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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References

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  1. C. W. Hellstrom, “Image Restoration by the Method of Least Squares,” J. Opt. Soc. Am.,  57, 297–303 (1967).
    [Crossref]
  2. B. L. McGlamery, “Restoration of Turbulence Degraded Images,” J. Opt. Soc. Am.,  57, 293–297 (1967).
    [Crossref]
  3. G. M. Robbins and T. S. Huang, “Inverse Filtering for Linear Shift-Variant Imaging Systems,” Proc. IEEE,  60, 862–872 (1972).
    [Crossref]
  4. A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE,  60, 854–861 (1972).
    [Crossref]
  5. M. M. Sondhi, “Image Restoration: The Removal of Spatially Invariant Degradations,” Proc. IEEE,  60, 842–853 (1972).
    [Crossref]
  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).
    [Crossref]
  7. A. A. Sawchuk, “Space-Variant Image Restoration by Coordinate Transformations,” J. Opt. Soc. Am.,  64, 138–144 (1974).
    [Crossref]
  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, June1973.
  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, August1974.
  11. T. G. Stockham, T. M. Cannon, and R. B. Ingebretsen, “Blind Deconvolution Through Digital Signal Processing,” Proc. IEEE,  63, 678–692 (1975).
    [Crossref]
  12. T. M. Cannon, “Blind Deconvolution of Spatially Invariant Image Blurs with Phase,” IEEE Trans. Acoust., Speech, Signal Process.,  24, 58–63 (1976).
    [Crossref]
  13. A. Filip, “Estimating the Impulse Response of Linear, Shift-Invariant Image Degrading Systems,” Ph.D. Thesis, M.I.T., October1972 (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, March1977, 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).
    [Crossref]
  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).
    [Crossref]
  18. E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta,  10, 33–40 (1963).
    [Crossref]
  19. A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta,  10, 41–49 (1963).
    [Crossref]
  20. P. Roman and A. S. Marathay, “Analyticity and Phase Retrieval,” Nuova Cimento,  30, 1452–1463 (1973).
    [Crossref]
  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).
    [Crossref]
  23. B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys.,  16, 1719–1725 (1975).
    [Crossref]
  24. R. A. Gonsalves, “Phase retrieval from modulus data,” J. Opt. Soc. Am.,  66, 961–964 (1976).
    [Crossref]
  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, April1978.

1976 (3)

1975 (2)

B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys.,  16, 1719–1725 (1975).
[Crossref]

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

1974 (2)

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

A. A. Sawchuk, “Space-Variant Image Restoration by Coordinate Transformations,” J. Opt. Soc. Am.,  64, 138–144 (1974).
[Crossref]

1973 (3)

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).
[Crossref]

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

P. Roman and A. S. Marathay, “Analyticity and Phase Retrieval,” Nuova Cimento,  30, 1452–1463 (1973).
[Crossref]

1972 (4)

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).

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

A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE,  60, 854–861 (1972).
[Crossref]

M. M. Sondhi, “Image Restoration: The Removal of Spatially Invariant Degradations,” Proc. IEEE,  60, 842–853 (1972).
[Crossref]

1967 (3)

1963 (2)

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta,  10, 33–40 (1963).
[Crossref]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta,  10, 41–49 (1963).
[Crossref]

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).
[Crossref]

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

T. M. Cannon, “Digital Image Deblurring by Nonlinear Homomorphic Filtering,” Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-091, August1974.

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, June1973.

Filip, A.

A. Filip, “Estimating the Impulse Response of Linear, Shift-Invariant Image Degrading Systems,” Ph.D. Thesis, M.I.T., October1972 (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).
[Crossref]

Huang, T. S.

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

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, March1977, 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).
[Crossref]

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).
[Crossref]

Knox, K. T.

K. T. Knox, “Image retrieval from astronomical speckle patterns,” J. Opt. Soc. Am.,  66, 1236–1239 (1976).
[Crossref]

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

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).
[Crossref]

McGlamery, B. L.

