pith. sign in

arxiv: 1907.11713 · v1 · pith:IJ2JILK4new · submitted 2019-07-26 · 📡 eess.IV · cs.LG

Learning to Synthesize: Robust Phase Retrieval at Low Photon counts

Mo Deng , Shuai Li , Alexandre Goy , Iksung Kang , George Barbastathis This is my paper

Pith reviewed 2026-05-24 15:24 UTC · model grok-4.3

classification 📡 eess.IV cs.LG
keywords phase retrievaldeep learninglow photon countfrequency synthesisinverse problemsnoise robustnessimagingspatial frequencies
0
0 comments X

The pith

A learning-to-synthesize method retrieves clean phase maps from low-photon images by handling frequency bands separately before combination.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a learning to synthesize approach for quantitative phase retrieval. Prior deep learning solutions suppressed high spatial frequencies that were rare in training data, while ad hoc pre-amplification of those frequencies created new artifacts and distortions. The proposed method trains separate models on low and high frequency bands, then learns a synthesis step to combine them into full-band outputs. This produces high-resolution reconstructions that stay free of artifacts and remain stable even when photon flux is very low and noise is high. The same separation-plus-synthesis idea is stated to apply to any inverse problem whose forward operator treats frequency bands unevenly.

Core claim

By learning to handle low and high spatial frequency bands separately and then synthesizing them, the method achieves phase reconstructions with high spatial resolution, free of artifacts, and resilient to high-noise conditions such as very low photon flux.

What carries the argument

The learning to synthesize (LS) procedure that trains on low and high frequency bands independently before learning their combination.

If this is right

  • Phase reconstructions maintain high spatial resolution without introducing high-frequency artifacts.
  • Low-frequency content remains undistorted even under high noise.
  • The approach stays effective at very low photon counts where conventional methods fail.
  • The same separation and synthesis strategy extends to other inverse problems whose forward operators are ill-posed for particular frequency bands.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The method may allow smaller or less diverse training sets because each band can be learned in isolation.
  • Analogous band-separation steps could be tested on other ill-posed imaging tasks such as limited-angle tomography.
  • Systematic sweeps of photon flux levels could quantify the exact noise threshold where the synthesis step begins to outperform single-network baselines.

Load-bearing premise

That training separate models on the two frequency bands and then synthesizing their outputs will avoid the high-frequency artifacts and low-frequency distortions produced by earlier pre-amplification strategies.

What would settle it

Side-by-side comparison of phase reconstructions from the same low-photon-count data using the LS method versus pre-amplification, measured against known ground-truth phase maps for the presence of artifacts or distortions.

Figures

Figures reproduced from arXiv: 1907.11713 by Alexandre Goy, George Barbastathis, Iksung Kang, Mo Deng, Shuai Li.

