Abstract

We describe a method for combining concentric logarithmic aspheric lenses in order to obtain an extended depth of field. Substantial improvement in extending the depth of field is obtained by carefully controlling the optical path difference among the concentric lenses so that their outputs combine incoherently. The system is analyzed through diffraction theory and the point spread function is shown to be highly invariant over a long range of object distances. After testing the image performance on a three-dimensional scene, we found that the incoherently combined logarithmic aspheres can provide a high-quality image over an axial distance corresponding to a defocus of ±14(λ/4). Studies of the images of two-point objects are presented to illustrate the resolution of these lenses.

© 2009 Optical Society of America

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    [CrossRef] [PubMed]
  2. W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875-877 (2001).
    [CrossRef]
  3. W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20, 2260-2273(2003).
    [CrossRef]
  4. J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. 30, 1647-1649 (2005).
    [CrossRef] [PubMed]
  5. E. Ben-Eliezer, E. Marom, N. Konforti, and Zeev Zalevsky, “Radial mask for imaging systems that exhibit high resolution and extended depths of field,” Appl. Opt. 45, 2001-2013(2006).
    [CrossRef] [PubMed]
  6. W. Chi, K. Chu, and N. George, “Polarization coded aperture,” Opt. Express 14, 6634-6642 (2006).
    [CrossRef] [PubMed]
  7. K. Chu, N. George, and W. Chi, “Extending the depth of field through unbalanced OPD,” Appl. Opt. 47, 6895-6903(2008).
    [CrossRef] [PubMed]
  8. N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157 (2003).
    [CrossRef]
  9. W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).
    [CrossRef]
  10. X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).
  11. H. J. Trussel and B. R. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1978), Vol. 3, pp. 196-198.
  12. J. G. Nagy and D. P. O'Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063-1082(1998).
    [CrossRef]
  13. T. P. Costello and W. B. Mikhael, “Efficient restoration of known space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1-22 (2003).
    [CrossRef]
  14. L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259-268(1992).
    [CrossRef]
  15. C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227-238 (1996).
    [CrossRef]
  16. A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
    [CrossRef]
  17. T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
    [CrossRef]
  18. D. Goldforb and W. Yin, “Second-order cone programming methods for total variation-based image restoration,” SIAM J. Sci. Comput. 27, 622-645 (2005).
    [CrossRef]
  19. Yilun Wang, Junfeng Yang, Wotao Yin, and Yin Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” Tech. Rep. TR07-10 (Rice University, Department of Computational and Applied Mathematics, 2007).
  20. Lord Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 430-432.
  21. W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2001).

2008

X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).

K. Chu, N. George, and W. Chi, “Extending the depth of field through unbalanced OPD,” Appl. Opt. 47, 6895-6903(2008).
[CrossRef] [PubMed]

2006

2005

J. Ojeda-Castaneda, J. E. A. Landgrave, and H. M. Escamilla, “Annular phase-only mask for high focal depth,” Opt. Lett. 30, 1647-1649 (2005).
[CrossRef] [PubMed]

D. Goldforb and W. Yin, “Second-order cone programming methods for total variation-based image restoration,” SIAM J. Sci. Comput. 27, 622-645 (2005).
[CrossRef]

W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).
[CrossRef]

2003

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157 (2003).
[CrossRef]

T. P. Costello and W. B. Mikhael, “Efficient restoration of known space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1-22 (2003).
[CrossRef]

W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20, 2260-2273(2003).
[CrossRef]

2001

1999

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

1998

J. G. Nagy and D. P. O'Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063-1082(1998).
[CrossRef]

1997

A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

1996

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227-238 (1996).
[CrossRef]

1995

1992

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259-268(1992).
[CrossRef]

Bakin, D.

X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).

Ben-Eliezer, E.

Cathey, W. T.

Chambolle, A.

A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

Chan, T. F.

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

Chen, X.

X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).

Chi, W.

Chu, K.

Costello, T. P.

T. P. Costello and W. B. Mikhael, “Efficient restoration of known space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1-22 (2003).
[CrossRef]

Dowski, E. R.

Escamilla, H. M.

