Abstract

Imaging with low-dose light is of importance in various fields, especially when minimizing radiation-induced damage onto samples is desirable. The raw image captured at the detector plane is then predominantly a Poisson random process with Gaussian noise added due to the quantum nature of photo-electric conversion. Under such noisy conditions, highly ill-posed problems such as phase retrieval from raw intensity measurements become prone to strong artifacts in the reconstructions; a situation that deep neural networks (DNNs) have already been shown to be useful at improving. Here, we demonstrate that random phase modulation on the optical field, also known as coherent modulation imaging (CMI), in conjunction with the phase extraction neural network (PhENN) and a Gerchberg-Saxton-Fienup (GSF) approximant, further improves resilience to noise of the phase-from-intensity imaging problem. We offer design guidelines for implementing the CMI hardware with the proposed computational reconstruction scheme and quantify reconstruction improvement as function of photon count.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2020 (4)

M. Deng, S. Li, A. Goy, I. Kang, and G. Barbastathis, “Learning to synthesize: Robust phase retrieval at low photon counts,” Light: Sci. Appl. 9(1), 36 (2020).
[Crossref]

J. Wu, H. Zhang, W. Zhang, G. Jin, L. Cao, and G. Barbastathis, “Single-shot lensless imaging with fresnel zone aperture and incoherent illumination,” Light: Sci. Appl. 9(1), 53 (2020).
[Crossref]

T.-A. Pham, E. Soubies, A. Ayoub, J. Lim, D. Psaltis, and M. Unser, “Three-dimensional optical diffraction tomography with lippmann-schwinger model,” IEEE Trans. Comput. Imaging 6, 727–738 (2020).
[Crossref]

M. Deng, A. Goy, S. Li, K. Arthur, and G. Barbastathis, “Probing shallower: perceptual loss trained phase extraction neural network (plt-phenn) for artifact-free reconstruction at low photon budget,” Opt. Express 28(2), 2511–2535 (2020).
[Crossref]

2019 (5)

G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6(8), 921–943 (2019).
[Crossref]

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 1–12 (2019).
[Crossref]

A. Goy, G. Rughoobur, S. Li, K. Arthur, A. I. Akinwande, and G. Barbastathis, “High-resolution limited-angle phase tomography of dense layered objects using deep neural networks,” Proc. Natl. Acad. Sci. U. S. A. 116(40), 19848–19856 (2019).
[Crossref]

W. Tang, J. Yang, W. Yi, Q. Nie, J. Zhu, M. Zhu, Y. Guo, M. Li, X. Li, and W. Wang, “Single-shot coherent power-spectrum imaging of objects hidden by opaque scattering media,” Appl. Opt. 58(4), 1033–1039 (2019).
[Crossref]

Y. Wu, Y. Luo, G. Chaudhari, Y. Rivenson, A. Calis, K. de Haan, and A. Ozcan, “Bright-field holography: cross-modality deep learning enables snapshot 3d imaging with bright-field contrast using a single hologram,” Light: Sci. Appl. 8(1), 25 (2019).
[Crossref]

2018 (5)

2017 (2)

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” Optica 4(9), 1117–1125 (2017).
[Crossref]

P. P. Laissue, R. A. Alghamdi, P. Tomancak, E. G. Reynaud, and H. Shroff, “Assessing phototoxicity in live fluorescence imaging,” Nat. Methods 14(7), 657–661 (2017).
[Crossref]

2016 (3)

2015 (2)

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6(1), 5913 (2015).
[Crossref]

M. Holler, A. Díaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3d resolution,” Sci. Rep. 4(1), 3857 (2015).
[Crossref]

2014 (3)

2012 (1)

J. Miao, R. L. Sandberg, and C. Song, “Coherent x-ray diffraction imaging,” IEEE J. Sel. Top. Quantum Electron. 18(1), 399–410 (2012).
[Crossref]

2011 (1)

2010 (5)

