Abstract

The technique of phase diversity has been used in traditional incoherent imaging systems to jointly estimate an object and optical system aberrations. This paper extends the technique of phase diversity to polarimetric imaging systems. Specifically, we describe penalized-likelihood methods for jointly estimating Stokes images and optical system aberrations from measurements that contain phase diversity. Jointly estimating Stokes images and optical system aberrations involves a large parameter space. A closed-form expression for the estimate of the Stokes images in terms of the aberration parameters is derived and used in a formulation that reduces the dimensionality of the search space to the number of aberration parameters only. We compare the performance of the joint estimator under both quadratic and edge-preserving regularization; we also compare the performance of the reduced parameter search strategy to the full parameter search strategy under quadratic regularization. The joint estimator with edge-preserving regularization yields higher fidelity polarization estimates than with quadratic regularization. With the reduced parameter search strategy, accurate aberration estimates can be obtained without recourse to regularization “tuning.”

© 2010 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
    [CrossRef] [PubMed]
  2. H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).
  3. W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144–152 (1998).
    [CrossRef]
  4. G. C. Giakos, “Multifusion, multispectral, optical polarimetric imaging sensing principles,” IEEE Trans. Instrum. Meas. 55, 1628–1633 (2006).
    [CrossRef]
  5. R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22.
  6. E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).
  7. J. R. Valenzuela and J. A. Fessler, “Joint reconstruction of Stokes images from polarimetric measurements,” J. Opt. Soc. Am. A 26, 962–968 (2009).
    [CrossRef]
  8. R. Paxman, T. Schulz, and J. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  9. D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramer–Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16, 1005–1015 (1999).
    [CrossRef]
  10. J. W. Goodman, Fourier Optics, 3rd ed. (Roberts, 2005).
  11. R. J. Knoll, “Zernike polynomials and atmospheric turbulence,” J. Opt. Soc. Am. 66, 207–211 (1976).
    [CrossRef]
  12. H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
    [CrossRef]
  13. R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).
  14. J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).
  15. D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
    [CrossRef]
  16. J. Fessler, Department of Electrical and Computer Engineering, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109–2122, USA, is preparing a book to be called Image Reconstruction: Algorithms and Analysis.

2009 (1)

2006 (2)

J. S. Tyo, D. L. Goldstein, D. B. Chenault, and J. A. Shaw, “Review of passive imaging polarimetry for remote sensing applications,” Appl. Opt. 45, 5453–5469 (2006).
[CrossRef] [PubMed]

G. C. Giakos, “Multifusion, multispectral, optical polarimetric imaging sensing principles,” IEEE Trans. Instrum. Meas. 55, 1628–1633 (2006).
[CrossRef]

2005 (1)

J. W. Goodman, Fourier Optics, 3rd ed. (Roberts, 2005).

2004 (1)

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

1999 (1)

1998 (2)

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144–152 (1998).
[CrossRef]

1996 (1)

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

1995 (1)

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22.

1992 (1)

1989 (1)

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

1979 (1)

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

1976 (1)

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Barrett, H.

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Chenault, D. B.

Chidlaw, R.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Chipman, R. A.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22.

Egan, W. G.

W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144–152 (1998).
[CrossRef]

Engl, H. W.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

Fessler, J.

J. Fessler, Department of Electrical and Computer Engineering, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109–2122, USA, is preparing a book to be called Image Reconstruction: Algorithms and Analysis.

Fessler, J. A.

Fienup, J.

Giakos, G. C.

G. C. Giakos, “Multifusion, multispectral, optical polarimetric imaging sensing principles,” IEEE Trans. Instrum. Meas. 55, 1628–1633 (2006).
[CrossRef]

Goldstein, D. L.

Gonsalves, R. A.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Goodman, J. W.

J. W. Goodman, Fourier Optics, 3rd ed. (Roberts, 2005).

Hanke, M.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

Hecht, E.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

Knoll, R. J.

Lee, D. J.

Liu, D.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Myers, K. J.

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

Nelder, J. A.

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

Neubauer, A.

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[CrossRef]

Nocedal, J.

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Paxman, R.

Roggemann, M. C.

Schulz, T.

Shaw, J. A.

Tyo, J. S.

Valenzuela, J. R.

Welsh, B. M.

Appl. Opt. (1)

Comput. J. (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7, 308–313 (1965).

IEEE Trans. Instrum. Meas. (1)

G. C. Giakos, “Multifusion, multispectral, optical polarimetric imaging sensing principles,” IEEE Trans. Instrum. Meas. 55, 1628–1633 (2006).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Math. Program. (1)

D. Liu and J. Nocedal, “On the limited memory BFGS method for large scale optimization,” Math. Program. 45, 503–528 (1989).
[CrossRef]

Proc. SPIE (2)

W. G. Egan, “Polarization and surface roughness,” Proc. SPIE 3426, 144–152 (1998).
[CrossRef]

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 207, 32–39 (1979).

Other (6)

H. Barrett and K. J. Myers, Foundations of Image Science (Wiley, 2004).

J. Fessler, Department of Electrical and Computer Engineering, University of Michigan, 1301 Beal Avenue, Ann Arbor, Michigan 48109–2122, USA, is preparing a book to be called Image Reconstruction: Algorithms and Analysis.

R. A. Chipman, “Polarimetry,” in Handbook of Optics, 2nd ed., M.Bass, ed. (McGraw-Hill, 1995), Vol. 2, Chap. 22.

E. Hecht, Optics, 3rd ed. (Addison-Wesley, 1998).

J. W. Goodman, Fourier Optics, 3rd ed. (Roberts, 2005).

H. W. Engl, M. Hanke, and A. Neubauer, Regularization of Inverse Problems (Kluwer Academic, 1996).
[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 (13)

Fig. 1
Fig. 1

Traditional phase-diversity imaging strategy.

