Abstract

The phase-gradient algorithm represents a powerful new signal-processing technique with applications to aperture-synthesis imaging. These include, for example, synthetic-aperture-radar phase correction and stellar-image reconstruction. The algorithm combines redundant information present in the data to arrive at an estimate of the phase derivative. We show that the estimator is in fact a linear, minimum-variance estimator of the phase derivative.

© 1989 Optical Society of America

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References

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  1. P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, Opt. Lett. 14, 1 (1989).
    [CrossRef] [PubMed]
  2. G. J. M. Aitken, R. Johnson, R. Houtman, Opt. Commun. 56, 379 (1986).
    [CrossRef]
  3. D. C. Ghiglia, G. A. Mastin, Opt. Lett. 14, 1104 (1989).
    [CrossRef] [PubMed]
  4. D. C. Ghiglia, L. A. Romero, Opt. Lett. 14, 1107 (1989).
    [CrossRef] [PubMed]
  5. J. W. Goodman, J. Opt. Soc. Am. 66, 1145 (1976).
    [CrossRef]
  6. P. H. Eichel, “The phase gradient autofocus algorithm: an optimal estimator of the phase derivative,” Rep. SAND89-0761 (Sandia National Laboratories, Albuquerque, N.M., 1989).
  7. J. Melsa, D. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

1989

1986

G. J. M. Aitken, R. Johnson, R. Houtman, Opt. Commun. 56, 379 (1986).
[CrossRef]

1976

Aitken, G. J. M.

G. J. M. Aitken, R. Johnson, R. Houtman, Opt. Commun. 56, 379 (1986).
[CrossRef]

Cohn, D.

J. Melsa, D. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Eichel, P. H.

P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, Opt. Lett. 14, 1 (1989).
[CrossRef] [PubMed]

P. H. Eichel, “The phase gradient autofocus algorithm: an optimal estimator of the phase derivative,” Rep. SAND89-0761 (Sandia National Laboratories, Albuquerque, N.M., 1989).

Ghiglia, D. C.

Goodman, J. W.

Houtman, R.

G. J. M. Aitken, R. Johnson, R. Houtman, Opt. Commun. 56, 379 (1986).
[CrossRef]

Jakowatz, C. V.

Johnson, R.

G. J. M. Aitken, R. Johnson, R. Houtman, Opt. Commun. 56, 379 (1986).
[CrossRef]

Mastin, G. A.

Melsa, J.

J. Melsa, D. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Romero, L. A.

J. Opt. Soc. Am.

Opt. Commun.

G. J. M. Aitken, R. Johnson, R. Houtman, Opt. Commun. 56, 379 (1986).
[CrossRef]

Opt. Lett.

Other

P. H. Eichel, “The phase gradient autofocus algorithm: an optimal estimator of the phase derivative,” Rep. SAND89-0761 (Sandia National Laboratories, Albuquerque, N.M., 1989).

J. Melsa, D. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

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Equations (24)

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g ( t ) = a exp { j [ ω 0 t + ϕ ( t ) ] } + n ( t ) ,
n ( t ) = m = M M b m exp [ j ( ω m t + θ m ) ] ,
ϕ ˙ k ( t ) = Im [ g ˙ k ( t ) g k * ( t ) ] | g k ( t ) | 2 .
n ( t ) = α ( t ) exp [ j ϕ n ( t ) ] ,
g ( t ) = a exp [ j ϕ ( t ) ] + α ( t ) exp [ j ϕ n ( t ) ] = a exp [ j ϕ ( t ) ] ( 1 + α ( t ) a exp { j [ ϕ n ( t ) ϕ ε ( t ) ] } ) .
ϕ ( t ) = g ( t ) = ϕ ( t ) + tan 1 [ α ( t ) a Im ( exp { j [ ϕ n ( t ) ϕ ( t ) ] } ) 1 + α ( t ) a Re ( exp { j [ ϕ n ( t ) ϕ ( t ) ] } ) ] .
ϕ ( t ) ϕ ( t ) + α ( t ) a Im ( exp { j [ ϕ n ( t ) ϕ ( t ) ] } ) .
V ( t ) α ( t ) a Im ( exp { j [ ϕ n ( t ) ϕ ( t ) ] } ) ,
ϕ ( t ) ϕ ( t ) + V ( t ) .
ϕ ˙ ( t ) ϕ ˙ ( t ) + V ˙ ( t ) .
E [ V ˙ ( t ) ] = 0 ,
E [ V ˙ 2 ( t ) ] = C 0 a 2 ,
E [ V ˙ ( t ) ϕ ˙ ( t ) ] = 0 ,
E [ ϕ ˙ ( t ) ] = E [ ϕ ˙ ( t ) ] ,
g k ( t ) = a k exp [ j ϕ ( t ) ] + n k ( t ) , k [ 1 , K ] .
ϕ ˙ k ( t ) = Im [ g ˙ k ( t ) g k * ( t ) ] | g k ( t ) | 2 , k [ 1 , K ] .
ϕ ˙ k ( t ) = Im [ g ˙ k ( t ) g k * ( t ) ] a k 2 = ϕ ˙ ( t ) + V ˙ k ( t ) , k [ 1 , K ] .
Φ ˙ = H ϕ ˙ + V ,
Φ ˙ = [ Im [ g ˙ 1 ( t ) g 1 * ( t ) ] / a 1 2 Im [ g ˙ K ( t ) g K * ( t ) ] / a K 2 ] is K × 1 , H = [ 1 1 ] is K × 1 , V = [ V ˙ ( t ) V ˙ ( t ) ] is K × 1 .
R V = C 0 [ 1 a 1 2 0 0 1 a K 2 ] .
ϕ ˙ ˆ ( t ) = ( H T R V 1 H + σ ϕ ˙ 2 ) 1 ( H T R V 1 Φ ˙ + σ ϕ ˙ 2 μ ϕ ˙ ) ,
σ ϕ ˙ 2 = 0 , μ ϕ ˙ = 0 ,
ϕ ˙ ˆ ( t ) = ( H T R V 1 H ) 1 H T R V 1 Φ ˙ .
ϕ ˙ ˆ ( t ) = [ [ 11 1 ] 1 C 0 [ a 1 2 0 0 a K 2 ] [ 1 1 1 ] ] 1 [ 11 1 ] 1 C 0 [ a 1 2 0 0 a K 2 ] [ Im [ g ˙ 1 ( t ) g 1 * ( t ) ] / a 1 2 Im [ g ˙ K ( t ) g K * ( t ) ] / a K 2 ] , = k = 1 K Im [ g ˙ k ( t ) g k * ( t ) ] k = 1 K | g k ( t ) | 2 ,

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