F. Zhang and J. Rodenburg, “Phase retrieval based on wave-front relay and modulation,” Phys. Rev. B 82(12), 121104 (2010).
[Crossref]

G. Williams, H. Quiney, A. Peele, and K. Nugent, “Fresnel coherent diffractive imaging: treatment and analysis of data,” New J. Phys. 12(3), 035020 (2010).
[Crossref]

K. A. Nugent, “Coherent methods in the x-ray sciences,” Adv. Phys. 59(1), 1–99 (2010).
[Crossref]

L. Waller, L. Tian, and G. Barbastathis, “Transport of intensity phase-amplitude imaging with higher order intensity derivatives,” Opt. Express 18(12), 12552–12561 (2010).
[Crossref]

L. Waller, S. S. Kou, C. J. Sheppard, and G. Barbastathis, “Phase from chromatic aberrations,” Opt. Express 18(22), 22817–22825 (2010).
[Crossref]

2009 (4)

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17(15), 13040–13049 (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]

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2(1), 183–202 (2009).
[Crossref]

C. Kohler, F. Zhang, and W. Osten, “Characterization of a spatial light modulator and its application in phase retrieval,” Appl. Opt. 48(20), 4003–4008 (2009).
[Crossref]

2008 (1)

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. De Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008).
[Crossref]

2007 (1)

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75(4), 043805 (2007).
[Crossref]

2006 (1)

G. Williams, H. Quiney, B. Dhal, C. Tran, K. A. Nugent, A. Peele, D. Paterson, and M. De Jonge, “Fresnel coherent diffractive imaging,” Phys. Rev. Lett. 97(2), 025506 (2006).
[Crossref]

2004 (2)

F. Zhang, I. Yamaguchi, and L. Yaroslavsky, “Algorithm for reconstruction of digital holograms with adjustable magnification,” Opt. Lett. 29(14), 1668–1670 (2004).
[Crossref]

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

2002 (1)

A. Strehl and J. Ghosh, “Cluster ensembles—a knowledge reuse framework for combining multiple partitions,” J. Mach. Learn. Res. 3, 583–617 (2002).

1999 (1)

G. K. Matsopoulos, N. A. Mouravliansky, K. K. Delibasis, and K. S. Nikita, “Automatic retinal image registration scheme using global optimization techniques,” IEEE Trans. Inf. Technol. Biomed. 3(1), 47–60 (1999).
[Crossref]

1998 (1)

1997 (1)

1996 (1)

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]

1995 (1)

A. M. Eskicioglu and P. S. Fisher, “Image quality measures and their performance,” IEEE Trans. Commun. 43(12), 2959–2965 (1995).
[Crossref]

1992 (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60(1-4), 259–268 (1992).
[Crossref]

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

1982 (1)

1981 (1)

1972 (1)

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

1965 (1)

J. A. Nelder and R. Mead, “A simplex method for function minimization,” Comput. J. 7(4), 308–313 (1965).
[Crossref]

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161(4098), 777–778 (1948).
[Crossref]

Abbey, B.

B. Abbey, K. A. Nugent, G. J. Williams, J. N. Clark, A. G. Peele, M. A. Pfeifer, M. De Jonge, and I. McNulty, “Keyhole coherent diffractive imaging,” Nat. Phys. 4(5), 394–398 (2008).
[Crossref]

Akinwande, A. I.

A. Goy, G. Rughoobur, S. Li, K. Arthur, A. I. Akinwande, and G. Barbastathis, “High-resolution limited-angle phase tomography of dense layered objects using deep neural networks,” Proc. Natl. Acad. Sci. U. S. A. 116(40), 19848–19856 (2019).
[Crossref]

Alahi, A.

J. Johnson, A. Alahi, and L. Fei-Fei, “Perceptual losses for real-time style transfer and super-resolution,” in European Conference on Computer Vision (ECCV), (Springer, 2016), pp. 694–711.

Alghamdi, R. A.

