Abstract

Several phase-modulation functions have been reported to decrease the aberration variance of the modulation-transfer-function (MTF) in aberration-tolerant hybrid imaging systems. The choice of this phase-modulation function is crucial for optimization of the overall system performance. To prevent a significant loss in signal-to-noise ratio, it is common to enforce restorability constraints on the MTF, requiring trade of aberration-tolerance and noise-gain. Instead of optimizing specific MTF characteristics, we directly minimize the expected imaging-error of the joint design. This method is used to compare commonly used phase-modulation functions: the antisymmetric generalized cubic polynomial and fourth-degree rotational symmetric phase-modulation. The analysis shows how optimal imaging performance is obtained using moderate phase-modulation, and more importantly, the relative merits of the above functions.

© 2010 OSA

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References

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2009

M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[CrossRef]

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]

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]

M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE 7429, 74290 (2009).
[CrossRef]

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 (2009).
[CrossRef]

2008

2007

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007).
[CrossRef]

2006

2005

K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE 5784, 146–156 (2005).
[CrossRef]

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]

2004

S. S. Sherif, W. T. Cathey, and E. R. Dowski, “Phase plate to extend the depth of field of incoherent hybrid imaging systems,” Appl. Opt. 43(13), 2709–2721 (2004).
[CrossRef] [PubMed]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[CrossRef] [PubMed]

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

2003

S. Mezouari and A. R. Harvey, “Combined amplitude and phase filters for increased tolerance to spherical aberration,” J. Mod. Opt. 50(11), 2213–2220 (2003).

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]

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

2001

1999

1996

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36(17), 2759–2770 (1996).
[CrossRef] [PubMed]

1995

1994

D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. 73(6), 814–817 (1994).
[CrossRef] [PubMed]

Andersson, M.

Bialek, W.

D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. 73(6), 814–817 (1994).
[CrossRef] [PubMed]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[CrossRef] [PubMed]

Caron, N.

Cathey, W. T.

Chi, W.

Chumachenko, V. V.

K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE 5784, 146–156 (2005).
[CrossRef]

Conchello, J. A.

Demenikov, M.

M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE 7429, 74290 (2009).
[CrossRef]

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]

Dowski, E. R.

Feng, G.

M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[CrossRef]

Findlay, E.

George, N.

Harvey, A. R.

Huckridge, D.

Kaveh, M.

Y. L. You and M. Kaveh, “Blind image restoration by anisotropic regularization,” IEEE Trans. Image Process. 8(3), 396–407 (1999).
[CrossRef] [PubMed]

Kubala, K. S.

K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE 5784, 146–156 (2005).
[CrossRef]

Liu, L.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007).
[CrossRef]

Markham, J.

Mezouari, S.

Muyo, G.

Pauca, V. P.

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

Robinson, M. D.

M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[CrossRef]

Ruderman, D. L.

D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. 73(6), 814–817 (1994).
[CrossRef] [PubMed]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[CrossRef] [PubMed]

Sheng, Y.

Sherif, S. S.

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[CrossRef] [PubMed]

Singh, A.

Stork, D.

M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[CrossRef]

Sun, J.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007).
[CrossRef]

Torgersen, T. C.

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

van der Schaaf, A.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36(17), 2759–2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36(17), 2759–2770 (1996).
[CrossRef] [PubMed]

Wach, H. B.

K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE 5784, 146–156 (2005).
[CrossRef]

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[CrossRef] [PubMed]

Wood, A.

Yang, Q.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007).
[CrossRef]

You, Y. L.

Y. L. You and M. Kaveh, “Blind image restoration by anisotropic regularization,” IEEE Trans. Image Process. 8(3), 396–407 (1999).
[CrossRef] [PubMed]

Appl. Opt.

IEEE Trans. Image Process.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13(4), 600–612 (2004).
[CrossRef] [PubMed]

Y. L. You and M. Kaveh, “Blind image restoration by anisotropic regularization,” IEEE Trans. Image Process. 8(3), 396–407 (1999).
[CrossRef] [PubMed]

J. Mod. Opt.

S. Mezouari and A. R. Harvey, “Combined amplitude and phase filters for increased tolerance to spherical aberration,” J. Mod. Opt. 50(11), 2213–2220 (2003).

