Abstract

Coded-illumination (CI) imaging is a feasible technique enabling resolution enhancement and high-dimensional information extraction in optical systems. It incorporates optical encoding and computational reconstruction together to help overcome physical limitations. Existing CI reconstruction methods suffer from a trade-off between noise robustness and low computational complexity, which are both requisite for practical applications. In this paper, we propose a novel noise-robust and low-complexity reconstruction scheme for CI imaging. The scheme runs in an iterative way, and each iteration consists of two phases. First, the measurements are input into a novel non-uniform and adaptive weighted solver, whose weight updates in each iteration. This enables effective identification and attenuation of various measurement noise from coarse to fine. Second, the preserved latent information enters an alternating projection optimization procedure, which reconstructs target image by imposing support constraints without matrix lifting. We have successfully applied the scheme to structured illumination imaging and Fourier ptychography. Both simulations and experiments demonstrate that the method obtains strong robustness, low computational complexity, and fast convergence. The scheme can be adopted for various incoherent and coherent CI imaging modalities with wide extensions.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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    [Crossref]
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    [Crossref]
  32. S. Dong, J. Liao, K. Guo, L. Bian, J. Suo, and G. Zheng, “Resolution doubling with a reduced number of image acquisitions,” Biomed. Opt. Express 6, 2946–2952 (2015).
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2018 (1)

2017 (1)

2016 (1)

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6, 27384 (2016).
[Crossref]

2015 (8)

2014 (2)

V. Elser, “Solution of the crystallographic phase problem by iterated projections,” Acta Crystallogr. Sect. A 59, 201–209 (2014).
[Crossref]

L. Ziji, T. Lei, L. Sijia, and W. Laura, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref]

2013 (4)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci 6, 199–225 (2013).
[Crossref]

2011 (1)

2008 (3)

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

2007 (1)

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev., Ser. B,  76, 3009–3014 (2007).
[Crossref]

2006 (1)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theor. 52, 1289–1306 (2006).
[Crossref]

2004 (3)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2004).
[Crossref]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

W. Zhou, B. Alan Conrad, S. Hamid Rahim, and S.P. Eero, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1982 (1)

1978 (1)

1972 (1)

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Agard, D. A.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Alan Conrad, B.

W. Zhou, B. Alan Conrad, S. Hamid Rahim, and S.P. Eero, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Bian, L.

Cande, W. Z.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Candès, E. J.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci 6, 199–225 (2013).
[Crossref]

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” in International Conference on Neural Information Processing Systems, (2015), pp. 739–747.

Carlton, P. M.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Charalambous, P.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Chen, C. C.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev., Ser. B,  76, 3009–3014 (2007).
[Crossref]

Chen, F.

Chen, M.

Chen, Y.

Y. Chen and E. J. Candès, “Solving random quadratic systems of equations is nearly as easy as solving linear systems,” in International Conference on Neural Information Processing Systems, (2015), pp. 739–747.

Chung, J.

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6, 27384 (2016).
[Crossref]

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

D’aspremont, A.

I. Waldspurger, A. D’aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Dai, Q.

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Ding, H.

Dong, J.

Dong, S.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theor. 52, 1289–1306 (2006).
[Crossref]

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Eero, S.P.

W. Zhou, B. Alan Conrad, S. Hamid Rahim, and S.P. Eero, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci 6, 199–225 (2013).
[Crossref]

Elser, V.

V. Elser, “Solution of the crystallographic phase problem by iterated projections,” Acta Crystallogr. Sect. A 59, 201–209 (2014).
[Crossref]

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Fienup, J. R.

Gerchberg, R. W.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Gillette, M. U.

Golubovskaya, I. N.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Guo, K.

Gustafsson, M. G. L.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Hamid Rahim, S.

W. Zhou, B. Alan Conrad, S. Hamid Rahim, and S.P. Eero, “Image quality assessment: from error visibility to structural similarity,” IEEE Trans. Image Process. 13, 600–612 (2004).
[Crossref]

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving least squares problems(Prentice-Hall, Inc.,, 1974). Prentice-Hall Series in Automatic Computation.

Horstmeyer, R.

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

Jiang, S.

Johnson, D.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Kirz, J.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
[Crossref]

Laura, W.

L. Ziji, T. Lei, L. Sijia, and W. Laura, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving least squares problems(Prentice-Hall, Inc.,, 1974). Prentice-Hall Series in Automatic Computation.

Lee, T. K.

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev., Ser. B,  76, 3009–3014 (2007).
[Crossref]

Lei, T.

L. Ziji, T. Lei, L. Sijia, and W. Laura, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref]

Li, P.

Li, X.

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Liao, J.

Luke, D. R.

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2004).
[Crossref]

Maiden, A.

Mallat, S.

I. Waldspurger, A. D’aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev., Ser. B,  76, 3009–3014 (2007).
[Crossref]

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Millet, L.

Mir, M.

Ou, X.

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6, 27384 (2016).
[Crossref]

Popescu, G.

Rice, J.

J. Rice, Mathematical Statistics and Data Analysis (Cengage Learning, 1988).

Rodenburg, J. M.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

Rodriguez, J. A.

J. A. Rodriguez, R. Xu, C. C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref] [PubMed]

Rogers, J.