Fatemi, E.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259-268(1992).
[CrossRef]

George, N.

X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).

K. Chu, N. George, and W. Chi, “Extending the depth of field through unbalanced OPD,” Appl. Opt. 47, 6895-6903(2008).
[CrossRef] [PubMed]

W. Chi, K. Chu, and N. George, “Polarization coded aperture,” Opt. Express 14, 6634-6642 (2006).
[CrossRef] [PubMed]

W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).
[CrossRef]

W. Chi and N. George, “Computational imaging with the logarithmic asphere: theory,” J. Opt. Soc. Am. A 20, 2260-2273(2003).
[CrossRef]

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157 (2003).
[CrossRef]

W. Chi and N. George, “Electronic imaging using a logarithmic asphere,” Opt. Lett. 26, 875-877 (2001).
[CrossRef]

Goldforb, D.

D. Goldforb and W. Yin, “Second-order cone programming methods for total variation-based image restoration,” SIAM J. Sci. Comput. 27, 622-645 (2005).
[CrossRef]

Golub, G. H.

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

Hunt, B. R.

H. J. Trussel and B. R. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1978), Vol. 3, pp. 196-198.

Konforti, N.

Landgrave, J. E. A.

Lions, P. L.

A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

Liu, C.

X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).

Marom, E.

Mikhael, W. B.

T. P. Costello and W. B. Mikhael, “Efficient restoration of known space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1-22 (2003).
[CrossRef]

Mulet, P.

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

Nagy, J. G.

J. G. Nagy and D. P. O'Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063-1082(1998).
[CrossRef]

Ojeda-Castaneda, J.

O'Leary, D. P.

J. G. Nagy and D. P. O'Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063-1082(1998).
[CrossRef]

Oman, M. E.

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227-238 (1996).
[CrossRef]

Osher, S.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259-268(1992).
[CrossRef]

Rayleigh, Lord

Lord Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 430-432.

Rudin, L.

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259-268(1992).
[CrossRef]

Smith, W. J.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2001).

Trussel, H. J.

H. J. Trussel and B. R. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1978), Vol. 3, pp. 196-198.

Vogel, C. R.

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227-238 (1996).
[CrossRef]

Wang, Yilun

Yilun Wang, Junfeng Yang, Wotao Yin, and Yin Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” Tech. Rep. TR07-10 (Rice University, Department of Computational and Applied Mathematics, 2007).

Yang, Junfeng

Yilun Wang, Junfeng Yang, Wotao Yin, and Yin Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” Tech. Rep. TR07-10 (Rice University, Department of Computational and Applied Mathematics, 2007).

Yin, W.

D. Goldforb and W. Yin, “Second-order cone programming methods for total variation-based image restoration,” SIAM J. Sci. Comput. 27, 622-645 (2005).
[CrossRef]

Yin, Wotao

Yilun Wang, Junfeng Yang, Wotao Yin, and Yin Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” Tech. Rep. TR07-10 (Rice University, Department of Computational and Applied Mathematics, 2007).

Zalevsky, Zeev

Zhang, Yin

Yilun Wang, Junfeng Yang, Wotao Yin, and Yin Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” Tech. Rep. TR07-10 (Rice University, Department of Computational and Applied Mathematics, 2007).

Appl. Opt.

Digital Signal Process.

T. P. Costello and W. B. Mikhael, “Efficient restoration of known space-variant blurs from physical optics by sectioning with modified Wiener filtering,” Digital Signal Process. 13, 1-22 (2003).
[CrossRef]

J. Opt. A Pure Appl. Opt.

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A Pure Appl. Opt. 5, S157 (2003).
[CrossRef]

J. Opt. Soc. Am. A

Numer. Math.

A. Chambolle and P. L. Lions, “Image recovery via total variation minimization and related problems,” Numer. Math. 76, 167-188 (1997).
[CrossRef]

Opt. Commun.

W. Chi and N. George, “Integrated imaging with a centrally obscured logarithmic asphere,” Opt. Commun. 245, 85-92(2005).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. D

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259-268(1992).
[CrossRef]

Proc. SPIE

X. Chen, D. Bakin, C. Liu, and N. George, “Optics optimization in high-resolution imaging module with extended depth of field,” Proc. SPIE 7061, 1-12 (2008).

