Abstract

Deep neural networks (DNNs) are efficient solvers for ill-posed problems and have been shown to outperform classical optimization techniques in several computational imaging problems. In supervised mode, DNNs are trained by minimizing a measure of the difference between their actual output and their desired output; the choice of measure, referred to as “loss function,” severely impacts performance and generalization ability. In a recent paper [A. Goy et al., Phys. Rev. Lett. 121(24), 243902 (2018)], we showed that DNNs trained with the negative Pearson correlation coefficient (NPCC) as the loss function are particularly fit for photon-starved phase-retrieval problems, though the reconstructions are manifestly deficient at high spatial frequencies. In this paper, we show that reconstructions by DNNs trained with default feature loss (defined at VGG layer ReLU-22) contain more fine details; however, grid-like artifacts appear and are enhanced as photon counts become very low. Two additional key findings related to these artifacts are presented here. First, the frequency signature of the artifacts depends on the VGG’s inner layer that perceptual loss is defined upon, halving with each MaxPooling2D layer deeper in the VGG. Second, VGG ReLU-12 outperforms all other layers as the defining layer for the perceptual loss.

© 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 (4)

G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6(8), 921–943 (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]

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[Crossref]

H. Wang, Y. Rivenson, Y. Jin, Z. Wei, R. Gao, H. Günaydın, L. Bentolila, C. Kural, and A. Ozcan, “Deep learning enables cross-modality super-resolution in fluorescence microscopy,” Nat. Methods 16(1), 103–110 (2019).
[Crossref]

2018 (6)

S. Li and G. Barbastathis, “Spectral pre-modulation of training examples enhances the spatial resolution of the phase extraction neural network (PhENN),” Opt. Express 26(22), 29340–29352 (2018).
[Crossref]

Z. D. C. Kemp, “Propagation based phase retrieval of simulated intensity measurements using artificial neural networks,” J. Opt. 20(4), 045606 (2018).
[Crossref]

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

S. Li, M. Deng, J. Lee, A. Sinha, and G. Barbastathis, “Imaging through glass diffusers using densely connected convolutional networks,” Optica 5(7), 803–813 (2018).
[Crossref]

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 Trans. Med. Imag. 37(6), 1440–1453 (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]

2017 (2)

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(9), 4509–4522 (2017).
[Crossref]

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” Optica 4(9), 1117–1125 (2017).
[Crossref]

2016 (1)

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. Comput. Imag. 2(1), 59–70 (2016).
[Crossref]

2015 (3)

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]

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

C. Dong, C. Loy, K. He, and X. Tang, “Image super-resolution using deep convolutional networks,” IEEE Trans. on Pattern Analysis Mach. Intell. 38, 295–307 (2015).

2014 (1)

2013 (1)

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

2010 (1)

R. Rubinstein, A. M. Bruckstein, and M. Elad, “Dictionaries for sparse representation modeling,” Proc. IEEE 98(6), 1045–1057 (2010).
[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)

2000 (1)

M. S. Lewicki and T. J. Sejnowski, “Learning overcomplete representations,” Neural Comput. 12(2), 337–365 (2000).
[Crossref]

1996 (1)

A. Van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36(17), 2759–2770 (1996).
[Crossref]

1986 (1)

1985 (1)

1984 (1)

N. Streibl, “Phase imaging by the transport equation of intensity,” Opt. Commun. 49(1), 6–10 (1984).
[Crossref]

1983 (1)

M. R. Teague, “Deterministic phase retrieval: a Green’s function solution,” J. Opt. Soc. Am. A 73(11), 1434–1441 (1983).
[Crossref]

1982 (1)

1978 (1)

1972 (1)

R. W. Gerchberg, “A practical algorithm for the determination of 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(3), 77–79 (1967).
[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 systems,” (2015). Software available from tensorflow.org.

Acosta, A.

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

Adarsh, V.

S. S. Khan, V. Adarsh, V. Boominathan, J. Tan, A. Veeraraghavan, and K. Mitra, “Towards photorealistic reconstruction of highly multiplexed lensless images,” in Proceedings of the IEEE International Conference on Computer Vision, (2019), pp. 7860–7869.

Agarwal, A.

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 systems,” (2015). Software available from tensorflow.org.

Aharon, M.

M. Elad and M. Aharon, “Image denoising via learned dictionaries and sparse representation,” in 2006 IEEE Computer Society Conference on Computer Vision and Pattern Recognition, vol. 1 (IEEE, 2006), pp. 895–900.

