Abstract

An effective method for reducing the sensitivity of phase diversity (PD) technique to Poisson noise is proposed. The denoising algorithm based on blocking-matching and 3D filtering is first introduced in the wavefront sensing field as a preprocessing stage. Then, the PD technique is applied to the denoised images. Results of the numerical simulations and experiments demonstrate that our approach is better than the traditional PD technique in terms of both the root-mean-square error (RMSE) of phase estimates and the structural similarity index metrics (SSIM). The RMSEs of phase estimates on synthetic data are decreased by approximately 40% across noise levels within the range of 58.7-18.8 dB in terms of peak signal-to-noise ratio (PSNR). Meanwhile, the overall decline range of SSIM is significantly decreased from 49% to 9%. The experiment and simulation results are in good agreement. The approach may be widely used in various domains, such as the measurements of intrinsic aberrations in optical systems and compensations for atmospheric turbulence.

© 2016 Optical Society of America

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References

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  1. R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 0207, 32–39 (1979).
    [Crossref]
  2. R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 215829 (1982).
    [Crossref]
  3. R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9(7), 1072–1085 (1992).
    [Crossref]
  4. J. J. Dolne, “Evaluation of the phase diversity algorithm for noise statistics error and diversity function combination,” Proc. SPIE 6307, 630708 (2006).
    [Crossref]
  5. R. L. Kendrick, D. S. Acton, and A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33(27), 6533–6546 (1994).
    [Crossref] [PubMed]
  6. B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
    [Crossref]
  7. J. H. Seldin and R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” Proc. SPIE 2302, 2302268 (1994).
    [Crossref]
  8. M. G. Löfdahl and G. B. Scharmer, “Application of phase-diversity to solar images,” Proc. SPIE 2302, 2302254 (1994).
    [Crossref]
  9. M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
    [Crossref]
  10. L. Meynadier, V. Michau, M. T. Velluet, J. M. Conan, L. M. Mugnier, and G. Rousset, “Noise propagation in wave-front sensing with phase diversity,” Appl. Opt. 38(23), 4967–4979 (1999).
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2011 (1)

M. Mäkitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20(1), 99–109 (2011).
[Crossref] [PubMed]

2007 (1)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

2006 (2)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3D filtering,” Proc. SPIE 6064, 606414 (2006).
[Crossref]

J. J. Dolne, “Evaluation of the phase diversity algorithm for noise statistics error and diversity function combination,” Proc. SPIE 6307, 630708 (2006).
[Crossref]

2003 (1)

1999 (2)

1998 (1)

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

1997 (1)

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

1994 (4)

J. H. Seldin and R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” Proc. SPIE 2302, 2302268 (1994).
[Crossref]

M. G. Löfdahl and G. B. Scharmer, “Application of phase-diversity to solar images,” Proc. SPIE 2302, 2302254 (1994).
[Crossref]

R. L. Kendrick, D. S. Acton, and A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33(27), 6533–6546 (1994).
[Crossref] [PubMed]

M. G. Löfdahl and G. B. Scharmer, “Wave-front sensing and image restoration from focused and defocused solar images,” Astrophys. Suppl. Ser. 107, 243–264 (1994).

1992 (1)

1982 (2)

1979 (1)

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 0207, 32–39 (1979).
[Crossref]

1976 (1)

Acton, D. S.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

R. L. Kendrick, D. S. Acton, and A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33(27), 6533–6546 (1994).
[Crossref] [PubMed]

Black, K. A.

Carrara, D. A.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

Chidlaw, R.

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 0207, 32–39 (1979).
[Crossref]

Conan, J. M.

Cunningham, P. R.

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3D filtering,” Proc. SPIE 6064, 606414 (2006).
[Crossref]

Deville, J. H.

Dolne, J. J.

Duncan, A. L.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

R. L. Kendrick, D. S. Acton, and A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33(27), 6533–6546 (1994).
[Crossref] [PubMed]

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3D filtering,” Proc. SPIE 6064, 606414 (2006).
[Crossref]

Ellerbroek, B. L.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

Fienup, J. R.

Foi, A.

M. Mäkitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20(1), 99–109 (2011).
[Crossref] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3D filtering,” Proc. SPIE 6064, 606414 (2006).
[Crossref]

Gonsalves, R. A.

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 215829 (1982).
[Crossref]

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 0207, 32–39 (1979).
[Crossref]

Harwit, A.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

Idell, P. S.

Katkovnik, V.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3D filtering,” Proc. SPIE 6064, 606414 (2006).
[Crossref]

Kendrick, R. L.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

R. L. Kendrick, D. S. Acton, and A. L. Duncan, “Phase-diversity wave-front sensor for imaging systems,” Appl. Opt. 33(27), 6533–6546 (1994).
[Crossref] [PubMed]

Lee, D. J.

D. J. Lee, M. C. Roggemann, and B. M. Welsh, “Cramér-Rao analysis of phase-diverse wave-front sensing,” J. Opt. Soc. Am. A 16(5), 1005–1015 (1999).
[Crossref]

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

Löfdahl, M. G.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

M. G. Löfdahl and G. B. Scharmer, “Application of phase-diversity to solar images,” Proc. SPIE 2302, 2302254 (1994).
[Crossref]

M. G. Löfdahl and G. B. Scharmer, “Wave-front sensing and image restoration from focused and defocused solar images,” Astrophys. Suppl. Ser. 107, 243–264 (1994).

Mahajan, V. N.

Mäkitalo, M.

M. Mäkitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20(1), 99–109 (2011).
[Crossref] [PubMed]

Meynadier, L.

Michau, V.

Mitchell, K. E.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

Mugnier, L. M.

Noll, R. J.