Figure 1
Figure 1. Figure 1: LS-DNN schematic. (a) Training stage, (b) Test stage. After DNN-L, DNN-H and DNN-S have been trained, they are combined in the LS system and operated as shown in Fig￾ure 1(b). The input ξ(x, y) is passed to DNN-L and DNN-H in parallel fashion, and the respective outputs ˆ f LF(x, y) and ˆ f HF(x, y) are passed to DNN-S, which produces the final esti￾mate ˆ f(x, y). It is worth noting that it is not valid t… view at source ↗
Figure 2
Figure 2. Figure 2: compares the 2D (log-scale) Fourier spectrum mag￾nitude of a ground truth image (from ImageNet [63]), Approxi￾mant (19) computed without noise, and Approximant (19) com￾puted from an input subject to Poisson statistics corresponding to average flux of one photon per pixel. We can see that although the single photon Approximant (which we will later use as the input to the LS-DNN) has a large support in its … view at source ↗
Figure 3
Figure 3. Figure 3: Optical apparatus for experiments. SF: spatial filter, CL: collimating lens, VND: variable neutral density filter. photons. Here, we report results for two levels of photon flux p = 9.8 ± 5% and p = 1.1 ± 5%, respectively quoted in the text as “10” and “1” photons. The data acquisition, training and testing procedures of the entire LS-DNN architecture were repeated separately for each value of p. B. Recons… view at source ↗
Figure 4
Figure 4. Figure 4: Reconstructions by LS-DNN (top: 1 photon/pixel/frame, bottom: 10 photons/pixel/frame); from left to right: Approximant (the input to the LS-DNN system), DNN-L reconstruction [39], DNN-H reconstruction (q = 0.5), DNN-S reconstruction, ground truth. made that choice because spatial resolution under highly noisy conditions becomes non-trivially coupled to the noise statistics, and a complete investigation wou… view at source ↗
Figure 5
Figure 5. Figure 5: Comparisons of LS-DNN reconstructions under different q 0 s for p = 1 photon/ pixel. Rows from top to bottom: DNN-L output; DNN-H output under different q’s; DNN-S under different q’s; 1D cross-section (along the dashed line in the row below) of DNN-S output under different q 0 s; ground truth. REFERENCES 1. P. Marquet, B. Rappaz, P. J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digit… view at source ↗
Figure 6
Figure 6. Figure 6: Comparisons of LS-DNN reconstructions under different q 0 s for p = 10 photon/ pixel. Rows from top to bottom: DNN-L output; DNN-H output under different q’s; DNN-S under different q’s; 1D cross-section (along the dashed line in the row below) of DNN-S output under different q 0 s; ground truth. cuits,” Nature 543, 402–406 (2017). 7. J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intens… view at source ↗
Figure 7
Figure 7. Figure 7: Fourier spectra of two test examples and their reconstructions from the components of the LS scheme. Donoho, and J. M. Pauly, “Recurrent generative adversarial networks for proximal learning and automated compres￾sive image recovery,” CoRR. (2017). 32. C.-Y. Yang, C. Ma, and M.-H. Yang, “Single-image super￾resolution: A benchmark,” in European Conference on Com￾puter Vision, (Springer, 2014), pp. 372–386. … view at source ↗
Figure 8
Figure 8. Figure 8: 1D diagonal cross-sections of the average 2D power spectral density (PSD) of 50 test images for p = 1 photon per pixel. The case of p = 10 photons per pixel is in the Supplementary Material. tica 6, 618–629 (2019). 44. U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2, 517–522 (2015). 45. A. Goy, G. Rughoobur, … view at source ↗
read the original abstract

The quality of inverse problem solutions obtained through deep learning [Barbastathis et al, 2019] is limited by the nature of the priors learned from examples presented during the training phase. In the case of quantitative phase retrieval [Sinha et al, 2017, Goy et al, 2019], in particular, spatial frequencies that are underrepresented in the training database, most often at the high band, tend to be suppressed in the reconstruction. Ad hoc solutions have been proposed, such as pre-amplifying the high spatial frequencies in the examples [Li et al, 2018]; however, while that strategy improves resolution, it also leads to high-frequency artifacts as well as low-frequency distortions in the reconstructions. Here, we present a new approach that learns separately how to handle the two frequency bands, low and high; and also learns how to synthesize these two bands into the full-band reconstructions. We show that this "learning to synthesize" (LS) method yields phase reconstructions of high spatial resolution and artifact-free; and it is also resilient to high-noise conditions, e.g. in the case of very low photon flux. In addition to the problem of quantitative phase retrieval, the LS method is applicable, in principle, to any inverse problem where the forward operator treats different frequency bands unevenly, i.e. is ill-posed.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The manuscript proposes a 'learning to synthesize' (LS) deep-learning method for quantitative phase retrieval. Separate networks are trained on low- and high-frequency bands of the training data; a third synthesis stage then combines the band-specific outputs into a full-band reconstruction. The central claim is that this frequency-separated training avoids both the high-frequency artifacts and low-frequency distortions produced by prior ad-hoc pre-amplification strategies, while remaining robust at very low photon flux.

