsey, Deconvolution of Absorption Spectra (Academic, 1981).

Calef, B.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).

Demoment, G.

G. Demoment, "Image reconstruction and restoration: overview of common estimation structures and problems," IEEE Trans. Acoust. Speech , Signal Process. 37, 2024-2036 (1989).
[CrossRef]

Frieden, B. R.

B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Processing, T.S.Huang, ed. (Springer-Verlag, 1975), pp. 177-248.

Gerwe, D. R.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).

Halsey, G. W.

W. E. Blass and G. W. Halsey, Deconvolution of Absorption Spectra (Academic, 1981).

Hatzinakos, D.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13(3), 43-64 (1996).
[CrossRef]

Hunt, B. R.

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

Jain, M.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).

Katsaggelos, A. K.

J. Mateos, R. Molina, and A. K. Katsaggelos, "Approximation of posterior distributions in blind deconvolution using variational methods," in Proceedings of IEEE ICIP (IEEE, 2005), pp. II-770-II-773.

Kopeika, N. S.

Kundur, D.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13(3), 43-64 (1996).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk, A. Tekal, and J. Biemond, "Maximum likelihood image and blur identification: a unifying approach," Opt. Eng. 29, 422-435 (1990).
[CrossRef]

Lam, E.

Luna, C.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).

Mateos, J.

J. Mateos, R. Molina, and A. K. Katsaggelos, "Approximation of posterior distributions in blind deconvolution using variational methods," in Proceedings of IEEE ICIP (IEEE, 2005), pp. II-770-II-773.

Molina, R.

J. Mateos, R. Molina, and A. K. Katsaggelos, "Approximation of posterior distributions in blind deconvolution using variational methods," in Proceedings of IEEE ICIP (IEEE, 2005), pp. II-770-II-773.

Sezan, M. I.

M. I. Sezan and A. M. Tekalp, "Tutorial review of recent developments in digital image restoration," in Visual Communications and Image Processing 1990, M. Kunt, ed., Proc. SPIE 1360, 1346-1359 (1990).
[CrossRef]

Soumekh, M.

M. Soumekh, Fourier Array Imaging (Prentice-Hall, 1994).

Stern, A.

Tekal, A.

R. L. Lagendijk, A. Tekal, and J. Biemond, "Maximum likelihood image and blur identification: a unifying approach," Opt. Eng. 29, 422-435 (1990).
[CrossRef]

Tekalp, A. M.

M. I. Sezan and A. M. Tekalp, "Tutorial review of recent developments in digital image restoration," in Visual Communications and Image Processing 1990, M. Kunt, ed., Proc. SPIE 1360, 1346-1359 (1990).
[CrossRef]

Young, S. S.

S. S. Young, "Alias-free image subsampling using Fourier-based windowing methods," Opt. Eng. 43, 843-855 (2004).

Appl. Opt.

IEEE Signal Process. Mag.

D. Kundur and D. Hatzinakos, "Blind image deconvolution," IEEE Signal Process. Mag. 13(3), 43-64 (1996).
[CrossRef]

IEEE Trans. Acoust. Speech

G. Demoment, "Image reconstruction and restoration: overview of common estimation structures and problems," IEEE Trans. Acoust. Speech , Signal Process. 37, 2024-2036 (1989).
[CrossRef]

Opt. Eng.

R. L. Lagendijk, A. Tekal, and J. Biemond, "Maximum likelihood image and blur identification: a unifying approach," Opt. Eng. 29, 422-435 (1990).
[CrossRef]

S. S. Young, "Alias-free image subsampling using Fourier-based windowing methods," Opt. Eng. 43, 843-855 (2004).

Proc. SPIE

M. I. Sezan and A. M. Tekalp, "Tutorial review of recent developments in digital image restoration," in Visual Communications and Image Processing 1990, M. Kunt, ed., Proc. SPIE 1360, 1346-1359 (1990).
[CrossRef]

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, "Regularization for nonlinear image restoration using a prior on the object power spectrum," in Unconventional Imaging,Proc. SPIE 5896, 1-15 (2005).

Other

J. Mateos, R. Molina, and A. K. Katsaggelos, "Approximation of posterior distributions in blind deconvolution using variational methods," in Proceedings of IEEE ICIP (IEEE, 2005), pp. II-770-II-773.

M. Soumekh, Fourier Array Imaging (Prentice-Hall, 1994).

H. C. Andrews and B. R. Hunt, Digital Image Restoration (Prentice-Hall, 1977).

B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Processing, T.S.Huang, ed. (Springer-Verlag, 1975), pp. 177-248.

W. E. Blass and G. W. Halsey, Deconvolution of Absorption Spectra (Academic, 1981).

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Figures (9)

Fig. 1
Fig. 1

(a) Image acquisition model; (b) noise spectrum support band.

Fig. 2
Fig. 2

P-deblurring filter properties.

Fig. 3
Fig. 3

Example of the direct design for a FLIR tank image. (a) Original and (b) P-deblurred images are illustrated in the spatial domain. (c) Gaussian and (d) P-deblurring filters are illustrated in the spatial-frequency domain.

