Abstract

We develop a maximum-likelihood (ML) algorithm for estimation and correction (autofocus) of phase errors induced in synthetic-aperture-radar (SAR) imagery. Here, M pulse vectors in the range-compressed domain are used as input for simultaneously estimating M − 1 phase values across the aperture. The solution involves an eigenvector of the sample covariance matrix of the range-compressed data. The estimator is then used within the basic structure of the phase gradient autofocus (PGA) algorithm, replacing the original phase-estimation kernel. We show that, in practice, the new algorithm provides excellent restorations to defocused SAR imagery, typically in only one or two iterations. The performance of the new phase estimator is demonstrated essentially to achieve the Cramér–Rao lower bound on estimation-error variance for all but small values of target-toclutter ratio. We also show that for the case in which M is equal to 2, the ML estimator is similar to that of the original PGA method but achieves better results in practice, owing to a bias inherent in the original PGA phase estimation kernel. Finally, we discuss the relationship of these algorithms to the shear-averaging and spatial correlation methods, two other phase-correction techniques that utilize the same phase-estimation kernel but that produce substantially poorer performance because they do not employ several fundamental signal-processing steps that are critical to the algorithms of the PGA class.

© 1993 Optical Society of America

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References

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  1. P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “A new phase correction method for synthetic aperture radar,” in Proceedings of the Digital Signal Processing Workshop at Stanford Sierra Lodge (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 2.12.1–2.12.2.
  2. P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Speckle processing method for synthetic-aperture-radar phase correction,” Opt. Lett. 14, 1–3 (1989).
    [CrossRef] [PubMed]
  3. P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, D. E. Wahl, “Phase-gradient autofocus for SAR phase correction: explanation and demonstration of algorithmic steps,” in Proceedings of the Digital Signal Processing Workshop at Starved Rock State Park (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 6.6.1–6.6.2.
    [CrossRef]
  4. D. E. Wahl, P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Phase gradient autofocus: a robust tool for high resolution SAR phase correction,”IEEE Trans. Aerosp. Electron. Syst. (to be published).
  5. D. A. Gray, W. O. Wolfe, J. L. Riley, “An eigenvector method for estimating the positions of the elements of an array of receivers,” in Proceedings of the Australian Symposium on Signal Processing Applications (Adelaide, Australia, 1989), pp. 391–393.
  6. D. A. Gray, J. L. Riley, “Maximum likelihood estimate and Cramér–Rao bound for a complex signal vector,” in Proceedings of the International Symposium on Signal Processing Applications (Gold Coast, Australia, 1990), pp. 352–355.
  7. Although this assumption is unreasonable for real SAR scenes, the PGA algorithm utilizes shifting of the strongest reflector of each range line to the scene center to approximate this condition.
  8. N. R. Goodman (Ref. 9) discusses the general conditions under which Eq. (3) is valid.
  9. N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Statist. Anal. 34, 152–177 (1963).
    [CrossRef]
  10. The relevant formula here is that of Sherman–Morrison.11
  11. G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, p. 3.
  12. H. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2, pp. 79–81.
  13. J. Fienup, “Phase error correction by shear averaging,” in Signal Recovery and Synthesis II, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 134–137.
  14. E. H. Attia, B. D. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,”IEEE Trans. Antennas Propagat. 37, 30–38 (1989).
    [CrossRef]
  15. P. H. Eichel, “The phase gradient autofocus algorithm: an optimal estimator of the phase derivative,” (Sandia National Laboratories, Albuquerque, N.M., 1989).

1989 (2)

E. H. Attia, B. D. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,”IEEE Trans. Antennas Propagat. 37, 30–38 (1989).
[CrossRef]

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Speckle processing method for synthetic-aperture-radar phase correction,” Opt. Lett. 14, 1–3 (1989).
[CrossRef] [PubMed]

1963 (1)

N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Statist. Anal. 34, 152–177 (1963).
[CrossRef]

Attia, E. H.

