Abstract

Direct methods for restoration of images blurred by motion are analyzed and compared. The term direct means that the considered methods are performed in a one-step fashion without any iterative technique. The blurring point-spread function is assumed to be unknown, and therefore the image restoration process is called blind deconvolution. What is believed to be a new direct method, here called the whitening method, was recently developed. This method and other existing direct methods such as the homomorphic and the cepstral techniques are studied and compared for a variety of motion types. Various criteria such as quality of restoration, sensitivity to noise, and computation requirements are considered. It appears that the recently developed method shows some improvements over other older methods. The research presented here clarifies the differences among the direct methods and offers an experimental basis for choosing which blind deconvolution method to use. In addition, some improvements on the methods are suggested.

© 1999 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  5. O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations, part IV: real-time numerical calculation of optical transfer function and experimental verification,” Opt. Eng. 33, 566–578 (1994).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  13. E. Cole, “The removal of unknown image blurs by homomorphic filtering,” Ph.D. dissertation (University of Utah, Salt Lake City, Utah, 1973).
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    [CrossRef]
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    [CrossRef] [PubMed]
  17. S. J. Wernecke, L. R. D’addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
    [CrossRef]
  18. M. I. Sezan, A. M. Teklap, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
    [CrossRef]
  19. R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
    [CrossRef]
  20. A. K. Katsaggelos, ed., Digital Image Restoration (Springer-Verlag, New York, 1991).
    [CrossRef]
  21. G. Pavlovic, A. M. Tekalp, “Maximum likelihood parametric blur identification based on a continuous spatial domain model,” IEEE Trans. Image Process. 1, 496–504 (1992).
    [CrossRef] [PubMed]
  22. Y. Yitzhaky, I. Mor, A. Lantzman, N. S. Kopeika, “A direct method for restoration of motion blurred images,” J. Opt. Soc. Am. A 15, 1512–1519 (1998).
    [CrossRef]
  23. A. E. Savakis, H. J. Trussell, “Blur identification by residual spectral matching,” IEEE Trans. Image Process. 2, 141–151 (1993).
    [CrossRef] [PubMed]

1998 (1)

1997 (1)

Y. Yitzhaky, N. S. Kopeika, “Identification of blur parameters from motion blurred images,” CVGIP. Graph. Models Image Process. 59, 321–332 (1997).

1994 (2)

O. Hadar, S. R. Rotman, N. S. Kopeika, “Target acquisition modeling of forward-motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations, part IV: real-time numerical calculation of optical transfer function and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

1993 (1)

A. E. Savakis, H. J. Trussell, “Blur identification by residual spectral matching,” IEEE Trans. Image Process. 2, 141–151 (1993).
[CrossRef] [PubMed]

1992 (1)

G. Pavlovic, A. M. Tekalp, “Maximum likelihood parametric blur identification based on a continuous spatial domain model,” IEEE Trans. Image Process. 1, 496–504 (1992).
[CrossRef] [PubMed]

1990 (2)

M. I. Sezan, A. M. Teklap, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

1977 (1)

S. J. Wernecke, L. R. D’addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
[CrossRef]

1976 (1)

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, (1976).

1975 (1)

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

1972 (2)

M. M. Sondhi, “Image restoration: the removal of spatially invariant degradations,” Proc. IEEE 60, 842–853 (1972).
[CrossRef]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
[CrossRef] [PubMed]

1971 (1)

1968 (1)

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1268–1291 (1968).
[CrossRef]

1967 (1)

D. Slepian, “Restoration of photographs blurred by image motion,” Bell Syst. Tech. J. 46, 2353–2362 (1967).
[CrossRef]

1966 (1)

Biemond, J.

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Cannon, M.

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, (1976).

Cannon, T. M.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Cole, E.

E. Cole, “The removal of unknown image blurs by homomorphic filtering,” Ph.D. dissertation (University of Utah, Salt Lake City, Utah, 1973).

D’addario, L. R.

S. J. Wernecke, L. R. D’addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
[CrossRef]

Dror, I.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations, part IV: real-time numerical calculation of optical transfer function and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

Frieden, B. R.

Hadar, O.

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations, part IV: real-time numerical calculation of optical transfer function and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

O. Hadar, S. R. Rotman, N. S. Kopeika, “Target acquisition modeling of forward-motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

Harris, L. J.

Ingebretsen, R. B.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Jain, A. K.

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

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1.

Kopeika, N. S.