Fig. 2
Fig. 2

Polarimetric phase-diversity strategy utilizing the division-of-focal-plane technique.

Fig. 3
Fig. 3

Polarimetric phase-diversity strategy utilizing the division-of-amplitude technique.

Fig. 4
Fig. 4

Phase of the generalized pupil function.

Fig. 5
Fig. 5

Image estimation results for SNR = 45 dB . From left to right: object, estimate using edge-preserving regularizer, estimate using quadratic regularizer, and the conventional estimate.

Fig. 6
Fig. 6

Image estimation results for SNR = 25 dB . From left to right: object, estimate using edge-preserving regularizer, estimate using quadratic regularizer, and the conventional estimate.

Fig. 7
Fig. 7

Minimum mean squared error as a function of defocus measured from peak to valley.

Fig. 8
Fig. 8

Data for SNR = 45 dB : from left to right: {0°, 45°, 90°, 135°}.

Fig. 9
Fig. 9

Data for SNR = 25 dB : from left to right: {0°, 45°, 90°, 135°}.

Fig. 10
Fig. 10

Cuts through a column of TPOL for the object and reconstructions with edge-preserving and quadratic regularization at SNR = 45 dB .

Fig. 11
Fig. 11

Cuts through a column of TPOL for the object and reconstructions with edge-preserving and quadratic regularization at SNR = 25 dB .

Fig. 12
Fig. 12

Residual wavefront errors for SNR = 45 dB . From left to right: edge-preserving regularization, quadratic regularization tuned for object estimation, and quadratic regularization tuned for aberration estimation.

Fig. 13
Fig. 13

Residual wavefront errors for SNR = 25 dB . From left to right: edge-preserving regularization, quadratic regularization tuned for object estimation, and quadratic regularization tuned for aberration estimation.

Tables (2)

Tables Icon

Table 1 RMS Error Percentages for SNR = 45 dB

Tables Icon

Table 2 RMS Error Percentages for SNR = 25 dB

Equations (25)

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

Γ ( θ ) = 1 2 [ S 0 + S 1 cos ( 2 θ ) + S 2 sin ( 2 θ ) ] .
[ Γ ( θ 1 ) Γ ( θ J ) ] = 1 2 [ 1 cos ( 2 θ 1 ) sin ( 2 θ 1 ) 1 cos ( 2 θ J ) sin ( 2 θ J ) ] [ S 0 S 1 S 2 ] .
Γ = ( T J × 3 I n p ) S , n p = N M ,
S ̂ conv = { [ ( T J × 3 T J × 3 ) 1 T J × 3 ] I n p } Γ ,
y j ( n , m ) = b j ( n , m ) Γ j ( n , m ) + ε j ( n , m ) n = 1 , , N , m = 1 , , M ,
y j = B j [ ( T J × 3 ) j I n p ] S + ε j , j = 1 , , J ,
y = B ( T J × 3 I n p ) S + ε ,
H ( u , v ) = A ( u , v ) exp [ ı W ( u , v ) ] ,
H ( u , v ; α ) = A ( u , v ) exp [ ı k = 1 K α k ϕ k ( u , v ) ] , where α = ( α 1 , , α K ) .
H j ( u , v ; α , φ j ) = A ( u , v ) exp { ı [ k = 1 K α k ϕ k ( u , v ) + φ j ( u , v ) ] } .
h j ( x , y ; α , φ j ) = c | F 1 [ H j ( u , v ; α , φ j ) ] | 2 ,
H j ( u , v ; α , φ j ) = c F [ | F 1 [ H j ( u , v ; α , φ j ) ] | 2 ] ,
B j ( α ) = Q Ω j ( α ) Q ,
L ( S , α ) = 1 2 σ 2 y B ( α ) ( T J × 3 I n p ) S 2 .
( S ̂ , α ̂ ) = argmin ( S , α ) { L ( S , α ) + R ( S ) } argmin ( S , α ) Ψ ( S , α ) ,
R ( S ) = l = 0 2 k = 1 2 n p β l ψ ( [ C S l ] k ; δ l ) ,
( S ̂ , α ̂ ) = argmin ( S , α ) 1 2 σ 2 y B ( α ) ( T J × 3 I n p ) S 2 + l = 0 2 k = 1 2 n p β l ψ ( [ C S l ] k ; δ l ) .
R ( S ) = 1 2 ( β 3 C ) S 2 ,
( S ̂ , α ̂ ) = argmin ( S , α ) 1 2 σ 2 y B ( α ) ( T J × 3 I n p ) S 2 + 1 2 ( β 3 C ) S 2 .
S ( α ) = [ ( T J × 3 I n p ) B ( α ) B ( α ) ( T J × 3 I n p ) + σ 2 β 3 C C ] 1 × ( T J × 3 I n p ) B ( α ) y .
( T J × 3 I n p ) Q Ω ( α ) Ω ( α ) u = ( T J × 3 I n p ) v ̌ 0 ,
α ̂ = argmin α { 1 2 σ 2 y B ( α ) ( T J × 3 I n p ) S ( α ) 2 + 1 2 ( β 3 C ) S ( α ) 2 } .
F = 1 σ 2 [ α μ ( α ) ] [ α μ ( α ) ] ,
l k ( α ) = [ B ( α ) B ( α ) + σ 2 β k C C ] 1 B ( α ) B ( α ) e k ,
β ̂ 0 = argmin β 0 FWHM [ l 0 ( α ) ] r 2 ,

Metrics