B. L. McGlamery, “Restoration of Turbulence Degraded Images,” J. Opt. Soc. Am.,  57, 293–297 (1967).
[Crossref]

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, April1978.

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, March1977, 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).
[Crossref]

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).
[Crossref]

Roman, P.

P. Roman and A. S. Marathay, “Analyticity and Phase Retrieval,” Nuova Cimento,  30, 1452–1463 (1973).
[Crossref]

Sawchuk, A. A.

A. A. Sawchuk, “Space-Variant Image Restoration by Coordinate Transformations,” J. Opt. Soc. Am.,  64, 138–144 (1974).
[Crossref]

A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE,  60, 854–861 (1972).
[Crossref]

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).
[Crossref]

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).
[Crossref]

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).
[Crossref]

Walther, A.

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta,  10, 33–40 (1963).
[Crossref]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta,  10, 41–49 (1963).
[Crossref]

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. (1)

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

IEEE Trans. Acoust., Speech, Signal Process. (2)

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

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 (1)

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

J. Math. Phys. (1)

B. J. Hoenders, “On the Solution of the Phase Retrieval Problem,” J. Math. Phys.,  16, 1719–1725 (1975).
[Crossref]

J. Opt. Soc. Am. (6)

Nuova Cimento (1)

P. Roman and A. S. Marathay, “Analyticity and Phase Retrieval,” Nuova Cimento,  30, 1452–1463 (1973).
[Crossref]

Opt. Acta (2)

E. L. O’Neill and A. Walther, “The Question of Phase in Image Formation,” Opt. Acta,  10, 33–40 (1963).
[Crossref]

A. Walther, “The Question of Phase Retrieval in Optics,” Opt. Acta,  10, 41–49 (1963).
[Crossref]

Optik(Stutts.) (1)

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).

Proc. IEEE (4)

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

A. A. Sawchuk, “Space-Variant Image Motion Degradation and Restoration,” Proc. IEEE,  60, 854–861 (1972).
[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, and R. B. Ingebretsen, “Blind Deconvolution Through Digital Signal Processing,” Proc. IEEE,  63, 678–692 (1975).
[Crossref]

Other (10)

J. B. Morton, “An Investigation Into An A Posteriori Method of Image Restoration,” USCIPI Report No. 810, University of Southern California Image Processing Institute, April1978.

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).

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, June1973.

T. M. Cannon, “Digital Image Deblurring by Nonlinear Homomorphic Filtering,” Dept. of Computer Science, University of Utah, ARPA Technical Report UTEC-CSC-74-091, August1974.

A. Filip, “Estimating the Impulse Response of Linear, Shift-Invariant Image Degrading Systems,” Ph.D. Thesis, M.I.T., October1972 (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, March1977, pp. 92–142 (unpublished).

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

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Figures (13)

FIG. 1
FIG. 1

Location of sample points.

FIG. 2
FIG. 2

Representation of H(u).

FIG. 3
FIG. 3

Example of phase bound for L = 3.

FIG. 4
FIG. 4

Schematic of restoration processing using interpolation in the Fourier domain.

FIG. 5
FIG. 5

Four statistically similar images.

FIG. 6
FIG. 6

Log10 of magnitude autocorrolation for Δu = 1, Δv = 0.

FIG. 7
FIG. 7

Log10 of magnitude autocorrolation for Δu = 0, Δυ = 1.

FIG. 8
FIG. 8

Estimates of magnitude and phase of OTF.

FIG. 9
FIG. 9

Blurred images and restorations.

FIG. 10
FIG. 10

Estimates of magnitude and phase of OTF.

FIG. 11
FIG. 11

Blurred images and restorations.

FIG. 12
FIG. 12

Estimates of magnitude and phase of OTF for third real world blurred images.

FIG. 13
FIG. 13

Blurred images and restorations.

