Abstract

Information theoretic bounds on the estimated Zernike coefficients for various diversity phase functions are presented. We show that, in certain cases, defocus diversity may yield a higher Cramer–Rao lower bound (CRLB) than some other diversity phase functions. Using simulated images to evaluate the performance of the phase-diversity algorithm, we find that, for an extended scene and defocus diversity, the phase-diversity algorithm achieves the CRLB for known objects. Furthermore, the phase-diversity algorithm achieves the CRLB by a factor of ~2 for unknown objects.

© 2005 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. R. Lucke, “Fundamentals of wide-field sparse-aperture imaging,” IEEE Aerospace Conf. Proc. 3, 3/1401–3/1419 (2001).
  2. R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
    [CrossRef]
  3. J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 43–47 (2000).
    [CrossRef]
  4. A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  5. R. R. Parenti, R. J. Sasiela, “Laser-guide-star systems for astronomical applications,” J. Opt. Soc. Am. A 11, 288–309 (1994).
    [CrossRef]
  6. M. C. Roggemann, B. Welsh, Imaging through Turbulence, (CRC Press, 1996).
  7. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
    [CrossRef]
  8. R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
    [CrossRef]
  9. R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. A 9, 1072–1085 (1992).
    [CrossRef]
  10. J. J. Dolne, R. T. Tansey, K. A. Black, J. H. Deville, P. R. Cunningham, K. C. Widen, P. S. Idell, “Practical issues in wave-front sensing by use of phase diversity,” Appl. Opt. 42, 5284–5289 (2003).
    [CrossRef] [PubMed]
  11. L. Meynadier, V. Michau, M. T. Velluet, J. M. Conan, L. M. Mugnier, G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38, 4967–4979 (1999).
    [CrossRef]
  12. D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. A 16, 1005–1015 (1999).
    [CrossRef]
  13. S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).
  14. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
  15. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).
  16. R. Noll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  17. J. J. Dolne, D. Gerwe, M. M. Johnson, “Performance of three reconstruction methods on blurred and noisy images of extended scenes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 164–175 (1999).
    [CrossRef]

2003

2002

R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
[CrossRef]

2001

R. Lucke, “Fundamentals of wide-field sparse-aperture imaging,” IEEE Aerospace Conf. Proc. 3, 3/1401–3/1419 (2001).

1999

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. A 16, 1005–1015 (1999).
[CrossRef]

L. Meynadier, V. Michau, M. T. Velluet, J. M. Conan, L. M. Mugnier, G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38, 4967–4979 (1999).
[CrossRef]

1994

1992

R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. A 9, 1072–1085 (1992).
[CrossRef]

1988

1982

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

1976

1970

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Black, K. A.

Calus, J. R.

R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
[CrossRef]

Conan, J. M.

Cunningham, P. R.

Deville, J. H.

Dolne, J. J.

J. J. Dolne, R. T. Tansey, K. A. Black, J. H. Deville, P. R. Cunningham, K. C. Widen, P. S. Idell, “Practical issues in wave-front sensing by use of phase diversity,” Appl. Opt. 42, 5284–5289 (2003).
[CrossRef] [PubMed]

J. J. Dolne, D. Gerwe, M. M. Johnson, “Performance of three reconstruction methods on blurred and noisy images of extended scenes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 164–175 (1999).
[CrossRef]

Fienup, J. R.

R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 43–47 (2000).
[CrossRef]

Fiete, R. D.

R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
[CrossRef]

Gerwe, D.

J. J. Dolne, D. Gerwe, M. M. Johnson, “Performance of three reconstruction methods on blurred and noisy images of extended scenes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 164–175 (1999).
[CrossRef]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

Idell, P. S.

Johnson, M. M.

J. J. Dolne, D. Gerwe, M. M. Johnson, “Performance of three reconstruction methods on blurred and noisy images of extended scenes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 164–175 (1999).
[CrossRef]

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lee, D. J.

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. A 16, 1005–1015 (1999).
[CrossRef]

Lucke, R.

R. Lucke, “Fundamentals of wide-field sparse-aperture imaging,” IEEE Aerospace Conf. Proc. 3, 3/1401–3/1419 (2001).

Meynadier, L.

Michau, V.

Mooney, J. A.

R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
[CrossRef]

Mugnier, L. M.

Noll, R.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Parenti, R. R.

Paxman, R. G.

R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. A 9, 1072–1085 (1992).
[CrossRef]

R. G. Paxman, J. R. Fienup, “Optical misalignment sensing and image reconstruction using phase diversity,” J. Opt. Soc. Am. A 5, 914–923 (1988).
[CrossRef]

Roggemann, M. C.

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. A 16, 1005–1015 (1999).
[CrossRef]

M. C. Roggemann, B. Welsh, Imaging through Turbulence, (CRC Press, 1996).

Rousset, G.

Sasiela, R. J.

Schultz, T. J.

R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. A 9, 1072–1085 (1992).
[CrossRef]

Tansey, R. T.