Aitken, A.

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

Alahi, A.

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

Arjovsky, M.

M. Arjovsky, S. Chintala, and L. Bottou, “Wasserstein generative adversarial networks,” in International conference on machine learning, (2017), pp. 214–223.

Arthur, K.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “The importance of physical pre-processors for quantitative phase retrieval under extremely low photon counts,” in Quantitative Phase Imaging V, vol. 10887 (International Society for Optics and Photonics, 2019), p. 108870S.

Ba, J.

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

Baburajan, R.

L. Boominathan, M. Maniparambil, H. Gupta, R. Baburajan, and K. Mitra, “Phase retrieval for fourier ptychography under varying amount of measurements,” CoRR abs/1805.03593 (2018).

Baraniuk, R.

C. Metzler, P. Schniter, A. Veeraraghavan, and R. Baraniuk, “Prdeep: Robust phase retrieval with flexible deep neural networks. arxiv 2018,” arXiv preprint arXiv:1803.00212 (2018).

Barbastathis, G.

G. Barbastathis, A. Ozcan, and G. Situ, “On the use of deep learning for computational imaging,” Optica 6(8), 921–943 (2019).
[Crossref]

S. Li, M. Deng, J. Lee, A. Sinha, and G. Barbastathis, “Imaging through glass diffusers using densely connected convolutional networks,” Optica 5(7), 803–813 (2018).
[Crossref]

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “Low photon count phase retrieval using deep learning,” Phys. Rev. Lett. 121(24), 243902 (2018).
[Crossref]

S. Li and G. Barbastathis, “Spectral pre-modulation of training examples enhances the spatial resolution of the phase extraction neural network (PhENN),” Opt. Express 26(22), 29340–29352 (2018).
[Crossref]

A. Sinha, J. Lee, S. Li, and G. Barbastathis, “Lensless computational imaging through deep learning,” Optica 4(9), 1117–1125 (2017).
[Crossref]

M. Deng, S. Li, A. Goy, I. Kang, and G. Barbastathis, “Learning to synthesize: Robust phase retrieval at low photon counts,” arXiv preprint arXiv:1907.11713 (2019).

S. Li, G. Barbastathis, and A. Goy, “Analysis of phase-extraction neural network (phenn) performance for lensless quantitative phase imaging,” in Quantitative Phase Imaging V, vol. 10887 (International Society for Optics and Photonics, 2019), p. 108870T.

A. Goy, K. Arthur, S. Li, and G. Barbastathis, “The importance of physical pre-processors for quantitative phase retrieval under extremely low photon counts,” in Quantitative Phase Imaging V, vol. 10887 (International Society for Optics and Photonics, 2019), p. 108870S.

Barham, P.

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 systems,” (2015). Software available from tensorflow.org.

Bauschke, H. H.

Bengio, Y.

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

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

Bentolila, L.

H. Wang, Y. Rivenson, Y. Jin, Z. Wei, R. Gao, H. Günaydın, L. Bentolila, C. Kural, and A. Ozcan, “Deep learning enables cross-modality super-resolution in fluorescence microscopy,” Nat. Methods 16(1), 103–110 (2019).
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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 systems,” (2015). Software available from tensorflow.org.

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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 systems,” (2015). Software available from tensorflow.org.

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Figures (14)