Saxton, W. O.

R. W. Gerchberg and W. O. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Sayre, D.

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of X-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

Sedat, J. W.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Segev, M.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Shao, L.

M. G. L. Gustafsson, L. Shao, P. M. Carlton, C. J. R. Wang, I. N. Golubovskaya, W. Z. Cande, D. A. Agard, and J. W. Sedat, “Three-dimensional resolution doubling in wide-field fluorescence microscopy by structured illumination,” Biophys. J. 94, 4957–4970 (2008).
[Crossref] [PubMed]

Shechtman, Y.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging,” IEEE Signal Process. Mag. 32, 87–109 (2015).
[Crossref]

Sijia, L.

L. Ziji, T. Lei, L. Sijia, and W. Laura, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
[Crossref]

Soltanolkotabi, M.

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of fourier ptychography phase retrieval algorithms,” Opt. Express 23, 33214–33240 (2015).
[Crossref]

E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
[Crossref]

Strohmer, T.

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E. J. Candès, T. Strohmer, and V. Voroninski, “Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming,” Commun. Pure Appl. Math. 66, 1241–1274 (2013).
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M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Signal Process. Mag. 25, 83–91 (2008).
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D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theor. 52, 1289–1306 (2006).
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E. J. Candès, X. Li, and M. Soltanolkotabi, “Phase retrieval via wirtinger flow: Theory and algorithms,” IEEE Trans. Inf. Theory 61, 1985–2007 (2015).
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D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2004).
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[Crossref] [PubMed]

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L. Ziji, T. Lei, L. Sijia, and W. Laura, “Real-time brightfield, darkfield, and phase contrast imaging in a light-emitting diode array microscope,” J. Biomed. Opt. 19, 106002 (2014).
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I. Waldspurger, A. D’aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149, 47–81 (2015).
[Crossref]

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 739–745 (2013).
[Crossref]

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Phys. Rev., Ser. B (1)

C. C. Chen, J. Miao, C. W. Wang, and T. K. Lee, “Application of optimization technique to noncrystalline X-ray diffraction microscopy: Guided hybrid input-output method,” Phys. Rev., Ser. B,  76, 3009–3014 (2007).
[Crossref]

Sci. Rep. (1)

L. Bian, J. Suo, J. Chung, X. Ou, C. Yang, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using Poisson maximum likelihood and truncated Wirtinger gradient,” Sci. Rep. 6, 27384 (2016).
[Crossref]

SIAM J. Imag. Sci (1)

E. J. Candès, Y. C. Eldar, T. Strohmer, and V. Voroninski, “Phase retrieval via matrix completion,” SIAM J. Imag. Sci 6, 199–225 (2013).
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Figures (7)

Fig. 1
Fig. 1 Two examplar image formation models of coded illumination imaging. (a) Incoherent structured illumination (SI) imaging. (b) Coherent Fourier ptychograpic (FP) imaging.
Fig. 2
Fig. 2 Flowchart of the proposed reconstruction scheme for coded-illumination imaging.
Fig. 3
Fig. 3 Different settings of k lead to different weight functions and different reconstruction quality.
Fig. 4
Fig. 4 Simulation results using the conventional AP method and the reported scheme for incoherent structured illumination imaging. (a) Examples of raw images. (b) Reconstruction comparison under different levels of Gaussian noise. (c) Reconstruction comparison under different levels of speckle noise.
Fig. 5
Fig. 5 Simulation results for coherent Fourier ptychograpic imaging. (a) The ground-truth amplitude and phase image. (b) Raw images corrupted by Gaussian and speckle noise. (c) Reconstruction comparison under different levels of Gaussian noise. (d) Reconstruction comparison under different levels of speckle noise.
Fig. 6
Fig. 6 SI experiment results. (a) Raw image captured under uniform illumination. (b) Reconstruction results at different exposure time.
Fig. 7
Fig. 7 FP experiment results. (a) Image captured at normal incidence. (b) Reconstruction results at different exposure time.

Equations (11)

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

I = | T ( P O ) | 2 ,
I = | Φ | 2 , Φ = T ( Ψ ) , Ψ = P O .
I P o i s s o n ( | Φ | 2 ) .
arg  max | Φ | log  e ( | Φ | 2 ) ( | Φ | 2 ) I I ! arg  max | Φ | | Φ | 2 + I log  | Φ | 2 .
min  L ( | Φ | ) = | Φ | 2 I log  | Φ | 2 + 1 γ ( | Φ | | T ( P O n ) | ) 2 ,
L | Φ | = 2 | Φ | 2 I | Φ | + 2 γ ( | Φ | | T ( P O n ) t | ) .
( 1 + 1 γ ) | Φ | 2 1 γ | T ( P O n ) | | Φ | I = 0 ,
| Φ n + 1 | = 1 γ | T ( P O n ) | + ( 1 γ | T ( P O n ) | ) 2 + 4 ( 1 + 1 γ ) I 2 ( 1 + 1 γ ) .
γ = 1 log k ( 1 + | I | T ( P O n ) | 2 | ) .
Φ n + 1 = | Φ n + 1 | Φ n | Φ n | .
O n + 1 = O n + α P | P | max  2 ( Ψ n + 1 Ψ n ) ,

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