Abstract

We present a tomographic imaging technique, termed Deep Prior Diffraction Tomography (DP-DT), to reconstruct the 3D refractive index (RI) of thick biological samples at high resolution from a sequence of low-resolution images collected under angularly varying illumination. DP-DT processes the multi-angle data using a phase retrieval algorithm that is extended by a deep image prior (DIP), which reparameterizes the 3D sample reconstruction with an untrained, deep generative 3D convolutional neural network (CNN). We show that DP-DT effectively addresses the missing cone problem, which otherwise degrades the resolution and quality of standard 3D reconstruction algorithms. As DP-DT does not require pre-captured data or pre-training, it is not biased towards any particular dataset. Hence, it is a general technique that can be applied to a wide variety of 3D samples, including scenarios in which large datasets for supervised training would be infeasible or expensive. We applied DP-DT to obtain 3D RI maps of bead phantoms and complex biological specimens, both in simulation and experiment, and show that DP-DT produces higher-quality results than standard regularization techniques. We further demonstrate the generality of DP-DT, using two different scattering models, the first Born and multi-slice models. Our results point to the potential benefits of DP-DT for other 3D imaging modalities, including X-ray computed tomography, magnetic resonance imaging, and electron microscopy.

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

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2019 (11)

J. Mertz, “Strategies for volumetric imaging with a fluorescence microscope,” Optica 6(10), 1261–1268 (2019).
[Crossref]

S. Chowdhury, M. Chen, R. Eckert, D. Ren, F. Wu, N. Repina, and L. Waller, “High-resolution 3d refractive index microscopy of multiple-scattering samples from intensity images,” Optica 6(9), 1211–1219 (2019).
[Crossref]

T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. C. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
[Crossref]

G. Ding, Y. Liu, R. Zhang, and H. L. Xin, “A joint deep learning model to recover information and reduce artifacts in missing-wedge sinograms for electron tomography and beyond,” Sci. Rep. 9(1), 12803 (2019).
[Crossref]

A. Dave, A. K. Vadathya, R. Subramanyam, R. Baburajan, and K. Mitra, “Solving inverse computational imaging problems using deep pixel-level prior,” IEEE Trans. Comput. Imaging 5(1), 37–51 (2019).
[Crossref]

Y. Jo, H. Cho, S. Y. Lee, G. Choi, G. Kim, H.-S. Min, and Y. Park, “Quantitative phase imaging and artificial intelligence: a review,” IEEE J. Sel. Top. Quantum Electron. 25(1), 1–14 (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]

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]

K. Gong, C. Catana, J. Qi, and Q. Li, “Pet image reconstruction using deep image prior,” IEEE Trans. Med. Imaging 38(7), 1655–1665 (2019).
[Crossref]

Ç. Isil, F. S. Oktem, and A. Koç, “Deep iterative reconstruction for phase retrieval,” Appl. Opt. 58(20), 5422–5431 (2019).
[Crossref]

K. C. Zhou, R. Qian, S. Degan, S. Farsiu, and J. A. Izatt, “Optical coherence refraction tomography,” Nat. Photonics 13(11), 794–802 (2019).
[Crossref]

2018 (9)

T. Nguyen, Y. Xue, Y. Li, L. Tian, and G. Nehmetallah, “Deep learning approach for Fourier ptychography microscopy,” Opt. Express 26(20), 26470–26484 (2018).
[Crossref]

H.-Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “Seagle: Sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

T. C. Nguyen, V. Bui, and G. Nehmetallah, “Computational optical tomography using 3-d deep convolutional neural networks,” Opt. Eng. 57(4), 043111 (2018).
[Crossref]

A. Lucas, M. Iliadis, R. Molina, and A. K. Katsaggelos, “Using deep neural networks for inverse problems in imaging: beyond analytical methods,” IEEE Signal Process. Mag. 35(1), 20–36 (2018).
[Crossref]

J. Lim, A. Goy, M. H. Shoreh, M. Unser, and D. Psaltis, “Learning tomography assessed using mie theory,” Phys. Rev. Appl. 9(3), 034027 (2018).
[Crossref]

K. He, X. Huang, X. Wang, S. Yoo, P. Ruiz, I. Gdor, N. J. Ferrier, N. Scherer, M. Hereld, A. K. Katsaggelos, and O. Cossairt, “Design and simulation of a snapshot multi-focal interferometric microscope,” Opt. Express 26(21), 27381–27402 (2018).
[Crossref]

R. Ling, W. Tahir, H.-Y. Lin, H. Lee, and L. Tian, “High-throughput intensity diffraction tomography with a computational microscope,” Biomed. Opt. Express 9(5), 2130–2141 (2018).
[Crossref]