P. P. Laissue, R. A. Alghamdi, P. Tomancak, E. G. Reynaud, and H. Shroff, “Assessing phototoxicity in live fluorescence imaging,” Nat. Methods 14(7), 657–661 (2017).
[Crossref]

Ambs, P.

Antoine, E. E.

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 1–12 (2019).
[Crossref]

Arthur, K.

M. Deng, A. Goy, S. Li, K. Arthur, and G. Barbastathis, “Probing shallower: perceptual loss trained phase extraction neural network (plt-phenn) for artifact-free reconstruction at low photon budget,” Opt. Express 28(2), 2511–2535 (2020).
[Crossref]

A. Goy, G. Rughoobur, S. Li, K. Arthur, A. I. Akinwande, and G. Barbastathis, “High-resolution limited-angle phase tomography of dense layered objects using deep neural networks,” Proc. Natl. Acad. Sci. U. S. A. 116(40), 19848–19856 (2019).
[Crossref]

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

Aspden, R. S.

P. A. Morris, R. S. Aspden, J. E. Bell, R. W. Boyd, and M. J. Padgett, “Imaging with a small number of photons,” Nat. Commun. 6(1), 5913 (2015).
[Crossref]

Ayoub, A.

T.-A. Pham, E. Soubies, A. Ayoub, J. Lim, D. Psaltis, and M. Unser, “Three-dimensional optical diffraction tomography with lippmann-schwinger model,” IEEE Trans. Comput. Imaging 6, 727–738 (2020).
[Crossref]

Ayoub, A. B.

J. Lim, A. B. Ayoub, E. E. Antoine, and D. Psaltis, “High-fidelity optical diffraction tomography of multiple scattering samples,” Light: Sci. Appl. 8(1), 1–12 (2019).
[Crossref]

Ba, J.

D. P. Kingma and J. Ba, “Adam: A method for stochastic optimization,” arXiv preprint arXiv:1412.6980 (2014).

Barbastathis, G.

M. Deng, A. Goy, S. Li, K. Arthur, and G. Barbastathis, “Probing shallower: perceptual loss trained phase extraction neural network (plt-phenn) for artifact-free reconstruction at low photon budget,” Opt. Express 28(2), 2511–2535 (2020).
[Crossref]

M. Deng, S. Li, A. Goy, I. Kang, and G. Barbastathis, “Learning to synthesize: Robust phase retrieval at low photon counts,” Light: Sci. Appl. 9(1), 36 (2020).
[Crossref]

J. Wu, H. Zhang, W. Zhang, G. Jin, L. Cao, and G. Barbastathis, “Single-shot lensless imaging with fresnel zone aperture and incoherent illumination,” Light: Sci. Appl. 9(1), 53 (2020).
[Crossref]

A. Goy, G. Rughoobur, S. Li, K. Arthur, A. I. Akinwande, and G. Barbastathis, “High-resolution limited-angle phase tomography of dense layered objects using deep neural networks,” Proc. Natl. Acad. Sci. U. S. A. 116(40), 19848–19856 (2019).
[Crossref]

G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6(8), 921–943 (2019).
[Crossref]

S. Li and G. Barbastathis, “Spectral pre-modulation of training examples enhances the spatial resolution of the phase extraction neural network (phenn),” Opt. Express 26(22), 29340–29352 (2018).
[Crossref]

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
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Figures (18)

