Abstract

Fourier ptychographic microscopy (FPM) is recently proposed as a computational imaging method to bypass the limitation of the space-bandwidth product of the traditional optical system. It employs a sequence of low-resolution images captured under angularly varying illumination and applies the phase retrieval algorithm to iteratively reconstruct a wide-field, high-resolution image. In current FPM imaging system, system uncertainties, such as the pupil aberration of the employed optics, may significantly degrade the quality of the reconstruction. In this paper, we develop and test a nonlinear optimization algorithm to improve the robustness of the FPM imaging system by simultaneously considering the reconstruction and the system imperfections. Analytical expressions for the gradient of a squared-error metric with respect to the object and illumination allow joint optimization of the object and system parameters. The algorithm achieves superior reconstructions when the system parameters are inaccurately known or in the presence of noise and corrects the pupil aberrations simultaneously. Experiments on both synthetic and real captured data validate the effectiveness of the proposed method.

© 2015 Optical Society of America

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References

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2015 (2)

2014 (6)

2013 (6)

2010 (1)

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” Proc. SPIE 772977291I (2010).
[Crossref]

2009 (2)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

2008 (2)

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

2004 (1)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

2003 (1)

2002 (1)

J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” Proc. SPIE 47674845 (2002).

1999 (1)

H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Phys. Lett. 38(13), 2800–2807 (1999).

1997 (1)

1996 (1)

1982 (1)

1978 (1)

Bian, Z.

Bunk, O.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Bunka, O.

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

Dai, Q.

Y. Zhang, W. Jiang, L. Tian, L. Waller, and Q. Dai, “Self-learning based Fourier ptychographic microscopy,” Opt. Express 23(14), 18471–18486 (2015).
[Crossref] [PubMed]

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

David, C.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Dierolf, M.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Dierolfa, M.

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

Dong, S.

Dorsch, R. G.

Faulkner, H. M. L.

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

Ferreira, C.

Fienup, J. R.

Guizar-Sicairos, M.

Guo, K.

Heil, J.

J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” Proc. SPIE 47674845 (2002).

Horstmeyer, R.

Humphry, M. J.

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” Proc. SPIE 772977291I (2010).
[Crossref]

Jiang, W.

Y. Zhang, W. Jiang, L. Tian, L. Waller, and Q. Dai, “Self-learning based Fourier ptychographic microscopy,” Opt. Express 23(14), 18471–18486 (2015).
[Crossref] [PubMed]

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

Li, X.

Lohmann, A. W.

Maiden, A. M.

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” Proc. SPIE 772977291I (2010).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

Marriott, P.

J. Marrison, L. Rty, P. Marriott, and P. O’Toole, “Ptychography-a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Marrison, J.

J. Marrison, L. Rty, P. Marriott, and P. O’Toole, “Ptychography-a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Mendlovic, D.

Menzel, A.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Menzela, A.

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

Miller, J. J.

Nanda, P.

Nomura, H.

H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Phys. Lett. 38(13), 2800–2807 (1999).

O’Toole, P.

J. Marrison, L. Rty, P. Marriott, and P. O’Toole, “Ptychography-a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Ou, X.

Pfeiffer, F.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Pfeiffera, F.

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

Ramchandran, K.

Rodenburg, J. M.

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” Proc. SPIE 772977291I (2010).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

Rty, L.

J. Marrison, L. Rty, P. Marriott, and P. O’Toole, “Ptychography-a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Sato, T.

H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Phys. Lett. 38(13), 2800–2807 (1999).

Shiradkar, R.

Sure, Th.

J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” Proc. SPIE 47674845 (2002).

Thibault, P.

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Thibaulta, P.

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

Tian, L.

Waller, L.

Wesner, J.

J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” Proc. SPIE 47674845 (2002).

Yang, C.

Zalevsky, Z.

Zhang, Y.

Y. Zhang, W. Jiang, L. Tian, L. Waller, and Q. Dai, “Self-learning based Fourier ptychographic microscopy,” Opt. Express 23(14), 18471–18486 (2015).
[Crossref] [PubMed]

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

Zheng, G.

Appl. Opt. (3)

Appl. Phys. Lett. (2)

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 1385–1391 (2004).
[Crossref]

H. Nomura and T. Sato, “Techniques for measuring aberrations in lenses used in photolithography with printed patterns,” Appl. Phys. Lett. 38(13), 2800–2807 (1999).

Biomed. Opt. Express (2)

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

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Opt. Express (8)

Opt. Lett. (2)

Proc. SPIE (3)

W. Jiang, Y. Zhang, and Q. Dai, “Multi-channel super-resolution with Fourier ptychographic microscopy,” Proc. SPIE 9273927336 (2014).
[Crossref]

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “A new method of high resolution, quantitative phase scanning microscopy,” Proc. SPIE 772977291I (2010).
[Crossref]

J. Wesner, J. Heil, and Th. Sure, “Reconstructing the pupil function of microscope objectives from the intensity PSF,” Proc. SPIE 47674845 (2002).

Sci. Rep. (1)

J. Marrison, L. Rty, P. Marriott, and P. O’Toole, “Ptychography-a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3, 2369 (2013).
[Crossref]

Science (1)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-Resolution scanning X-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref] [PubMed]

Ultramicroscopy (2)

P. Thibaulta, M. Dierolfa, O. Bunka, A. Menzela, and F. Pfeiffera, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref] [PubMed]

Other (2)

University of South California, “SIPI image database,” http://sipi.usc.edu/database/

The computational image lab at University of California Berkeley, “LED array Fourier ptychography dataset,” http://www.laurawaller.com/opensource/ .

