Abstract

We study the problem of extracting target features by means of synthetic-aperture radar (SAR) in the presence of uncompensated aperture motion errors. A parametric data model for a spotlight-mode SAR system is established. The Cramér–Rao bounds (CRB’s) for the parameters of the data model are also derived. The CRB analysis shows that the unknown motion errors can significantly affect the accuracy of a common shift of the scatterer positions in the cross-range direction but have little effect on other target parameters, including the accuracy of the relative positions in the range direction. A relaxation-based motion compensation RELAX (MCRELAX) algorithm for estimating both target features and motion errors is devised. Simulation results show that the mean squared errors of the parameter estimates obtained by using the MCRELAX algorithm can approach the corresponding CRB’s. We also use a couple of examples to show that MCRELAX can simply be used for motion compensation only and can give good motion-compensation results.

© 1998 Optical Society of America

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References

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  1. P. H. Eichel, D. C. Ghiglia, C. V. Jakowatz, “Speckle processing methods for synthetic-aperture-radar phase correction,” Opt. Lett. 14, 1–3 (1989).
    [CrossRef] [PubMed]
  2. P. H. Eichel, C. V. Jakowatz, “Phase-gradient algorithm as an optimal estimator of the phase derivative,” Opt. Lett. 14, 1101–1103 (1989).
    [CrossRef] [PubMed]
  3. C. V. Jakowatz, D. E. Wahl, “Eigenvector method for maximum-likelihood estimation of phase errors in synthetic-aperture-radar imagery,” J. Opt. Soc. Am. A 10, 2539–2546 (1993).
    [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. 30, 827–835 (1994).
    [CrossRef]
  5. C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).
  6. J. Li, P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE Trans. Signal Process. 44, 281–295 (1996).
    [CrossRef]
  7. D. C. Munson, J. D. O’Brien, W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE 71, 917–925 (1983).
    [CrossRef]
  8. V. G. Karmanov, Programmation Mathematique (Mir, Moscow, 1977).
  9. W. Bangs, “Array processing with generalized beamformers,” Ph.D. dissertation (Yale University, New Haven, Conn., 1971).
  10. P. Stoica, R. L. Moses, Introduction to Spectral Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  11. D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

1996 (1)

J. Li, P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE Trans. Signal Process. 44, 281–295 (1996).
[CrossRef]

1994 (2)

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

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. 30, 827–835 (1994).
[CrossRef]

1993 (1)

1989 (2)

1983 (1)

D. C. Munson, J. D. O’Brien, W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE 71, 917–925 (1983).
[CrossRef]

Andersh, D. J.

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

Bangs, W.

W. Bangs, “Array processing with generalized beamformers,” Ph.D. dissertation (Yale University, New Haven, Conn., 1971).

Chu, Y.

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

Eichel, P. H.

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. 30, 827–835 (1994).
[CrossRef]

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

P. H. Eichel, C. V. Jakowatz, “Phase-gradient algorithm as an optimal estimator of the phase derivative,” Opt. Lett. 14, 1101–1103 (1989).
[CrossRef] [PubMed]

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).

Ghiglia, D. C.

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. 30, 827–835 (1994).
[CrossRef]

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

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).

Hazlett, M.

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

Jakowatz, C. V.

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. 30, 827–835 (1994).
[CrossRef]

C. V. Jakowatz, D. E. Wahl, “Eigenvector method for maximum-likelihood estimation of phase errors in synthetic-aperture-radar imagery,” J. Opt. Soc. Am. A 10, 2539–2546 (1993).
[CrossRef]

P. H. Eichel, C. V. Jakowatz, “Phase-gradient algorithm as an optimal estimator of the phase derivative,” Opt. Lett. 14, 1101–1103 (1989).
[CrossRef] [PubMed]

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

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).

Jenkins, W. K.

D. C. Munson, J. D. O’Brien, W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE 71, 917–925 (1983).
[CrossRef]

Karmanov, V. G.