Fig. 1.
Fig. 1. Schematic of coherent modulation imaging (CMI) setup. This setup involves three physical planes, i.e. an object plane, modulation plane, and a detector plane.
Fig. 2.
Fig. 2. Comparison in measurements in the spatial domain between with and without the random phase modulation. In these experiments, $z_1=490\ \textrm {mm}$ and $z_2= 48.5\ \textrm {mm}$ for CMI and the propagation distance is $z_1+z_2 = 538.5\ \textrm {mm}$ for CDI.
Fig. 3.
Fig. 3. Power spectral densities (PSDs) of the corresponding $500$ intensity measurements from Fig. 2. (Asymmetric periodic artifacts that are clearly visible in the PSDs are due to unwanted fixed-pattern noise (FPN) in our EM-CCD.)
Fig. 4.
Fig. 4. Inverse algorithm for reconstruction using GSF algorithm and deep neural network (DNN). The dashed box indicates the Approximant and GSF parts of our overall computational platform. Please refer to Appendix A for additional details.
Fig. 5.
Fig. 5. Phase Extraction Neural Network (PhENN) architecture [24].
Fig. 6.
Fig. 6. Photon arrival level for all figures is set to be $\mathbf {1}$ photon per pixel. For (a-c), perceptual loss from VGG network and for (d-f), Pearson correlation coefficient (PCC) were used to derive a design criterion of the location of the random phase modulation. Yellow asterisk marks our design for actual experiments, $\textit {i.e.}$ $(z_1, z_2) = \left (490\textrm {~mm},\ 48.5\textrm {~mm}\right )$ . Black dashed lines are the loci $z_1+z_2=\textrm {constant}$ , i.e. $\textrm {NA}=\textrm {constant}$ .
Fig. 7.
Fig. 7. All figures are based on $\textbf {perceptual loss}$ . To assess the effectiveness of DNN over GSF under the photon-limited condition, two different photon levels are assumed, $\textit {i.e}$ $1$ photon per pixel for (a-c) and $10$ photons per pixel for (d-f). Yellow asterisks and black dashed lines follow the same convention as in Fig. 6.
Fig. 8.
Fig. 8. Same comparisons as in Fig. 7 but now according to the PCC metric. Photon arrival rates per pixel are $\textit {i.e}$ $1$ photon per pixel for (a-c) and $10$ photons per pixel for (d-f). Yellow asterisks and black dashed lines follow the same convention as in Fig. 6.
Fig. 9.
Fig. 9. Proposed optical apparatus. VND: variable neutral density filter, HWP: half-wave plate, OBJ: objective lens, F: spatial filter, L: lens, A: aperture, POL: polarizer / analyzer, SLM: spatial light modulator, NPBS: non-polarizing beamsplitter and EM-CCD: electron-multiplying charge coupled device.
Fig. 10.
Fig. 10. ImageNet - Results with and without the random phase modulation over different photon levels. Two images were selected from ImageNet dataset to qualitatively illustrate how the random phase modulation deals with the noise-corrupted measurements due to the low-photon condition. In the experiments, $z_1 = 490\ \textrm {mm},\ z_2 = 48.5\ \textrm {mm}$ .
Fig. 11.
Fig. 11. IC layout - Results with and without the random phase modulation over different photon levels.
Fig. 12.
Fig. 12. ImageNet - Quantitative comparison using various metrics on DNN reconstructions from experimental measurements under different photon levels. Bar graphs denote the mean and standard deviation as error bars. PCC: Pearson correlation coefficient, SSIM: structural similarity index, NRMSE: normalized root mean-squared error, and PSNR: peak signal-to-noise ratio.
Fig. 13.
Fig. 13. IC layout - Quantitative comparison using various metrics on DNN reconstructions from experimental measurements under different photon levels. Bar graphs denote the mean and standard deviation with error bars.
Fig. 14.
Fig. 14. (a) Power spectral density (PSD) curves, which were circularly averaged and displayed in a logarithmic scale [59]. (b) Ratiometric comparison was made on two PSD curves with and without the random phase modulation under the condition of $1000$ photons per pixel. (c) Same as (b) but under the condition of $1$ photon per pixel.
Fig. 15.
Fig. 15. Qualitative results of cross-domain generalization. Columns indicate if the neural network was trained with either ImageNet or IC layout database, and rows denote if testing inputs are sampled from either ImageNet or IC layout database. Images are based on experimental intensity measurements under CMI scheme with the mean of photons per pixel of $1$ .
Fig. 16.
Fig. 16. SLM calibration curves. (a) Phase modulation of SLM $1$ . (b) Coupled amplitude modulation of SLM $1$ . (c) Phase modulation of SLM $2$ . Horizontal axes of all curves are $8$ -bit grayscale values.
Fig. 17.
Fig. 17. Various quantitative metrics based on different values of the size of interpolation kernel. Each box shows the median in the middle, and $25^\textrm{th}$ and $75^\textrm{th}$ quantiles at the bottom and the top, respectively. (a) NRMSE (normalized root-mean squared error). (b) PCC (Pearson correlation coefficient). (c) SSIM (structural similarity index). (d) PSNR (peak signal-to-noise ratio).
Fig. 18.
Fig. 18. (a) Original power spectral density (PSD) curves and (b) the ratio of each PSD curve to the reference. The case without any random phase modulation (or CDI) was set to be the reference. All cases are the results trained with NPCC as a loss function.