Tables (1)

Tables Icon

TABLE I Correlation coefficients between average phase differences of pairs of images of Fig. 5.

Equations (66)

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

g ( x , y ) = h ( x ξ , y η ) f ( ξ , η ) d ξ d η + n ( x , y ) .
G ( u , υ ) = H ( u , υ ) F ( u , υ ) + N ( u , υ ) ,
g i ( x , y ) i h ( x ξ , y η ) f i ( ξ , η ) d ξ d η ,
G i ( u , υ ) H ( u , υ ) F i ( u , υ ) .
| G i ( u , υ ) | e j θ G i ( u , υ ) | H ( u , υ ) | e j θ H ( u , υ ) | F i ( u , υ ) | e j θ F i ( u , υ ) .
| G i ( u , υ ) | | H ( u , υ ) F i ( u , υ ) | , ln | G i ( u , υ ) | ln | H ( u , υ ) | + ln | F i ( u , υ ) | ,
1 N i = 1 N ln | G i ( u , υ ) | ln | H ( u , υ ) | + 1 N i = 1 N ln | F i ( u , υ ) | .
1 N i = 1 N ln | P i ( u , υ ) | 1 N i = 1 N ln | F i ( u , υ ) | ,
ln | H ( u , υ ) | 1 N i = 1 N ln | G i ( u , υ ) | 1 N i = 1 N ln | P i ( u , υ ) | .
ϕ g ( u , υ ) = ϕ f ( u , υ ) | H ( u , υ ) | 2 + ϕ n ( u , υ )
| H ( u , υ ) | = [ ( ϕ g ( u , υ ) / ϕ f ( u , υ ) ) ] 1 / 2 .
θ G i ( u , υ ) θ H ( u , υ ) + θ F i ( u , υ ) .
θ ¯ G ( u , υ ) θ H ( u , υ ) + θ ¯ F ( u , υ ) .
g i ( x , y ) = h i ( x , y ) * f ( x , y ) ,
G i ( u , υ ) = H i ( u , υ ) F ( u , υ ) .
G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) = H i ( u , υ ) H i * ( u + Δ u , υ + Δ υ ) × F ( u , υ ) F * ( u + Δ u , υ + Δ υ ) ,
[ G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) ] = [ H i ( u , υ ) H i * ( u + Δ u , υ + Δ υ ) ] × F ( u , υ ) F * ( u + Δ u , υ + Δ υ ) .
phase { G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) } = phase { H i ( u , υ ) H i * ( u + Δ u , υ + Δ υ ) } + phase { F ( u , υ ) F * ( u + Δ u , υ + Δ υ ) } .
θ F ( u , υ ) θ F ( u + Δ u , υ + Δ υ ) phase { G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) }
G i ( u , υ ) = H ( u , υ ) F i ( u , υ ) .
G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) = H ( u , υ ) H * ( u + Δ u , υ + Δ υ ) × F i ( u , υ ) F i * ( u + Δ u , υ + Δ υ ) .
1 N i = 1 N G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) = H ( u , υ ) H * ( u + Δ u , υ + Δ υ ) 1 N i = 1 N F i ( u , υ ) × F i * ( u + Δ u , υ + Δ υ ) .
1 N i = 1 N G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ )
1 N i = 1 N F i ( u , υ ) F i * ( u + Δ u , υ + Δ υ )
H ( 0 , 0 ) = 1 .
H * ( u + Δ u , υ + Δ υ ) = 1 N i = 1 N G i ( u , υ ) G i * ( u + Δ u , υ + Δ υ ) H ( u , υ ) 1 N i = 1 N F i ( u , υ ) F i * ( u + Δ u , υ + Δ υ ) .
| G i ( u , υ ) | 2 = | H ( u , υ ) | 2 | F i ( u , υ ) | 2 .