V. G. Karmanov, Programmation Mathematique (Mir, Moscow, 1977).

Lee, S. W.

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

Li, J.

J. Li, P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE Trans. Signal Process. 44, 281–295 (1996).
[CrossRef]

Moses, R. L.

P. Stoica, R. L. Moses, Introduction to Spectral Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Munson, D. C.

D. C. Munson, J. D. O’Brien, W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE 71, 917–925 (1983).
[CrossRef]

O’Brien, J. D.

D. C. Munson, J. D. O’Brien, W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE 71, 917–925 (1983).
[CrossRef]

Reeves, D. D.

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

Stoica, P.

J. Li, P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE Trans. Signal Process. 44, 281–295 (1996).
[CrossRef]

P. Stoica, R. L. Moses, Introduction to Spectral Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Sullivan, D. P.

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

Thompson, P. A.

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).

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. 30, 827–835 (1994).
[CrossRef]

C. V. Jakowatz, D. E. Wahl, “Eigenvector method for maximum-likelihood estimation of phase errors in synthetic-aperture-radar imagery,” J. Opt. Soc. Am. A 10, 2539–2546 (1993).
[CrossRef]

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).

IEEE Trans. Aerosp. Electron. Syst. (1)

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. 30, 827–835 (1994).
[CrossRef]

IEEE Trans. Antennas Propag. Mag. (1)

D. J. Andersh, M. Hazlett, S. W. Lee, D. D. Reeves, D. P. Sullivan, Y. Chu, “XPATCH: a high-frequency electromagnetic scattering prediction code and environment for complex three-dimensional objects,” IEEE Trans. Antennas Propag. Mag. 36, 65–69 (1994).

IEEE Trans. Signal Process. (1)

J. Li, P. Stoica, “Efficient mixed-spectrum estimation with applications to target feature extraction,” IEEE Trans. Signal Process. 44, 281–295 (1996).
[CrossRef]

J. Opt. Soc. Am. A (1)

Opt. Lett. (2)

Proc. IEEE (1)

D. C. Munson, J. D. O’Brien, W. K. Jenkins, “A tomographic formulation of spotlight-mode synthetic aperture radar,” Proc. IEEE 71, 917–925 (1983).
[CrossRef]

Other (4)

V. G. Karmanov, Programmation Mathematique (Mir, Moscow, 1977).

W. Bangs, “Array processing with generalized beamformers,” Ph.D. dissertation (Yale University, New Haven, Conn., 1971).

P. Stoica, R. L. Moses, Introduction to Spectral Analysis (Prentice-Hall, Englewood Cliffs, N.J., 1977).

C. V. Jakowatz, D. E. Wahl, P. H. Eichel, D. C. Ghiglia, P. A. Thompson, Spotlight-Mode Synthetic Aperture Radar: A Signal Processing Approach (Kluwer Academic, Norwell, Mass., 1996).

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Figures (7)

Fig. 1
Fig. 1

Data collection geometry in a spotlight-mode SAR system.

Fig. 2
Fig. 2

Flow chart of the MCRELAX algorithm.

Fig. 3
Fig. 3

Comparison of (a) the original image with (b) the image with added motion errors before motion compensation and the images with added motion errors after motion compensation obtained by (c) the PGA algorithm and (d) the MCRELAX algorithm by using the dashed curves in Fig. 4 below.

Fig. 4
Fig. 4

Comparison of the true phase errors (solid curves) with their estimates (dashed curves) obtained by (a) PGA and (b) MCRELAX. The dashed–dotted curves are for {ψ¯m¯-mβ¯}m¯=2M¯-1, where β is manually adjusted to best fit the dashed–dotted curves to the solid curves.

Fig. 5
Fig. 5

Comparison of (a) the original image (with three embedded point scatterers) with (b) the image with added motion errors before motion compensation and the images with added motion errors after motion compensation obtained by (c) the PGA algorithm and (d) the MCRELAX algorithm by using the dashed curves in Fig. 6 below.