Tables (4)

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Table 1. Specifications on important parameters for experiments, pre-processing steps, and training process. Here, the photon count is the effective number of photons or photoelectrons per pixel. N TV is the number of iteration of optional TV denoising applied to φ ^ approx .

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Table 2. Quantitative results of cross-domain generalization in terms of SSIM. Mean and standard deviation are shown. Values are based on experimental intensity measurements under CMI scheme with the mean of photons per pixel of 1 .

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Table 3. Supplement to Fig. 6 - ImageNet database was used for both simulation and experiment. Values displayed are medians.

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Table 4. Supplement to Figs. 7 and 8 - Results from simulation and experiment are quantitatively compared using perceptual loss and PCC. ImageNet database was used for both simulation and experiment. Values displayed are medians. (*) In the simulation, 10 photons per pixel are the high photon count; in the experiment, the high photon count was 1000 photons per pixel.

Equations (16)

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ψ det ( x , y ) = e i 2 π z 2 / λ i λ z 2 ψ M ( x , y ) exp { i π ( x x ) 2 + ( y y ) 2 λ z 2 } d x d y ,
where ψ M ( x , y ) = ψ m ( x , y ) exp { i Φ ( x , y ) } and
ψ m ( x , y ) = e i 2 π z 1 / λ i λ z 1 ψ obj ( x , y ) exp { i π ( x x ) 2 + ( y y ) 2 λ z 1 } d x d y .
I det ( x , y ) = | ψ det ( x , y ) | 2 .
a k l = e i 2 π z 1 / λ i λ z 1 exp { i π ( x k x l ) 2 λ z 1 } , b k l = e i 2 π z 1 / λ i λ z 1 exp { i π ( y k y l ) 2 λ z 1 } , and
c k l = e i 2 π z 2 / λ i λ z 2 exp { i π ( x k x l ) 2 λ z 2 } , d k l = e i 2 π z 2 / λ i λ z 2 exp { i π ( y k y l ) 2 λ z 2 } ,
Ψ M , k k = exp { i Φ M , k k } ,   k k = 1 , , N 2 ,
Ψ det = [ ( C D ) Ψ M ( A B ) ] Ψ obj H z 1 , z 2 Ψ obj ,
I det = | Ψ det | 2 = | H z 1 , z 2 Ψ obj | 2 ,
φ ^ DNN = DNN ( φ ^ approx ) ,
w D N N = argmin w n ζ [ φ n , D N N ( φ ^ n , approx ( φ n ) ) ] ,
ζ NPCC ( f , g ) x , y ( f ( x , y ) f ) ( g ( x , y ) g ) x , y ( f ( x , y ) f ) 2 x , y ( g ( x , y ) g ) 2
ζ SSIM ( f , g ) ( 2 f g + c 1 ) ( 2 σ f g + c 2 ) ( f 2 + g 2 + c 1 ) ( σ f 2 + σ g 2 + c 2 ) .
γ metric = metric DNN,CMI metric DNN,CDI , where     metric = (perceptual loss) 1 ,   PCC ,
δ metric = metric DNN, CMI metric GSF, CMI , where     metric = (perceptual loss) 1 ,   PCC ,
M = T R S h S c ,