Abstract

Fourier ptychography is a recently developed imaging approach for large field-of-view and high-resolution microscopy. Here we model the Fourier ptychographic forward imaging process using a convolutional neural network (CNN) and recover the complex object information in a network training process. In this approach, the input of the network is the point spread function in the spatial domain or the coherent transfer function in the Fourier domain. The object is treated as 2D learnable weights of a convolutional or a multiplication layer. The output of the network is modeled as the loss function we aim to minimize. The batch size of the network corresponds to the number of captured low-resolution images in one forward/backward pass. We use a popular open-source machine learning library, TensorFlow, for setting up the network and conducting the optimization process. We analyze the performance of different learning rates, different solvers, and different batch sizes. It is shown that a large batch size with the Adam optimizer achieves the best performance in general. To accelerate the phase retrieval process, we also discuss a strategy to implement Fourier-magnitude projection using a multiplication neural network model. Since convolution and multiplication are the two most-common operations in imaging modeling, the reported approach may provide a new perspective to examine many coherent and incoherent systems. As a demonstration, we discuss the extensions of the reported networks for modeling single-pixel imaging and structured illumination microscopy (SIM). 4-frame resolution doubling is demonstrated using a neural network for SIM. The link between imaging systems and neural network modeling may enable the use of machine-learning hardware such as neural engine and tensor processing unit for accelerating the image reconstruction process. We have made our implementation code open-source for researchers.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2018 (3)

2017 (2)

2016 (1)

K. Guo, S. Dong, and G. Zheng, “Fourier ptychography for brightfield, phase, darkfield, reflective, multi-slice, and fluorescence imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 1–12 (2016).
[Crossref]

2015 (6)

2014 (7)

2013 (5)

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

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via Fourier ptychographic microscopy,” Opt. Lett. 38(22), 4845–4848 (2013).
[Crossref] [PubMed]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013).
[Crossref] [PubMed]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87(5), 053850 (2013).
[Crossref]

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

2012 (1)

2010 (2)

2009 (3)

T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperture Fourier holographic optical microscopy,” Opt. Express 17(10), 7873–7892 (2009).
[Crossref] [PubMed]

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. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

2008 (4)

J. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imaging Electron Phys. 150, 87–184 (2008).
[Crossref]

J. Di, J. Zhao, H. Jiang, P. Zhang, Q. Fan, and W. Sun, “High resolution digital holographic microscopy with a wide field of view based on a synthetic aperture technique and use of linear CCD scanning,” Appl. Opt. 47(30), 5654–5659 (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(2), 83–91 (2008).
[Crossref]

M. G. Gustafsson, L. Shao, P. M. Carlton, C. J. 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(12), 4957–4970 (2008).
[Crossref] [PubMed]

2007 (1)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

2006 (1)

2005 (1)

M. G. Gustafsson, “Nonlinear structured-illumination microscopy: wide-field fluorescence imaging with theoretically unlimited resolution,” Proc. Natl. Acad. Sci. U.S.A. 102(37), 13081–13086 (2005).
[Crossref] [PubMed]

2004 (1)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[Crossref] [PubMed]

2003 (1)

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(Pt 2), 82–87 (2000).
[Crossref] [PubMed]

1995 (1)

T. Turpin, L. Gesell, J. Lapides, and C. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).
[Crossref]

1987 (1)

J. R. Fienup, “Reconstruction of a complex-valued object from the modulus of its Fourier transform using a support constraint,” JOSA A 4(1), 118–123 (1987).
[Crossref]

1982 (1)

1976 (1)

R. Gonsalves, “Phase retrieval from modulus data,” JOSA 66(9), 961–964 (1976).
[Crossref]

1970 (1)

1969 (1)

W. Hoppe and G. Strube, “Diffraction in inhomogeneous primary wave fields. 2. Optical experiments for phase determination of lattice interferences,” Acta Crystallogr. A 25, 502–507 (1969).
[Crossref]

Agard, D. A.

M. G. Gustafsson, L. Shao, P. M. Carlton, C. J. 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(12), 4957–4970 (2008).
[Crossref] [PubMed]

Agrawal, G.

N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, and A. Borchers, “In-datacenter performance analysis of a tensor processing unit,” in Proceedings of the 44th Annual International Symposium on Computer Architecture (ACM, 2017), 1–12.
[Crossref]

Alexandrov, S. A.

Bajwa, R.

N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, and A. Borchers, “In-datacenter performance analysis of a tensor processing unit,” in Proceedings of the 44th Annual International Symposium on Computer Architecture (ACM, 2017), 1–12.
[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(2), 83–91 (2008).
[Crossref]

Bates, S.