Paxman, R. G.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

J. H. Seldin and R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” Proc. SPIE 2302, 2302268 (1994).
[Crossref]

R. G. Paxman, T. J. Schulz, and J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9(7), 1072–1085 (1992).
[Crossref]

Roggemann, M. C.

Rousset, G.

Scharmer, G. B.

M. G. Löfdahl and G. B. Scharmer, “Application of phase-diversity to solar images,” Proc. SPIE 2302, 2302254 (1994).
[Crossref]

M. G. Löfdahl and G. B. Scharmer, “Wave-front sensing and image restoration from focused and defocused solar images,” Astrophys. Suppl. Ser. 107, 243–264 (1994).

Schulz, T. J.

Seldin, J. H.

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

J. H. Seldin and R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” Proc. SPIE 2302, 2302268 (1994).
[Crossref]

Tansey, R. J.

Thelen, B. J.

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

Velluet, M. T.

Welsh, B. M.

Widen, K. C.

Appl. Opt. (3)

Astrophys. Suppl. Ser. (1)

M. G. Löfdahl and G. B. Scharmer, “Wave-front sensing and image restoration from focused and defocused solar images,” Astrophys. Suppl. Ser. 107, 243–264 (1994).

IEEE Trans. Image Process. (2)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16(8), 2080–2095 (2007).
[Crossref] [PubMed]

M. Mäkitalo and A. Foi, “Optimal inversion of the Anscombe transformation in low-count Poisson image denoising,” IEEE Trans. Image Process. 20(1), 99–109 (2011).
[Crossref] [PubMed]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Eng. (1)

R. A. Gonsalves, “Phase retrieval and diversity in adaptive optics,” Opt. Eng. 21(5), 215829 (1982).
[Crossref]

Proc. SPIE (7)

J. J. Dolne, “Evaluation of the phase diversity algorithm for noise statistics error and diversity function combination,” Proc. SPIE 6307, 630708 (2006).
[Crossref]

B. L. Ellerbroek, B. J. Thelen, D. J. Lee, D. A. Carrara, and R. G. Paxman, “Comparison of Shack-Hartmann wavefront sensing and phase-diverse phase retrieval,” Proc. SPIE 3126, 3126307 (1997).
[Crossref]

J. H. Seldin and R. G. Paxman, “Phase-diverse speckle reconstruction of solar data,” Proc. SPIE 2302, 2302268 (1994).
[Crossref]

M. G. Löfdahl and G. B. Scharmer, “Application of phase-diversity to solar images,” Proc. SPIE 2302, 2302254 (1994).
[Crossref]

M. G. Löfdahl, R. L. Kendrick, A. Harwit, K. E. Mitchell, A. L. Duncan, J. H. Seldin, R. G. Paxman, and D. S. Acton, “Phase diversity experiment to measure piston misalignment on the segmented primary mirror of the Keck II Telescope,” Proc. SPIE 3356, 33561190 (1998).
[Crossref]

R. A. Gonsalves and R. Chidlaw, “Wavefront sensing by phase retrieval,” Proc. SPIE 0207, 32–39 (1979).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising with block-matching and 3D filtering,” Proc. SPIE 6064, 606414 (2006).
[Crossref]

Other (1)

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

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Figures (6)

Fig. 1
Fig. 1 (a) Original object and (b) known aberration form
Fig. 2
Fig. 2 Plots (a), (c), and (e): simulation results in terms of the RMSEs of phase estimates for aberrations a, b, and c, respectively. Plots (b), (d), and (f): simulation results in terms of SSIM. The RMSEs and SSIM are both as functions of the PSNR of the noisy focused image.
Fig. 3
Fig. 3 Images for aberrations a, b, and c are denoted by (a), (b), and (c), respectively. Row 1: noisy focus-plane images degraded by aberrations at the PSNR levels of 58 dB and 25 dB. Row 2: reconstructed images using the traditional PD. Row 3: reconstructed images using the improved PD.
Fig. 4
Fig. 4 Optical layout of the proposed verification experiment
Fig. 5
Fig. 5 Row 1: focused images captured by CCD. Row 2: images restored using the traditional PD technique. Row 3: images reconstructed using the improved PD technique.
Fig. 6
Fig. 6 Experimental results: SSIM of the reconstructed images as a function of the PSNR of the noisy focused images.

Equations (12)

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g k (x)=f s k (x),
s k (x)= | F 1 { p e i( ϕ t + θ k ) } | 2 ,
ϕ t = j=1 J α j ϕ j ,
d k (x)= g k (x)+ n k (x).
E(α)= u | D 1 (u) S 1 (u) D 2 (u) S 2 (u) | 2 | S 1 (u) | 2 + | S 2 (u) | 2 ,
F(u)= D 1 (u) S 1 (u)+ D 2 (u) S 2 * (u) | S 1 (u) | 2 + | S 2 (u) | 2 ,
E(α)= u | Y 1 (u) S 1 (u) Y 2 (u) S 2 (u) | 2 | S 1 (u) | 2 + | S 2 (u) | 2 ,
F(u)= Y 1 (u) S 1 (u)+ Y 2 (u) S 2 (u) | S 1 (u) | 2 + | S 2 (u) | 2 ,
SRexp[ ( 2π σ w λ ) 2 ].
PSNR=10 log 10 ( max (d(j)) 2 j=1 N (d(j) d noisy (j)) 2 /N ),
RMSE( ϕ ^ )= j=1 N pupil [ϕ(j) ϕ ^ (j)] 2 / N pupil ,
SSIM(f, f ^ )= (2 μ f μ f ^ + c 1 )(2 σ f, f ^ + c 2 ) ( μ f 2 + μ f ^ 2 + c 1 )( σ f 2 + σ f ^ 2 + c 2 ) ,

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