Abstract

We present a method that can efficiently restore large images, blurred possibly nonuniformly and contaminated with noise, by use of a scanning singular-value-decomposition (SVD) method. Such an approach bypasses the prohibitive storage and speed limitations of the SVD method, thus, to our knowledge for the first time, making possible the restoration of reasonably sized images. We make use of the linear and local nature of the point spread function (PSF) to scan the image and restore it in the same raster without incurring blocking effects that are due to the overlap in neighboring reconstruction areas. The increase in speed compared with the conventional SVD approach can be many orders of magnitude, depending on the ratio of the point-spread blur to the image size. For example, if the linear extent of the PSF is one-eighth that of the image, a speed-up factor greater than 106 is achieved. A similar but less accurate solution to the problem of spatially variant blur by use of scanning Fourier transforms, which allows an even faster solution, is also described.

© 1996 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: A general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics 10: Signal Processing and Its Applications, N. K. Bose, C. R. Rao, eds. (North-Holland, Amsterdam, 1993), pp. 1–46.
  2. A. N. Tikhonov, N. Y Arsenin, Solutions of Ill-posed Problems (Winston/Wiley, Washington, 1977).
  3. M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: A singular value analysis I—The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
    [CrossRef]
  4. M. Bertero, P. Boccacci, E. R. Pike, “Resolution in diffraction limited imaging: A singular value analysis II—The case of incoherent illumination,” Opt. Acta 29, 1599–1611 (1982).
    [CrossRef]
  5. L. Landweber, “An iterative formula for Fredholm integral equations of the first kind,” Amer. J. Math. 73, 615–624 (1951).
    [CrossRef]
  6. M. Bertero, F. Maggio, E. R. Pike, D. A. Fish, “Assessment of methods used for reconstructing HST images,” in The Restoration of HST Images and Spectra II, R. Hamish, R. L. White, eds. (NASA, Space Telescope Science Institute, Baltimore, Md., 1994), pp. 300–307.
  7. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  8. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  9. A. A. Sawchuk, “Space-variant restoration by co-ordinate transformation,” J. Opt. Soc. Am. 64, 138–144 (1974).
    [CrossRef]
  10. S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography—Restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
    [CrossRef]
  11. H. J. Trussell, S. Fogel, “Identification and restoration of spatially variant motion blurs in sequential images,” IEEE Trans. Imag. Proc. 1, 123–126 (1992).
    [CrossRef]
  12. A. DeSantis, A. Germani, L. Jetto, “Space-variant recursive restoration of noisy images,” IEEE Trans. Circuits Syst. Analog Digital Signal Proc. 41, 249–261 (1994).
    [CrossRef]
  13. S. J. Reeves, “Optimal space-varying regularization in iterative image restoration,” IEEE Trans. Imag. Proc. 3, 319–324 (1994).
    [CrossRef]
  14. S. Koch, H. Kaufman, J. Biemond, “Restoration of spatially varying blurred images using multiple model-based extended Kalman filters,” IEEE Trans. Imag. Proc. 4, 520–523 (1995).
    [CrossRef]
  15. J. B. Abbiss, J. C. Allen, H. J. Whitehouse, “Computational issues in regularized restoration using large imaging operators,” Titan Spectron Report No. 94-3341-01, ONR Contract No. N00014-93-C-0109 (Tital Spectron Inc., Santa Ana, Calif., 1994).
  16. Lenna Sjoobloom, Playboy centerfold, November1972.

1995

S. Koch, H. Kaufman, J. Biemond, “Restoration of spatially varying blurred images using multiple model-based extended Kalman filters,” IEEE Trans. Imag. Proc. 4, 520–523 (1995).
[CrossRef]

1994

A. DeSantis, A. Germani, L. Jetto, “Space-variant recursive restoration of noisy images,” IEEE Trans. Circuits Syst. Analog Digital Signal Proc. 41, 249–261 (1994).
[CrossRef]

S. J. Reeves, “Optimal space-varying regularization in iterative image restoration,” IEEE Trans. Imag. Proc. 3, 319–324 (1994).
[CrossRef]

1992

S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography—Restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
[CrossRef]