N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, and A. Borchers, “In-datacenter performance analysis of a tensor processing unit,” in Proceedings of the 44th Annual International Symposium on Computer Architecture (ACM, 2017), 1–12.
[Crossref]

Batey, D. J.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

T. B. Edo, D. J. Batey, A. M. Maiden, C. Rau, U. Wagner, Z. D. Pešić, T. A. Waigh, and J. M. Rodenburg, “Sampling in x-ray ptychography,” Phys. Rev. A 87(5), 053850 (2013).
[Crossref]

Bhatia, S.

N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, and A. Borchers, “In-datacenter performance analysis of a tensor processing unit,” in Proceedings of the 44th Annual International Symposium on Computer Architecture (ACM, 2017), 1–12.
[Crossref]

Bian, L.

Bian, Z.

Boden, N.

N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, and A. Borchers, “In-datacenter performance analysis of a tensor processing unit,” in Proceedings of the 44th Annual International Symposium on Computer Architecture (ACM, 2017), 1–12.
[Crossref]

Borchers, A.

N. P. Jouppi, C. Young, N. Patil, D. Patterson, G. Agrawal, R. Bajwa, S. Bates, S. Bhatia, N. Boden, and A. Borchers, “In-datacenter performance analysis of a tensor processing unit,” in Proceedings of the 44th Annual International Symposium on Computer Architecture (ACM, 2017), 1–12.
[Crossref]

Bowman, A.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Bowman, R.

B. Sun, M. P. Edgar, R. Bowman, L. E. Vittert, S. Welsh, A. Bowman, and M. J. Padgett, “3D computational imaging with single-pixel detectors,” Science 340(6134), 844–847 (2013).
[Crossref] [PubMed]

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Cande, W. Z.

M. G. Gustafsson, L. Shao, P. M. Carlton, C. J. 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(12), 4957–4970 (2008).
[Crossref] [PubMed]

Carlton, P. M.

M. G. Gustafsson, L. Shao, P. M. Carlton, C. J. 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(12), 4957–4970 (2008).
[Crossref] [PubMed]

Ceylan Koydemir, H.

Y. Rivenson, H. Ceylan Koydemir, H. Wang, Z. Wei, Z. Ren, H. Günaydın, Y. Zhang, Z. Göröcs, K. Liang, D. Tseng, and A. Ozcan, “Deep learning enhanced mobile-phone microscopy,” ACS Photonicsacsphotonics.8b00146 (2018).
[Crossref]

Chen, F.

Chung, J.

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: a gigapixel superscope for biomedicine,” Optics and Photonics News 4, 26–33 (2014)

Claus, D.

D. J. Batey, D. Claus, and J. M. Rodenburg, “Information multiplexing in ptychography,” Ultramicroscopy 138, 13–21 (2014).
[Crossref] [PubMed]

Cullis, A. G.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

d’Aspremont, A.

I. Waldspurger, A. d’Aspremont, and S. Mallat, “Phase recovery, maxcut and complex semidefinite programming,” Math. Program. 149(1-2), 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(2), 83–91 (2008).
[Crossref]

David, C.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Di, J.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[Crossref] [PubMed]

Dobson, B. R.

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref] [PubMed]

Dong, S.

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(2), 83–91 (2008).
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Supplementary Material (1)

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» Code 1       Neural network models for Fourier Ptychography

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Figures (11)

