Abstract

It has been shown many times that using different versions of a scene perturbed with different blurs improved the quality of a restored image compared with using a single blurred image. We focus on large defocus blurs, and we first consider a case in which two different blurring kernels are used. We analyze with numerical simulations the influence of the relative diameter of both kernels on the quality of restoration. We then quantitatively evaluate how the two-kernel approach improves the robustness of restoration to a difference between the kernels used in designing the algorithm and the actual kernels that have perturbed the image. We finally show that using three different kernels may not improve the restoration performance compared with the two-kernel approach but still improves the robustness to kernel estimation.

© 2000 Optical Society of America

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References

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    [Crossref]
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    [Crossref]
  3. C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operators—an overview, plus performance characterizations for imaging sensors,” Proc. IEEE 78, 723–734 (1990).
    [Crossref]
  4. L. P. Yaroslavsky, H. J. Caulfield, “Deconvolution of multiple images of the same object,” Appl. Opt. 33, 2157–2162 (1994).
    [Crossref] [PubMed]
  5. M. Piana, M. Bertero, “Regularized deconvolution of multiple images of the same object,” J. Opt. Soc. Am. A 13, 1516–1523 (1996).
    [Crossref]
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  7. S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
    [Crossref]
  8. A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall Information and System Sciences Series (Prentice-Hall, Englewood Cliffs, N.J., 1989).
  9. P. A. Stokseth, “Properties of a defocused optical system,” J. Opt. Soc. Am. 59, 1314–1321 (1969).
    [Crossref]
  10. H. C. Lee, “Review of image-blur models in a photographic system using the principles of optics,” Opt. Eng. 29, 405–421 (1990).
    [Crossref]
  11. M. Subbarao, T. C. Wei, G. Surya, “Focused image recovery from two defocused images recorded with different camera settings,” IEEE Trans. Image Process. 12, 1613–1628 (1995).
    [Crossref]
  12. G. H. Golub, M. Health, G. Wabha, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [Crossref]

1996 (1)

1995 (1)

M. Subbarao, T. C. Wei, G. Surya, “Focused image recovery from two defocused images recorded with different camera settings,” IEEE Trans. Image Process. 12, 1613–1628 (1995).
[Crossref]

1994 (1)

1990 (3)

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operators—an overview, plus performance characterizations for imaging sensors,” Proc. IEEE 78, 723–734 (1990).
[Crossref]

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[Crossref]

H. C. Lee, “Review of image-blur models in a photographic system using the principles of optics,” Opt. Eng. 29, 405–421 (1990).
[Crossref]

1984 (1)

1982 (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

1979 (1)

G. H. Golub, M. Health, G. Wabha, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

1969 (1)

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Berenstein, C. A.

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operators—an overview, plus performance characterizations for imaging sensors,” Proc. IEEE 78, 723–734 (1990).
[Crossref]

Bertero, M.

Caulfield, H. J.

Ghiglia, D. C.

Golub, G. H.

G. H. Golub, M. Health, G. Wabha, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

Health, M.

G. H. Golub, M. Health, G. Wabha, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall Information and System Sciences Series (Prentice-Hall, Englewood Cliffs, N.J., 1989).

Lee, H. C.

H. C. Lee, “Review of image-blur models in a photographic system using the principles of optics,” Opt. Eng. 29, 405–421 (1990).
[Crossref]

Mersereau, R. M.

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[Crossref]

Patrick, E. V.

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operators—an overview, plus performance characterizations for imaging sensors,” Proc. IEEE 78, 723–734 (1990).
[Crossref]

Piana, M.

Reeves, S. J.

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[Crossref]

Stokseth, P. A.

Subbarao, M.

M. Subbarao, T. C. Wei, G. Surya, “Focused image recovery from two defocused images recorded with different camera settings,” IEEE Trans. Image Process. 12, 1613–1628 (1995).
[Crossref]

Surya, G.

M. Subbarao, T. C. Wei, G. Surya, “Focused image recovery from two defocused images recorded with different camera settings,” IEEE Trans. Image Process. 12, 1613–1628 (1995).
[Crossref]

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Wabha, G.

G. H. Golub, M. Health, G. Wabha, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Wei, T. C.

M. Subbarao, T. C. Wei, G. Surya, “Focused image recovery from two defocused images recorded with different camera settings,” IEEE Trans. Image Process. 12, 1613–1628 (1995).
[Crossref]

Yaroslavsky, L. P.