Fig. 6
Fig. 6

Comparison of the true phase errors (solid curves) with their estimates (dashed curves) obtained by (a) PGA and (b) MCRELAX. The dashed–dotted curve in (b) is obtained in the same way as those in Fig. 4.

Fig. 7
Fig. 7

Simulated tank example with M=M¯=32 and σ2=20: (a) true |αk| versus fk and f¯k, (b) |αˆ(f, f¯)| with added motion errors versus f and f¯, (c) |αˆ(f, f¯)| versus f and f¯ obtained by PGA, (d) |αˆk| versus fˆk and f¯ˆk obtained by PGA-RELAX, (e) |αˆk| versus fˆk and f¯ˆk obtained by MCRELAX.

Tables (3)

Tables Icon

Table 1 True Parameter Values of the Simulated Tank

Tables Icon

Table 2 Comparison of the CRB’s (in decibels) of the Target Parameters for the Simulated Tank Example

Tables Icon

Table 3 Comparison of the CRB’s (in decibels) of the Parameters of the Simulated Tank with the MSE’s Obtained by PGA-RELAX (MSE1) and MCRELAX (MSE2)

Equations (67)

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s(t)=cos[(2πf0t+γt2)],|t|T0/2,
r(t)=δτ exp{-j[2πf0(t-τ)+γ(t-τ)2]},
d˜(t)=δτ exp[j(2πf0-2γτ0)(τ-τ0)]×exp[-jγ(τ-τ0)2]exp[j2γ(τ-τ0)t].
d˜(t)=δτ exp[j(2πf0-2γτ0)(τ-τ0)]×exp[j2γ(τ-τ0)t],
R=[(R0 cos θ cos ϕ-x)2+(R0 sin θ cos ϕ-y)2+(R0 sin ϕ-z)2]1/2.
R=R01-2 xR0cos θ cos ϕ-2 yR0sin θ cos ϕ-2 zR0sin ϕ+x2+y2+z2R021/2
R01-xR0cos θ cos ϕ-yR0sin θ cos ϕ-zR0sin ϕ+x2+y2+z2-(x cos θ cos ϕ+y sin θ cos ϕ+z sin ϕ)22R02=R0-x cos θ cos ϕ-y sin θ sin ϕ-z sin ϕ+x2+y2+z2-(x cos θ cos ϕ+y sin θ cos ϕ+z sin ϕ)22R0.
x2+y2+z2-(x cos θ cos ϕ+y sin θ cos ϕ+z sin ϕ)2x2sin2 ϕ+y2+z2cos2 ϕ-xy sin 2θ cos2 ϕ-xz cos θ sin 2ϕ-yz sin θ sin 2ϕx2sin2 ϕ+y2+z2cos2 ϕ-xz sin 2ϕcos ϕ×cos θ cos ϕ-2(xy cos ϕ+yz sin ϕ)sin θ cos ϕ,
z sin ϕ(z tan ϕ)cos θ cos ϕ.
RR0-x˜ cos θ cos ϕ-y˜ sin θ cos ϕ,