SIAM J. Sci. Comput.

T. F. Chan, G. H. Golub, and P. Mulet, “A nonlinear primal-dual method for total variation-based image restoration,” SIAM J. Sci. Comput. 20, 1964-1977 (1999).
[CrossRef]

D. Goldforb and W. Yin, “Second-order cone programming methods for total variation-based image restoration,” SIAM J. Sci. Comput. 27, 622-645 (2005).
[CrossRef]

C. R. Vogel and M. E. Oman, “Iterative methods for total variation denoising,” SIAM J. Sci. Comput. 17, 227-238 (1996).
[CrossRef]

J. G. Nagy and D. P. O'Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063-1082(1998).
[CrossRef]

Other

Yilun Wang, Junfeng Yang, Wotao Yin, and Yin Zhang, “A new alternating minimization algorithm for total variation image reconstruction,” Tech. Rep. TR07-10 (Rice University, Department of Computational and Applied Mathematics, 2007).

Lord Rayleigh, “On the accuracy of focus necessary for sensibly perfect definition,” in Scientific Papers (Cambridge U. Press, 1899), Vol. 1, pp. 430-432.

W. J. Smith, Modern Optical Engineering, 3rd ed. (McGraw-Hill, 2001).

H. J. Trussel and B. R. Hunt, “Image restoration of space variant blurs by sectioned methods,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing (IEEE, 1978), Vol. 3, pp. 196-198.

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Figures (12)

Fig. 1
Fig. 1

Logarithmic lens designed to focus between ( s 1 , s 2 ).

Fig. 2
Fig. 2

PSFs from a diffraction-limited lens with various defocus amounts.

Fig. 3
Fig. 3

PSFs from a single logarithmic aspheric lens with various defocus amounts.

Fig. 4
Fig. 4

Lens with the inner phase mask ( P M 1 ) focusing between ( s 11 , s 12 ) and outer phase mask ( P M 2 ) focusing at ( s 21 , s 22 ). The thin glass plate in the pupil plane will create enough OPD to uncorrelate these two phase masks.

Fig. 5
Fig. 5

PSFs from the incoherently combined logarithmic aspheres with various defocus amounts. (a) PSFs of the lens with the same defocus values as in Figs. 2, 3 (b) PSFs of the lens with even larger defocus.

Fig. 6
Fig. 6

Diagram of the total-variation-based algorithm.

Fig. 7
Fig. 7

Three sharp-edged objects.

Fig. 8
Fig. 8

Scene seen by the detector array.

Fig. 9
Fig. 9

Images taken with the diffraction-limited lens. (a) Object position, ( 5 , 0 , 5 ) mm ; (b) object position, ( 10 , 0 , 10 ) mm ; (c) object position, ( 15 , 0 , 15 ) mm ; and (d) edge spread functions for those positions.

Fig. 10
Fig. 10

Images taken with the incoherently combined logarithmic aspheres. (a) Object position, ( 10 , 0 , 10 ) mm ; (b) object position, ( 40 , 0 , 40 ) mm ; (c) object position: ( 70 , 0 , 70 ) mm ; and (d) edge spread functions for these positions.

Fig. 11
Fig. 11

Reconstruction from the images shown in Fig. 10. (a) Object position, ( 10 , 0 , 10 ) mm ; (b) object position, ( 40 , 0 , 40 ) mm ; (c) object position, ( 70 , 0 , 70 ) mm ; and (d) edge spread functions for these positions.

Fig. 12
Fig. 12

Lateral resolution study with a two-point object at various object distances from the focus plane, as labeled. (a) Images from a diffraction-limited lens (images beyond 20 mm are too blurry to show here). (b) Blurry images taken by the incoherently combined logarithmic aspheres and the reconstructed images.