E. H. Attia, B. D. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,”IEEE Trans. Antennas Propagat. 37, 30–38 (1989).
[CrossRef]

Eichel, P. H.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Speckle processing method for synthetic-aperture-radar phase correction,” Opt. Lett. 14, 1–3 (1989).
[CrossRef] [PubMed]

D. E. Wahl, P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Phase gradient autofocus: a robust tool for high resolution SAR phase correction,”IEEE Trans. Aerosp. Electron. Syst. (to be published).

P. H. Eichel, “The phase gradient autofocus algorithm: an optimal estimator of the phase derivative,” (Sandia National Laboratories, Albuquerque, N.M., 1989).

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “A new phase correction method for synthetic aperture radar,” in Proceedings of the Digital Signal Processing Workshop at Stanford Sierra Lodge (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 2.12.1–2.12.2.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, D. E. Wahl, “Phase-gradient autofocus for SAR phase correction: explanation and demonstration of algorithmic steps,” in Proceedings of the Digital Signal Processing Workshop at Starved Rock State Park (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 6.6.1–6.6.2.
[CrossRef]

Fienup, J.

J. Fienup, “Phase error correction by shear averaging,” in Signal Recovery and Synthesis II, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 134–137.

Ghiglia, D. C.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Speckle processing method for synthetic-aperture-radar phase correction,” Opt. Lett. 14, 1–3 (1989).
[CrossRef] [PubMed]

D. E. Wahl, P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Phase gradient autofocus: a robust tool for high resolution SAR phase correction,”IEEE Trans. Aerosp. Electron. Syst. (to be published).

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “A new phase correction method for synthetic aperture radar,” in Proceedings of the Digital Signal Processing Workshop at Stanford Sierra Lodge (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 2.12.1–2.12.2.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, D. E. Wahl, “Phase-gradient autofocus for SAR phase correction: explanation and demonstration of algorithmic steps,” in Proceedings of the Digital Signal Processing Workshop at Starved Rock State Park (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 6.6.1–6.6.2.
[CrossRef]

Golub, G. H.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, p. 3.

Goodman, N. R.

N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Statist. Anal. 34, 152–177 (1963).
[CrossRef]

Gray, D. A.

D. A. Gray, W. O. Wolfe, J. L. Riley, “An eigenvector method for estimating the positions of the elements of an array of receivers,” in Proceedings of the Australian Symposium on Signal Processing Applications (Adelaide, Australia, 1989), pp. 391–393.

D. A. Gray, J. L. Riley, “Maximum likelihood estimate and Cramér–Rao bound for a complex signal vector,” in Proceedings of the International Symposium on Signal Processing Applications (Gold Coast, Australia, 1990), pp. 352–355.

Jakowatz, C. V.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Speckle processing method for synthetic-aperture-radar phase correction,” Opt. Lett. 14, 1–3 (1989).
[CrossRef] [PubMed]

D. E. Wahl, P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Phase gradient autofocus: a robust tool for high resolution SAR phase correction,”IEEE Trans. Aerosp. Electron. Syst. (to be published).

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “A new phase correction method for synthetic aperture radar,” in Proceedings of the Digital Signal Processing Workshop at Stanford Sierra Lodge (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 2.12.1–2.12.2.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, D. E. Wahl, “Phase-gradient autofocus for SAR phase correction: explanation and demonstration of algorithmic steps,” in Proceedings of the Digital Signal Processing Workshop at Starved Rock State Park (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 6.6.1–6.6.2.
[CrossRef]

Riley, J. L.

D. A. Gray, J. L. Riley, “Maximum likelihood estimate and Cramér–Rao bound for a complex signal vector,” in Proceedings of the International Symposium on Signal Processing Applications (Gold Coast, Australia, 1990), pp. 352–355.

D. A. Gray, W. O. Wolfe, J. L. Riley, “An eigenvector method for estimating the positions of the elements of an array of receivers,” in Proceedings of the Australian Symposium on Signal Processing Applications (Adelaide, Australia, 1989), pp. 391–393.

Steinberg, B. D.