Tantalo, T. A.

R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
[CrossRef]

Velluet, M. T.

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence, (CRC Press, 1996).

Welsh, B. M.

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. A 16, 1005–1015 (1999).
[CrossRef]

Widen, K. C.

Appl. Opt.

Astron. Astrophys.

A. Labeyrie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

IEEE Aerospace Conf. Proc.

R. Lucke, “Fundamentals of wide-field sparse-aperture imaging,” IEEE Aerospace Conf. Proc. 3, 3/1401–3/1419 (2001).

J. Opt. Soc. A

R. G. Paxman, T. J. Schultz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. A 9, 1072–1085 (1992).
[CrossRef]

D. J. Lee, M. C. Roggemann, B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. A 16, 1005–1015 (1999).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Opt. Eng.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21, 829–832 (1982).
[CrossRef]

R. D. Fiete, T. A. Tantalo, J. R. Calus, J. A. Mooney, “Image quality of sparse-aperture designs for remote sensing,” Opt. Eng. 41, 1957–1969 (2002).
[CrossRef]

Other

J. R. Fienup, “MTF and integration time versus fill factor for sparse-aperture imaging systems,” in Imaging Technology and Telescopes, J. W. Bilbro, J. B. Breckinridge, R. A. Carreras, S. R. Czyzak, M. J. Eckart, R. D. Fiete, P. S. Idell, eds., Proc. SPIE4091, 43–47 (2000).
[CrossRef]

M. C. Roggemann, B. Welsh, Imaging through Turbulence, (CRC Press, 1996).

S. M. Kay, Fundamentals of Statistical Signal Processing: Estimation Theory (Prentice-Hall, 1993).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, 1968).

J. J. Dolne, D. Gerwe, M. M. Johnson, “Performance of three reconstruction methods on blurred and noisy images of extended scenes,” in Digital Image Recovery and Synthesis IV, T. J. Schulz, P. S. Idell, eds., Proc. SPIE3815, 164–175 (1999).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1

128 × 128 pixel pristine scene used in CRLB evaluation. The image is sampled at the Nyquist value.

Fig. 2
Fig. 2

Combined CRLB on aberration parameters for a point source for aberration sets A, B, and C for (a) Poisson and (b) Gaussian noise. Average number of total photocounts, K = 1; σ = 1/128; X = Y = 128; fd = 0.5.

Fig. 3
Fig. 3

Combined CRLB on aberration parameters for an extended source for aberration sets A, B, and C for (a) Poisson and (b) Gaussian noise. Average number of total photocounts, K = 1; σ = 1/128; X = Y = 128; fd = 0.5.

Fig. 4
Fig. 4

CRLB on aberration parameters for an extended source as a function of Zernike term number for various diversity polynomials for (a) Poisson and (b) Gaussian noise for aberration set A. Average number of total photocounts, K = 1; σ = 1/128; X = Y = 128; fd = 0.5.

Fig. 5
Fig. 5

CRLB on aberration parameters for an extended source as a function of Zernike term number for various diversity polynomials for (a) Poisson and (b) Gaussian noise for aberration set B. Average number of total photocounts, K = 1; σ = 1/128; X = Y = 128; fd = 0.5.

Fig. 6
Fig. 6

CRLB on aberration parameters for an extended source as a function of Zernike term number for various diversity polynomials for (a) Poisson and (b) Gaussian noise for aberration set C. Average number of total photocounts, K = 1; σ = 1/128; X = Y = 128; fd = 0.5.

Fig. 7
Fig. 7

Combined CRLB and rss wavefront error from the phase-diversity algorithm for known and unknown objects. SNRpix = 40.

Tables (1)

Tables Icon

Table 1 Aberration Sets Used in Phase-Diversity Evaluationa

Equations (30)

