Abstract

Algorithms are developed which address the problem of estimating the magnitude and phase of the optical transfer function associated with a blurred image. The primary focus of the paper is on the estimate of the phase of the optical transfer function. Once an estimate of the optical transfer function has been made, the corresponding blurred imagery is Wiener filtered to estimate the original unblurred object. Results are demonstrated on real world photographically induced blurs. Included is a mathematical bound on the phase of the optical transfer function.

© 1979 Optical Society of America

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  1. C. W. Hellstrom, "Image Restoration by the Method of Least Squares," J. Opt. Soc. Am., 57, 297–303 (1967).
  2. B. L.McGlamery, "Restoration of Turbulence Degraded Images," J. Opt. Soc. Am., 57, 293–297 (1967).
  3. G. M. Robbins and T. S. Huang, "Inverse Filtering for Linear Shift-Variant Imaging Systems," Proc. IEEE, 60, 862–872 (1972).
  4. A. A. Sawchuk, "Space-Variant Image Motion Degradation and Restoration," Proc. IEEE, 60, 854–861 (1972).
  5. M. M. Sondhi, "Image Restoration: The Removal of Spatially Invariant Degradations," Proc. IEEE, 60, 842–853 (1972).
  6. B. R. Hunt, "The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer," IEEE Trans. Computers, C-22, 805–812 (1973).
  7. A. A. Sawchuk, "Space-Variant Image Restoration by Coordinate Transformations," J. Opt. Soc. Am., 64, 138–144 (1974).
  8. H. C. Andrews and B. R. Hunt, Digital Image Restoration, (Prentice-Hall, Englewood Cliffs, New Jersey, 1977).
  9. E. R. Cole, "The Removal of Unknown Image Blurs by Homomorphic Filtering," Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-029, June 1973.
  10. T. M. Cannon, "Digital Image Deblurring by Nonlinear Homomorphic Filtering," Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-091, August 1974.
  11. T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, "Blind Deconvolution Through Digital Signal Processing," Proc. IEEE, 63, 678–692 (1975).
  12. T. M. Cannon, "Blind Deconvolution of Spatially Invariant Image Blurs with Phase," IEEE Trans. Acoust., Speech, Signal Process., 24, 58–63 (1976).
  13. A. Filip, "Estimating the Impulse Response of Linear, Shift-Invariant Image Degrading Systems," Ph.D. Thesis, M.I.T., October 1972 (unpublished).
  14. B. O'Connor and T. S. Huang, "Phase Unwrapping with Applications to Stability and Picture Deblurring," Image Understanding and Information Extraction, School of Electrical Engineering, Purdue U., Report TR-EE-16, March 1977, pp. 92–142 (unpublished).
  15. K. T. Knox and B. J. Thompson, "Recovery of Images from Atmospherically Degraded Short-Exposure Photographs," Astrophys. J., 193, L45–L48 (1974).
  16. K. T. Knox, "Diffraction-Limited Imaging with Astronomical Telescopes," Ph.D. Dissertation, U. of Rochester, N.Y., (1975) (unpublished).
  17. K. T. Knox, "Image retrieval from astronomical speckle patterns," J. Opt. Soc. Am., 66, 1236–1239 (1976).
  18. E. L. O'Neill and A. Walther, "The Question of Phase in Image Formation," Opt. Acta, 10, 33–40 (1963).
  19. A. Walther, "The Question of Phase Retrieval in Optics," Opt. Acta, 10, 41–49 (1963).
  20. P. Roman and A. S. Marathay, "Analyticity and Phase Retrieval," Nuova Cimento, 30, 1452–1463 (1973).
  21. R. W. Gerchberg and W. O. Saxton, "A Practical Algorithm for the Determination of Phase from Image Diffraction Plane Pictures," Optik (Stutts.), 35, 237–246 (1972).
  22. D. Kohler and L. Mandel, "Source reconstruction from the modulus of the correlation function: A practical approach to the phase problem of optical coherence theory," J. Opt. Soc. Am., 63, 126–134 (1973).
  23. B. J. Hoenders, "On the Solution of the Phase Retrieval Problem," J. Math. Phys., 16, 1719–1725 (1975).
  24. R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am., 66, 961–964 (1976).
  25. B. L. McGlamery, "Image Restoration Techniques Applied to Astronomical Photography," in Astronomical Use of Television-type Image Sensors, Report No. N71-28509-525 of the National Technical Information Service of the U.S. Dept. of Commerce, 1971.
  26. I. P. Natanson, Konstruktive Funktionentheorie, (Akademie-Verlag, Berlin, 1955).
  27. A. Papoulis, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill, New York, 1965).
  28. P. D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra," IEEE Trans. Acoust., Speech, Signal Process., 15, 70–73 (1967).
  29. J. B. Morton, "An Investigation Into An A Posteriori Method of Image Restoration," USCIPI Report No. 810, University of Southern California Image Processing Institute, April 1978.