Equations (23)

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Φ log ( ρ ; s 1 , s 2 ) = k { t 2 + ρ 2 t + a ( t 2 + ρ 2 ) 2 ln [ A ( t 2 + ρ 2 ) e ] a t 2 2 ln [ A t 2 e ] } + Φ ( 0 ) ,
a = 1 s 1 1 R 2 + s 2 2 ln R 2 + t 2 t 2 ,
A = 1 t 2 exp [ s 2 2 + R 2 R 2 + s 2 2 s 1 ln ( R 2 + t 2 t 2 ) ] .
1 s 1 + 1 s 2 = 2 s 0
Δ s near = s 0 s 1 ,
Δ s far = s 2 s 0 ,
PSF ( ρ 2 , s ; ν ) = | 0 R e i [ k ( s 2 + ρ 2 + t 2 + ρ 2 + ρ 2 2 ) + Φ ( ρ ) ] k 2 t 2 + ρ 2 + ρ 2 2 s 2 + ρ 2 × J 0 ( k ρ 2 ρ t 2 + ρ 2 + ρ 2 2 ) ρ d ρ | 2 ,
Φ log ( i ) ( ρ ; s i 1 , s i 2 ) = k { ( t 2 + ρ 2 t ) + a i ( t 2 + ρ 2 ) 2 ln [ A i ( t 2 + ρ 2 ) e ] a i t 2 2 ln ( A i t 2 e ) } + Φ i ( 0 ) ,
a 1 = 1 s 11 1 R 2 / 2 + s 12 2 ln R 2 / 2 + t 2 t 2 ,
A 1 = 1 t 2 exp [ s 12 2 + R 2 / 2 R 2 / 2 + s 12 2 s 11 ln ( R 2 / 2 + t 2 t 2 ) ] ,
a 2 = 1 R 2 / 2 + s 21 2 1 R 2 + s 22 2 ln R 2 + t 2 δ R 2 + t 2 ,
A 2 = 1 R 2 / 2 + t 2 exp [ s 22 2 + R 2 R 2 + s 22 2 R 2 / 2 + s 21 2 ln ( R 2 + t 2 R 2 / 2 + t 2 ) ] .
PSF whole ( ρ 2 , s ) = S ( ν ) { PSF 1 ( ρ 2 , s ; ν ) + PSF 2 ( ρ 2 , s ; ν ) } d ν ,
PSF 1 ( ρ 2 , s ; ν ) = | 0 R / 2 d ρ ρ e i [ k ( s 2 + ρ 2 + t 2 + ρ 2 + ρ 2 2 ) + Φ log ( 1 ) + ( n 1 ) d n ρ 2 2 s 2 ] k 2 t 2 + ρ 2 + ρ 2 2 s 2 + ρ 2 × J 0 ( k ρ 2 ρ t 2 + ρ 2 + ρ 2 2 ) | 2 ,
PSF 2 ( ρ 2 , s ; ν ) = | R / 2 R d ρ ρ e i [ k ( s 2 + ρ 2 + t 2 + ρ 2 + ρ 2 2 ) + Φ log ( 2 ) ] k 2 t 2 + ρ 2 + ρ 2 2 s 2 + ρ 2 × J 0 ( k ρ 2 ρ t 2 + ρ 2 + ρ 2 2 ) | 2
I ( x , y ) = m O m ( x , y , s m ) * PSF ( x , y , s m ) + n ( x , y ) ,
min O ^ ( O ^ x ) 2 + ( O ^ y ) 2 d x d y + μ 2 | O ^ * PSF I | 2 d x d y ,
min W , O ^ w 1 2 + w 2 2 d x d y + β 2 [ ( w 1 O ^ x ) 2 + ( w 2 - O ^ y ) 2 ] d x d y + μ 2 | O ^ * PSF I | 2 d x d y .
[ O ^ x , O ^ y ] ,
min W w 1 2 + w 2 2 d x d y + β 2 [ ( w 1 O ^ x ) 2 + ( w 2 O ^ y ) 2 ] d x d y .
min O ^ β 2 [ ( w 1 O ^ x ) 2 + ( w 2 - O ^ y ) 2 ] d x d y + μ 2 | O ^ * PSF I | 2 d x d y .
PSF ¯ ( ρ 2 ) = PSF ( ρ 2 , s ) ¯ , s ( s 1 , s 2 ) ,
d = 1.22 λ f / .

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