Appl. Opt. (1)

IEEE Trans. Image Process. (1)

M. Subbarao, T. C. Wei, G. Surya, “Focused image recovery from two defocused images recorded with different camera settings,” IEEE Trans. Image Process. 12, 1613–1628 (1995).
[Crossref]

J. Opt. Soc. Am. (1)

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

Opt. Eng. (3)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[Crossref]

H. C. Lee, “Review of image-blur models in a photographic system using the principles of optics,” Opt. Eng. 29, 405–421 (1990).
[Crossref]

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[Crossref]

Proc. IEEE (1)

C. A. Berenstein, E. V. Patrick, “Exact deconvolution for multiple convolution operators—an overview, plus performance characterizations for imaging sensors,” Proc. IEEE 78, 723–734 (1990).
[Crossref]

Technometrics (1)

G. H. Golub, M. Health, G. Wabha, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[Crossref]

Other (2)

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

A. K. Jain, Fundamentals of Digital Image Processing, Prentice-Hall Information and System Sciences Series (Prentice-Hall, Englewood Cliffs, N.J., 1989).

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Figures (9)

Fig. 1
Fig. 1

(a) Impulse response of a defocus blur with a diameter of 10 pixels imbedded in a 256 × 256 pixel image. (b) Same impulse response enlarged. (c) Cross section of the modulus of the Fourier transform of the filter whose impulse response is (a).

Fig. 2
Fig. 2

Logarithm of the cross section of the modulus of the transfer functions of the equivalent kernels [Eq. (11)] computed from the following configurations of two kernels: solid curve, {Φ[h (1)] = 10, Φ[h (2)] = 11.3}; long-dashed curve, {Φ[h (1)] = 10, Φ[h (2)] = 10}; short-dashed curve, {Φ[h (1)] = 11.3, Φ[h (2)] = 11.3}; Φ(h), diameter of kernel h in pixels.

Fig. 3
Fig. 3

SNRoutopt) as a function of the diameter of kernel h (2). The sharp image is represented in Fig. 4(a). The diameter of kernel h (1) is 10 pixels. The restoration algorithm is in Eq. (4): diamonds, SNRin = 40 dB; crosses, SNRin = 30 dB. The horizontal lines represent the value of SNRoutopt) reached when h (1) = h (2) for SNRin: dashed line, 40 dB; dotted line, 30 dB.

Fig. 4
Fig. 4

Sharp image (256 × 256 pixels). (b) Sharp image perturbed by a defocus blur with a diameter of 10 pixels and SNRin = 40 dB. (c) Restoration of image (b) with the algorithm in Eq. (4), Φ[h (1)] = Φ[h (2)] = 10 pixels. The obtained SNRoutopt) = 15.8 dB. (d) Restoration of image (b) with the algorithm in Eq. (4), Φ[h (1)] = 10 and Φ[h (2)] = 11.3. The obtained SNRoutopt) = 17.2 dB. Φ(h) represents the diameter of the kernel h.

Fig. 5
Fig. 5

(a) Goldhill image (256 × 256 pixels). (b) Lena image (256 × 256 pixels).

Fig. 6
Fig. 6

SNRoutopt) as a function of α, SNRin = 40 dB. The sharp image is represented in Fig. 4(a). α is the difference in diameter (in pixels) between the kernels perturbing the image and the kernels used for designing the restoration filter in Eq. (4): crosses, two-kernel algorithm, Φ[h (1)] = Φ[h (2)] = 10 pixels; diamonds, two-kernel algorithm, Φ[h (1)] = 10 pixels and Φ[h (2)] = 11.3 pixels. Φ(h) represents the diameter of the kernel h.

Fig. 7
Fig. 7

Influence of α on the restoration quality, SNRin = 40 dB. Left column: two-kernel algorithm [see Eq. (4)], Φ[h (1)] = Φ[h (2)] = 10 pixels. Right column: two-kernel algorithm [see Eq. (4)], Φ[h (1)] = 10 pixels and Φ[h (2)] = 11.3 pixels. Φ(h) represents the diameter of the kernel h.

Fig. 8
Fig. 8

SNRoutopt) obtained with the three-kernel approach as a function of the diameters of the kernels h (2) and h (3). The sharp image is represented in Fig. 3(a). The diameter of kernel h (1) is 10 pixels. The restoration algorithm is in Eq. (4). The numbers inside the graph correspond to the values of SNRoutopt) on the corresponding contour.

