Abstract

Deep learning has been proven to yield reliably generalizable solutions to numerous classification and decision tasks. Here, we demonstrate for the first time to our knowledge that deep neural networks (DNNs) can be trained to solve end-to-end inverse problems in computational imaging. We experimentally built and tested a lensless imaging system where a DNN was trained to recover phase objects given their propagated intensity diffraction patterns.

© 2017 Optical Society of America

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References

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2017 (2)

A. Dosovitskiy, J. T. Springenberg, M. Tatarchenko, and T. Brox, “Learning to generate chairs, tables and cars with convolutional networks,” IEEE Trans. Pattern Anal. Mach. Intell. 39, 692–705 (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]

2016 (2)

R. Horisaki, R. Takagi, and J. Tanida, “Learning-based imaging through scattering media,” Opt. Express 24, 13738–13743 (2016).
[Crossref]

D. Silver, A. Huang, C. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of go with deep neural networks and tree search,” Nature 529, 484–489 (2016).
[Crossref]

2015 (4)

V. Mnih, K. Kavukcuoglu, D. Silver, A. Rusu, J. Veness, M. Bellemare, A. Graves, M. Riedmiller, A. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” Nature 518, 529–533 (2015).
[Crossref]

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

O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, and M. Bernstein, “Imagenet large scale visual recognition challenge,” Int. J. Comput. Vis. 115, 211–252 (2015).
[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]

2012 (1)

H. Ji and K. Wang, “Robust image deblurring with an inaccurate blur kernel,” IEEE Trans. Image Process. 21, 1624–1634 (2012).
[Crossref]

2010 (1)

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

2008 (1)

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

2003 (1)

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[Crossref]

1988 (1)

1986 (1)

D. E. Rumelhart, G. E. Hinton, and R. J. Williams, “Learning representations by back-propagating errors,” Nature 5323, 533–536 (1986).
[Crossref]

1984 (1)

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

1983 (1)

1982 (1)

J. J. Hopfield, “Neural networks and physical systems with emergent collective computational abilities,” Proc. Natl. Acad. Sci. USA 79, 2554–2558 (1982).
[Crossref]

1978 (1)

1975 (1)

K. Fukushima, “Cognitron: a self-organizing multilayered neural network,” Biol. Cybernet. 20, 121–136 (1975).
[Crossref]

1972 (1)

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

1967 (1)

J. W. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[Crossref]

1943 (1)

A. N. Tikhonov, “On the stability of inverse problems,” Dokl. Akad. Nauk SSSR 39, 195–198 (1943).

Anguelov, D.

C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 1–9.

Antonoglou, I.

D. Silver, A. Huang, C. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of go with deep neural networks and tree search,” Nature 529, 484–489 (2016).
[Crossref]

V. Mnih, K. Kavukcuoglu, D. Silver, A. Rusu, J. Veness, M. Bellemare, A. Graves, M. Riedmiller, A. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” Nature 518, 529–533 (2015).
[Crossref]

Barbastathis, G.

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” arXiv:1702.08516 (2017).

Beattie, C.

V. Mnih, K. Kavukcuoglu, D. Silver, A. Rusu, J. Veness, M. Bellemare, A. Graves, M. Riedmiller, A. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” Nature 518, 529–533 (2015).
[Crossref]

Bellemare, M.

V. Mnih, K. Kavukcuoglu, D. Silver, A. Rusu, J. Veness, M. Bellemare, A. Graves, M. Riedmiller, A. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” Nature 518, 529–533 (2015).
[Crossref]

Bengio, Y.

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

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

Berg, T.

G. B. Huang, M. Ramesh, T. Berg, and E. Learned-Miller, “Labeled faces in the wild: a database for studying face recognition in unconstrained environments,” (University of Massachusetts, 2007).

Bernstein, M.

O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, and M. Bernstein, “Imagenet large scale visual recognition challenge,” Int. J. Comput. Vis. 115, 211–252 (2015).
[Crossref]

Bertero, M.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC Press, 1998).