T.-A. Pham, E. Soubies, A. Goy, J. Lim, F. Soulez, D. Psaltis, and M. Unser, “Versatile reconstruction framework for diffraction tomography with intensity measurements and multiple scattering,” Opt. Express 26(3), 2749–2763 (2018).
[Crossref]

Y. Park, C. Depeursinge, and G. Popescu, “Quantitative phase imaging in biomedicine,” Nat. Photonics 12(10), 578–589 (2018).
[Crossref]

2017 (3)

S. Chowdhury, W. J. Eldridge, A. Wax, and J. Izatt, “Refractive index tomography with structured illumination,” Optica 4(5), 537–545 (2017).
[Crossref]

M. T. McCann, K. H. Jin, and M. Unser, “Convolutional neural networks for inverse problems in imaging: A review,” IEEE Signal Process. Mag. 34(6), 85–95 (2017).
[Crossref]

K. H. Jin, M. T. McCann, E. Froustey, and M. Unser, “Deep convolutional neural network for inverse problems in imaging,” IEEE Trans. on Image Process. 26(9), 4509–4522 (2017).
[Crossref]

2016 (5)

2015 (4)

2014 (1)

G. Zheng, X. Ou, R. Horstmeyer, J. Chung, and C. Yang, “Fourier ptychographic microscopy: A gigapixel superscope for biomedicine,” Opt. Photonics News 25(4), 26–33 (2014).
[Crossref]

2013 (1)

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

2012 (3)

Y. Sung, W. Choi, N. Lue, R. R. Dasari, and Z. Yaqoob, “Stain-free quantification of chromosomes in live cells using regularized tomographic phase microscopy,” PLoS One 7(11), e49502 (2012).
[Crossref]

B. Goris, W. Van den Broek, K. J. Batenburg, H. H. Mezerji, and S. Bals, “Electron tomography based on a total variation minimization reconstruction technique,” Ultramicroscopy 113, 120–130 (2012).
[Crossref]

A. Lucchi, K. Smith, R. Achanta, G. Knott, and P. Fua, “Supervoxel-Based Segmentation of Mitochondria in EM Image Stacks With Learned Shape Features,” IEEE Trans. Med. Imaging 31(2), 474–486 (2012).
[Crossref]

2011 (1)

2010 (1)

O. Haeberle, K. Belkebir, H. Giovaninni, and A. Sentenac, “Tomographic diffractive microscopy: basics, techniques and perspectives,” J. Mod. Opt. 57(9), 686–699 (2010).
[Crossref]

2009 (2)

2008 (1)

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy 108(5), 481–487 (2008).
[Crossref]

2007 (1)

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. R. Dasari, and M. S. Feld, “Tomographic phase microscopy,” Nat. Methods 4(9), 717–719 (2007).
[Crossref]

2004 (1)

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)

V. Lauer, “New approach to optical diffraction tomography yielding a vector equation of diffraction tomography and a novel tomographic microscope,” J. Microsc. 205(2), 165–176 (2002).
[Crossref]

1998 (1)

A. H. Delaney and Y. Bresler, “Globally convergent edge-preserving regularized reconstruction: an application to limited-angle tomography,” IEEE Trans. on Image Process. 7(2), 204–221 (1998).
[Crossref]

1985 (1)

B. A. Roberts and A. C. Kak, “Reflection mode diffraction tomography,” Ultrason. Imaging 7(4), 300–320 (1985).
[Crossref]

1981 (1)

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1(4), 153–156 (1969).
[Crossref]

Abadi, M.

M. Abadi, A. Agarwal, P. Barham, E. Brevdo, Z. Chen, C. Citro, G. S. Corrado, A. Davis, J. Dean, M. Devin, S. Ghemawat, I. Goodfellow, A. Harp, G. Irving, M. Isard, Y. Jia, R. Jozefowicz, L. Kaiser, M. Kudlur, J. Levenberg, D. Mané, R. Monga, S. Moore, D. Murray, C. Olah, M. Schuster, J. Shlens, B. Steiner, I. Sutskever, K. Talwar, P. Tucker, V. Vanhoucke, V. Vasudevan, F. Viégas, O. Vinyals, P. Warden, M. Wattenberg, M. Wicke, Y. Yu, and X. Zheng, “Tensorflow: Large-scale machine learning on heterogeneous distributed systems,” arXiv preprint arXiv:1603.04467 (2016).

Abbas, F.

F. Shamshad, F. Abbas, and A. Ahmed, “Deep ptych: Subsampled fourier ptychography using generative priors,” in ICASSP 2019-2019 IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP), (IEEE, 2019), pp. 7720–7724.

Achanta, R.

A. Lucchi, K. Smith, R. Achanta, G. Knott, and P. Fua, “Supervoxel-Based Segmentation of Mitochondria in EM Image Stacks With Learned Shape Features,” IEEE Trans. Med. Imaging 31(2), 474–486 (2012).
[Crossref]

Aditya Mohan, K.

