Abstract

Image reconstruction under multiple light scattering is crucial in a number of applications such as diffraction tomography. The reconstruction problem is often formulated as a nonconvex optimization, where a nonlinear measurement model is used to account for multiple scattering and regularization is used to enforce prior constraints on the object. In this paper, we propose a powerful alternative to this optimization-based view of image reconstruction by designing and training a deep convolutional neural network that can invert multiple scattered measurements to produce a high-quality image of the refractive index. Our results on both simulated and experimental datasets show that the proposed approach is substantially faster and achieves higher imaging quality compared to the state-of-the-art methods based on optimization.

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

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2018 (3)

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, 73–86 (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, 2749–2763 (2018).
[Crossref] [PubMed]

J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general deep learning framework for inverse problems,” SIAM J. Imaging Sci. 11, 991–1048 (2018).
[Crossref]

2017 (5)

U. S. Kamilov, H. Mansour, and B. Wohlberg, “A plug-and-play priors approach for solving nonlinear imaging inverse problems,” IEEE Signal. Proc. Let. 24, 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 Trans. Image Process. 26, 4509–4522 (2017).
[Crossref]

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

S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comp. Imag. 3, 84–98 (2017).
[Crossref]

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

2016 (5)

T. Zhang, C. Godavarthi, P. C. Chaumet, G. Maire, H. Giovannini, A. Talneau, M. Allain, K. Belkebir, and A. Sentenac, “Far-field diffraction microscopy at λ/10 resolution,” Optica 3, 609–612 (2016).
[Crossref]

U. S. Kamilov and H. Mansour, “Learning optimal nonlinearities for iterative thresholding algorithms,” IEEE Signal Process. Lett. 23, 747–751 (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 Trans. Comp. Imag. 2, 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, 1052–1056 (2016).
[Crossref]

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

2015 (3)

2012 (1)

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inv. Probl. 28, 065007 (2012).
[Crossref]

2011 (1)

2010 (1)

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref] [PubMed]

2009 (3)

2008 (1)

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25, 84–99 (2008).
[Crossref]

2007 (2)

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

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref] [PubMed]

2006 (2)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (2)

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inv. Probl. 21, S117–S130 (2005).
[Crossref]

K. Belkebir, P. C. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

2003 (1)

1998 (1)

1992 (1)

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

1981 (1)

1969 (1)

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

Afonso, M. V.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref] [PubMed]

Allain, M.

Ba, J.

D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in International Conference on Learning Representations (ICLR), (San Diego, 2015).

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, 266–277 (2009).
[Crossref] [PubMed]

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

Baraniuk, R. G.

A. Mousavi, A. B. Patel, and R. G. Baraniuk, “A deep learning approach to structured signal recovery,” in Proceedings of the Allerton Conference on Communication, Control, and Computing, (Allerton Park, IL, USA, 2015), pp. 1336–1343.

Barbastathis, G.

Beck, A.

A. Beck and M. Teboulle, “Fast gradient-based algorithm for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref] [PubMed]

Belkebir, K.

Bioucas-Dias, J. M.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref] [PubMed]

Borgerding, M.

M. Borgerding and P. Schniter, “Onsanger-corrected deep networks for sparse linear inverse problems,” in Proceedings of the IEEE Global Conference on Signal Processing and Information Processing (GlobalSIP), (Washington, DC, USA, 2016).

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2003), pp. 695–734.

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 Trans. Comput. Imaging 4, 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, 1052–1056 (2016).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in Proceedings of the IEEE Int. Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), (Calgary, Canada, 2018).

Bouman, C. A.

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

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

Brady, D. J.

Brox, T.

O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in “Medical Image Computing and Computer-Assisted Intervention (MICCAI),”, vol. 9351 of LNCS (Springer, 2015), pp. 234–241.

Buzzard, G. T.

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

Candès, E. J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Cha, E.

J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general deep learning framework for inverse problems,” SIAM J. Imaging Sci. 11, 991–1048 (2018).
[Crossref]

Chan, S. H.

S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comp. Imag. 3, 84–98 (2017).
[Crossref]

Chang, W.

E. Kang, W. Chang, J. Yoo, and J. C. Ye, “Deep convolutional framelet denosing for low-dose ct via wavelet residual network,” IEEE Trans. Med. Imaging, (in press) (2018).
[Crossref]

Chaumet, P. C.