H. J. Trussell, S. Fogel, “Identification and restoration of spatially variant motion blurs in sequential images,” IEEE Trans. Imag. Proc. 1, 123–126 (1992).
[CrossRef]

1982

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: A singular value analysis I—The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

M. Bertero, P. Boccacci, E. R. Pike, “Resolution in diffraction limited imaging: A singular value analysis II—The case of incoherent illumination,” Opt. Acta 29, 1599–1611 (1982).
[CrossRef]

1974

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

A. A. Sawchuk, “Space-variant restoration by co-ordinate transformation,” J. Opt. Soc. Am. 64, 138–144 (1974).
[CrossRef]

1972

1951

L. Landweber, “An iterative formula for Fredholm integral equations of the first kind,” Amer. J. Math. 73, 615–624 (1951).
[CrossRef]

Abbiss, J. B.

J. B. Abbiss, J. C. Allen, H. J. Whitehouse, “Computational issues in regularized restoration using large imaging operators,” Titan Spectron Report No. 94-3341-01, ONR Contract No. N00014-93-C-0109 (Tital Spectron Inc., Santa Ana, Calif., 1994).

Allen, J. C.

J. B. Abbiss, J. C. Allen, H. J. Whitehouse, “Computational issues in regularized restoration using large imaging operators,” Titan Spectron Report No. 94-3341-01, ONR Contract No. N00014-93-C-0109 (Tital Spectron Inc., Santa Ana, Calif., 1994).

Arsenin, N. Y

A. N. Tikhonov, N. Y Arsenin, Solutions of Ill-posed Problems (Winston/Wiley, Washington, 1977).

Bertero, M.

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: A singular value analysis I—The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

M. Bertero, P. Boccacci, E. R. Pike, “Resolution in diffraction limited imaging: A singular value analysis II—The case of incoherent illumination,” Opt. Acta 29, 1599–1611 (1982).
[CrossRef]

M. Bertero, F. Maggio, E. R. Pike, D. A. Fish, “Assessment of methods used for reconstructing HST images,” in The Restoration of HST Images and Spectra II, R. Hamish, R. L. White, eds. (NASA, Space Telescope Science Institute, Baltimore, Md., 1994), pp. 300–307.

M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: A general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics 10: Signal Processing and Its Applications, N. K. Bose, C. R. Rao, eds. (North-Holland, Amsterdam, 1993), pp. 1–46.

Biemond, J.

S. Koch, H. Kaufman, J. Biemond, “Restoration of spatially varying blurred images using multiple model-based extended Kalman filters,” IEEE Trans. Imag. Proc. 4, 520–523 (1995).
[CrossRef]

Boccacci, P.

M. Bertero, P. Boccacci, E. R. Pike, “Resolution in diffraction limited imaging: A singular value analysis II—The case of incoherent illumination,” Opt. Acta 29, 1599–1611 (1982).
[CrossRef]

DeSantis, A.

A. DeSantis, A. Germani, L. Jetto, “Space-variant recursive restoration of noisy images,” IEEE Trans. Circuits Syst. Analog Digital Signal Proc. 41, 249–261 (1994).
[CrossRef]

Fish, D. A.

M. Bertero, F. Maggio, E. R. Pike, D. A. Fish, “Assessment of methods used for reconstructing HST images,” in The Restoration of HST Images and Spectra II, R. Hamish, R. L. White, eds. (NASA, Space Telescope Science Institute, Baltimore, Md., 1994), pp. 300–307.

Fogel, S.

H. J. Trussell, S. Fogel, “Identification and restoration of spatially variant motion blurs in sequential images,” IEEE Trans. Imag. Proc. 1, 123–126 (1992).
[CrossRef]

Germani, A.

A. DeSantis, A. Germani, L. Jetto, “Space-variant recursive restoration of noisy images,” IEEE Trans. Circuits Syst. Analog Digital Signal Proc. 41, 249–261 (1994).
[CrossRef]

Jetto, L.

A. DeSantis, A. Germani, L. Jetto, “Space-variant recursive restoration of noisy images,” IEEE Trans. Circuits Syst. Analog Digital Signal Proc. 41, 249–261 (1994).
[CrossRef]

Kaufman, H.