Significance. If the empirical results hold, the LS strategy offers a principled way to mitigate frequency-dependent bias in learned priors for ill-posed inverse problems. The approach is directly relevant to photon-limited coherent imaging and, in principle, to any linear inverse problem whose forward operator attenuates high frequencies.

minor comments (3)
  1. The abstract states that LS yields 'high spatial resolution and artifact-free' reconstructions, yet the manuscript provides no quantitative comparison (e.g., RMSE, SSIM, or resolution metrics) against the pre-amplification baseline of Li et al. (2018) on the same test set; this comparison should be added to §4 or a new results table.
  2. Notation for the low- and high-frequency operators (presumably defined in §3) is used without an explicit equation reference in the synthesis-stage description; adding an equation label would improve readability.
  3. The training database composition (number of examples, spatial-frequency content) is described only qualitatively; a short table or paragraph quantifying the frequency distribution would strengthen the motivation for band separation.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive summary of our work, the recognition of its potential significance for frequency-dependent bias in learned priors, and the recommendation for minor revision. The referee's description of the LS method is accurate. No major comments were listed in the report, so we have no specific points requiring response or revision at this stage.

Circularity Check

0 steps flagged

Minor self-citations present but not load-bearing; no circularity in core method

full rationale

The paper introduces a new 'learning to synthesize' strategy that separates low- and high-frequency band learning before synthesis for phase retrieval. It references prior DL work via [Barbastathis et al, 2019] and pre-amplification via [Li et al, 2018], both involving overlapping authors, but these serve only as background contrast rather than load-bearing justification for the central claim. The LS method's claimed advantages (artifact-free high-resolution reconstructions at low photon flux) rest on the described architecture and experimental outcomes, without any self-definitional reduction, fitted input renamed as prediction, or uniqueness theorem imported from self-citations. The derivation chain remains self-contained with independent content.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Based on the abstract alone, the central claim rests on the domain assumption that the forward operator in phase retrieval treats frequency bands unevenly and that deep learning can learn the bands separately. No specific free parameters or invented entities are named.

axioms (1)
  • domain assumption The forward operator in phase retrieval treats different frequency bands unevenly (i.e., is ill-posed).
    Explicitly stated as the condition under which the LS method is applicable.

pith-pipeline@v0.9.0 · 5786 in / 1209 out tokens · 50500 ms · 2026-05-24T15:24:59.874765+00:00 · methodology

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

87 extracted references · 87 canonical work pages · 3 internal anchors

  1. [1]

    Digital holographic mi- croscopy: a noninvasive contrast imaging technique allow- ing quantitative visualization of living cells with subwave- length axial accuracy,

    P . Marquet, B. Rappaz, P . J. Magistretti, E. Cuche, Y. Emery, T. Colomb, and C. Depeursinge, “Digital holographic mi- croscopy: a noninvasive contrast imaging technique allow- ing quantitative visualization of living cells with subwave- length axial accuracy,” Opt. Lett. 30, 468–470 (2005)

    work page 2005

  2. [2]

    Diffraction phase microscopy for quantifying cell struc- ture and dynamics,

    G. Popescu, T. Ikeda, R. R. Dasari, and M. S. Feld, “Diffraction phase microscopy for quantifying cell struc- ture and dynamics,” Opt. letters 31, 775–777 (2006)

    work page 2006

  3. [3]

    X-ray phase-contrast microscopy and microtomography,

    S. C. Mayo, T. J. Davis, T. E. Gureyev, P . R. Miller, D. Pa- ganin, A. Pogany, A. W. Stevenson, and S. W. Wilkins, “X-ray phase-contrast microscopy and microtomography,” Opt. Express 11, 2289–2302 (2003)

    work page 2003

  4. [4]

    Phase retrieval and differential phase-contrast imaging with low- brilliance X-ray sources,

    F. Pfeiffer, T. Weitkamp, O. Bunk, and C. David, “Phase retrieval and differential phase-contrast imaging with low- brilliance X-ray sources,” Nat. Phys. 2, 258–261 (2006)

    work page 2006

  5. [5]

    Contrast enhancement in x-ray phase contrast tomography,

    A. Pan, Ling Xu, J. C. Petruccelli, R. Gupta, B. Singh, and G. Barbastathis, “Contrast enhancement in x-ray phase contrast tomography,” Opt. express 22, 18020–18026 (2014)

    work page 2014

  6. [6]