Chen, B.

Chen, Y.

Y. Chen, W. Yu, and T. Pock, “On learning optimized reaction diffuction processes for effective image restoration,” in Proceedings of te IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Boston, MA, USA, 2015), pp. 5261–5269.

Choi, K.

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, 266–277 (2009).
[Crossref] [PubMed]

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

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref] [PubMed]

Dasari, R. R.

Devaney, A. J.

Dong, C.

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in Proceedings of ECCV, (Zurich, Switzerland, 2014), pp. 184–199.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Drummy, L. F.

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

Egiazarian, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref] [PubMed]

Elgendy, O. A.

S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comp. Imag. 3, 84–98 (2017).
[Crossref]

Eyraud, C.

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inv. Probl. 21, S117–S130 (2005).
[Crossref]

Fang-Yen, C.

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, 266–277 (2009).
[Crossref] [PubMed]

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

Fatemi, E.

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

Feld, M. S.

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, 266–277 (2009).
[Crossref] [PubMed]

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

Figueiredo, M. A. T.

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref] [PubMed]

Fischer, P.

O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in “Medical Image Computing and Computer-Assisted Intervention (MICCAI),”, vol. 9351 of LNCS (Springer, 2015), pp. 234–241.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref] [PubMed]

Froustey, E.

K. H. 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).
[Crossref]

Geffrin, J.-M.

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inv. Probl. 21, S117–S130 (2005).
[Crossref]

Giovannini, H.

Godavarthi, C.

Goy, A.

Han, Y.

J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general deep learning framework for inverse problems,” SIAM J. Imaging Sci. 11, 991–1048 (2018).
[Crossref]

He, K.

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in Proceedings of ECCV, (Zurich, Switzerland, 2014), pp. 184–199.

Horisaki, R.

Jin, K. H.

Kamilov, U. S.

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, 73–86 (2018).
[Crossref]

U. S. Kamilov, H. Mansour, and B. Wohlberg, “A plug-and-play priors approach for solving nonlinear imaging inverse problems,” IEEE Signal. Proc. Let. 24, 1872–1876 (2017).
[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, 1052–1056 (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 Trans. Comp. Imag. 2, 59–70, (2016).
[Crossref]

U. S. Kamilov and H. Mansour, “Learning optimal nonlinearities for iterative thresholding algorithms,” IEEE Signal Process. Lett. 23, 747–751 (2016).
[Crossref]

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).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in Proceedings of the IEEE Int. Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), (Calgary, Canada, 2018).

Kang, E.

E. Kang, W. Chang, J. Yoo, and J. C. Ye, “Deep convolutional framelet denosing for low-dose ct via wavelet residual network,” IEEE Trans. Med. Imaging, (in press) (2018).
[Crossref]

Katkovnik, V.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref] [PubMed]

Kingma, D.

D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in International Conference on Learning Representations (ICLR), (San Diego, 2015).

Lee, J.

Lee, K. R.

Lee, S. E.

Li, S.

Lim, J.

Lim, J. W.

Lim, S.

Liu, D.

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, 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, 1052–1056 (2016).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in Proceedings of the IEEE Int. Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), (Calgary, Canada, 2018).

Liu, H.-Y.

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, 73–86 (2018).
[Crossref]

Liu, Z.

Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings of the International Conference on Computer Vision (ICCV), (Santiago, 2015).

Loy, C. C.

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in Proceedings of ECCV, (Zurich, Switzerland, 2014), pp. 184–199.

Lue, N.

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

Luo, P.

Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings of the International Conference on Computer Vision (ICCV), (Santiago, 2015).

Ma, Y.

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in Proceedings of the IEEE Int. Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), (Calgary, Canada, 2018).

Maire, G.

Mansour, H.

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, 73–86 (2018).
[Crossref]

U. S. Kamilov, H. Mansour, and B. Wohlberg, “A plug-and-play priors approach for solving nonlinear imaging inverse problems,” IEEE Signal. Proc. Let. 24, 1872–1876 (2017).
[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, 1052–1056 (2016).
[Crossref]

U. S. Kamilov and H. Mansour, “Learning optimal nonlinearities for iterative thresholding algorithms,” IEEE Signal Process. Lett. 23, 747–751 (2016).
[Crossref]

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in Proceedings of the IEEE Int. Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), (Calgary, Canada, 2018).

Marks, D. L.

