Abstract

The technique of maximizing sharpness metrics has been used to estimate and compensate for aberrations with adaptive optics, to correct phase errors in synthetic-aperture radar, and to restore images. The largest class of sharpness metrics is the sum over a nonlinear point transformation of the image intensity. How the second derivative of the point nonlinearity varies with image intensity determines the effects of various metrics on the imagery. Some metrics emphasize making shadows darker, and other emphasize making bright points brighter. One can determine the image content needed to pick the best metric by computing the statistics of the image autocorrelation or of the Fourier magnitude, either of which is independent of the phase error. Computationally efficient, closed-form expressions for the gradient make possible efficient search algorithms to maximize sharpness.

© 2003 Optical Society of America

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References

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    [CrossRef]
  2. R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion ,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
    [CrossRef]
  3. F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–91 (1996).
    [CrossRef]
  4. J. R. Fienup, “Synthetic-aperture radar autofocus by maximizing sharpness,” Opt. Lett. 25, 221–223 (2000).
    [CrossRef]
  5. J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
    [CrossRef]
  6. B. C. Flores, “A robust method for the motion compensation of ISAR Imagery ,” in Intelligent Robots and Computer Vision X: Algorithms and Techniques, D. P. Casasent, ed., Proc. SPIE1607, 512–517 (1991).
    [CrossRef]
  7. S Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, New York, 1975), p. 249.
  8. H. L. Royden, Real Analysis, 2nd ed. (MacMillan, New York, 1968), p. 110.
  9. B. C. Flores, A Martinez, J Hammer, “Optimization of high-resolution-radar motion compensation via entropy-like functions,” in Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1906–1909.
  10. C. V. Jakowatz, D. E. Wahl, “An eigenvector method for maximum likelihood estimation of phase errors in SAR imagery,” J. Opt. Soc. Am. A 10, 2539–2546 (1993).
    [CrossRef]
  11. J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
    [CrossRef]
  12. J. R. Fienup, J. J. Miller, “Generalized image sharpness metrics for correcting phase errors,” in Signal Recovery and Synthesis, Vol. 67 of Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 84–86.

2000 (1)

1997 (1)

1996 (1)

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–91 (1996).
[CrossRef]

1993 (1)

1980 (1)

J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
[CrossRef]

1974 (1)

Berizzi, F.

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–91 (1996).
[CrossRef]

Buffington, A.

Corsini, G.

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–91 (1996).
[CrossRef]

Fienup, J. R.

J. R. Fienup, “Synthetic-aperture radar autofocus by maximizing sharpness,” Opt. Lett. 25, 221–223 (2000).
[CrossRef]

J. R. Fienup, “Invariant error metrics for image reconstruction,” Appl. Opt. 36, 8352–8357 (1997).
[CrossRef]

J. R. Fienup, J. J. Miller, “Generalized image sharpness metrics for correcting phase errors,” in Signal Recovery and Synthesis, Vol. 67 of Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 84–86.

Flores, B. C.

B. C. Flores, “A robust method for the motion compensation of ISAR Imagery ,” in Intelligent Robots and Computer Vision X: Algorithms and Techniques, D. P. Casasent, ed., Proc. SPIE1607, 512–517 (1991).
[CrossRef]

B. C. Flores, A Martinez, J Hammer, “Optimization of high-resolution-radar motion compensation via entropy-like functions,” in Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1906–1909.

Hammer, J

B. C. Flores, A Martinez, J Hammer, “Optimization of high-resolution-radar motion compensation via entropy-like functions,” in Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1906–1909.

Jakowatz, C. V.

Karlin, S

S Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, New York, 1975), p. 249.

Marron, J. C.

R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion ,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
[CrossRef]

Martinez, A

B. C. Flores, A Martinez, J Hammer, “Optimization of high-resolution-radar motion compensation via entropy-like functions,” in Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1906–1909.

Miller, J. J.

J. R. Fienup, J. J. Miller, “Generalized image sharpness metrics for correcting phase errors,” in Signal Recovery and Synthesis, Vol. 67 of Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 84–86.

Muller, R. A.

Paxman, R. G.

R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion ,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
[CrossRef]

Royden, H. L.

H. L. Royden, Real Analysis, 2nd ed. (MacMillan, New York, 1968), p. 110.

Taylor, H. M.

S Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, New York, 1975), p. 249.

Wahl, D. E.

Walker, J. L.