Y. Yitzhaky, I. Mor, A. Lantzman, N. S. Kopeika, “A direct method for restoration of motion blurred images,” J. Opt. Soc. Am. A 15, 1512–1519 (1998).
[CrossRef]

Y. Yitzhaky, N. S. Kopeika, “Identification of blur parameters from motion blurred images,” CVGIP. Graph. Models Image Process. 59, 321–332 (1997).

O. Hadar, S. R. Rotman, N. S. Kopeika, “Target acquisition modeling of forward-motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations, part IV: real-time numerical calculation of optical transfer function and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

N. S. Kopeika, A System Engineering Approach to Imaging (SPIE Press, Bellingham, Wash., 1998).

Kunt, M.

M. Kunt, Digital Signal Processing (Artech House, Norwood, Mass., 1986), Chap. 7.

Lagendijk, R. L.

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Lantzman, A.

Mor, I.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1268–1291 (1968).
[CrossRef]

Pavlovic, G.

G. Pavlovic, A. M. Tekalp, “Maximum likelihood parametric blur identification based on a continuous spatial domain model,” IEEE Trans. Image Process. 1, 496–504 (1992).
[CrossRef] [PubMed]

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1.

Rotman, S. R.

O. Hadar, S. R. Rotman, N. S. Kopeika, “Target acquisition modeling of forward-motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

Savakis, A. E.

A. E. Savakis, H. J. Trussell, “Blur identification by residual spectral matching,” IEEE Trans. Image Process. 2, 141–151 (1993).
[CrossRef] [PubMed]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1268–1291 (1968).
[CrossRef]

Sezan, M. I.

M. I. Sezan, A. M. Teklap, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Slepian, D.

D. Slepian, “Restoration of photographs blurred by image motion,” Bell Syst. Tech. J. 46, 2353–2362 (1967).
[CrossRef]

Som, S. C.

Sondhi, M. M.

M. M. Sondhi, “Image restoration: the removal of spatially invariant degradations,” Proc. IEEE 60, 842–853 (1972).
[CrossRef]

Stockham, T. G.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1268–1291 (1968).
[CrossRef]

Tekalp, A. M.

G. Pavlovic, A. M. Tekalp, “Maximum likelihood parametric blur identification based on a continuous spatial domain model,” IEEE Trans. Image Process. 1, 496–504 (1992).
[CrossRef] [PubMed]

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

Teklap, A. M.

M. I. Sezan, A. M. Teklap, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Trussell, H. J.

A. E. Savakis, H. J. Trussell, “Blur identification by residual spectral matching,” IEEE Trans. Image Process. 2, 141–151 (1993).
[CrossRef] [PubMed]

Wernecke, S. J.

S. J. Wernecke, L. R. D’addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
[CrossRef]

Yitzhaky, Y.

Y. Yitzhaky, I. Mor, A. Lantzman, N. S. Kopeika, “A direct method for restoration of motion blurred images,” J. Opt. Soc. Am. A 15, 1512–1519 (1998).
[CrossRef]

Y. Yitzhaky, N. S. Kopeika, “Identification of blur parameters from motion blurred images,” CVGIP. Graph. Models Image Process. 59, 321–332 (1997).

Bell Syst. Tech. J. (1)

D. Slepian, “Restoration of photographs blurred by image motion,” Bell Syst. Tech. J. 46, 2353–2362 (1967).
[CrossRef]

CVGIP. Graph. Models Image Process. (1)

Y. Yitzhaky, N. S. Kopeika, “Identification of blur parameters from motion blurred images,” CVGIP. Graph. Models Image Process. 59, 321–332 (1997).

IEEE Trans. Acoust. Speech Signal Process (1)

M. Cannon, “Blind deconvolution of spatially invariant image blurs with phase,” IEEE Trans. Acoust. Speech Signal Process. ASSP-24, (1976).

IEEE Trans. Comput. (1)

S. J. Wernecke, L. R. D’addario, “Maximum entropy image reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1977).
[CrossRef]

IEEE Trans. Image Process. (2)

G. Pavlovic, A. M. Tekalp, “Maximum likelihood parametric blur identification based on a continuous spatial domain model,” IEEE Trans. Image Process. 1, 496–504 (1992).
[CrossRef] [PubMed]

A. E. Savakis, H. J. Trussell, “Blur identification by residual spectral matching,” IEEE Trans. Image Process. 2, 141–151 (1993).
[CrossRef] [PubMed]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (1)

Opt. Eng. (4)

M. I. Sezan, A. M. Teklap, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

R. L. Lagendijk, A. M. Tekalp, J. Biemond, “Maximum likelihood image and blur identification: a unifying approach,” Opt. Eng. 29, 422–435 (1990).
[CrossRef]