    Holler, M

    M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non- desctructive three-dimensional analysis of integrated cir- Research Article Vol. X, No. X / April 2016 / Optica 10 Fig. 6. Comparisons of LS-DNN reconstructions under different q′s for p = 10 photon/ pixel. Rows from top to bottom: DNN-L outp...

    work page 2016

  7. [7]

    The transport of intensity equation for optical path length recovery using partially coherent illumination,

    J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Exp. 21, 14430 (2013)

    work page 2013

  8. [8]

    Digital image formation from electronically detected holograms,

    J. W. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. physics letters 11, 77–79 (1967)

    work page 1967

  9. [9]

    Phase-shifting speckle interferometry,

    K. Creath, “Phase-shifting speckle interferometry,” Appl. Opt. 24, 3053–3058 (1985)

    work page 1985

  10. [10]

    A practical algo- rithm for the determination of the phase from image and diffraction plane pictures,

    R. W. Gerchberg and W. O. Saxton, “A practical algo- rithm for the determination of the phase from image and diffraction plane pictures,” Optik. 35, 237–250 (1972)

    work page 1972

  11. [11]

    Reconstruction of an object from the modulus of its Fourier transform,

    J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. letters 3, 27–29 (1978)

    work page 1978

  12. [12]

    Wide-field, high- resolution fourier ptychographic microscopy,

    G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high- resolution fourier ptychographic microscopy,” Nat. pho- tonics 7, 739 (2013)

    work page 2013

  13. [13]

    Multiplexed coded illumination for fourier ptychography with an led array microscope,

    L. Tian, X. Li, K. Ramchandran, and L. Waller, “Multiplexed coded illumination for fourier ptychography with an led array microscope,” Biomed. optics express 5, 2376–2389 (2014)

    work page 2014

  14. [14]

    Deterministic phase retrieval: a Green’s function solution,

    M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” JOSA 73, 1434–1441 (1983)

    work page 1983

  15. [15]

    Streibl, “Phase imaging by the transport-equation of Research Article Vol

    N. Streibl, “Phase imaging by the transport-equation of Research Article Vol. X, No. X / April 2016 / Optica 11 Average PCC± std.dev Average PSNR± std.dev (dB) Average SSIM± std.dev p = 1 p = 10 p = 1 p = 10 p = 1 p = 10 Approximant ˆf∗ 0.149± 0.067 0.181± 0.082 8.082± 3.979 8.098± 3.986 0.220± 0.105 0.222± 0.106 DNN-L output ˆf LF 0.808± 0.099 0.880± 0.0...

    work page 2016

  16. [16]

    Phase retrieval algorithms – a comparison,

    J. R. Fienup, “Phase retrieval algorithms – a comparison,” Appl. Opt. 21, 2758–2769 (1982)

    work page 1982

  17. [17]

    Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,

    H. H. Bauschke, P . L. Combettes, and D. R. Luke, “Phase retrieval, error reduction algorithm, and fienup variants: a view from convex optimization,” JOSA A 19, 1334–1345 (2002)

    work page 2002

  18. [18]

    The lock problem in the Gerchberg- Saxton algorithm for phase retrieval,

    R. W. Gerchberg, “The lock problem in the Gerchberg- Saxton algorithm for phase retrieval,” Optik 74, 91–93 (1986)

    work page 1986

  19. [19]

    Phase-retrieval stagnation problems and solutions,

    J. Fienup and C. Wackerman, “Phase-retrieval stagnation problems and solutions,” JOSA A 3, 1897–1907 (1986)

    work page 1907

  20. [20]

    Decoding by linear programming,

    E. Candès and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005)

    work page 2005

  21. [21]

    Robust uncertainty principles: Exact signal reconstruction from highly incom- plete frequency information,

    E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incom- plete frequency information,” IEEE Transactions on infor- mation theory 52, 489–509 (2006)

    work page 2006

  22. [22]

    Compressed sensing,

    D. L. Donoho, “Compressed sensing,” IEEE Transactions on information theory 52, 1289–1306 (2006)

    work page 2006

  23. [23]

    Stable signal recovery from incomplete and inaccurate measurements,

    E. Candès, J. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure Appl. Math. 59, 1207–1223 (2006)

    work page 2006

  24. [24]