| H ( u , υ ) | 2 = 1 N i = 1 N | G i ( u , υ ) | 2 / 1 N i = 1 N | F i ( u , υ ) | 2 .
θ G i ( u , υ ) θ G i ( u + Δ u , υ + Δ υ ) = θ H ( u , υ ) θ H ( u + Δ u , υ + Δ υ ) + θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ,
θ H ( u + Δ u , υ + Δ υ ) = θ H ( u , υ ) [ θ G i ( u , υ ) θ G i ( u + Δ u , υ + Δ υ ) ] av + [ θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ] av ,
1 N i = 1 N | F i ( u , υ ) | 2
1 N i = 1 N | F i ( u , υ ) F i * ( u + Δ u , υ + Δ υ ) |
[ θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ] av
H ( u ) = k = N / 2 N / 2 1 h ( k ) e ( j 2 π k u / N ) = k = L L h ( k ) e ( j 2 π k u / N ) = k = L L h ( k ) cos ( 2 π k u N ) j k = L L h ( k ) sin ( 2 π k u N ) = R ( u ) j I ( u ) .
θ ( u ) = tan 1 [ I ( u ) / R ( u ) ] .
T ( u ) = k = 0 L a k cos ( 2 π k u N ) + b k sin ( 2 π k u N )
R ( u ) = k = L L h k cos ( 2 π k u N ) .
R ( u ) = k = 0 L a k cos ( 2 π k u N ) ,
I ( u ) = k = 0 L b k sin ( 2 π k u N ) ,
I ( u ) = I ( u ) .
L π < θ ( N / 2 ) < L π
L π < θ ( N / 2 ) < L π .
L π θ ( N / 2 ) L π
L π θ ( N / 2 ) L π .
L π θ ( N / 2 , 0 ) L π , L π θ ( N / 2 , 0 ) L π .
L π θ ( 0 , N / 2 ) L π , L π θ ( 0 , N / 2 ) L π .
G ( u , υ ) = H ( u , υ ) F ( u , υ ) .
F ̂ ( u , υ ) = G ( u , υ ) / Ĥ ( u , υ ) ,
R ( u , υ ) = H * ( u , υ ) ϕ f ( u , υ ) | H ( u , υ ) | 2 ϕ f ( u , υ ) + ϕ n ( u , υ ) ,
ϕ g ( u , υ ) = | H ( u , υ ) | 2 ϕ f ( u , υ ) + ϕ n ( u , υ ) .
1 N i = 1 N | F i ( u , υ ) | 2
[ θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ] av ;
1 N i = 1 N | F i ( u , υ ) F i * ( u + Δ u , υ + Δ υ ) |
[ θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ] av .
1 N i = 1 N | F i ( u , υ ) |
1 N i = 1 N | F i k ( u , υ ) F i k * ( u + Δ u , υ + Δ υ ) |
1 N i = 1 N | F i ( u , υ ) F i * ( u + Δ u , υ + Δ υ ) |
[ θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ] av
[ θ F i ( u , υ ) θ F i ( u + Δ u , υ + Δ υ ) ] av
θ G i ( u , υ ) θ H ( u , υ ) + θ F i ( u , υ )
G ( u , υ ) * W ( u , υ ) = [ H ( u , υ ) F ( u , υ ) ] * W ( u , υ ) ,
G i ( u , υ ) * W ( u , υ ) [ H ( u , υ ) F i ( u , υ ) ] * W ( u , υ )
G i ( u , υ ) * W ( u , υ ) H ( u , υ ) [ F i ( u , υ ) ] * W ( u , υ )
| G i ( u , υ ) * W ( u , υ ) | | H ( u , υ ) F i ( u , υ ) * W ( u , υ ) | .
θ G i * W ( u , υ ) θ H ( u , υ ) + θ F i * W ( u , υ ) ,
θ H ( u + Δ u , υ + Δ υ ) = θ H ( u , υ ) [ θ G i * W ( u , υ ) θ G i * W ( u + Δ u , υ + Δ υ ) ] av + [ θ F i * W ( u , υ ) θ F i * W ( u + Δ u , υ + Δ υ ) ] av .