Boccacci, P.

M. Bertero and P. Boccacci, Introduction to Inverse Problems in Imaging (CRC Press, 1998).

Boccara, A.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Boccara, A. C.

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” arXiv:1005.0532 (2010).

Bottou, L.

Y. A. LeCun, L. Bottou, G. B. Orr, and K.-R. Müller, “Efficient backprop,” in Neural Networks: Tricks of the Trade (Springer, 2012), pp. 9–48.

Brady, D. J.

D. J. Brady, Optical Imaging and Spectroscopy (Wiley, 2009).

Brox, T.

A. Dosovitskiy, J. T. Springenberg, M. Tatarchenko, and T. Brox, “Learning to generate chairs, tables and cars with convolutional networks,” IEEE Trans. Pattern Anal. Mach. Intell. 39, 692–705 (2017).
[Crossref]

Candès, E. J.

E. J. Candès and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[Crossref]

Cao, W.

J. Sun, W. Cao, Z. Xu, and J. Ponce, “Learning a convolutional neural network for non-uniform motion blur removal,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 769–777.

Carminati, R.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

Chen, E.

J. Xie, L. Xu, and E. Chen, “Image denoising and inpainting with deep neural networks,” in Advances in Neural Information Processing Systems (2012), pp. 341–349.

Cheng, Z.

Z. Cheng, Q. Yang, and B. Sheng, “Deep colorization,” in Proceedings of the IEEE International Conference on Computer Vision (2015), pp. 415–423.

Choi, M. C.

Y. Jo, S. Park, J. Jung, J. Yoon, H. Joo, M.-H. Kim, S.-J. Jo, M. C. Choi, S. Y. Lee, and Y. Park, “Holographic deep learning for rapid optical screening of anthrax spores,” BioRxiv (2017), 109108.

Courville, A.

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

Deng, J.

O. Russakovsky, J. Deng, H. Su, J. Krause, S. Satheesh, S. Ma, Z. Huang, A. Karpathy, A. Khosla, and M. Bernstein, “Imagenet large scale visual recognition challenge,” Int. J. Comput. Vis. 115, 211–252 (2015).
[Crossref]

Dieleman, S.

D. Silver, A. Huang, C. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of go with deep neural networks and tree search,” Nature 529, 484–489 (2016).
[Crossref]

Dong, C.

C. Dong, C. C. Loy, K. He, and X. Tang, “Learning a deep convolutional network for image super-resolution,” in European Conference on Computer Vision (Springer, 2014), pp. 184–199.

Donoho, D. L.

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[Crossref]

Dosovitskiy, A.

A. Dosovitskiy, J. T. Springenberg, M. Tatarchenko, and T. Brox, “Learning to generate chairs, tables and cars with convolutional networks,” IEEE Trans. Pattern Anal. Mach. Intell. 39, 692–705 (2017).
[Crossref]

Elad, M.

D. L. Donoho and M. Elad, “Optimally sparse representation in general (nonorthogonal) dictionaries via l1 minimization,” Proc. Natl. Acad. Sci. USA 100, 2197–2202 (2003).
[Crossref]

Erhan, D.

C. Szegedy, W. Liu, Y. Jia, P. Sermanet, S. Reed, D. Anguelov, D. Erhan, V. Vanhoucke, and A. Rabinovich, “Going deeper with convolutions,” in Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (2015), pp. 1–9.

Fergus, R.

M. D. Zeiler and R. Fergus, “Visualizing and understanding convolutional networks,” in European Conference on Computer Vision (Springer, 2014), pp. 818–833.

Fidjeland, A.

V. Mnih, K. Kavukcuoglu, D. Silver, A. Rusu, J. Veness, M. Bellemare, A. Graves, M. Riedmiller, A. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” Nature 518, 529–533 (2015).
[Crossref]

Fienup, J. R.

Fink, M.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” arXiv:1005.0532 (2010).