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Figures (14)

Fig. 1.
Fig. 1. Summary of Deep Prior Diffraction Tomography (DP-DT). Microscope captures variably-illuminated image $k$ -stack. Standard tomographic reconstruction methods suffer from artifacts caused by a missing cone in $k$ -space. In this work, we propose the use of a deep image prior to help account for missing cone artifacts to improve 3D image reconstruction.
Fig. 2.
Fig. 2. The missing cone problem and its effect on the 3D SBP. (a) Simulations of the effects of transfer functions containing missing cones for a variety of illumination (vertical) and collection (horizontal) NAs. The odd columns are the $k$ -space supports of the transfer functions and the even columns are the spatial domain representations of 0.8-µm-diameter bead ( $n$ =1.35) immersed in water ( $n$ =1.33), which have been filtered by these transfer functions. (b) 3D SBPs in gigavoxels; 2D fields of view, collection NAs, and magnifications taken from [44]; arbitrarily assumes a 20-µm axial range; please note the semilog scale.
Fig. 3.
Fig. 3. The physical forward model and reconstruction algorithm for DP-DT, under the first Born or Rytov approximation.
Fig. 4.
Fig. 4. Comparison of 3D reconstruction quality for phaseless DT using several regularizers. (a) 1D traces through simulations of two beads spaced axially. Rows show different bead axial separations, while columns show different bead sizes (imaging NA = 0.4, illumination NA = 0.5). Each curve corresponds to a different regularization technique (i.e., none, DIP, positivity (+), and TV) and ground truth. Scale bar corresponds to the Nyquist period. (b) The RI RMSEs from the ground truths for each regularizer. Each of the four plots corresponds to a different bead size, and each curve corresponds to a different edge-to-edge bead separation, where $z_0=$ 0.75 µm.
Fig. 5.
Fig. 5. 2D cross-sections of select simulated bead pairs. (a) The first row is the through-origin $k_xk_y$ cross-sections of the scattering potential spectra of the reconstructions containing all the bead pairs, under illumination NA = 0.4, imaging NA = 0.5. The second row shows through-center $xz$ cross sections at two different separations. (b) The same information as (a), but under imaging NA = 0.3. (c) The same information as (a), but under imaging NA = 0.1.
Fig. 6.
Fig. 6. Simulated biological sample, reconstructed under various regularizers. The first row shows an $xy$ cross-section of the reconstruction volume, the second row shows and $xz$ cross-section, and the third row shows the through-origin $k_xk_y$ cross-section of the scattering potential spectrum. The fourth row shows 2D histograms, where the vertical axis is the RI error from the ground truth and the horizontal axis is the ground truth RI. Scale bars, 15 µm.
Fig. 7.
Fig. 7. Experimental 2-layer, 2-µm bead results, using the first-Born model. (a) Example raw LED images. (b) Comparison of performance of different regularizers (columns). The first two rows are $xy$ slices through the two bead layers, and the last two rows are two axial slices whose positions are indicated by the red and magenta lines in the upper left plot. Please note the double-headed arrows, indicating axial elongation due to the missing cone when no regularization is used. Scale bars, 5 µm. (c) 1D traces through regions indicated in the first panel of the $yz$ slice row. The expected RI of the beads is 1.59.
Fig. 8.
Fig. 8. Experimental 800-nm bead sample. (a) Example raw LED images. (b) Comparison of different regularizers, with the positivity replaced with negativity regularization because the RI is below that of the medium. The first row is $xy$ cross-sections and the second row is $xz$ cross-sections indicated by the red line. The third row shows a close-up view of the first row, indicated by the purple box. Scale bars, 5 µm (top two rows of (b)), 1 µm (bottom row of (b)). (c) 1D axial traces through the vertical dotted lines in the second row of (b).
Fig. 9.
Fig. 9. Experimental stacked starfish embryo results. (a) Example raw LED images. (b) $xy$ cross-sections at various depths. (c) $xz$ cross-sections at positions indicated by the horizontal red lines in the upper left panel of (b). Scale bars, 10 µm.
Fig. 10.
Fig. 10. Experimental 2-layer, 2-µm bead results, using the multi-slice model. (a) $xy$ slices at the two bead layers. (b) Axial slices indicated by the lines in the upper left panel of (a). (c) 1D axial traces through the dotted lines in (b) (expected RI=1.59). Scale bars, 5 µm.
Fig. 11.
Fig. 11. The DIP architecture used for the DP-DT reconstructions in this paper.
Fig. 12.
Fig. 12. Comparison of the quality of 3D reconstruction from intensity measurements, using several regularizers for various bead sizes and separations (illumination NA = 0.4, imaging NA = 0.1). Figure layout is analogous to that of Fig. 4. (a) 1D traces through simulations of two beads spaced axially. Rows show different bead separations, while columns show different bead sizes. Each curve corresponds to a different regularization technique (i.e., none, DIP, positivity (+), and TV) as well as the ground truth. Scale bar corresponds to the Nyquist period. (b) The RI RMSEs from the ground truths for each regularizer. Each of the four plots corresponds to a different bead size, and each curve corresponds to a different edge-to-edge bead separation, where ${z_{0}}=$ 0.75 µm.
Fig. 13.
Fig. 13. Comparison of the quality of 3D reconstruction from intensity measurements, using several regularizers for various bead sizes and separations (illumination NA = 0.4, imaging NA = 0.3). See caption for Fig. 12.
Fig. 14.
Fig. 14. The $xy$ - (left) and $xz$ - (right) cross-sections of the 3D DP-DT reconstruction of the starfish sample (Fig. 9). The top row shows the mean reconstruction of 20 independent initializations and the second row shows the standard deviation. The cross-sections correspond to the second column of Fig. 9.

