Abstract

Optical diffraction tomography is an effective tool to estimate the refractive indices of unknown objects. It proceeds by solving an ill-posed inverse problem for which the wave equation governs the scattering events. The solution has traditionally been derived by the minimization of an objective function in which the data-fidelity term encourages measurement consistency while the regularization term enforces prior constraints. In this work, we propose to train a convolutional neural network (CNN) as the projector in a projected-gradient-descent method. We iteratively produce high-quality estimates and ensure measurement consistency, thus keeping the best of CNN-based and regularization-based worlds. Our experiments on two-dimensional-simulated and real data show an improvement over other conventional or deep-learning-based methods. Furthermore, our trained CNN projector is general enough to accommodate various forward models for the handling of multiple-scattering events.

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

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References

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2019 (3)

H. K. Aggarwal, M. P. Mani, and M. Jacob, “MODL: Model-based deep learning architecture for inverse problems,” IEEE Transactions on Med. Imaging 38(2), 394–405 (2019).
[Crossref]

L. Li, L. G. Wang, F. L. Teixeira, C. Liu, A. Nehorai, and T. J. Cui, “Deep NIS: Deep neural network for nonlinear electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. 67(3), 1819–1825 (2019).
[Crossref]

E. Soubies, F. Soulez, M. McCann, T.-a. Pham, L. Donati, T. Debarre, D. Sage, and M. Unser, “Pocket guide to solve inverse problems with Global Bio Im,” Inverse Probl. 35(10), 104006 (2019).
[Crossref]

2018 (4)

H. Gupta, K. H. Jin, H. Q. Nguyen, M. T. McCann, and M. Unser, “CNN-based projected gradient descent for consistent CT image reconstruction,” IEEE Transactions on Med. Imaging 37(6), 1440–1453 (2018).
[Crossref]

H. Qiao, J. Wu, X. Li, M. H. Shoreh, J. Fan, and Q. Dai, “GPU-based deep convolutional neural network for tomographic phase microscopy with l1 fitting and regularization,” J. Biomed. Opt. 23(06), 1 (2018).
[Crossref]

Y. Sun, Z. Xia, and U. S. Kamilov, “Efficient and accurate inversion of multiple scattering with deep learning,” Opt. Express 26(11), 14678–14688 (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 Transactions on Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

2017 (5)

E. Soubies, T.-a. Pham, and M. Unser, “Efficient inversion of multiple-scattering model for optical diffraction tomography,” Opt. Express 25(18), 21786–21800 (2017).
[Crossref]

U. S. Kamilov, H. Mansour, and B. Wohlberg, “A plug-and-play priors approach for solving nonlinear imaging inverse problems,” IEEE Signal Process. Lett. 24(12), 1872–1876 (2017).
[Crossref]

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

J. Adler and O. Öktem, “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Probl. 33(12), 124007 (2017).
[Crossref]

2016 (4)

C. Dong, C. C. Loy, K. He, and X. Tang, “Image super-resolution using deep convolutional networks,” IEEE Transactions on Pattern Analysis Mach. Intell. 38(2), 295–307 (2016).
[Crossref]

P. Liu, L. Chin, W. Ser, H. Chen, C.-M. Hsieh, C.-H. Lee, K.-B. Sung, T. Ayi, P. Yap, B. Liedberg, and K. Wang, “Cell refractive index for cell biology and disease diagnosis: Past, present and future,” Lab Chip 16(4), 634–644 (2016).
[Crossref]

U. S. Kamilov, I. N. Papadopoulos, M. H. Shoreh, A. Goy, C. Vonesch, M. Unser, and D. Psaltis, “Optical tomographic image reconstruction based on beam propagation and sparse regularization,” IEEE Transactions on Comput. Imaging 2(1), 59–70 (2016).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23(8), 1052–1056 (2016).
[Crossref]

2015 (4)

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(6), 517–522 (2015).
[Crossref]

M. Schürmann, J. Scholze, P. Müller, C. Chan, A. Ekpenyong, K. Chalut, and J. Guck, “Refractive index measurements of single spherical cells using digital holographic microscopy,” Methods Cell Biol. 125, 143–159 (2015).
[Crossref]

J. Lim, K. Lee, K. H. Jin, S. Shin, S. Lee, Y. Park, and J. C. Ye, “Comparative study of iterative reconstruction algorithms for missing cone problems in optical diffraction tomography,” Opt. Express 23(13), 16933–16948 (2015).
[Crossref]

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref]

2011 (1)

2009 (2)

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25(2), 024003 (2009).
[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]

2005 (2)

2002 (1)

A. Abubakar and P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18(2), 495–510 (2002).
[Crossref]

2001 (1)

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, “A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints,” Signal Process. 81(5), 945–974 (2001).
[Crossref]

1995 (1)

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]

1981 (1)

1970 (1)

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1(7), 323–328 (1970).
[Crossref]

1969 (1)

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

Abubakar, A.