O. Hadar, S. R. Rotman, N. S. Kopeika, “Target acquisition modeling of forward-motion considerations for airborne reconnaissance over hostile territory,” Opt. Eng. 33, 3106–3117 (1994).
[CrossRef]

O. Hadar, I. Dror, N. S. Kopeika, “Image resolution limits resulting from mechanical vibrations, part IV: real-time numerical calculation of optical transfer function and experimental verification,” Opt. Eng. 33, 566–578 (1994).
[CrossRef]

Proc. IEEE (3)

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1268–1291 (1968).
[CrossRef]

M. M. Sondhi, “Image restoration: the removal of spatially invariant degradations,” Proc. IEEE 60, 842–853 (1972).
[CrossRef]

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1975).
[CrossRef]

Other (6)

M. Kunt, Digital Signal Processing (Artech House, Norwood, Mass., 1986), Chap. 7.

N. S. Kopeika, A System Engineering Approach to Imaging (SPIE Press, Bellingham, Wash., 1998).

A. K. Katsaggelos, ed., Digital Image Restoration (Springer-Verlag, New York, 1991).
[CrossRef]

E. Cole, “The removal of unknown image blurs by homomorphic filtering,” Ph.D. dissertation (University of Utah, Salt Lake City, Utah, 1973).

A. Rosenfeld, A. C. Kak, Digital Picture Processing (Academic, New York, 1982), Vol. 1.

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

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Figures (9)

Fig. 1
Fig. 1

Sample of images used in the blur identification tests: (a) IMG1, (b) IMG2, (c) IMG3, (d) IMG4.

Fig. 2
Fig. 2

Comparison of blur identification results. (a) IMG1 blurred by uniform velocity motion (R = ∞) and 10-pixel blur extent. (b) PSF identification results: solid line, true PSF; dotted curve, identification by the homomorphic technique; dashed curve, identification by the whitening technique.

Fig. 3
Fig. 3

Restoration results for Fig. 2 (R = ∞). (a) Restoration by a Wiener filter with the homomorphic PSF. (b) Restoration by a Wiener filter with the whitening PSF.

Fig. 4
Fig. 4

Comparison of blur identification results for IMG2 for R = 10. (a) Blurred image. (b) Identification results.

Fig. 6
Fig. 6

Comparison of blur identification results for IMG4 for R = 1. (a) Blurred image. (b) Identification results.

Fig. 8
Fig. 8

Comparison of blur identification results for IMG4 for R = 0.1. (a) Blurred image. (b) Identification results.

Fig. 5
Fig. 5

Restoration results for Fig. 4 (R = 10). (a) Restoration by a Wiener filter with the homomorphic PSF. (b) Restoration by a Wiener filter with the whitening PSF.

Fig. 7
Fig. 7

Restoration results for Fig. 4 (R = 1). (a) Restoration by a Wiener filter with the homomorphic PSF. (b) Restoration by a Wiener filter with the whitening PSF.

Fig. 9
Fig. 9

Restoration results for Fig. 4 (R = 0.1). (a) Restoration by a Wiener filter with the homomorphic PSF. (b) Restoration by a Wiener filter with the whitening PSF.

Tables (4)

Tables Icon

Table 1 Comparison of Blur Extent Identifications between the Cepstral Method and the Whitening Method for a Square-Pulse Horizontal Blur Effect (Perfect Uniform Velocity Motion) with Various Extents and SNR’sa

Tables Icon

Table 2 Same as Table 1, but for IMG2

Tables Icon

Table 3 Same as Table 1, but for IMG3

Tables Icon

Table 4 Comparison of the MSE’s between the True and the Estimated PSF’s for Different Values of Ra

Equations (17)

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

gi, j=mn hi-m, j-nfm, n+ni, j,
hi, j=1/d,ifi2+j21/2d/2,j/i=tan θ0elsewhere.
Gu, v=Hu, vFu, v+Nu, v,
log Gu, v=log Hu, v+log Fu, v.
1Ni=1Nlog Giu, vlog Hu, v+1Ni=1Nlog Fiu, v.
K  N  L
ΔGu, v=Gu, vWvWu,
S¯ΔGSΔPSF,
SΔPSF=|HWu|2,
MTFuSΔGu1/2|Wu|.
PTFu=-12π02πlnMTFαcotu-α2dα.
H=MTF expjPTF.
SNR=10 log10variance of the blurred imagevariance of the noise dB.
LSFx=1tev02+2ax1/2,
R=v02/a,
 PSF=1.
Wiener filter=H*|H|2+γ,

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