Abstract

We present the first analytical analysis of image artifacts in defocused hybrid imaging systems that employ a cubic phase-modulation function. We show that defocus artifacts have the form of image replications and are caused by a net phase modulation of the optical transfer function. Both numerical simulations and experimental images are presented that exhibit replication artifacts that are compatible with the analytical expressions.

©2010 Optical Society of America

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References

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  1. J. E. R. Dowski and W. T. Cathey, “Extended depth of field through wave-front coding,” Appl. Opt. 34(11), 1859 (1995).
    [Crossref] [PubMed]
  2. G. Muyo and A. R. Harvey, “Decomposition of the optical transfer function: wavefront coding imaging systems,” Opt. Lett. 30(20), 2715–2717 (2005).
    [Crossref] [PubMed]
  3. S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
    [Crossref]
  4. S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
    [Crossref]
  5. D. S. Barwick, “Efficient metric for pupil-phase engineering,” Appl. Opt. 46(29), 7258–7261 (2007).
    [Crossref] [PubMed]
  6. D. S. Barwick, “Increasing the information acquisition volume in iris recognition systems,” Appl. Opt. 47(26), 4684–4691 (2008).
    [Crossref] [PubMed]
  7. P. Mouroulis, “Depth of field extension with spherical optics,” Opt. Express 16(17), 12995–13004 (2008).
    [Crossref] [PubMed]
  8. G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. 11(5), 054002–054010 (2009).
    [Crossref]
  9. M. Demenikov, E. Findlay, and A. R. Harvey, “Miniaturization of zoom lenses with a single moving element,” Opt. Express 17(8), 6118–6127 (2009).
    [Crossref] [PubMed]
  10. G. Muyo, A. Singh, M. Andersson, D. Huckridge, A. Wood, and A. R. Harvey, “Infrared imaging with a wavefront-coded singlet lens,” Opt. Express 17(23), 21118–21123 (2009).
    [Crossref] [PubMed]
  11. S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003).
    [Crossref] [PubMed]
  12. N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A, Pure Appl. Opt. 5(5), S157–S163 (2003).
    [Crossref]
  13. S. Mezouari, G. Muyo, and A. Harvey, “Circularly symmetric phase filters for control of primary third-order aberrations: coma and astigmatism,” J. Opt. Soc. Am. A 23(5), 1058–1062 (2006).
    [Crossref]
  14. J. van der Gracht, J. Nagy, V. Pauca, and R. Plemmons, “Iterative restoration of wavefront coded imagery for focus invariance,” in Integrated Computational Imaging Systems, OSA Technical Digest Series (Optical Society of America, 2001).

2009 (3)

2008 (2)

2007 (1)

2006 (1)

2005 (1)

2004 (1)

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

2003 (3)

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A, Pure Appl. Opt. 5(5), S157–S163 (2003).
[Crossref]

S. Mezouari and A. R. Harvey, “Phase pupil functions for reduction of defocus and spherical aberrations,” Opt. Lett. 28(10), 771–773 (2003).
[Crossref] [PubMed]

1995 (1)

Andersson, M.

Barwick, D. S.

Cathey, W. T.

Chi, W.

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A, Pure Appl. Opt. 5(5), S157–S163 (2003).
[Crossref]

Demenikov, M.

Dowski, J. E. R.

Findlay, E.

George, N.

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A, Pure Appl. Opt. 5(5), S157–S163 (2003).
[Crossref]

Harvey, A.

Harvey, A. R.

Huckridge, D.

Mezouari, S.

Mouroulis, P.

Muyo, G.

Pauca, V. P.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Plemmons, R. J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Prasad, S.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Singh, A.

Torgersen, T. C.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

van der Gracht, J.

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

van der Gratch, J.

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

Wood, A.

Appl. Opt. (3)

J. Opt. A, Pure Appl. Opt. (2)

G. Muyo and A. R. Harvey, “The effect of detector sampling in wavefront-coded imaging systems,” J. Opt. A, Pure Appl. Opt. 11(5), 054002–054010 (2009).
[Crossref]

N. George and W. Chi, “Extended depth of field using a logarithmic asphere,” J. Opt. A, Pure Appl. Opt. 5(5), S157–S163 (2003).
[Crossref]

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

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (2)

S. Prasad, T. C. Torgersen, V. P. Pauca, R. J. Plemmons, and J. van der Gratch, “Engineering the pupil phase to improve image quality,” Proc. SPIE 5108, 1–12 (2003).
[Crossref]

S. Prasad, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, and J. van der Gracht, “Pupil-phase optimization or extended focus, aberration corrected imaging systems,” Proc. SPIE 5559, 335–345 (2004).
[Crossref]

Other (1)

J. van der Gracht, J. Nagy, V. Pauca, and R. Plemmons, “Iterative restoration of wavefront coded imagery for focus invariance,” in Integrated Computational Imaging Systems, OSA Technical Digest Series (Optical Society of America, 2001).

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Figures (3)

Fig. 1
Fig. 1 (a) Cameraman image. Restored cameraman image with cubic phase mask of α = 5λ, restoration kernel at W 20,0 = 2λ and defocus of (b) W 20 = 0λ, (c) W 20 = 1λ, (d) W 20 = 2λ, (e) W 20 = 3λ and (f) W 20 = 4λ.

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Fig. 2
Fig. 2 Restored images with W 20,0 = 0λ, W 20 = 1λ, and (a) α = 2.5λ, (b) α = 5λ and (c) α = 10λ.

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Fig. 3
Fig. 3 Images (a-c) acquired at 1.5 m with a conventional imaging system, focusing at (a) 1m, (b) 1.5m and (c) infinity. Images (d-f) and (g-i) acquired at 1.5 m with a hybrid imaging system with a cubic phase mask with α = 5λ and α = 10λ, respectively, focusing at (d, g) 1m, (e, h) 1.5m and (f, i) infinity, and restored with kernel at 1.5m. The hybrid imaging system has extended depth of field, because the restored images (d-i) are less blurry than images (a-c), but the restored images (d, f, g, i) have very distinct replication artifacts.

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Equations (7)

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θ ( W 20 ) ≈ − 4 π v ( − α v 2 + W 20 2 3 α ) + 3 α 2 π v [ W 20 − | W 20 | max ] sin [ 4 π v 3 α [ W 20 − | W 20 | max ] 2 + π 4 ] ,
I ' r e s ( v ) = exp [ i Δ θ ( v ) ] I d i f f ( v ) ,
Δ θ ( v ) = 4 π v ( W 20 , 0 2 − W 20 2 ) 3 α + 3 α 2 π v ( sin [ 4 π v ( Δ W 20 ) 2 / ( 3 α ) ] Δ W 20 − sin [ 4 π v ( Δ W 20 , 0 ) 2 / ( 3 α ) ] Δ W 20 , 0 ) ,
I ' r e s ( v ) ≈ exp [ i ( A sin [ B v ] + C sin [ − D v ] ) ] I d i f f ( v ) ,
exp [ i A sin ( B v ) ] = ∑ n = − ∞ ∞ J n ( A ) exp [ i n B v ] ,
exp [ i ( A sin [ B v ] + C sin [ − D v ] ) ] = ∑ n = − ∞ ∞ ( ∑ m = − ∞ ∞ J n ( A ) J m ( C ) exp [ i ( n B − m D ) v ] ) ,
I ' r e s ( v ) ≈ ∑ n = − ∞ ∞ ( ∑ m = − ∞ ∞ J n ( A ) J m ( C ) exp [ i ( n B − m D ) v ] ) I d i f f ( v ) .

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