J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
[CrossRef]

Appl. Opt. (1)

IEEE Trans. Aerosp. Electron. Syst. (2)

F. Berizzi, G. Corsini, “Autofocusing of inverse synthetic aperture radar images using contrast optimization,” IEEE Trans. Aerosp. Electron. Syst. 32, 1185–91 (1996).
[CrossRef]

J. L. Walker, “Range-Doppler imaging of rotating objects,” IEEE Trans. Aerosp. Electron. Syst. AES-16, 23–52 (1980).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Lett. (1)

Other (6)

J. R. Fienup, J. J. Miller, “Generalized image sharpness metrics for correcting phase errors,” in Signal Recovery and Synthesis, Vol. 67 of Topics in Optics and Photonics Series (Optical Society of America, Washington, D.C., 2001), pp. 84–86.

B. C. Flores, “A robust method for the motion compensation of ISAR Imagery ,” in Intelligent Robots and Computer Vision X: Algorithms and Techniques, D. P. Casasent, ed., Proc. SPIE1607, 512–517 (1991).
[CrossRef]

S Karlin, H. M. Taylor, A First Course in Stochastic Processes, 2nd ed. (Academic, New York, 1975), p. 249.

H. L. Royden, Real Analysis, 2nd ed. (MacMillan, New York, 1968), p. 110.

B. C. Flores, A Martinez, J Hammer, “Optimization of high-resolution-radar motion compensation via entropy-like functions,” in Antennas and Propagation Society International Symposium Digest (Institute of Electrical and Electronics Engineers, New York, 1993), pp. 1906–1909.

R. G. Paxman, J. C. Marron, “Aberration correction of speckled imagery with an image-sharpness criterion ,” in Statistical Optics, G. M. Morris, ed., Proc. SPIE976, 37–47 (1988).
[CrossRef]

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Figures (16)

Fig. 1
Fig. 1

Simulated images of point targets plus random-clutter background. Ratio of energy of point targets to energy of background: (a) 0, (b) 0.1, (c) 1.

Fig. 2
Fig. 2

Simulated point targets plus random-clutter background plus no-return area.

Fig. 3
Fig. 3

Sixth-order phase error used for simulation experiments.

Fig. 4
Fig. 4

Plots of sharpness metrics versus rms phase error α (in radians) for a few bright points plus clutter background, with target/background ratio of 1.0. All are for a sixth-order polynomial phase error and for point nonlinearity Γ [ I ] = I β . (a) β = 0.5 , (b) β = 2.0 , (c) β = 5 , (d)–(f) expanded versions of (a)–(c), respectively.

Fig. 5
Fig. 5

Same as Fig. 4, but with target/background ratio of 0.01.

Fig. 6
Fig. 6

Same as Fig. 4, but with no bright points and with a 128 × 128 no-return area embedded in a 384 × 384 clutter background.

Fig. 7
Fig. 7

Γ [ I ] , the point nonlinearity of the intensity. If Γ [ I ] is positive, then the sharpness, the sum over Γ [ I ] , will increase as the values of I are spread out.

Fig. 8
Fig. 8

(a) Four point nonlinearities and (b) their second derivatives.

Fig. 9
Fig. 9

Examples of designer metrics and their second derivatives. (a) Γ D 1 , (b) Γ D 1 , (c) Γ D 2 , (d) Γ D 2 , (e) Γ D 3 , (f) Γ D 3 (defined in the text).

Fig. 10
Fig. 10

Real SAR images used in simulations, taken with Veridian/ERIM’s DCS: (a) Stadium, (b) Target, (c) Shadow, (d) Trees.

Fig. 11
Fig. 11

Image-focusing example. (a) Image of Trees smeared by 20-rad rms of sixth-order phase error, (b) image focused after 140 iterations of image sharpening with β = 2 power-law metric.

Fig. 12
Fig. 12

Scaled objective function versus iteration number for Trees.

Fig. 13
Fig. 13

Normalized rms error versus iteration number for Trees.

Fig. 14
Fig. 14

Color plot showing performance of various metrics (see text). (a) Target, with quadratic phase error; (b) Trees, with quadratic phase error; (c) Clutter Plus Shadow, with sixth-order phase error; (d) Clutter Plus Points, with target/clutter ratio 0.1 and quadratic phase error. Metrics for (a) and (b) (left to right): Power law β = 0.45 ,   0.6 ,   0.95 ,   1.1 ,   2 ,   2.5 ,   3 ,   4 ,   5 ; D1, D2, and D3, γ = 0.001 ,   0.01 ,   0.1 ,   1 ,   10 ,   100 ,   1000 ; Shannon entropy, exponential entropy. Metrics for (c) (left to right): Power law β = 1.1 ,   2 ,   2.5 ,   4 ; D1, D2, and D3, γ = 1 ,   10 ,   100 ,   1000 ; 2500 ,   5000 ; Shannon entropy, exponential entropy.Metrics for (d) (left to right): Power law β = 1.1 ,   2 ,   2.5 ,   4 ; D1, D2 and D3, γ = 100 ,   1000 ;   2500 ,   5000 ; Shannon entropy.