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]

Fukushima, K.

K. Fukushima, “Cognitron: a self-organizing multilayered neural network,” Biol. Cybernet. 20, 121–136 (1975).
[Crossref]

Gerchberg, R.

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

Gigan, S.

S. Popoff, G. Lerosey, R. Carminati, M. Fink, A. Boccara, and S. Gigan, “Measuring the transmission matrix in optics: an approach to the study and control of light propagation in disordered media,” Phys. Rev. Lett. 104, 100601 (2010).
[Crossref]

S. Popoff, G. Lerosey, M. Fink, A. C. Boccara, and S. Gigan, “Image transmission through an opaque material,” arXiv:1005.0532 (2010).

Goodfellow, I.

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

Goodman, J. W.

J. W. Goodman and R. Lawrence, “Digital image formation from electronically detected holograms,” Appl. Phys. Lett. 11, 77–79 (1967).
[Crossref]

Gorocs, Z.

Y. Rivenson, Z. Gorocs, H. Gunaydin, Y. Zhang, H. Wang, and A. Ozcan, “Deep learning microscopy,” arXiv:1705.04709 (2017).

Goy, A.

Graepel, T.

D. Silver, A. Huang, C. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of go with deep neural networks and tree search,” Nature 529, 484–489 (2016).
[Crossref]

Graves, A.

V. Mnih, K. Kavukcuoglu, D. Silver, A. Rusu, J. Veness, M. Bellemare, A. Graves, M. Riedmiller, A. Fidjeland, G. Ostrovski, S. Petersen, C. Beattie, A. Sadik, I. Antonoglou, H. King, D. Kumaran, D. Wierstra, S. Legg, and D. Hassabis, “Human-level control through deep reinforcement learning,” Nature 518, 529–533 (2015).
[Crossref]

Grewe, D.

D. Silver, A. Huang, C. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of go with deep neural networks and tree search,” Nature 529, 484–489 (2016).
[Crossref]

Guez, A.

D. Silver, A. Huang, C. Maddison, A. Guez, L. Sifre, G. van den Driessche, J. Schrittwieser, I. Antonoglou, V. Panneershelvam, M. Lanctot, S. Dieleman, D. Grewe, J. Nham, N. Kalchbrenner, I. Sutskever, T. Lillicrap, M. Leach, K. Kavukcuoglu, T. Graepel, and D. Hassabis, “Mastering the game of go with deep neural networks and tree search,” Nature 529, 484–489 (2016).
[Crossref]

Gunaydin, H.

Y. Rivenson, Z. Gorocs, H. Gunaydin, Y. Zhang, H. Wang, and A. Ozcan, “Deep learning microscopy,” arXiv:1705.04709 (2017).

Y. Rivenson, Y. Zhang, H. Gunaydin, D. Teng, and A. Ozcan, “Phase recovery and holographic image reconstruction using deep learning in neural networks,” arXiv:1705.04286 (2017).

Hassabis, D.

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Supplementary Material (1)

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Figures (13)

Fig. 1.
Fig. 1.

Experimental arrangement. SF, spatial filter; CL, collimating lens; M, mirror; POL, linear polarizer; BS, beam splitter; SLM, spatial light modulator.

Fig. 2.
Fig. 2.

DNN training. Rows (a) and (b) denote the networks trained on Faces-LFW and ImageNet datasets, respectively. (i) Randomly selected example drawn from the database; (ii) calibrated phase image of the drawn sample; (iii) diffraction pattern generated on the CMOS by the same sample; (iv) DNN output before training (i.e., with randomly initialized weights); (v) DNN output after training.

Fig. 3.
Fig. 3.

Detailed schematic of our DNN architecture, indicating the number of layers, nodes in each layer, etc.

Fig. 4.
Fig. 4.