Equations (27)

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V ( r ) = k 2 4 π ( n ( r ) 2 n 0 2 ) ,
V ~ ( k ) = F 3 D ( V ( r ) ) ,
E ( S m o d e l , ϕ p u p i l , u 0 ) = 1 M i , j , p ( I p r e d m o d e l [ i , j , p ] I d a t a [ i , j , p ] ) 2 ,
L ( S m o d e l , ϕ p u p i l , u 0 ) = E ( S m o d e l , ϕ p u p i l , u 0 ) + R e g ( S m o d e l ) .
arg min S m o d e l , ϕ p u p i l , u 0 L ( S m o d e l , ϕ p u p i l , u 0 ) .
arg min θ , ϕ p u p i l , u 0 L B o r n ( V ~ = F 3 D ( G ( θ ) ) , ϕ p u p i l , u 0 ) ,
arg min θ , ϕ p u p i l , u 0 L M S ( δ o b j = G ( θ ) , ϕ p u p i l , u 0 ) ,
R T V ( S ( r ) ) = r | x S ( r ) | 2 + | y S ( r ) | 2 + | z S ( r ) | 2 ,
R + ( n ( r ) ) = r min ( Re { n ( r ) } n 0 , 0 ) 2 .
L = E + λ T V R T V + λ + R + ,
k c a p [ i , j ] = ( k x c a p [ i , j ] , k y c a p [ i , j ] , k z c a p [ i , j ] ) ,
k i l l [ p ] = ( k x i l l [ p ] , k y i l l [ p ] , k z i l l [ p ] )
u ~ [ i , j , p ] = V ~ ( k c a p [ i , j ] k i l l [ p ] ) 4 π 1 i k z c a p [ i , j ]
u [ , , p ] = F 2 1 ( u ~ [ , , p ] A [ , ] e x p ( 1 i ϕ p u p i l [ , ] ) )
u b a c k [ , , p ] = u 0 [ p ] e x p ( 1 i ( k x i l l [ p ] x [ , ] + k y i l l [ p ] y [ , ] ) )
I p r e d B o r n [ , , p ] = | u b a c k [ , , p ] + u [ , , p ] | 2 .
I p r e d R y t o v [ , , p ] = | u b a c k [ , , p ] e x p ( u [ , , p ] / u b a c k [ , , p ] ) | 2 .
θ o b j [ , , r ] = k δ n o b j [ , , r ] δ z
u k p [ , , 0 ] = u b a c k [ , , p ]
u k p [ , , r + 1 ] = F 2 1 ( F 2 ( u k p [ , , r ] ) D ( δ z ) [ , ] ) e x p ( 1 i θ o b j [ , , r ] )
D ( z ) [ , ] = e x p ( 1 i   ( k x 2 + k y 2 ) z k n 0 + k 2 n 0 2 k x 2 k y 2 )
u k p d e t [ , ] = F 2 1 ( F 2 ( u k p [ , , Δ z / δ z 1 ] ) × D ( Δ z f Δ z / 2 ) [ , ] A [ , ] e x p ( 1 i ϕ p u p i l [ , ] ) )
u k p a p o d [ , , 0 ] = u b a c k [ , , p ] e x p ( ( x x p ) 2 + ( y y p ) 2 2 σ 2 )
x p = ( Δ z f + Δ z / 2 ) k x i l l [ p ] k z i l l [ p ]
y p = ( Δ z f + Δ z / 2 ) k y i l l [ p ] k z i l l [ p ]
I p r e d M S [ , , p ] = | u k p d e t [ , ] | 2 .
I d a t a [ , , p ] I d a t a [ , , p ] e x p ( ( x x p ) 2 + ( y y p ) 2 2 σ 2 ) .

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