Fig. 1
Fig. 1 The CNN model for the Fourier ptychographic imaging process. The input of the network is the nth PSF. The object is treated as a two-channel learnable filter of a convolutional layer. We use a stride of 4 for the convolutional layer, and thus, the pixel size of the output intensity images is 4 times larger than that of the object. The output of the network represents the captured FP image. The optimization process of the network is to minimize Eq. (6).
Fig. 2
Fig. 2 (a) High-resolution amplitude and phase images for simulation. (b) The output of the CNN based on (a) and different wave vector ( k xn , k yn )s.
Fig. 3
Fig. 3 Different learning rates of the Adam optimizer in TensorFlow for the training process. (a)-(d) The recovered complex object images with learning rates ranging from 0.003 to 10. (e) The L1 loss (in log scale) as a function of epochs. A higher learning rate can decay the loss faster but gets stuck at a worse value of loss. This is because there is too much ‘energy’ in the optimization process and the learnable weights are bouncing around chaotically, unable to settle in a nice spot in the optimization landscape.
Fig. 4
Fig. 4 Performance of different solvers in TensorFlow: (a)Adam, (b) RMSprop, (c) SGD, and (d) SGDM. We use 5 by 5 plane waves for sample illumination and the step size for k xn and k yn is 0.15 in this simulation. The recovered amplitude ((a1)-(d1)) and phase ((a2)-(d2)) with 500 epochs (the best learning rates are chosen in this simulation). Different color curves in (a3)-(d3) represent different learning rates and the loss is in log scale. Adam gives the best performance overall. Batch size is 1 in this simulation.
Fig. 5
Fig. 5 Performance of different batch sizes as a function of epoch. (a)-(d) The recovered object for different batch sizes and with 20 epochs. (e) The loss (in log scale) with different batch sizes.
Fig. 6
Fig. 6 Performance of different batch sizes as a function of the updating times. (a)-(d) The recovered object for different batch sizes and with 20 updating times. (e) The loss curves (in log scale) with different batch sizes.
Fig. 7
Fig. 7 The multiplication neural network model for the Fourier ptychographic imaging process in Eq. (9). The input of the network is the nth CTF and the measured data I n . The object’s Fourier spectrum is modeled as learnable weights of a multiplication layer. We use a λ-layer to define the Fourier-magnitude-projection operation. The output of the network is 0, and thus, the training process of the network minimizes the loss function defined in Eq. (9).
Fig. 8
Fig. 8 Comparison of different cases with batch size = 1. The learning rates are chosen based on the fastest loss decay in 10 epochs. The Adam optimizer is used for all cases in this simulation study. (a) Minimizing the loss function in Eq. (6) with L2-norm, termed ‘L2 intensity’. (b) Minimizing the loss function in Eq. (6) with L1-norm, termed ‘L1 intensity’. (c) Minimizing the loss function in Eq. (9) with L2-norm, termed ‘L2 exit wave’. (d) Minimizing the loss function in Eq. (9) with L1-norm, termed ‘L1 exit wave’. The resolution improvement is more obvious for the exit-wave cases. (e) The performances of different approaches are quantified using MSE.
Fig. 9
Fig. 9 Test the L2-norm exit-wave network with experimental data. We use Adam optimizer with 20 epochs in this experiment and the batch size is 1. (a) The experimental setup with a 2X, 0.1 NA objective lens and a 3.45 µm pixel size camera, which is the same as the simulation setting. We test three different samples: (b) a blood smear, (c) a brain slide, and (d) a tissue section stained by immunohistochemistry methodology. (b1)-(d1) show the captured raw images using the 2X objective lens. The recovered intensity images using neural network are shown in (b2)-(d2) and the recovered phase images are shown in (b3)-(d3). As a comparison, (b4)-(d4) and (b5)-(d5) show the standard FPM reconstructions for intensity and phase [19].
Fig. 10
Fig. 10 CNN models for (a) single-pixel imaging and (b) structured illumination microscopy. For single-pixel imaging, the input is the illumination pattern P n (x,y), the object is modeled as learnable weights in a valid convolutional layer, and the output is the predicted single-pixel measurement. Similarly, for structured illumination microscopy, the object is modeled as learnable weights for a multiplication layer and the output is the predicted 2D image.
Fig. 11
Fig. 11 4-frame resolution doubling using the CNN-based SIM model. (a) The object image under uniform illumination. (b) The 4 SIM measurements using 4 sinusoidal patterns for sample illumination. The resolution doubled image recovered by the training process of the CNN in Fig. 10(b).

Equations (13)

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I n ( x,y )= | (O( x,y ) e i( k xn x+ k yn y) )*PSF(x,y) | 2 ,
I n ( x,y )= | O( x,y )*(PSF(x,y) e i( k xn x+ k yn y) ) | 2 = | O( x,y )*PS F n ( x,y ) | 2 ,
O( x,y )=  O r ( x,y )+ O i ( x,y ),PS F n ( x,y )= PS F nr ( x,y )+PS F ni ( x,y ),
I n ( x,y )= ( PS F nr * O r  PS F ni * O i ) 2 + ( PS F ni * O r + PS F nr * O i ) 2 ,
I n_predict ( x,y )= ( PS F nr * O r  PS F ni * O i ) 2 + ( PS F ni * O r + PS F nr * O i ) 2 ,
loss=diff( I n , I n predict )= n=i i1+batchSize | I n I n_predict |,
φ n ^ ( k x , k y )= O ^ ( k x , k y ) CTF n ( k x , k y ),
φ n update ^ ( k x , k y )=FT{ I n O( x,y )*PS F n ( x,y ) | O( x,y )*PS F n ( x,y ) |  }
loss=diff( φ n update ^ ( k x , k y ),  O ^ ( k x , k y ) CTF n ( k x , k y ) )                                                    = n=i i1+batchSize | φ n update ^ O ^ CTF n | 2 ,
I n = x,y O(x,y) P n (x,y),
loss=diff( I n , x,y O(x,y) P n (x,y)= n=i i1+batchSize | I n x,y O(x,y) P n (x,y) | ,
I n ( x,y )=( O( x,y ) P n ( x,y ) )*PSF,
loss=diff( I n ,( O( x,y ) P n ( x,y ) )*PSF )                                 = n=i i1+batchSize | I n ( O( x,y ) P n ( x,y ) )*PSF |,

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