    Y. C. Eldar and G. Kutyniok, Compressed Sensing: Theory and Applications (Cambridge University Press, 2012)

    work page 2012

  25. [25]

    Image denoising via sparse and redundant representations over learned dictionaries,

    M. Elad and M. Aharon, “Image denoising via sparse and redundant representations over learned dictionaries,” IEEE Transactions on Image processing 15, 3736–3745 (2006)

    work page 2006

  26. [26]

    K-svd: An algo- rithm for designing overcomplete dictionaries for sparse representation,

    M. Aharon, M. Elad, and A. Bruckstein, “K-svd: An algo- rithm for designing overcomplete dictionaries for sparse representation,” IEEE Transactions on signal processing 54, 4311 (2006)

    work page 2006

  27. [27]

    Dictionaries for sparse representation modeling,

    R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98, 1045– 1057 (2010)

    work page 2010

  28. [28]

    Dictionary learning and sparse coding: algorithms and convergence analysis,

    Changlong Bao, Hui Ji, Yuhui Quan, and Zuowei Shen, “Dictionary learning and sparse coding: algorithms and convergence analysis,” IEEE Tras. Patt. Anal. Mach. Intel. 38, 1356–1369 (2016)

    work page 2016

  29. [29]

    Learning fast approximations of sparse coding,

    K. Gregor and Y. LeCun, “Learning fast approximations of sparse coding,” in Proceedings of the 27th International Conference on International Conference on Machine Learning, (Omnipress, 2010), pp. 399–406

    work page 2010

  30. [30]

    Generative Adversarial Nets,

    I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, Bing Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio, “Generative Adversarial Nets,” in Neural Information Pro- cessing Systems (NIPS), , vol. 27 (2014)

    work page 2014

  31. [31]

    Recurrent generative adversarial networks for proximal learning and automated compres- sive image recovery,

    M. Mardani, H. Monajemi, V . Papyan, S. Vasanawala, D. L. Research Article Vol. X, No. X / April 2016 / Optica 12 Fig. 7. Fourier spectra of two test examples and their reconstructions from the components of the LS scheme. Donoho, and J. M. Pauly, “Recurrent generative adversarial networks for proximal learning and automated compres- sive image recovery,”...

    work page 2016

  32. [32]

    Single-image super- resolution: A benchmark,

    C.-Y. Yang, C. Ma, and M.-H. Yang, “Single-image super- resolution: A benchmark,” in European Conference on Com- puter Vision, (Springer, 2014), pp. 372–386

    work page 2014

  33. [33]

    Learning a deep convolutional neural network for image super-resolution,

    Chao Dong, Chen Loy, Kaiming He, and Xiaoou Tang, “Learning a deep convolutional neural network for image super-resolution,” in European Conference on Computer Vi- sion (ECCV) / Lecture Notes on Computer Science Part IV ,, vol. 8692 (2014), pp. 184–199

    work page 2014

  34. [34]

    Im- age super-resolution using deep convolutional networks,

    Chao Dong, Chen Loy, Kaiming He, and Xiaoou Tang, “Im- age super-resolution using deep convolutional networks,” pami. 38, 295–307 (2015)

    work page 2015

  35. [35]

    Deep convolutional neural network for inverse problems in imaging,

    Kyong Hwan Jin, M. T. McCann, E. Froustey, and M. Unser, “Deep convolutional neural network for inverse problems in imaging,” IEEE Trans. Image Process. 26, 4509–4522 (2017)

    work page 2017

  36. [36]

    Convolu- tional neural networks for inverse problems in imaging,

    M. T. McCann, Kyong Hwan Jin, and M. Unser, “Convolu- tional neural networks for inverse problems in imaging,” IEEE Sig. Process. Mag. pp. 85–95 (2017)

    work page 2017

  37. [37]

    Lens- less computational imaging through deep learning,

    A. Sinha, Justin Lee, Shuai Li, and G. Barbastathis, “Lens- less computational imaging through deep learning,” Op- tica. 4, 1117–1125 (2017)

    work page 2017

  38. [38]

    Spectral pre-modulation of training examples enhances the spatial resolution of the phase extraction neural network (PhENN),