J. Opt. A, Pure Appl. Opt.

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 (2009).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Commun.

Q. Yang, L. Liu, and J. Sun, “Optimized phase pupil masks for extended depth of field,” Opt. Commun. 272(1), 56–66 (2007).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

D. L. Ruderman and W. Bialek, “Statistics of natural images: Scaling in the woods,” Phys. Rev. Lett. 73(6), 814–817 (1994).
[CrossRef] [PubMed]

Proc. SPIE

M. D. Robinson, G. Feng, and D. Stork, “Spherical coded imagers: improving lens speed, depth-of-field, and manufacturing yield through enhanced spherical aberration and compensating image processing,” Proc. SPIE 7429, 74290M (2009).
[CrossRef]

K. S. Kubala, H. B. Wach, V. V. Chumachenko, and E. R. Dowski, “Increasing the depth of field in an LWIR system for improved object identification,” Proc. SPIE 5784, 146–156 (2005).
[CrossRef]

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

M. Demenikov and A. R. Harvey, “A technique to remove image artefacts in optical systems with wavefront coding,” Proc. SPIE 7429, 74290 (2009).
[CrossRef]

Vision Res.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36(17), 2759–2770 (1996).
[CrossRef] [PubMed]

Other

P. A. Jansson, Deconvolution of images and spectra (2nd ed.) (Ac. Press, Inc., Orlando, FL, USA, 1996).

E. P. Simoncelli, “Statistical Modeling of Photographic Images,” in Handbook of Image and Video Processing(Ac. Press, 2005), pp. 431–441.

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Figures (5)

Fig. 1
Fig. 1

False-color coding of error magnitudes of simulated images recorded with pure-cubic pupil phase-modulation (β ≡ 0) for various values of α and W 20. The colors indicate pixel value differences between the restored image and the ideal, noise-less diffraction-limited image, normalized to the dynamic range. The optimal modulation depth, α opt,cubic of the pure-cubic for a defocus tolerance |W 20| ≤ 5λ is 2.87λ. The two left-most images in Fig. 3c, of the spoke target and the cameraman, are used as test-scenes.

Fig. 2
Fig. 2

(a) The square root of the expected mean-square imaging-error normalized to the dynamic range, ε, is shown as a function of defocus for various cubic phase-modulation depths α; (b) shows in solid-blue line the average imaging-error as a function of α for |W 20 | ≤ 5λ. For comparison, the dashed-green line and dot-dashed-red line show the average imaging error of respectively the generalized cubic mask with β = −3α, and the quartic phase-modulation with optimal δ = −0.7, as a function of their respective parameters α and γ.

Fig. 3
Fig. 3

(a) ε ¯ [dB] for generalized cubic phase-masks as described by Eq. (1) calculated using a statistical model of the scene. (b) The structural similarity of the same hybrid imaging systems averaged for the test-scenes shown in c), see section 5 for further discussion.

Fig. 4
Fig. 4

(a) The global optimal phase-modulation parameters α and β as a function of the required defocus tolerance. (b) The secondary optimal phase-modulation parameters as a function of the required defocus tolerance. (c) The root of the expected mean-square imaging-error for both profile types as function of the maximum defocus W 20 ,max .

Fig. 5
Fig. 5

False-color hybrid imaging error for various phase-modulations and defocus. The secondary optimum and multiples of the generalized cubic phase-modulation (α opt,gen = 1.29λ, β = −3α), are used for the error magnitudes shown in the top three rows. The quartic with optimal parameters γ opt = 2.8λ and δ opt = −0.7 and multiples are used for the error magnitudes in the bottom three rows.

Equations (5)

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

θ ( x , y ) = α     ( x 3 + y 3 ) + β     ( x 2 y + x y 2 ) ,
θ ( ρ ) = γ     ( ρ 2 + δ ) 2 ,
ε 2 = E ( | I r I d l | 2 ) f X , f Y = E ( | ( H W H a b H d l ) S + H W N | 2 ) f X , f Y .
ε ¯ = ε 2 a b
ε 2 = | H W H a b H d l | 2 P S f X , f Y + | H W | 2 P N f X , f Y .

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