McCann, M. T.

K. H. 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).
[Crossref]

Mousavi, A.

A. Mousavi, A. B. Patel, and R. G. Baraniuk, “A deep learning approach to structured signal recovery,” in Proceedings of the Allerton Conference on Communication, Control, and Computing, (Allerton Park, IL, USA, 2015), pp. 1336–1343.

Mudry, E.

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inv. Probl. 28, 065007 (2012).
[Crossref]

Nesterov, Y.

Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course (Kluwer Academic Publishers, 2004).
[Crossref]

Oh, S.

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

Osher, S.

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

Papadopoulos, I. N.

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 Trans. Comp. Imag. 2, 59–70, (2016).
[Crossref]

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).
[Crossref]

Park, Y. K.

Patel, A. B.

A. Mousavi, A. B. Patel, and R. G. Baraniuk, “A deep learning approach to structured signal recovery,” in Proceedings of the Allerton Conference on Communication, Control, and Computing, (Allerton Park, IL, USA, 2015), pp. 1336–1343.

Pham, T.-A.

Pock, T.

Y. Chen, W. Yu, and T. Pock, “On learning optimized reaction diffuction processes for effective image restoration,” in Proceedings of te IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Boston, MA, USA, 2015), pp. 5261–5269.

Psaltis, D.

Ribés, A.

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25, 84–99 (2008).
[Crossref]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Ronneberger, O.

O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in “Medical Image Computing and Computer-Assisted Intervention (MICCAI),”, vol. 9351 of LNCS (Springer, 2015), pp. 234–241.

Roth, S.

U. Schmidt and S. Roth, “Shrinkage fields for effective image restoration,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Columbus, OH, USA, 2014), pp. 2774–2781.

Rudin, L. I.

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

Sabouroux, P.

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inv. Probl. 21, S117–S130 (2005).
[Crossref]

Schmidt, U.

U. Schmidt and S. Roth, “Shrinkage fields for effective image restoration,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Columbus, OH, USA, 2014), pp. 2774–2781.

Schmitt, F.

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25, 84–99 (2008).
[Crossref]

Schniter, P.

M. Borgerding and P. Schniter, “Onsanger-corrected deep networks for sparse linear inverse problems,” in Proceedings of the IEEE Global Conference on Signal Processing and Information Processing (GlobalSIP), (Washington, DC, USA, 2016).

Sentenac, A.

Shin, S.

Shoreh, M. H.

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 Trans. Comp. Imag. 2, 59–70, (2016).
[Crossref]

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).
[Crossref]

Simmons, J. P.

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

Sinha, A.

Soubies, E.

Soulez, F.

Sreehari, S.

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

Stamnes, J. J.

Sung, Y.

Talneau, A.

Tang, X.

Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings of the International Conference on Computer Vision (ICCV), (Santiago, 2015).

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in Proceedings of ECCV, (Zurich, Switzerland, 2014), pp. 184–199.

Tao, T.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Teboulle, M.

A. Beck and M. Teboulle, “Fast gradient-based algorithm for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref] [PubMed]

Tian, L.

Unser, M.

Venkatakrishnan, S. V.

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

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

Vonesch, C.

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 Trans. Comp. Imag. 2, 59–70, (2016).
[Crossref]

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).
[Crossref]

Waller, L.

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, 73–86 (2018).
[Crossref]

L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]

Wang, X.

S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comp. Imag. 3, 84–98 (2017).
[Crossref]

Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings of the International Conference on Computer Vision (ICCV), (Santiago, 2015).

Wohlberg, B.

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

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

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

Wolf, E.

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

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2003), pp. 695–734.

Ye, J. C.

J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general deep learning framework for inverse problems,” SIAM J. Imaging Sci. 11, 991–1048 (2018).
[Crossref]

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

E. Kang, W. Chang, J. Yoo, and J. C. Ye, “Deep convolutional framelet denosing for low-dose ct via wavelet residual network,” IEEE Trans. Med. Imaging, (in press) (2018).
[Crossref]

Yoo, J.

E. Kang, W. Chang, J. Yoo, and J. C. Ye, “Deep convolutional framelet denosing for low-dose ct via wavelet residual network,” IEEE Trans. Med. Imaging, (in press) (2018).
[Crossref]

Yu, W.