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

    work page 2018

  39. [39]

    Low photon count phase retrieval using deep learning,

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

    work page 2018

  40. [40]

    Phase recovery and holographic image reconstruc- tion using deep learning in neural networks,

    Y. Rivenson, Yibo Zhang, H. Gunaydin, Da Teng, and A. Oz- can, “Phase recovery and holographic image reconstruc- tion using deep learning in neural networks,” Light. Sci. 7, 17141 (2018)

    work page 2018

  41. [41]

    Extended depth-of-field in holo- graphic imaging using deep-learning-based autofocusing and phase recovery,

    Y. Wu, Y. Rivenson, Y. Zhang, Z. Wei, H. Günaydin, X. Lin, and A. Ozcan, “Extended depth-of-field in holo- graphic imaging using deep-learning-based autofocusing and phase recovery,” Optica. 5, 704–710 (2018)

    work page 2018

  42. [42]

    Deep learning approach for fourier ptychography mi- croscopy,

    T. Nguyen, Y. Xue, Y. Li, L. Tian, and G. Nehmetallah, “Deep learning approach for fourier ptychography mi- croscopy,” Opt. express 26, 26470–26484 (2018)

    work page 2018

  43. [43]

    Reliable deep-learning- based phase imaging with uncertainty quantification,

    Y. Xue, S. Cheng, Y. Li, and L. Tian, “Reliable deep-learning- based phase imaging with uncertainty quantification,” Op- Research Article Vol. X, No. X / April 2016 / Optica 13 Fig. 8. 1D diagonal cross-sections of the average 2D power spectral density (PSD) of 50 test images for p = 1 photon per pixel. The case of p = 10 photons per pixel is in the Supple...

    work page 2016

  44. [44]

    Learning approach to optical tomography,

    U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Learning approach to optical tomography,” Optica 2, 517–522 (2015)

    work page 2015

  45. [45]

    High-resolution limited-angle phase tomography of dense layered objects using deep neural networks,

    A. Goy, G. Rughoobur, Shuai Li, K. Arthur, A. Akinwande, and G. Barbastathis, “High-resolution limited-angle phase tomography of dense layered objects using deep neural networks,” Proc. Nat. Acad. Sci. ((accepted) 2019)

    work page 2019

  46. [46]

    Quanti- tative phase imaging and artificial intelligence: A review,

    YoungJu Jo, Hyungjoo Cho, Sang Yun Lee, Gunho Choi, Geon Kim, Hyun-Seok Min, and YongKeun Park, “Quanti- tative phase imaging and artificial intelligence: A review,” IEEE J. Sel. Top. Quant. Electron. 25 (2019)

    work page 2019

  47. [47]

    On the use of deep learning for computational imaging,

    G. Barbastathis, A. Ozcan, and Guohai Situ, “On the use of deep learning for computational imaging,” Optica. (2019)

    work page 2019

  48. [48]

    Orthonormal bases of compactly supported wavelets,

    I. Daubechies, “Orthonormal bases of compactly supported wavelets,” Comm. Pure Appl. Math. 41, 909–996 (1988)

    work page 1988

  49. [49]

    Daubechies, Ten lectures on wavelets(Soc

    I. Daubechies, Ten lectures on wavelets(Soc. Ind. Appl. Math., 1992)

    work page 1992

  50. [50]

    Translation invariant de- noising,

    R. Coifman and D. L. Donoho, “Translation invariant de- noising,” in Wavelets and statistics, , vol. 103 of Lecture Notes in Statistics (Springer-Verlag, 1995), pp. 120–150

    work page 1995

  51. [51]

    Strang and Truong Nguyen, Wavelets and filter banks (Wellesley-Cambridge Press, 1996), 2nd ed

    G. Strang and Truong Nguyen, Wavelets and filter banks (Wellesley-Cambridge Press, 1996), 2nd ed

    work page 1996

  52. [52]

    Wavelet algorithms for high-resolution image reconstruc- tion,

    Raymond Chan, Tony Chan, Lixin Shen, and Zuowei Shen, “Wavelet algorithms for high-resolution image reconstruc- tion,” SIAM J. Sci. Comp.. 24, 1408–1432 (2003)

    work page 2003

  53. [53]