Fig. 4
Fig. 4

Cross sections of images in Fig. 3.

Fig. 5
Fig. 5

Ensemble magnitude spectrum of a FLIR image.

Fig. 6
Fig. 6

Thermal image examples for 12 target sets.

Fig. 7
Fig. 7

Gaussian function in frequency domain.

Fig. 8
Fig. 8

Sample examples of blurred, blurred with noise, and P-deblurred images.

Fig. 9
Fig. 9

Perception results. (a) Noise 0 Blur versus Noise 0 Deblur; (b) Noise 1 Blur versus Noise 1 Deblur; (c) Noise 2 Blur versus Noise 2 Deblur.

Tables (1)

Tables Icon

Table 1 Experimental Matrix Layout and Naming Convention

Equations (126)

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

r ( x , y )
g ( x , y )
n ( x , y )
r ( x , y )
g ( x , y )
f ( x , y )
r ( x , y ) = f ( x , y ) g ( x , y ) + n ( x , y ) .
( k x , k y )
R ( k x , k y ) = F ( k x , k y ) G ( k x , k y ) + N ( k x , k y ) ,
R ( k x , k y )
F ( k x , k y )
G ( k x , k y )
N ( k x , k y )
r ( x , y )
f ( x , y )
g ( x , y )
n ( x , y )
G ( k x , k y )
ρ = ± k x 2 + k y 2 ,
G ( ρ )
G ( ρ )
n ( x , y )
| N ( ρ ) | 2
f ( x , y )
r ( x , y )
h ( x , y )
R ( ρ ) G ( ρ ) = F ( ρ ) + N ( ρ ) G ( ρ ) .
| F ( ρ ) |
| N ( ρ ) |
[ f ( x , y ) ]
f ( x , y )
f ( x , y ) = arg max p [ f ( x , y ) | r ( x , y ) ] .
H Wiener ( ρ ) = G ( ρ ) * | G ( ρ ) | 2 + S n ( ρ ) / S f ( ρ ) ,
S f ( ρ )
S n ( ρ )
[ S n ( ρ ) / S f ( ρ ) 1 ]
1 / G ( ρ )
G ( ρ )
G ( ρ )
G ( ρ )
1 / G ( ρ )
[ S n ( ρ ) / S f ( ρ ) 1 ]
G ( ρ ) * / [ S n ( ρ ) / S f ( ρ ) ]
H W i e n e r ( ρ ) = G ( ρ ) * | G ( ρ ) | 2 + γ S n ( ρ ) / S f ( ρ ) .
( γ > 1 )
( γ < 1 )
G ( ρ )
P ( ρ )
P ( ρ ) = 1 G ( ρ ) W ( ρ ) ,
W ( ρ )
W ( ρ ) = exp ( α ρ n ) ,
( α , n )
P ( ρ ) = W ( ρ ) G ( ρ ) = exp ( α ρ n + ρ 2 2 σ 2 ) ,
P ( ρ ) ρ | ρ p = 0 ρ p = ( α n σ 2 ) 1 / ( 2 n ) .
ρ p
( α , n , σ )
ρ n
ρ n
ρ n
ρ n
ρ n
P ( ρ ) = { > 1 , if   ρ < ρ n = 1 , if   ρ = ρ n < 1 , if   ρ > ρ n .
ρ n
( α , n , σ )
ρ n
P ( ρ n ) = 1
ρ 0
( α , n )
ρ g
ρ g
1 G ( ρ g ) = m g ρ g = σ 2 ln ( m g ) ,
ρ 0
ρ g
Δ x
Δ y
N x Δ x Δ k x = 2 π ,
N y Δ y Δ k y = 2 π ,
N x
N y
Δ k x
Δ k y
Δ k x
Δ k y
± K x 0 = ± π Δ x ,
± K y 0 = ± π Δ y
σ = 0.25 ρ max , m g = 10 ,
ρ max
f ( x , y )
F x ( k x , y ) = ( x ) [ f ( x , y ) ] .
f ( x , y )
f i j , i = 1 , . . . , N ,
j = 1 , . . . , M
F x ( k x , y ) : F i j ( i ) , i = 1 , . . . N , 
j = 1 , . . . M
S = [ s 1 , . . . s N ]
s i = 1 M j = 1 M | F i j ( i ) | 2 , i = 1 , . . .   ,   N .
S M = [ s m 1 , . . . , s m N ] .
[ 1 1 1 ] / 3
G i = s m i + 1 s m i 1 .
ρ n
ρ n = σ 2 ln ( SNR ) ,
( α , n , σ )
P ( ρ )
( α , n )
P ρ 1 = ρ n = 1 ,
P ρ 2 = 0.8 ρ n = W 2 .
W 2 ( W 2 > 1 )
W 2
W 2 < exp ( ρ 2 2 2 σ 2 ) .
W 2
( α , n )
P i d = P P g P s P g , P g = 1 12 , P s = 9 10
P i d
P i d
( 60 % 90 % )
P i d
( 10 % 30 % )
( 38 % 65 % )
P i d
P i d
P i d
P i d
P i d
P i d
P i d
P i d

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