Y. Chen, W. Yu, and T. Pock, “On learning optimized reaction diffuction processes for effective image restoration,” in Proceedings of te IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Boston, MA, USA, 2015), pp. 5261–5269.

Zhang, T.

Appl. Opt. (1)

IEEE Signal Process. Lett. (2)

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

U. S. Kamilov and H. Mansour, “Learning optimal nonlinearities for iterative thresholding algorithms,” IEEE Signal Process. Lett. 23, 747–751 (2016).
[Crossref]

IEEE Signal Process. Mag. (1)

A. Ribés and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25, 84–99 (2008).
[Crossref]

IEEE Signal. Proc. Let. (1)

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

IEEE Trans. Comp. Imag. (3)

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 Trans. Comp. Imag. 2, 59–70, (2016).
[Crossref]

S. Sreehari, S. V. Venkatakrishnan, B. Wohlberg, G. T. Buzzard, L. F. Drummy, J. P. Simmons, and C. A. Bouman, “Plug-and-play priors for bright field electron tomography and sparse interpolation,” IEEE Trans. Comp. Imag. 2, 408–423 (2016).

S. H. Chan, X. Wang, and O. A. Elgendy, “Plug-and-play ADMM for image restoration: Fixed-point convergence and applications,” IEEE Trans. Comp. Imag. 3, 84–98 (2017).
[Crossref]

IEEE Trans. Comput. Imaging (1)

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, 73–86 (2018).
[Crossref]

IEEE Trans. Image Process. (4)

K. H. 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).
[Crossref]

A. Beck and M. Teboulle, “Fast gradient-based algorithm for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref] [PubMed]

M. V. Afonso, J. M. Bioucas-Dias, and M. A. T. Figueiredo, “Fast image recovery using variable splitting and constrained optimization,” IEEE Trans. Image Process. 19, 2345–2356 (2010).
[Crossref] [PubMed]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-D transform-domain collaborative filtering,” IEEE Trans. Image Process. 16, 2080–2095 (2007).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (2)

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Inv. Probl. (2)

J.-M. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inv. Probl. 21, S117–S130 (2005).
[Crossref]

E. Mudry, P. C. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inv. Probl. 28, 065007 (2012).
[Crossref]

J. Opt. Soc. Am. A (3)

Nat. Methods (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, 717–719 (2007).
[Crossref] [PubMed]

Opt. Commun. (1)

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

Opt. Express (5)

Opt. Lett. (1)

Optica (4)

Phys. D (1)

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

SIAM J. Imaging Sci. (1)

J. C. Ye, Y. Han, and E. Cha, “Deep convolutional framelets: A general deep learning framework for inverse problems,” SIAM J. Imaging Sci. 11, 991–1048 (2018).
[Crossref]

Other (15)

E. Kang, W. Chang, J. Yoo, and J. C. Ye, “Deep convolutional framelet denosing for low-dose ct via wavelet residual network,” IEEE Trans. Med. Imaging, (in press) (2018).
[Crossref]

J. Yoo, S. Sabir, D. Heo, K. H. Kim, A. Wahab, Y. Choi, S.-I. Lee, E. Y. Chae, H. H. Kim, Y. M. Bae, Y.-W. Choi, S. Cho, and J. C. Ye, “Deep learning can reverse photon migration for diffuse optical tomography,” https://arxiv.org/abs/171200912 .

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, 2003), pp. 695–734.

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in Proceedings of ECCV, (Zurich, Switzerland, 2014), pp. 184–199.

U. Schmidt and S. Roth, “Shrinkage fields for effective image restoration,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Columbus, OH, USA, 2014), pp. 2774–2781.

A. Mousavi, A. B. Patel, and R. G. Baraniuk, “A deep learning approach to structured signal recovery,” in Proceedings of the Allerton Conference on Communication, Control, and Computing, (Allerton Park, IL, USA, 2015), pp. 1336–1343.

Y. Chen, W. Yu, and T. Pock, “On learning optimized reaction diffuction processes for effective image restoration,” in Proceedings of te IEEE Conference on Computer Vision and Pattern Recognition (CVPR), (Boston, MA, USA, 2015), pp. 5261–5269.

M. Borgerding and P. Schniter, “Onsanger-corrected deep networks for sparse linear inverse problems,” in Proceedings of the IEEE Global Conference on Signal Processing and Information Processing (GlobalSIP), (Washington, DC, USA, 2016).