    An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,

    I. Daubechies, M. Defrise, and C. D. Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Comm. Pure Appl. Math. 57, 1413– 1457 (2004)

    work page 2004

  54. [54]

    An EM algorithm for wavelet-based image restoration,

    M. A. T. Figueiredo and R. Nowak, “An EM algorithm for wavelet-based image restoration,” IEEE Trans. Image Proc. 12, 906–916 (2003)

    work page 2003

  55. [55]

    Mallat, A wavelet tour of signal processing: the sparse way (Academic Press, 2008)

    S. Mallat, A wavelet tour of signal processing: the sparse way (Academic Press, 2008)

    work page 2008

  56. [56]

    Wide-field flu- orescence sectioning with hybrid speckle and uniform- illumination microscopy,

    D. Lim, Kengye K. Chu, and J. Mertz, “Wide-field flu- orescence sectioning with hybrid speckle and uniform- illumination microscopy,” Opt. Lett. 33, 1819–1821 (2008)

    work page 2008

  57. [57]

    Optical sectioning microscopy with planar or structured illumination,

    J. Mertz, “Optical sectioning microscopy with planar or structured illumination,” Nat. Methods 8, 811–819 (2011)

    work page 2011

  58. [58]

    Three dimensional HiLo-based structured illumination for a digital scanned laser sheet microscopy (DSLM) in thick tissue imaging,

    D. Bhattacharya, V . R. Singh, Chen Zhi, P . T. C. So, P . Matsu- daira, and G. Barbastathis, “Three dimensional HiLo-based structured illumination for a digital scanned laser sheet microscopy (DSLM) in thick tissue imaging,” Opt. Express 20, 27337–27347 (2012)

    work page 2012

  59. [59]

    Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,

    Y. Yunhui, A. Shanker, Lei Tian, L. Waller, and G. Barbas- tathis, “Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,” Opt. express 22, 26696–26711 (2014)

    work page 2014

  60. [60]

    Learning dual convolutional neural networks for low-level vision,

    J. Pan, S. Liu, D. Sun, J. Zhang, Y. Liu, J. Ren, Z. Li, J. Tang, H. Lu, Y.-W. Taiet al., “Learning dual convolutional neural networks for low-level vision,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition,(2018), pp. 3070–3079

    work page 2018

  61. [61]

    On the solution of ill-posed problems and the method of regularization,

    A. N. Tikhonov, “On the solution of ill-posed problems and the method of regularization,” Dokl. Acad. Nauk SSSR 151, 501–504 (1963)

    work page 1963

  62. [62]

    On the regularization of ill-posed prob- lems,

    A. N. Tikhonov, “On the regularization of ill-posed prob- lems,” Dokl. Acad. Nauk SSSR 153, 49–52 (1963)

    work page 1963

  63. [63]

    ImageNet large scale visual recognition chal- lenge,

    O. Russakovsky, Jia Deng, Hao Su, J. Krause, S. Satheesh, Sean Ma, Zhiheng Huang, A. Karpathy, A. Khosla, and M. Bernstein, “ImageNet large scale visual recognition chal- lenge,” Int. J. Comput. Vis. 115, 211–252 (2015)

    work page 2015

  64. [64]

    MNIST hand- written digit database,

    Y. LeCun, C. Cortes, and C. J. Burges, “MNIST hand- written digit database,” AT&T Labs [Online]. Available: http://yann. lecun. com/exdb/mnist 2 (2010)

    work page 2010

  65. [65]

    Imaging through glass diffusers using densely con- nected convolutional networks,

    Shuai Li, Mo Deng, Justin Lee, A. Sinha, and G. Barbas- tathis, “Imaging through glass diffusers using densely con- nected convolutional networks,” Optica 5, 803–813 (2018)

    work page 2018

  66. [66]

    Emergence of simple- cell receptive field properties by learning a sparse code for natural images,

    B. A. Olshausen and D. J. Field, “Emergence of simple- cell receptive field properties by learning a sparse code for natural images,” Nature 381, 607–609 (1996)

    work page 1996

  67. [67]