A. Abubakar and P. M. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18(2), 495–510 (2002).
[Crossref]

Adler, A.

A. Adler, D. Boublil, and M. Zibulevsky, “Block-based compressed sensing of images via deep learning,” in IEEE 19th International Workshop on Multimedia Signal Process. (MMSP) (IEEE, 2017), pp. 1–6.

Adler, J.

J. Adler and O. Öktem, “Solving ill-posed inverse problems using iterative deep neural networks,” Inverse Probl. 33(12), 124007 (2017).
[Crossref]

Aggarwal, H. K.

H. K. Aggarwal, M. P. Mani, and M. Jacob, “MODL: Model-based deep learning architecture for inverse problems,” IEEE Transactions on Med. Imaging 38(2), 394–405 (2019).
[Crossref]

Aime, C.

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, “A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints,” Signal Process. 81(5), 945–974 (2001).
[Crossref]

Ayi, T.

P. Liu, L. Chin, W. Ser, H. Chen, C.-M. Hsieh, C.-H. Lee, K.-B. Sung, T. Ayi, P. Yap, B. Liedberg, and K. Wang, “Cell refractive index for cell biology and disease diagnosis: Past, present and future,” Lab Chip 16(4), 634–644 (2016).
[Crossref]

Ba, J.

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

Badizadegan, K.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

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]

Belkebir, K.

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25(2), 024003 (2009).
[Crossref]

Bengio, Y.

Y. LeCun, Y. Bengio, and G. Hinton, “Deep learning,” Nature 521(7553), 436–444 (2015).
[Crossref]

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

Bertsekas, D. P.

D. P. Bertsekas, Convex optimization algorithms (Athena Scientific, Belmont, Massachusetts, 2015).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th (expanded) edition, vol. 461 (United Kingdom: Press Syndicate of the University of Cambridge, 1999).

Boublil, D.

A. Adler, D. Boublil, and M. Zibulevsky, “Block-based compressed sensing of images via deep learning,” in IEEE 19th International Workshop on Multimedia Signal Process. (MMSP) (IEEE, 2017), pp. 1–6.

Boufounos, P. T.

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 Transactions on Comput. Imaging 4(1), 73–86 (2018).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23(8), 1052–1056 (2016).
[Crossref]

Bouman, C. A.

S. V. Venkatakrishnan, C. A. Bouman, and B. Wohlberg, “Plug-and-play priors for model based reconstruction,” in IEEE Global Conference on Signal and Information Processing, (IEEE, 2013), pp. 945–948.

Brox, T.

O. Ronneberger, P. Fischer, and T. Brox, “U-Net: Convolutional networks for biomedical image segmentation,” in International Conference on Medical Image Computing and Computer-Assisted Intervention, (Springer, 2015), pp. 234–241.

Bui, V.

T. Nguyen, V. Bui, and G. Nehmetallah, “3D optical diffraction tomography using deep learning,” in Digital Holography and Three-Dimensional Imaging (Optical Society of America, 2018), pp. DW2F–4.

Caballero, J.

J. Schlemper, J. Caballero, J. V. Hajnal, A. Price, and D. Rueckert, “A deep cascade of convolutional neural networks for MR image reconstruction,” in International Conference on Information Processing in Medical Imaging (Springer, 2017), pp. 647–658.

Chalut, K.

M. Schürmann, J. Scholze, P. Müller, C. Chan, A. Ekpenyong, K. Chalut, and J. Guck, “Refractive index measurements of single spherical cells using digital holographic microscopy,” Methods Cell Biol. 125, 143–159 (2015).
[Crossref]

Chan, C.

M. Schürmann, J. Scholze, P. Müller, C. Chan, A. Ekpenyong, K. Chalut, and J. Guck, “Refractive index measurements of single spherical cells using digital holographic microscopy,” Methods Cell Biol. 125, 143–159 (2015).
[Crossref]

Chaumet, P. C.

P. C. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25(2), 024003 (2009).
[Crossref]

Chen, H.