S. Koch, H. Kaufman, J. Biemond, “Restoration of spatially varying blurred images using multiple model-based extended Kalman filters,” IEEE Trans. Imag. Proc. 4, 520–523 (1995).
[CrossRef]

Koch, S.

S. Koch, H. Kaufman, J. Biemond, “Restoration of spatially varying blurred images using multiple model-based extended Kalman filters,” IEEE Trans. Imag. Proc. 4, 520–523 (1995).
[CrossRef]

Koles, Z. J.

S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography—Restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
[CrossRef]

Landweber, L.

L. Landweber, “An iterative formula for Fredholm integral equations of the first kind,” Amer. J. Math. 73, 615–624 (1951).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

Maggio, F.

M. Bertero, F. Maggio, E. R. Pike, D. A. Fish, “Assessment of methods used for reconstructing HST images,” in The Restoration of HST Images and Spectra II, R. Hamish, R. L. White, eds. (NASA, Space Telescope Science Institute, Baltimore, Md., 1994), pp. 300–307.

Overton, T. R.

S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography—Restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
[CrossRef]

Pike, E. R.

M. Bertero, P. Boccacci, E. R. Pike, “Resolution in diffraction limited imaging: A singular value analysis II—The case of incoherent illumination,” Opt. Acta 29, 1599–1611 (1982).
[CrossRef]

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: A singular value analysis I—The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: A general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics 10: Signal Processing and Its Applications, N. K. Bose, C. R. Rao, eds. (North-Holland, Amsterdam, 1993), pp. 1–46.

M. Bertero, F. Maggio, E. R. Pike, D. A. Fish, “Assessment of methods used for reconstructing HST images,” in The Restoration of HST Images and Spectra II, R. Hamish, R. L. White, eds. (NASA, Space Telescope Science Institute, Baltimore, Md., 1994), pp. 300–307.

Rathee, S.

S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography—Restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
[CrossRef]

Reeves, S. J.

S. J. Reeves, “Optimal space-varying regularization in iterative image restoration,” IEEE Trans. Imag. Proc. 3, 319–324 (1994).
[CrossRef]

Richardson, W. H.

Sawchuk, A. A.

Sjoobloom, Lenna

Lenna Sjoobloom, Playboy centerfold, November1972.

Tikhonov, A. N.

A. N. Tikhonov, N. Y Arsenin, Solutions of Ill-posed Problems (Winston/Wiley, Washington, 1977).

Trussell, H. J.

H. J. Trussell, S. Fogel, “Identification and restoration of spatially variant motion blurs in sequential images,” IEEE Trans. Imag. Proc. 1, 123–126 (1992).
[CrossRef]

Whitehouse, H. J.

J. B. Abbiss, J. C. Allen, H. J. Whitehouse, “Computational issues in regularized restoration using large imaging operators,” Titan Spectron Report No. 94-3341-01, ONR Contract No. N00014-93-C-0109 (Tital Spectron Inc., Santa Ana, Calif., 1994).

Amer. J. Math.

L. Landweber, “An iterative formula for Fredholm integral equations of the first kind,” Amer. J. Math. 73, 615–624 (1951).
[CrossRef]

Astron. J.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[CrossRef]

IEEE Trans. Circuits Syst. Analog Digital Signal Proc.

A. DeSantis, A. Germani, L. Jetto, “Space-variant recursive restoration of noisy images,” IEEE Trans. Circuits Syst. Analog Digital Signal Proc. 41, 249–261 (1994).
[CrossRef]

IEEE Trans. Imag. Proc.

S. J. Reeves, “Optimal space-varying regularization in iterative image restoration,” IEEE Trans. Imag. Proc. 3, 319–324 (1994).
[CrossRef]

S. Koch, H. Kaufman, J. Biemond, “Restoration of spatially varying blurred images using multiple model-based extended Kalman filters,” IEEE Trans. Imag. Proc. 4, 520–523 (1995).
[CrossRef]

H. J. Trussell, S. Fogel, “Identification and restoration of spatially variant motion blurs in sequential images,” IEEE Trans. Imag. Proc. 1, 123–126 (1992).
[CrossRef]

IEEE Trans. Med. Imag.