1976

T. M. Cannon, "Blind Deconvolution of Spatially Invariant Image Blurs with Phase," IEEE Trans. Acoust., Speech, Signal Process., 24, 58–63 (1976).

R. A. Gonsalves, "Phase retrieval from modulus data," J. Opt. Soc. Am., 66, 961–964 (1976).

K. T. Knox, "Image retrieval from astronomical speckle patterns," J. Opt. Soc. Am., 66, 1236–1239 (1976).

1975

B. J. Hoenders, "On the Solution of the Phase Retrieval Problem," J. Math. Phys., 16, 1719–1725 (1975).

1974

K. T. Knox and B. J. Thompson, "Recovery of Images from Atmospherically Degraded Short-Exposure Photographs," Astrophys. J., 193, L45–L48 (1974).

A. A. Sawchuk, "Space-Variant Image Restoration by Coordinate Transformations," J. Opt. Soc. Am., 64, 138–144 (1974).

1973

D. Kohler and L. Mandel, "Source reconstruction from the modulus of the correlation function: A practical approach to the phase problem of optical coherence theory," J. Opt. Soc. Am., 63, 126–134 (1973).

B. R. Hunt, "The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer," IEEE Trans. Computers, C-22, 805–812 (1973).

P. Roman and A. S. Marathay, "Analyticity and Phase Retrieval," Nuova Cimento, 30, 1452–1463 (1973).

1972

R. W. Gerchberg and W. O. Saxton, "A Practical Algorithm for the Determination of Phase from Image Diffraction Plane Pictures," Optik (Stutts.), 35, 237–246 (1972).

1967

P. D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra," IEEE Trans. Acoust., Speech, Signal Process., 15, 70–73 (1967).

B. L.McGlamery, "Restoration of Turbulence Degraded Images," J. Opt. Soc. Am., 57, 293–297 (1967).

C. W. Hellstrom, "Image Restoration by the Method of Least Squares," J. Opt. Soc. Am., 57, 297–303 (1967).

1963

E. L. O'Neill and A. Walther, "The Question of Phase in Image Formation," Opt. Acta, 10, 33–40 (1963).

A. Walther, "The Question of Phase Retrieval in Optics," Opt. Acta, 10, 41–49 (1963).

Andrews, H. C.

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

Cannon, T. M.

T. M. Cannon, "Blind Deconvolution of Spatially Invariant Image Blurs with Phase," IEEE Trans. Acoust., Speech, Signal Process., 24, 58–63 (1976).

T. M. Cannon, "Digital Image Deblurring by Nonlinear Homomorphic Filtering," Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-091, August 1974.

T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, "Blind Deconvolution Through Digital Signal Processing," Proc. IEEE, 63, 678–692 (1975).

Cole, E. R.

E. R. Cole, "The Removal of Unknown Image Blurs by Homomorphic Filtering," Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-029, June 1973.

Filip, A.

A. Filip, "Estimating the Impulse Response of Linear, Shift-Invariant Image Degrading Systems," Ph.D. Thesis, M.I.T., October 1972 (unpublished).

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, "A Practical Algorithm for the Determination of Phase from Image Diffraction Plane Pictures," Optik (Stutts.), 35, 237–246 (1972).

Gonsalves, R. A.