P. Liu, L. Chin, W. Ser, H. Chen, C.-M. Hsieh, C.-H. Lee, K.-B. Sung, T. Ayi, P. Yap, B. Liedberg, and K. Wang, “Cell refractive index for cell biology and disease diagnosis: Past, present and future,” Lab Chip 16(4), 634–644 (2016).
[Crossref]

Chin, L.

P. Liu, L. Chin, W. Ser, H. Chen, C.-M. Hsieh, C.-H. Lee, K.-B. Sung, T. Ayi, P. Yap, B. Liedberg, and K. Wang, “Cell refractive index for cell biology and disease diagnosis: Past, present and future,” Lab Chip 16(4), 634–644 (2016).
[Crossref]

Choi, W.

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17(1), 266–277 (2009).
[Crossref]

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]

Combettes, P. L.

P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Model. Simul. 4(4), 1168–1200 (2005).
[Crossref]

Courville, A.

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

Cuche, E.

Cuevas, O.

H. Lantéri, M. Roche, O. Cuevas, and C. Aime, “A general method to devise maximum-likelihood signal restoration multiplicative algorithms with non-negativity constraints,” Signal Process. 81(5), 945–974 (2001).
[Crossref]

Cui, T. J.

L. Li, L. G. Wang, F. L. Teixeira, C. Liu, A. Nehorai, and T. J. Cui, “Deep NIS: Deep neural network for nonlinear electromagnetic inverse scattering,” IEEE Trans. Antennas Propag. 67(3), 1819–1825 (2019).
[Crossref]

Dai, Q.

H. Qiao, J. Wu, X. Li, M. H. Shoreh, J. Fan, and Q. Dai, “GPU-based deep convolutional neural network for tomographic phase microscopy with l1 fitting and regularization,” J. Biomed. Opt. 23(06), 1 (2018).
[Crossref]

Dändliker, R.

R. Dändliker and K. Weiss, “Reconstruction of the three-dimensional refractive index from scattered waves,” Opt. Commun. 1(7), 323–328 (1970).
[Crossref]

Dasari, R. R.

Debarre, T.

E. Soubies, F. Soulez, M. McCann, T.-a. Pham, L. Donati, T. Debarre, D. Sage, and M. Unser, “Pocket guide to solve inverse problems with Global Bio Im,” Inverse Probl. 35(10), 104006 (2019).
[Crossref]

Depeursinge, C.

Devaney, A.

Donati, L.

E. Soubies, F. Soulez, M. McCann, T.-a. Pham, L. Donati, T. Debarre, D. Sage, and M. Unser, “Pocket guide to solve inverse problems with Global Bio Im,” Inverse Probl. 35(10), 104006 (2019).
[Crossref]

Dong, C.

C. Dong, C. C. Loy, K. He, and X. Tang, “Image super-resolution using deep convolutional networks,” IEEE Transactions on Pattern Analysis Mach. Intell. 38(2), 295–307 (2016).
[Crossref]

Ekpenyong, A.

M. Schürmann, J. Scholze, P. Müller, C. Chan, A. Ekpenyong, K. Chalut, and J. Guck, “Refractive index measurements of single spherical cells using digital holographic microscopy,” Methods Cell Biol. 125, 143–159 (2015).
[Crossref]

Emery, Y.

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H. Qiao, J. Wu, X. Li, M. H. Shoreh, J. Fan, and Q. Dai, “GPU-based deep convolutional neural network for tomographic phase microscopy with l1 fitting and regularization,” J. Biomed. Opt. 23(06), 1 (2018).
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Figures (11)