“Deep residual learning for compressed sensing CT reconstruction via persistent homology analysis,” https://arxiv.org/abs/161106391 .

Y. Ma, H. Mansour, D. Liu, P. T. Boufounos, and U. S. Kamilov, “Accelerated image reconstruction for nonlinear diffractive imaging,” in Proceedings of the IEEE Int. Conference on Acoustics, Speech and Signal Processing (ICASSP 2018), (Calgary, Canada, 2018).

Z. Liu, P. Luo, X. Wang, and X. Tang, “Deep learning face attributes in the wild,” in Proceedings of the International Conference on Computer Vision (ICCV), (Santiago, 2015).

D. Kingma and J. Ba, “Adam: A method for stochastic optimization,” in International Conference on Learning Representations (ICLR), (San Diego, 2015).

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

O. Ronneberger, P. Fischer, and T. Brox, “U-net: Convolutional networks for biomedical image segmentation,” in “Medical Image Computing and Computer-Assisted Intervention (MICCAI),”, vol. 9351 of LNCS (Springer, 2015), pp. 234–241.

Y. Nesterov, Introductory Lectures on Convex Optimization: A Basic Course (Kluwer Academic Publishers, 2004).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 We consider an object of a scattering potential f(r), illuminated with an input wave uin, which interacts with the object, and leads to the scattered field usc at the sensors.
Fig. 2
Fig. 2 The overview of the proposed approach that first backpropagates the data into a complex valued image and then maps this image the into final image with a ConvNet.
Fig. 3
Fig. 3 Visual illustration of the proposed learning architecture based on U-Net [45]. The input consists of two channels for the real and imaginary parts of the backpropagated vector w ∈ ℂN. The output is a single image of the scattering potential x ∈ ℝN.
Fig. 4
Fig. 4 Illustration of the convergence of the training on the dataset of piecewise-smooth objects. The left figure shows the training loss and the right figure shows the validation loss. The horizontal lines on the right show the losses of other algorithms on the same data.
Fig. 5
Fig. 5 Simulated Datasets: Visual comparison of the reconstructed images using the linear model with the first Born approximation regularized by imposing non-negativity [46] (FB-NN, column 2) and the total variation [17] (FB-TV, column 4), the non-linear method in [14] regularized by imposing non-negativity (LS-NN, column 3) and the total variation (LS-TV, column 5), the reconstruction by using BM3D as a plug-and-play prior (LS-BM3D, column 6). The values above images show the SNR (dB) of the reconstruction. The first column shows the true images. Each row corresponds to a different scattering scenario, which is denoted above the leading true image.
Fig. 6
Fig. 6 The illustration of the performance of ScaDec under strong scattering, trained at 20 dB noise SNR, but tested at noise levels in the SNR range of 15–25 dB.
Fig. 7
Fig. 7 Illustration of five randomly-generated images used for training ScaDec to reconstruct from experimental measurements: FoamDielExtTM and FoamDielIntTM.
Fig. 8
Fig. 8 Experimental Dataset: Reconstructed images obtained by ScaDec, LS-TV and FB-TV from the data of 2D experimental measurements. The first and second row relate to the setting of FoamDielExtTM and FoamDielIntTM, respectively. The first column shows the ground truth of each setting. The size of all reconstructed images are 128 × 128 pixels. Note that the colormap for FB-TV is different from the rest because the permittivity contrast was extremely underestimated by FB.

Tables (1)

Tables Icon

Table 1 SNR (dB) comparison of six methods on two datasets

Equations (11)

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

u ( r ) = u in ( r ) + Ω g ( r r ) f ( r ) u ( r ) d r , ( r Ω )
g ( r ) j 4 H 0 ( 1 ) ( k b r 2 )
u sc ( r ) = Ω g ( r r ) f ( r ) u ( r ) d r , ( r Γ )
u = u in + G ( u x )
y = S ( u x ) + e ,
H ( x ) S ( u ( x ) x ) where u ( x ) arg min u N { 1 2 u C ( u x ) u in 2 2 } ,
y = H ( x ) + e ,
x ^ = arg min x N { 1 2 y H ( x ) 2 2 + ( x ) } ,
z k = P k y k with P k diag ( u in , k * ) S H ,
w = k = 1 K z k = k = 1 K P k y k ,
SNR ( x , x ^ ) max a , b { 10 log 10 ( x 2 2 x a x ^ + b 2 2 ) } ,

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