x˜=x+z tan ϕ-x2sin2 ϕ+y2+z2cos2 ϕ-xz sin 2ϕ2R0 cos ϕ,
y˜=y+xy cos ϕ+yz sin ϕR0.
d(t, θ)=δx, y,z exp[j(x˜tx+y˜ty)],
tx=-4[πf0+γ(t-τ0)]cos ϕccos θ,
ty=-4[πf0+γ(t-τ0)]cos ϕcsin θ.
s˜(m, m¯)=k=1Kα˜k exp[j2π(mfk+m¯f˜k)],
m=0, 1, , M-1,m¯=0, 1, , M¯-1,
y(m, m¯)=s˜(m, m¯)exp(jψ˜m¯)+e(m, m¯),
αk=α˜k exp(jψ˜0),k=1, 2, , K,
ψm¯=ψ˜m¯-ψ˜0-(ψ˜1-ψ˜0)m¯,m¯=0, 1, , M¯-1,
f¯k=f˜k+ψ˜1-ψ˜02π.
y(m, m¯)=s(m, m¯)exp(jψm¯)+e(m, m¯),
s(m, m¯)=k=1Kαk exp[j2π(mfk+m¯f¯k)].
P=diag(1, 1, exp(jψ2),  , exp(jψM¯-1)),
ωM(fk)=(1 exp(j2πfk) exp[j2π(M-1)fk])T,
Y=k=1KαkωM(fk)ωM¯T(f¯k)P+E.
C1({αk,fk,f¯k}k=1K, {ψm¯}m¯=2M¯-1)
=Y-k=1KαkωM(fk)ωM¯T(f¯k)PF2,
C2({ψm¯}m¯=2M¯-1)=m¯=2M¯-1ym¯-sˆm¯ exp(jψm¯)2,
ψ¯m¯=angle(sˆm¯Hym¯),m¯=2, 3, , M¯-1,
Z=YPˆ-1.
C3({αk, fk, f¯k}k=1K)=Z-k=1KαkωM(fk)ωM¯T(f¯k)F2.
Zk=Z-i=1,ikK¯αˆiωM(fˆi)ωM¯T(f¯ˆi).
(fˆk, f¯ˆk)=max(fk, f¯k)|ωMH(fk)ZkωM¯*(f¯k)|2,
αˆk=ωMH(fk)ZkωM¯*(f¯k)MM¯fk=fˆk,f¯k=f¯ˆk.
y=vec(Y),
e=vec(E),
y=k=1Kαk[PωM¯(f¯k)]ωM(fk)+eΩα+e,
Ω=[[PωM¯(f¯1)]ωM(f1)[PωM¯(f¯K)]ωM(fK)],
α=(α1α2αK)T.
(FIM)ij=tr(Q-1QiQ-1Qj)+2 Re[(αHΩH)iQ-1(Ωα)j],
η=([Re(α)]T[Im(α)]TfTf¯TψT)T,
f=(f1f2fK)T,
f¯=(f¯1f¯2f¯K)T,
ψ=(ψ2ψ3ψM¯-1)T.
F=[ΩjΩDfDf¯Dψ],
CRB(η)=[2 Re(FHQ-1F)]-1.
f˘k=fk-δ,
k=1Kf˘k=0.
δ=1Kk=1Kfk.
f˘=Tf,
T=1KK-1-1-1-1K-1-1-1-1K-1.
f¯˘=Tf¯,
δ¯=1Kk=1Kf¯k.
CRB(f˘)=T CRB(f)TT,
CRB(δ)=1K21T CRB(f)1,
1=(111)T.
CRB(f¯˘)=T CRB(f¯)TT,
CRB(δ¯)=1K21T CRB(f¯)1.
|vm,m¯|=max(|vm,0|, |vm,1|, , |vm,M¯-1|),
v˜m=(v˜m,0v˜m,1v˜m,M¯-1)=(vm,m¯vm,m¯+1vm,M¯-1vm,0vm,1vm,m¯-1).
v˘m=(v˘m,0v˘m,1v˘m,M¯-1)=(v˜m,0v˜m,d00v˜m,M¯-dv˜m,M¯-1).
sm¯=m=0M-1|v˜m,m¯|2,m¯=0, 1, , M¯-1,
Δϕˆm¯=Δϕ˜m¯-Δϕ˜1,m¯=1, 2, , M¯-1,
Δϕ˜m¯=angle(gm¯-1Hgm¯).
yˆ(m, m¯)=y(m, m¯)exp(-jϕˆm¯),
m=0, 1, , M-1,m¯=0, 1, , M¯-1.

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