Hellstrom, C. W.

Hoenders, B. J.

B. J. Hoenders, "On the Solution of the Phase Retrieval Problem," J. Math. Phys., 16, 1719–1725 (1975).

Huang, T. S.

G. M. Robbins and T. S. Huang, "Inverse Filtering for Linear Shift-Variant Imaging Systems," Proc. IEEE, 60, 862–872 (1972).

B. O'Connor and T. S. Huang, "Phase Unwrapping with Applications to Stability and Picture Deblurring," Image Understanding and Information Extraction, School of Electrical Engineering, Purdue U., Report TR-EE-16, March 1977, pp. 92–142 (unpublished).

Hunt, B. R.

B. R. Hunt, "The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer," IEEE Trans. Computers, C-22, 805–812 (1973).

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

Ingebretsen, R. B.

T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, "Blind Deconvolution Through Digital Signal Processing," Proc. IEEE, 63, 678–692 (1975).

Knox, K. T.

K. T. Knox, "Image retrieval from astronomical speckle patterns," J. Opt. Soc. Am., 66, 1236–1239 (1976).

K. T. Knox and B. J. Thompson, "Recovery of Images from Atmospherically Degraded Short-Exposure Photographs," Astrophys. J., 193, L45–L48 (1974).

K. T. Knox, "Diffraction-Limited Imaging with Astronomical Telescopes," Ph.D. Dissertation, U. of Rochester, N.Y., (1975) (unpublished).

Kohler, D.

Mandel, L.

Marathay, A. S.

P. Roman and A. S. Marathay, "Analyticity and Phase Retrieval," Nuova Cimento, 30, 1452–1463 (1973).

McGlamery, B. L

McGlamery, B. L.

B. L. McGlamery, "Image Restoration Techniques Applied to Astronomical Photography," in Astronomical Use of Television-type Image Sensors, Report No. N71-28509-525 of the National Technical Information Service of the U.S. Dept. of Commerce, 1971.

Morton, J. B.

J. B. Morton, "An Investigation Into An A Posteriori Method of Image Restoration," USCIPI Report No. 810, University of Southern California Image Processing Institute, April 1978.

Natanson, I. P.

I. P. Natanson, Konstruktive Funktionentheorie, (Akademie-Verlag, Berlin, 1955).

O'Connor, B.

B. O'Connor and T. S. Huang, "Phase Unwrapping with Applications to Stability and Picture Deblurring," Image Understanding and Information Extraction, School of Electrical Engineering, Purdue U., Report TR-EE-16, March 1977, pp. 92–142 (unpublished).

O'Neill, E. L.

E. L. O'Neill and A. Walther, "The Question of Phase in Image Formation," Opt. Acta, 10, 33–40 (1963).

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill, New York, 1965).

Robbins, G. M.

G. M. Robbins and T. S. Huang, "Inverse Filtering for Linear Shift-Variant Imaging Systems," Proc. IEEE, 60, 862–872 (1972).

Roman, P.

P. Roman and A. S. Marathay, "Analyticity and Phase Retrieval," Nuova Cimento, 30, 1452–1463 (1973).

Sawchuk, A. A.

A. A. Sawchuk, "Space-Variant Image Restoration by Coordinate Transformations," J. Opt. Soc. Am., 64, 138–144 (1974).

A. A. Sawchuk, "Space-Variant Image Motion Degradation and Restoration," Proc. IEEE, 60, 854–861 (1972).

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, "A Practical Algorithm for the Determination of Phase from Image Diffraction Plane Pictures," Optik (Stutts.), 35, 237–246 (1972).

Sondhi, M. M.

M. M. Sondhi, "Image Restoration: The Removal of Spatially Invariant Degradations," Proc. IEEE, 60, 842–853 (1972).

Stockham, T. G.

T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, "Blind Deconvolution Through Digital Signal Processing," Proc. IEEE, 63, 678–692 (1975).

Thompson, B. J.

K. T. Knox and B. J. Thompson, "Recovery of Images from Atmospherically Degraded Short-Exposure Photographs," Astrophys. J., 193, L45–L48 (1974).