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

i d ( x , y ) = f d o h d ( x , y ) + n d ( x , y ) ,
h ( x , y ) = F - 1 [ H ( u , v ) ] ,
H ( u , v ) = P ( u , v ) exp [ i 2 π Θ ( u , v ) ] P ( u , v ) exp [ i 2 π Θ ( u , v ) ] , H d ( u , v ) = P ( u , v ) exp { i 2 π [ Θ ( u , v ) + Φ ( u , v ) ] } P ( u , v ) exp { i 2 π [ Θ ( u , v ) + Φ ( u , v ) ] } ,
h ( x , y ) = a ( x , y ) 2 ,             h d ( x , y ) = a d ( x , y ) 2 .
a d ( x , y ) = F - 1 ( P ( u , v ) exp { i 2 π [ Θ ( u , v ) + Φ d ( u , v ) ] } ) .
I ( u , v ) = F [ i ( x , y ) ] , I d ( u , v ) = F [ i d ( x , y ) ] , O ( u , v ) = F [ o ( x , y ) ] , H ( u , v ) = F [ h ( x , y ) ] , H d ( u , v ) = F [ h d ( x , y ) ] , N ( u , v ) = F [ n ( x , y ) ] , N d ( u , v ) = F [ n d ( x , y ) ] .
l [ o , h i 1 ( 1 , 1 ) , , i D ( X , Y ) ] = d = 1 , x = 1 , y = 1 D , X , Y 1 2 π σ 2 × exp { - [ i d ( x , y ) - f d o h d ( x , y ) ] 2 w ( x , y ) 2 σ 2 } ,
L = - 1 2 σ 2 d , x , y [ i d ( x , y ) - f d o h d ( x , y ) ] 2 w ( x , y ) .
E = 1 2 σ 2 d , x , y [ i d ( x , y ) - f d o h d ( x , y ) ] 2 w ( x , y ) .
O ^ ( u , v ) = f 1 H ^ * ( u , v ) I ( u , v ) + f d H ^ d * ( u , v ) I d ( u , v ) f 1 2 H ^ ( u , v ) 2 + f d 2 H ^ d ( u , v ) 2 ,
E = u , v f d I ( u , v ) H ^ d ( u , v ) - f 1 I d ( u , v ) H ^ ( u , v ) 2 f 1 2 H ^ ( u , v ) 2 + f d 2 H ^ d ( u , v ) 2 .
E = 1 2 σ 2 d , x , y [ i d ( x , y ) - f d o h d ( x , y ) ] 2 w ( x , y ) .
F τ k = - E { 2 L c τ c k } ,
L c k = 2 2 σ 2 x , y , d [ i d - f d o h d ( x , y ) ] × [ f d o h d ( x , y ) c k ] w ( x , y ) .
h d ( x , y ) c k = a d * ( x , y ) a d ( x , y ) c k + c . c . ,
h d ( x , y ) c k = - 2 imag { a d * ( x , y ) × F - 1 [ 2 π Z k ( u , v ) C d ( u , v ) ] } ,
C d ( u , v ) = P ( u , v ) exp { i 2 π [ Θ ( u , v ) + Φ d ( u , v ) ] } .
2 L c τ c k = 1 σ 2 x , y , d { [ i d ( x , y ) - f d o h d ( x , y ) ] [ f d o 2 h d ( x , y ) c τ c k ] - f d 2 τ k = τ , k [ o h d ( x , y ) c τ k ] } w ( x , y ) ,
τ k = τ , k [ o h d ( x , y ) c τ k ] = [ o h d ( x , y ) c τ ] [ o h d ( x , y ) c k ] .
F τ k = 1 σ 2 x , y , d { f d 2 [ x , y o ( x , y ) h d ( x - x , y - y ) c τ ] × [ a , b o ( a , b ) h d ( x - a , y - b ) c k ] } w ( x , y ) .
F τ k = K 2 σ 2 x , y , d { f d 2 [ x , y o n ( x , y ) h d ( x - x , y - y ) c τ ] × [ a , b o n ( a , b ) h d ( x - a , y - b ) c k ] } w ( x , y ) .
F τ k = ( SNR pix X Y ) 2 x , y , d f d 2 × [ x , y o n ( x , y ) h d ( x - x , y - y ) c τ ] × [ a , b o n ( a , b ) h d ( x - a , y - b ) c k ] w ( x , y ) ,
F τ k = 4 ( SNR pix X Y ) 2 x , y , d f d 2 ( o n imag { a d * ( x , y ) × F - 1 [ 2 π Z τ ( u , v ) C d ( u , v ) ] } ) × ( o n imag { a d * ( x , y ) F - 1 [ 2 π Z k ( u , v ) × C d ( u , v ) ] } ) w ( x , y ) .
F τ k = K x , y , d { f d o n h d ( x , y ) × [ x , y o n ( x , y ) h d ( x - x , y - y ) c τ ] × [ a , b o n ( a , b ) h d ( x - a , y - b ) c k ] } w ( x , y ) .
F τ k = SNR pix 2 X X x , y , d f d o n h d ( x , y ) × [ x , y o n ( x , y ) h d ( x - x , y - y ) c τ ] × [ a , b o n ( a , b ) h d ( x - a , y - b ) c k ] w ( x , y ) ,
F τ k = 4 ( SNR pix ) 2 X Y x , y , d f d o n h d ( x , y ) × ( o n imag { a d * ( x , y ) F - 1 [ 2 π Z τ ( u , v ) C d ( u , v ) ] } ) × ( o n imag { a d * ( x , y ) F - 1 [ 2 π Z k ( u , v ) C d ( u , v ) ] } ) × w ( x , y ) .
[ o h d ( x , y ) c τ ] [ o h d ( x , y ) c k ]
[ o h d ( x , y ) c τ ] [ o h d ( x , y ) c k ]
{ E [ ( c i - c ^ i ) 2 ] } 1 / 2 ,
σ i 2 = E [ ( c i - c ^ i ) 2 ] = 1 N k = 1 N ( c i - c ^ i , k ) 2 .

Metrics