Fig. 15
Fig. 15

Images and their autocorrelations. (a) SAR image of cultural objects, (b) autocorrelation of (a), (c) SAR image of Trees, (d) autocorrelation of (c).

Fig. 16
Fig. 16

Autocorrelation statistics versus image sharpness.  *  Kolmogorov–Smirnov; x, maximum value; ○, variance.

Equations (24)

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G d ( x ,   ν ) = F ( x ,   ν ) exp [ i ϕ e ( ν ) ] ,
g d ( x ,   y ) = FT - 1 [ G d ( x ,   ν ) ] = ( 1 / N ) x G d ( x ,   ν ) exp ( i 2 π ν y / N ) ,
G ( x ,   ν ) = G d ( x ,   ν ) exp [ - i ϕ ( ν ) ] ,
g ( x ,   y ) = FT - 1 [ G ( x ,   ν ) ] .
I ( x ,   y ) = | g ( x ,   y ) | 2 .
S Γ = x , y w ( x ) Γ [ I ( x ,   y ) ] .
Γ [ I ( x ,   y ) ] = [ I ( x ,   y ) ] β .
S Γ ϕ ( ν ) = x , y w ( x )   Γ [ I ( x ,   y ) ] I ( x ,   y )   I ( x ,   y ) ϕ ( ν ) ,
I ( x ,   y ) ϕ ( ν ) = | g ( x ,   y ) | 2 ϕ ( ν ) = ( 2 / N ) Im [ g * ( x ,   y ) G ( x ,   ν ) exp ( i 2 π ν y / N ) ]
S Γ ϕ ( ν ) = ( 2 / N ) x w ( x ) × Im G ( x ,   ν ) FT g ( x ,   y )   Γ [ I ( x ,   y ) ] I ( x ,   y ) * .
Γ [ I ( x ,   y ) ] I ( x ,   y ) = β [ I ( x ,   y ) ] β - 1
Γ [ I ( x ,   y ) ] I ( x ,   y ) = ln [ I ( x ,   y ) ] + 1
ϕ ( ν ) = j = 1 J a j L j ( ν ) ,
S Γ a j = x , y w ( x )   Γ [ I ( x ,   t ) ] I ( x ,   y )   ν   I ( x ,   y ) ϕ ( ν )   ϕ ( ν ) a j = ( 2 / N ) ν L j ( ν ) x w ( x ) × Im G ( x ,   ν ) FT g ( x ,   y )   Γ [ I ( x ,   y ) ] I ( x ,   y ) * = ( 2 / N ) ν L j ( ν )   S Γ ϕ ( ν ) ,
Γ [ I ] = 2 Γ I 2 .
S Γ = x , y Γ [ I ( x ,   y ) ] > MN Γ [ I ¯ ] .
Γ D 1 [ I ] = ( I - γ I 0 ) 2 ,
Γ D 1 [ I ] = ( 1 / 12 ) ( I - γ I 0 ) 4 ,
Γ D 2 [ I ] = ( I - γ I 0 ) 2 I + ,
Γ D 3 [ I ] = ( I - γ I 0 ) 4 I + ,
E S = S - S o S i - S o ,
G d ( x 0 ,   ν ) = F ( x 0 , ν ) exp [ i ϕ e ( ν ) ] = | F ( x 0 ,   ν ) | exp [ i ϕ e ( ν ) + i ψ ( x 0 ,   ν ) ] ,
ϕ ( ν ) = - [ ϕ e ( ν ) + ψ ( x 0 ,   ν ) ]
E 2 = min a , x 0 , y 0 | e ia g ( x - x 0 ,   y - y 0 ) - f ( x ,   y ) | 2 | f ( x ,   y ) | 2 = r gg ( 0 ,   0 ) + r ff ( 0 ,   0 ) - 2   max x 0 , y 0 | r fg ( x 0 ,   y 0 ) | r ff ( 0 ,   0 ) ,

Metrics