Walther, A.

A. Walther, "The Question of Phase Retrieval in Optics," Opt. Acta, 10, 41–49 (1963).

E. L. O'Neill and A. Walther, "The Question of Phase in Image Formation," Opt. Acta, 10, 33–40 (1963).

Welch, P. D.

P. D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra," IEEE Trans. Acoust., Speech, Signal Process., 15, 70–73 (1967).

Astrophys. J.

K. T. Knox and B. J. Thompson, "Recovery of Images from Atmospherically Degraded Short-Exposure Photographs," Astrophys. J., 193, L45–L48 (1974).

IEEE Trans. Acoust., Speech, Signal Process.

T. M. Cannon, "Blind Deconvolution of Spatially Invariant Image Blurs with Phase," IEEE Trans. Acoust., Speech, Signal Process., 24, 58–63 (1976).

P. D. Welch, "The Use of Fast Fourier Transform for the Estimation of Power Spectra," IEEE Trans. Acoust., Speech, Signal Process., 15, 70–73 (1967).

IEEE Trans. Computers

B. R. Hunt, "The Application of Constrained Least Squares Estimation to Image Restoration by Digital Computer," IEEE Trans. Computers, C-22, 805–812 (1973).

J. Math. Phys.

B. J. Hoenders, "On the Solution of the Phase Retrieval Problem," J. Math. Phys., 16, 1719–1725 (1975).

J. Opt. Soc. Am.

Nuova Cimento

P. Roman and A. S. Marathay, "Analyticity and Phase Retrieval," Nuova Cimento, 30, 1452–1463 (1973).

Opt. Acta

E. L. O'Neill and A. Walther, "The Question of Phase in Image Formation," Opt. Acta, 10, 33–40 (1963).

A. Walther, "The Question of Phase Retrieval in Optics," Opt. Acta, 10, 41–49 (1963).

Optik

R. W. Gerchberg and W. O. Saxton, "A Practical Algorithm for the Determination of Phase from Image Diffraction Plane Pictures," Optik (Stutts.), 35, 237–246 (1972).

Other

J. B. Morton, "An Investigation Into An A Posteriori Method of Image Restoration," USCIPI Report No. 810, University of Southern California Image Processing Institute, April 1978.

A. Filip, "Estimating the Impulse Response of Linear, Shift-Invariant Image Degrading Systems," Ph.D. Thesis, M.I.T., October 1972 (unpublished).

B. O'Connor and T. S. Huang, "Phase Unwrapping with Applications to Stability and Picture Deblurring," Image Understanding and Information Extraction, School of Electrical Engineering, Purdue U., Report TR-EE-16, March 1977, pp. 92–142 (unpublished).

B. L. McGlamery, "Image Restoration Techniques Applied to Astronomical Photography," in Astronomical Use of Television-type Image Sensors, Report No. N71-28509-525 of the National Technical Information Service of the U.S. Dept. of Commerce, 1971.

I. P. Natanson, Konstruktive Funktionentheorie, (Akademie-Verlag, Berlin, 1955).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, (McGraw-Hill, New York, 1965).

K. T. Knox, "Diffraction-Limited Imaging with Astronomical Telescopes," Ph.D. Dissertation, U. of Rochester, N.Y., (1975) (unpublished).

G. M. Robbins and T. S. Huang, "Inverse Filtering for Linear Shift-Variant Imaging Systems," Proc. IEEE, 60, 862–872 (1972).

A. A. Sawchuk, "Space-Variant Image Motion Degradation and Restoration," Proc. IEEE, 60, 854–861 (1972).

M. M. Sondhi, "Image Restoration: The Removal of Spatially Invariant Degradations," Proc. IEEE, 60, 842–853 (1972).

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

E. R. Cole, "The Removal of Unknown Image Blurs by Homomorphic Filtering," Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-029, June 1973.

T. M. Cannon, "Digital Image Deblurring by Nonlinear Homomorphic Filtering," Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-091, August 1974.

T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, "Blind Deconvolution Through Digital Signal Processing," Proc. IEEE, 63, 678–692 (1975).

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