Quantitative analysis of our trained deep neural networks for three object-to-sensor distances, (a) z1, (b) z2, and (c) z3, for the DNNs trained on Faces-LFW (blue) and ImageNet (red) on seven datasets. (d) The training and testing error curves for the network trained on ImageNet at distance z3 over 20 epochs.

Fig. 5.
Fig. 5.

Qualitative analysis of our trained deep neural networks for combinations of object-to-sensor distances z and training datasets. (i) Ground truth pixel value inputs to the SLM. (ii) Corresponding phase images calibrated by SLM response curve. (iii) Raw intensity images captured by CMOS detector at distance z1. (iv) DNN reconstruction from raw images when trained using Faces-LFW [43] dataset. (v) DNN reconstruction when trained using ImageNet [42] dataset. Columns (vi–viii) and (ix–xi) follow the same sequence as (iii–v), but in these sets the CMOS is placed at a distance of z2 and z3, respectively. Rows (a)–(f) correspond to the dataset from which the test image is drawn: (a) Faces-LFW, (b) ImageNet, (c) Characters, (d) MNIST Digits, (e) Faces-ATT [46], or (f) CIFAR [45], respectively.

Fig. 6.
Fig. 6.

Quantitative analysis of the sensitivity of the trained deep convolutional neural network to the object-to-sensor distance. The network was trained on (a) Faces-LFW database and (b) ImageNet, and tested on disjoint Faces-LFW and ImageNet sets, respectively. The nominal depths of field for the three corresponding training distances z1, z2, z3 are: (DOF)1=1.18±0.1  mm, (DOF)2=3.82±0.2  mm, and (DOF)3=7.97±0.3  mm, respectively.

Fig. 7.
Fig. 7.

Quantitative analysis of the sensitivity of the trained deep convolutional neural network to laterally shifted images on the SLM. The network was trained on (a) Faces-LFW database, (b) ImageNet, and tested on disjoint Faces-LFW and ImageNet sets, respectively.

Fig. 8.
Fig. 8.

Quantitative analysis of the sensitivity of the trained deep convolutional neural network to rotation of images on the SLM. The baseline distance on which the network was trained is (a) z1, (b) z2, and (c) z3, respectively.

Fig. 9.
Fig. 9.

Qualitative analysis of the sensitivity of the trained deep convolutional neural network to the object-to-sensor distance. The baseline distance on which the network was trained is z1.

Fig. 10.
Fig. 10.

Qualitative analysis of the sensitivity of the trained deep convolutional neural network to lateral shifts of images on the SLM. The baseline distance on which the network was trained is z1.

Fig. 11.
Fig. 11.

Qualitative analysis of the sensitivity of the trained deep convolutional neural network to rotation of images in steps of 90. The baseline distance on which the network was trained is z1.

Fig. 12.
Fig. 12.

Failure cases on networks trained on Faces-LFW (row a) and ImageNet (row b) datasets. (i) Ground truth input; (ii) calibrated phase input to SLM; (iii) raw image on camera; (iv) reconstruction by DNN trained on images at distance z1 between SLM and camera and tested on images at distance 107.5 cm; (v) raw image on camera; and (vi) reconstruction by network trained on images at distance z3 between SLM and camera and tested on images at distance 27.5 cm.

Fig. 13.
Fig. 13.

(1) 16×16 inputs that maximally activate the last set of 16 convolutional filters in layer 1 of our phase-retrieval network trained on ImageNet at distance of z1, a deblurring network, and an ImageNet classification network. The deblurring network was trained on images undergoing motion blur in a random angle within the range [0, 180] degrees and a random blur length in the range [10, 100] pixels. The image is downsampled by a factor of 2 in this layer. (2) 32×32 inputs that maximally activate the last set of 16 randomly chosen convolutional filters in layer 3 of our network, the same deblurring network, and the ImageNet classification network. The raw image is downsampled by a factor of 8 in this layer.

Equations (3)

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ϕ^(x,y)=HinvI(x,y)
ϕ^(x,y)=argminϕHϕI2+αϕ(ϕ).
min1whYG1.