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Figures (9)

Fig. 1
Fig. 1 Reconstructions of the conventional FPM method and our method (γ = 0.15) on the simulated dataset of boat. (a) The ground truth of the sample intensity. (b–c) Reconstructed intensity with the conventional FPM method and our method respectively, each takes 20 iterations. (d) The ground truth of the pupil function. The phase of the pupil function is set as zero for simplicity. (e) The ground truth of the sample phase. (f–g) Reconstructed intensity with the conventional FPM method and our method respectively. (h) The reconstructed pupil function with our method (amplified by 4 times for better visualization).
Fig. 2
Fig. 2 Reconstructions of the conventional FPM method and our method (γ = 0.2) on the simulated dataset of the pathological slide. (a) The ground truth of the sample intensity. (b–c) Reconstructed intensity with the conventional FPM method and our method respectively, each takes 20 iterations. (d) The ground truth of the pupil function. The phase of the pupil function is set as zero for simplicity. (e) The ground truth of the sample phase. (f–g) Reconstructed intensity with the conventional FPM method and our method respectively. (h) The reconstructed pupil function with our method (amplified by 4 times for better visualization).
Fig. 3
Fig. 3 Convergence of algorithms on simulated datasets. (a) The convergence of the conventional FPM method and our method (γ = 0.15) on the dataset of boat. The NIF-RMSE in Eqs. (12)(13) is used to evaluate the quality of reconstructions. (b) The convergence of the conventional FPM method and our method (γ = 0.2) on the dataset of pathological slide.
Fig. 4
Fig. 4 Comparisons of the convergence speed between different γ. We use the normalized error to evaluate the convergence of reconstructions.
Fig. 5
Fig. 5 Convergence error and the NIF-RMSE value of reconstructions according to different γ. Each reconstruction takes 100 iterations to assure convergence. (a) The convergence error according to different γ, here, the squared-error metric in Eq. (7) is used to evaluate the convergence error. (b) The quality of reconstructions according to different γ with the metric of NIF-RMSE.
Fig. 6
Fig. 6 A comparison between reconstructions with different γ on the dataset of boat. All reconstructions are recovered by the proposed algorithm (20 iterations), including intensity reconstruction and phase reconstruction. (amplified by 4 times for better visualization).
Fig. 7
Fig. 7 A comparison between reconstructions with different γ on the dataset of pathological slide. All reconstructions are recovered by the proposed algorithm with 20 iterations, including intensity reconstruction and phase reconstruction. (amplified by 4 times for better visualization).
Fig. 8
Fig. 8 Experimental reconstructions of the conventional FPM method and our method (γ = 0.1) using FPM blood smear dataset. (a) is the raw data captured under the illumination of the central LED. (b) and (e) are the reconstructed sample intensity and phase with the conventional FPM method respectively. (c) and (f) are the reconstructed sample intensity and phase with our method respectively. (d) and (g) are the intensity and phase of the reconstructed pupil function with our method respectively.
Fig. 9
Fig. 9 Experimental reconstructions of the conventional FPM method and our method (γ = 0.1) using FPM USAF dataset. (a) Raw image of a USAF chart and the bottom image is a magnified view of the central part of the top image (closed by the red rectangle). (b) Reconstruction with the conventional FPM method. (c) Reconstruction with our method.

Equations (13)

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I n = | e ( r ) p ( r ) | 2 = | 1 { [ e ( r ) ] [ p ( r ) ] } | 2 = | 1 { S ( k k n ) P ( k ) } | 2 ,
I s , n e i φ s = 1 { S g ( j ) ( k k n ) P g ( j ) ( k ) } .
I c , n e i φ c = I n I s , n e i φ s I s , n = I n e i φ s ,
S g ( j + 1 ) ( k ) = [ P g ( j ) ( k + k n ) ] * | P g ( j ) ( k + k n ) | max 2 { I c , n e i φ c } .
P g ( j + 1 ) ( k ) = P g ( j ) ( k ) .
C M = n = 1 L x , y W n ( x , y ) ( I s , n ( x , y ) I n ( x , y ) ) 2 ,
ε = n = 1 L x , y W n ( x , y ) { [ I s , n ( x , y ) + δ ] γ [ I n ( x , y ) + δ ] γ } 2 ,
S ( j ) = ε S g , R ( j ) ( k ) + i ε S g , I ( j ) ( k ) = 4 n = 1 L [ P g ( j ) ( k + k n ) ] * { W n [ ( I s , n + δ ) γ ( I n + δ ) γ ] × γ ( I s , n + δ ) γ 1 I s , n e i ( φ s 2 π ( x u n M + y v n N ) ) } ,
P ( j ) = ε P g , R ( j ) ( k ) + i ε P g , I ( j ) ( k ) = 4 n = 1 L [ S g ( j ) ( k k n ) ] * { W n [ ( I s , n + δ ) γ ( I n + δ ) γ ] × γ ( I s , n + δ ) γ 1 I s , n e i φ s .
S g ( j + 1 ) ( k ) = S g ( j ) ( k ) + α | P g ( j ) ( k + k n ) max 2 | S ( j ) ,
P g ( j + 1 ) ( k ) = P g ( j ) ( k ) + β | S g ( j ) ( k k n ) max 2 | P ( j ) ,
E 2 = 1 L n = 1 L [ min ρ n Σ k | ρ n S g ( k k n ) P g ( k ) S t ( k k n ) P t ( k ) | 2 Σ k | S t ( k k n ) P t ( k ) | 2 ] ,
ρ n = Σ k S t ( k k n ) P t ( k ) [ S g ( k k n ) P g ( k ) ] * Σ k | S g ( k k n ) P g ( k ) | 2 .

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