    Natural image statistics and efficient coding,

    B. A. Olshausen and D. J. Field, “Natural image statistics and efficient coding,” Netw. Comp. Neural Syst. 7, 333–339 Research Article Vol. X, No. X / April 2016 / Optica 14 (1996)

    work page 2016

  68. [68]

    Modelling the power spectra of natural images: statistics and informa- tion,

    A. Van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and informa- tion,” Vis. Res. 36, 2759–2770 (1996)

    work page 1996

  69. [69]

    Sparse coding with an overcomplete basis set: a strategy employed by V1?

    B. A. Olshausen and D. J. Field, “Sparse coding with an overcomplete basis set: a strategy employed by V1?” Vis. Res. 37, 3311–3325 (1997)

    work page 1997

  70. [70]

    A probabilistic frame- work for the adaptation and comparison of image codes,

    M. S. Lewicki and B. A. Olshausen, “A probabilistic frame- work for the adaptation and comparison of image codes,” J. Opt. Soc. A 16, 1587–1601 (1999)

    work page 1999

  71. [71]

    Learning overcomplete representations,

    M. S. Lewicki and T. J. Sejnowski, “Learning overcomplete representations,” Neural Comput. 12, 337–365 (2000)

    work page 2000

  72. [72]

    Learning to synthesize: splitting and recombining low and high spatial frequencies for image recovery

    M. Deng, S. Li, and G. Barbastathis, “Learning to syn- thesize: splitting and recombining low and high spa- tial frequencies for image recovery,” arXiv preprint arXiv:1811.07945 (2018)

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  73. [73]

    U-Net: Convolutional Networks for Biomedical Image Segmentation

    O. Ronneberger, P .Fischer, and T. Brox, “U-net: Convolu- tional networks for biomedical image segmentation,” in Medical Image Computing and Computer-Assisted Intervention (MICCAI), , vol. 9351 of LNCS (Springer, 2015), pp. 234–241. (available on arXiv:1505.04597 [cs.CV])

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  74. [74]

    Large-scale machine learning with stochas- tic gradient descent,

    L. Bottou, “Large-scale machine learning with stochas- tic gradient descent,” in Proceedings of COMPSTAT’2010, (Springer, 2010), pp. 177–186

    work page 2010

  75. [75]

    Adam: A Method for Stochastic Optimization

    D. P . Kingma and J. Ba, “Adam: A method for stochastic optimization,” CoRR abs/1412.6980 (2015)

    work page internal anchor Pith review Pith/arXiv arXiv 2015

  76. [76]

    Learning translation invariant recognition in massively parallel networks,

    G. E. Hinton, “Learning translation invariant recognition in massively parallel networks,” in Proceedings P ARLE Confer- ence on Parallel Architectures and Languages Europe,(Springer- Verlag, 1987), pp. 1–13

    work page 1987

  77. [77]

    C. M. Bishop, Pattern recognition and machine learning (Springer, 2006)

    work page 2006

  78. [78]

    Perceptual losses for real-time style transfer and super-resolution,

    J. Johnson, A. Alahi, and Li Fei-Fei, “Perceptual losses for real-time style transfer and super-resolution,” in European Conference on Computer Vision (ECCV) / Lecture Notes on Computer Science, , vol. 9906 B. Leide, J. Matas, N. Sebe, and M. Welling, eds. (2016), pp. 694–711

    work page 2016

  79. [79]

    Goodfellow, Y

    I. Goodfellow, Y. Bengio, and A. Courville,Deep Learning (MIT Press, 2016)

    work page 2016

  80. [80]

    Photo-realistic single image super-resolution using a Generative Adversarial Network,

    C. Ledig, L. Theis, F. Huczar, J. Caballero, A. Cunningham, A. Acosta, A. Aitken, A. Tejani, J. Totz, Zehan Wang, and Wenshe Shi, “Photo-realistic single image super-resolution using a Generative Adversarial Network,” in The IEEE Con- ference on Computer Vision and Pattern Recognition (CVPR), (2017), pp. 4681–4690

    work page 2017

Showing first 80 references.