Fig. 1.
Fig. 1. Optical diffraction tomography.
Fig. 2.
Fig. 2. One iteration of the RPGD for ODT. The forward model simulates the physical process from the current estimate $\mathbf {f}_k$. The vector $\mathbf {g}$ is obtained by gradient descent from the data-fidelity term. Then, $\mathbf {g}$ is fed into the (trained) CNN projector to obtain $\mathbf {z}_k$. The next estimate $\mathbf {f}_{k+1}$ is the sum of $\mathbf {z}_k$ and $\mathbf {f}_k$ weighted by $\alpha$ and $(1 - \alpha )$, respectively. Note that the initial guess $\mathbf {f}_0$, which is obtained with a direct inversion method (Rytov model), is directly fed into the (trained) CNN projector.
Fig. 3.
Fig. 3. Architecture of the network used for RI reconstruction. Each dark and light cube corresponds to a multichannel feature map. The number of channels is shown at the top of the cube. The x-z size is provided at the lower-left edge of the cube (example shown for 16$\times$16 in the coarsest resolution). The arrows denote the different operations, and the size of the corresponding parameter set is provided in the explanatory frame.
Fig. 4.
Fig. 4. Reconstructions of simulated data for one disk. Our method is RytovCNN. From top to bottom, the SNR of the measurement is 20dB, 15dB, and 10dB. The corresponding RI variation is $(-0.088)$, $(-0.060)$, and $(-0.060)$, respectively. The ground-truth images are presented in the last column.
Fig. 5.
Fig. 5. Reconstructions of simulated data for several shapes. The SNR of the measurement is 20dB. From top to bottom, the object is a square, a cell-like sample, and two non-overlapping disks with independent radius and RI variation. The corresponding RI variation is $(-0.046)$, $(-0.049)$, and $(-0.072)$, respectively. The ground-truth images are presented in the last column.
Fig. 6.
Fig. 6. Reconstructions of simulated data for two disks. Our method is RytovCNN. From top to bottom, the SNR of the measurement is 20dB, 15dB, and 10dB. The corresponding RI variation is $(-0.052)$, $(-0.040)$, and $(-0.061)$, respectively. The ground-truth images are presented in the last column.
Fig. 7.
Fig. 7. Sixteen examples of the images randomly generated to train the CNN projector to reconstruct complex objects.
Fig. 8.
Fig. 8. Reconstructions of complex simulated data. The SNR of the measurement is 20dB. The corresponding RI variation is $0.086$, $0.059$, and $0.053$, respectively. The ground-truth images are presented in the last column.
Fig. 9.
Fig. 9. Reconstructions of experimental data. From first to third row, the line that joins the center of the fibers is angled by $0^{\mathrm {o}}$, $45^{\mathrm {o}}$, and $90^{\mathrm {o}}$ with respect to the vertical dimension, respectively. The variation of the fibers is $(-0.055)$.
Fig. 10.
Fig. 10. Reconstructions with a nonlinear forward model applied to synthetic data. From top to bottom, the value of the RI variation is $0.088$, $0.171$, and $0.078$, respectively. The ground-truth images are presented in the last column. The training was done with simpler objects characterized by two embedded inclusions only.
Fig. 11.
Fig. 11. Reconstructions with a nonlinear forward model applied to experimental data. The value of the RI variation of the fibers is $(-0.055)$. The training was done with one disks.

Tables (6)

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Table 1. RI reconstructions.

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Table 2. ERROR of RI reconstructions in the presence of noise, for one disk.

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Table 3. SSIM of RI reconstructions in the presence of noise, for one disk.

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Table 4. ERROR of RI reconstructions in the presence of noise, for two disks.

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Table 5. SSIM of RI reconstructions in the presence of noise, for two disks.

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Table 6. ERROR and SSIM of RI reconstructions in the presence of 20 dB noise, for complex objects. RytovCNN outperforms the other methods (the BPM-based methods are not considered).

Equations (15)

Equations on this page are rendered with MathJax. Learn more.

2 u ( x ) + k 0 2 n 2 ( x ) u ( x ) = 0 ,
u ( x ) = u i n ( x ) + Ω g ( x z ) f ( z ) u ( z ) d z , x Ω ,
g ( x ) = j 4 H 0 ( 1 ) ( k 0 n b x ) , d = 2 ,
u B s c ( x ) = Ω g ( x z ) f ( z ) u i n ( z ) d z .
y p R y t = S p F f , p = 1 , , P ,
δ n m = n b ( f m k 2 + 1 1 ) .
f ^ = arg min f S 1 2 P p = 1 P y p R y t H p R y t ( f ) 2 2 ,
f k + 1 = P S ( f k γ H T H f k + γ H T y ) ,
P S ( g ) = C N N ( P C ( g ) ) ,
f ~ q , 1 = B ( H L i p ( f q ) )
f ~ q , 2 = C N N θ t 1 ( f ~ q , 1 )
f ~ q , 3 = f q ,
L ( θ ) = m = 1 M q = 1 Q f q C N N ( f ~ q , m ) 2 2 .
E R R O R ( n , n ^ ) = n n ^ 2 n 2
S S I M ( n , n ^ ) = ( 2 μ n μ n ^ + c 1 ) ( 2 σ n σ n ^ + c 2 ) ( μ n 2 + μ n ^ 2 + c 1 ) ( σ n 2 + σ n ^ 2 + c 2 ) ,

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