Fig. 9
Fig. 9

SNRoutopt) as a function of α, SNRin = 40 dB, the sharp image is represented in Fig. 4(a). α is the difference in diameter (in pixels) between the kernels perturbing the image and the kernels used for designing the restoration filter in Eq. (4): crosses, three-kernel algorithm, Φ[h (1)] = Φ[h (2)] = Φ[h (3)] = 10 pixels; diamonds, three-kernel algorithm, Φ[h (1)] = 10 pixels and Φ[h (2)] = Φ[h (3)] = 11.3 pixels; squares, three-kernel algorithm, Φ[h (1)] = 10, Φ[h (2)] = 11.3, and Φ[h (3)] = 13 pixels. Φ(h) represents the diameter of the kernel h.

Tables (1)

Tables Icon

Table 1 Optimal Value of the Second Diameter (Φ2 max) and the difference in SNRoutopt) between the Identical Kernel and the Two-Kernel Approach by Use of Φ2 max

Equations (33)

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

yk=hk*x+nk,
Fu=k=1K yk-hk*u2+λl*u2,
x2=1Ni=0N-1 xi2.
uˆν=k=1K hˆνk*yˆνkk=1K |hˆνk|2+λ|lˆν|2,
uˆν=hˆν*yˆνk|hˆν|2+λ|lˆν|2,
SNRout=10 log10varxEout,
varx=1/N i=0N-1xi-x¯2, x¯=1/N i=0N-1 xi.
Eout=mina,m1Ni=0N-1xi-aui-m2,
SNRin=10 log10varhk*xσk2,
Eout=1Nν=1N-1|xˆν|2-ν=1N-1k=1K|hˆνk|2|xˆν|2k=1K|hˆνk|2+λ|lˆν|22ν=1N-1k=1K|hˆνk|22|xˆν|2+k=1K|hˆνk|2σk2k=1K|hˆνk|2+λ|lˆν|22,
Eout=1Nν=1N-1|xˆν|2-ν=1N-1|hˆν|2|xˆν|2|hˆν|2+λ|lˆν|22ν=1N-1|hˆν|4|xˆν|2+|hˆν|2σ2|hˆν|2+λ|lˆν|22.
hˆνeq=k=1k|hˆνk|21/2
Fu=k=1Kν=0N-1 |yˆνk-hˆνkuˆν|2+λ|lˆνuˆν|2.
-k=1K hˆνk*yˆνk+k=1K |hˆνk|2|uˆν|2+λ|lˆν|2|uˆν|2=0.
eouta, m=1Ni=0N-1xi-aui-m2.
Eout=mina,meouta, m.
eouta, m=ν=0N-1 |xˆν-a uˆν-mNδν|2
=ν=1N-1 |xˆν-a uˆν|2+|xˆ0-auˆ0-mN|2
=Fa+Ga, m,
Fa=0  =ν=1N-1 uˆν*xˆνν=1N-1 |uˆν|2.
emin=mina,meouta, m=ν=1N-1 |xˆν|2-ν=1N-1 uˆν*xˆν2ν=1N-1 |uˆν|2.
Eoutν=1N-1 |xˆν|2-ν=1N-1 uˆν*xˆν2ν=1N-1 |uˆν|2.
uˆν=k=1K dˆνk* yˆνkk=1K |dˆνk|2+λ|lˆν|2=k=1K dˆνk* hνk xˆν+nˆνkk=1K |dˆνk|2+λ|lˆν|2.
uˆν*=k=1K dˆνk hˆνk* xˆν*k=1K |dˆνk|2+λ|lˆν|2,
|uˆν|2=k=1K dˆνk hˆνk*2|xˆν|2+k=1K |dˆνk|2σk2k=1K|dˆνk|2+λ|lˆν|22.
Eout1Nν=1N-1|xˆν|2-ν=1N-1 gˆν|xˆν|22ν=1N-1 |gˆν|2|xˆν|2+bˆν,
gˆν=k=1K dˆνk hˆνk*k=1K |dˆνk|2+λ|lˆν|2
bˆν=k=1K |dˆνk|2σk2k=1K|dˆνk|2+λ|lˆν|22.
gˆν=k=1K |hˆνk|2k=1K |hˆνk|2+λ|lˆν|2,
bˆν=k=1K |hˆνk|2σk2k=1K|hˆνk|2+λ|hˆνk|2+|lˆν|2.
gˆν=|hˆν|2|hˆν|2+λ|lˆν|2,
bˆν=|hˆν|2σ2|hˆν|2+λ|lˆν|22.
hˆνeq=k=1k|hˆνk|21/2

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