E. H. Attia, B. D. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,”IEEE Trans. Antennas Propagat. 37, 30–38 (1989).
[CrossRef]

Van Loan, C. F.

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, p. 3.

Van Trees, H.

H. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2, pp. 79–81.

Wahl, D. E.

D. E. Wahl, P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Phase gradient autofocus: a robust tool for high resolution SAR phase correction,”IEEE Trans. Aerosp. Electron. Syst. (to be published).

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, D. E. Wahl, “Phase-gradient autofocus for SAR phase correction: explanation and demonstration of algorithmic steps,” in Proceedings of the Digital Signal Processing Workshop at Starved Rock State Park (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 6.6.1–6.6.2.
[CrossRef]

Wolfe, W. O.

D. A. Gray, W. O. Wolfe, J. L. Riley, “An eigenvector method for estimating the positions of the elements of an array of receivers,” in Proceedings of the Australian Symposium on Signal Processing Applications (Adelaide, Australia, 1989), pp. 391–393.

Ann. Statist. Anal. (1)

N. R. Goodman, “Statistical analysis based on a certain multivariate complex Gaussian distribution (an introduction),” Ann. Statist. Anal. 34, 152–177 (1963).
[CrossRef]

IEEE Trans. Antennas Propagat. (1)

E. H. Attia, B. D. Steinberg, “Self-cohering large antenna arrays using the spatial correlation properties of radar clutter,”IEEE Trans. Antennas Propagat. 37, 30–38 (1989).
[CrossRef]

Opt. Lett. (1)

Other (12)

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, D. E. Wahl, “Phase-gradient autofocus for SAR phase correction: explanation and demonstration of algorithmic steps,” in Proceedings of the Digital Signal Processing Workshop at Starved Rock State Park (Institute of Electrical and Electronics Engineers, New York, 1992), pp. 6.6.1–6.6.2.
[CrossRef]

D. E. Wahl, P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Phase gradient autofocus: a robust tool for high resolution SAR phase correction,”IEEE Trans. Aerosp. Electron. Syst. (to be published).

D. A. Gray, W. O. Wolfe, J. L. Riley, “An eigenvector method for estimating the positions of the elements of an array of receivers,” in Proceedings of the Australian Symposium on Signal Processing Applications (Adelaide, Australia, 1989), pp. 391–393.

D. A. Gray, J. L. Riley, “Maximum likelihood estimate and Cramér–Rao bound for a complex signal vector,” in Proceedings of the International Symposium on Signal Processing Applications (Gold Coast, Australia, 1990), pp. 352–355.

Although this assumption is unreasonable for real SAR scenes, the PGA algorithm utilizes shifting of the strongest reflector of each range line to the scene center to approximate this condition.

N. R. Goodman (Ref. 9) discusses the general conditions under which Eq. (3) is valid.

The relevant formula here is that of Sherman–Morrison.11

G. H. Golub, C. F. Van Loan, Matrix Computations (Johns Hopkins U. Press, Baltimore, Md., 1983), Chap. 1, p. 3.

H. Van Trees, Detection, Estimation, and Modulation Theory, Part I (Wiley, New York, 1968), Chap. 2, pp. 79–81.

J. Fienup, “Phase error correction by shear averaging,” in Signal Recovery and Synthesis II, Vol. 15 of 1989 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 134–137.

P. H. Eichel, “The phase gradient autofocus algorithm: an optimal estimator of the phase derivative,” (Sandia National Laboratories, Albuquerque, N.M., 1989).

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “A new phase correction method for synthetic aperture radar,” in Proceedings of the Digital Signal Processing Workshop at Stanford Sierra Lodge (Institute of Electrical and Electronics Engineers, New York, 1988), pp. 2.12.1–2.12.2.

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Figures (9)

Fig. 1
Fig. 1

Algorithmic steps of the PGA.

Fig. 2
Fig. 2

Estimation-error variance versus Cramér–Rao lower bound (CRLB). SNR, signal-to-noise ratio; MSE, mean-square error.