S. Rathee, Z. J. Koles, T. R. Overton, “Image restoration in computed tomography—Restoration of experimental CT images,” IEEE Trans. Med. Imag. 11, 546–553 (1992).
[CrossRef]

J. Opt. Soc. Am.

Opt. Acta

M. Bertero, E. R. Pike, “Resolution in diffraction-limited imaging: A singular value analysis I—The case of coherent illumination,” Opt. Acta 29, 727–746 (1982).
[CrossRef]

M. Bertero, P. Boccacci, E. R. Pike, “Resolution in diffraction limited imaging: A singular value analysis II—The case of incoherent illumination,” Opt. Acta 29, 1599–1611 (1982).
[CrossRef]

Other

M. Bertero, E. R. Pike, “Signal processing for linear instrumental systems with noise: A general theory with illustrations from optical imaging and light scattering problems,” in Handbook of Statistics 10: Signal Processing and Its Applications, N. K. Bose, C. R. Rao, eds. (North-Holland, Amsterdam, 1993), pp. 1–46.

A. N. Tikhonov, N. Y Arsenin, Solutions of Ill-posed Problems (Winston/Wiley, Washington, 1977).

M. Bertero, F. Maggio, E. R. Pike, D. A. Fish, “Assessment of methods used for reconstructing HST images,” in The Restoration of HST Images and Spectra II, R. Hamish, R. L. White, eds. (NASA, Space Telescope Science Institute, Baltimore, Md., 1994), pp. 300–307.

J. B. Abbiss, J. C. Allen, H. J. Whitehouse, “Computational issues in regularized restoration using large imaging operators,” Titan Spectron Report No. 94-3341-01, ONR Contract No. N00014-93-C-0109 (Tital Spectron Inc., Santa Ana, Calif., 1994).

Lenna Sjoobloom, Playboy centerfold, November1972.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (4)

Fig. 1
Fig. 1

Variation of reconstruction error with reconstruction-box size: (a) reconstruction box 16 × 16 pixels, 43.0% error; (b) reconstruction box 14 × 14 pixels, 6.0% error; (c) reconstruction box 12 × 12 pixels, 5.0% error; (d) reconstruction box 10 × 10 pixels, 5.0% error.

Fig. 2
Fig. 2

(a) Original image of Lenna; (b) image with Lorentzian PSF of radius 1.0 pixel, condition number 138, and 5.0% Gaussian noise; (c) reconstruction 30.0% in error with the true image. The reconstruction was performed with a scanning box of 32 × 32 pixels and a reconstruction box of 2 × 2 pixels with 157 singular values.

Fig. 3
Fig. 3

(a) Original image. (b) An image formed with a spatially varying PSF. The PSF is Lorentzian in form with radius 1.0 pixel at the center increasing linearly to 3.0 pixels at the edge. (c) Reconstructed image made with the approximate Fourier scanning method: scanning box 32 × 32 pixels, reconstruction box 2 × 2 pixels.

Fig. 4
Fig. 4

(a) Image of Lenna with a spatially variant blur; (b) reconstructed image made with the scanning SVD method. Scanning box 32 × 32 pixels, reconstruction box 2 × 2 pixels. The PSF is Lorentzian in form with half-widths increasing from 0.3 pixel at the upper and lower edge of the picture to 1 pixel at the center.

Tables (1)

Tables Icon

Table 1 Errors in Reconstruction of a Cross Image with Reconstruction-Box Size

Equations (17)

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

g ( x ) = ( H f ) ( x ) = h ( x , x ) f ( x ) d x ,             f X g Y ,
g ( x ) = ( H f ) ( x ) = h ( x - x ) f ( x ) d x .
g = H f ,
H u k = σ k v k ,
H * v k = σ k u k .
H f = k σ k ( u k , f ) X v k ,
H * g = k σ k ( v k , g ) Y u k ,
f + = k = 0 R - 1 1 σ k ( v k , g ) Y u k ,
δ f X f X cond ( H ) δ g Y g Y ,
g = H f + η ,
α = η Y / f X .
σ 0 / σ k α
f ˜ = k = 0 K - 1 1 σ k ( v k , g ) Y u k .
f ˜ = k = 0 R - 1 W k u k ,
W k = σ k σ k 2 + γ k 2 ( u k , g ) Y ,
W x = { x : x S x } ,
g i = H i f i + η i .

Metrics