Fig. 3
Fig. 3

a, Original image; b, degraded image; c, 20 pulses, 1 iteration; d, 2 pulses, 1 iteration; e, 2 pulses, 3 iterations.

Fig. 4
Fig. 4

High-order phase-error function.

Fig. 5
Fig. 5

Restored impulse response plot from image corrupted with high-order phase-error function.

Fig. 6
Fig. 6

a, Degraded image; b, 2 pulses, 1 iteration; c, 2 pulses, 6 iterations; d, 20 pulses, 1 iteration; e, 20 pulses, 2 iterations.

Fig. 7
Fig. 7

Low-order phase-error function.

Fig. 8
Fig. 8

a, Degraded image; b, ML, 1 iteration; c, original PGA, 1 iteration; d, ML, 3 iterations; e, original PGA, 5 iterations.

Fig. 9
Fig. 9

Restored impulse response plot from image corrupted with low-order phase-error function.

Equations (35)

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

g k 1 = a k + n k 1 , g k 2 = a k e j ψ 2 + n k 2 , g k M = a k e j ψ m + n k M ,
β = σ a 2 σ n 2 .
x k t = [ g k 1 , g k 2 , g k M ] .
ln p ( X Ψ ) = - N ln [ π M C ] - k = 1 N x k H C - 1 x k ,
Ψ t = [ 0 , ψ 2 , ψ M ]
C = σ n 2 I + σ a 2 v v H .
v = [ 1 e j ψ 2 e j ψ M ] .
λ 1 = M σ a 2 + σ n 2 , λ 2 = σ n 2 , λ M = σ n 2 .
C = σ n 2 M ( 1 + M β ) ,
Q 1 = - k = M x k H C - 1 x k ,
C - 1 = a 1 I + a 2 v v H ,
a 1 = 1 σ n 2 , a 2 = - β σ n 2 + M σ a 2 .
Q 2 = k = N x k H v v H x k = k = 1 N v H x k x k H v = v H ( k = N x k x k H ) v .
C ^ = 1 N k = 1 N x k x k H ,
Q 3 = v H C ^ v ,
v 2 = v H v = M ,
var [ ψ ^ i ( X ) - ψ i ] J i i ,
J i j = E [ ln p ( X Ψ ) ψ i ln p ( X Ψ ) ψ j ] = - E [ 2 ln p ( X Ψ ) ψ i ψ j ] .
J = - 2 a 2 N σ a 2 [ M - 1 - 1 - 1 - 1 M - 1 - 1 - 1 - 1 - 1 - 1 M - 1 ] .
J = - 2 a 2 N σ a 2 [ M I - 1 1 T ] ,
1 T = [ 1 , 1 , 1 ] .
J - 1 = b 1 [ I + 1 1 T ] ,
b 1 = - 1 2 M N a 2 σ a 2 .
var [ ψ ^ i ( X ) - ψ i ] 1 + M β M N β 2 .
Q 3 = 1 N [ 1 - e - j ψ ] [ k = 1 N g k l 2 k = 1 N g k l g k , l + 1 * k = 1 N g k l * g k , l + 1 k = 1 N g k , l + 1 2 ] [ 1 e j ψ ] .
k = 1 N ( g k l g k , l + 1 * e j ψ + g k , l + 1 * e - j ψ ) = | k = 1 N g k l * g k , l + 1 | cos [ Δ ψ - ( k = 1 N g k l * g k , l + 1 ) ] .
ψ ^ M L = ( k = 1 N g k l * g k , l + 1 ) ,
ψ ^ ˙ PGA ( t ) = k = N Im [ g ˙ k ( t ) g k * ( t ) ] k = 1 N g k ( t ) 2 ,
Q 3 = v H C ^ v ,
v 2 = v H v = M
C ^ = P Λ P H ,
Q 3 = v H P Λ P H v .
Q 3 = z H Λ z .
z 2 = z H z = v H P P H v = v H v = v 2 .
Q 3 = k = 1 N λ k z k 2 ,

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