Fig. 1.
Fig. 1. VGG architecture.
Fig. 2.
Fig. 2. VGG-based feature loss as the training loss.
Fig. 3.
Fig. 3. Experimental Apparatus.
Fig. 4.
Fig. 4. Comparison of reconstructions from PhENN trained with perceptual loss vs. NPCC for 1 and 10-photon levels. The scaled up images show that some details are not rendered by the NPCC-trained PhENN whereas they become clearly identifiable with the perceptual loss function.
Fig. 5.
Fig. 5. (a) Reconstruction by the VGG16 ReLU-22 feature-loss trained DNN for the 1-photon level. (b) Log-scale magnitude of the 2D Fourier Transform of the reconstruction shown in (a). The artifact contributes in the modes indicated by the arrows. (c) Cross sections of the log-scale magnitude of the Fourier Transform of the perceptual loss reconstruction (blue), corresponding to image (b), ground truth (black) and the NPCC-trained DNN reconstruction (red).
Fig. 6.
Fig. 6. (a) Log-magnitude of the power spectral density of the test set of reconstructions $\hat {f}$ clearly showing the signature of the artifact, which is perceived in the reconstructions as a prominent network of horizontal and vertical strips, e.g. Figs. 4 and 5. (b) Horizontal profile of (a). (c) Vertical profile of (a). (d) Diagonal profile of (a).
Fig. 7.
Fig. 7. Reconstructions with feature loss defined at various layers of VGG16 for $p=1$. Row (i): Approximant and ground truth from a representative sample; rows (ii) to (v) each contain layers before 1-Pooling, after 1-Pooling, 2-Pooling and 3-Pooling, respectively.
Fig. 8.
Fig. 8. Scaled-up reconstructions; the region is indicated by the red square in the ground truth image of Fig. 7, row (i). The rows correspond to those in Fig. 7.
Fig. 9.
Fig. 9. Log-scale of PSDs of reconstructions, based on the entire test set of 50 randomly drawn samples. The rows correspond to those in Fig. 7.
Fig. 10.
Fig. 10. comparison of reconstructions from PhENN trained with feature loss vs. NPCC for 1, 10, 100 and 1000-photon levels. In some areas, as shown by the scaled up images, some details are only visible in the feature loss reconstruction.
Fig. 11.
Fig. 11. Dependence of VGG16 loss on the frequency of the noise. (a) Diagram showing the scanning scheme in the Fourier domain. The noise $n$ is added on at a single frequency and made Hermitian, i.e. $n(\nu _x, \nu _y) = n(-\nu _x, -\nu _y)^{*}$. (b) Loss as a function of frequency for the horizontal scan and five examples from the test set, for a noise amplitude of $A = 0.1$. (c) Loss as a function of frequency for the vertical scan for the same five examples. (d) Loss as a function of frequency for the diagonal scan for the same five examples. (e) Absolute value of the derivative of the loss with respect to frequency. The values are averaged over the 50 examples of the test set and plotted for the horizontal, vertical and diagonal scans. The ellipses in (c) and (d) indicate where strong non-smoothness can be observed in the loss curves. The position of the spikes correspond to artifact features observed in the spectrum of the average reconstruction.
Fig. 12.
Fig. 12. Change in the frequency components of an image through the minimization operation of Eq. (13). The frequency $\nu$ refers to position $(\nu _x, \nu _y)=(\nu , \nu )$ in the Fourier domain (diagonal scan). The value plotted is the difference of the modulus of the spectrum of the noisy image $\tilde {f}$ (defined in Eq. (10)) and the spectrum of $\hat {f}$, the result of optimization Eq. (13) starting from $\tilde {f}$.
Fig. 13.
Fig. 13. Reconstructions by VGG19 feature loss trained PhENN
Fig. 14.
Fig. 14. Reconstructions and PSDs produced by mixed loss defined at various layers of VGG19

Tables (3)

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Table 1. Quantitative assessment of reconstructions by feature loss PLT-PhENN defined at various VGG16 layers. Each entry takes the form of average ± standard deviation.

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Table 2. Quantitative assessment of reconstructions by feature loss PLT-PhENN defined at various VGG19 layers. Each entry takes the form of average ± standard deviation.

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Table 3. Quantitative assessment of reconstructions by mixed-loss PLT-PhENN defined at various VGG19 layers. Each entry takes the form of average ± standard deviation.

Equations (15)

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L feat ( i , j ) ( f , f ^ ) = 1 n f e a t N x N y k = 1 n f e a t | | VGG k ( i , j ) ( f ) VGG k ( i , j ) ( f ^ ) | | 2 2 ,
L mixed ( i , j ) ( f , f ^ ) = L image ( f , f ^ ) + α feat L feat ( i , j ) ( f , f ^ ) = | | f f ^ | | 2 2 + α feat 1 n f e a t N x N y k = 1 n f e a t | | VGG k ( i , j ) ( f ) VGG k ( i , j ) ( f ^ ) | | 2 2 ,
ψ obj ( x , y ) = t ( x , y ) e i f ( x , y )
g 0 ( x , y ) = | F z [ ψ inc ( x , y ) ψ obj ( x , y ) ] | 2 H 0 f ( x , y ) ,
g 0 ( x , y ) = | F z [ exp { i f ( x , y ) } ] | 2 = H 0 f ( x , y ) ,
g ( x , y ) = P { p H 0 f ( x , y ) H 0 f } + N H f ( x , y ) ,
f ^ = argmin f { D ( H 0 f , g ) + β Φ ( f ) } .
f ~ = arg { F 1 ( g arg { F ( u i n c ) } ) } ,
f ^ = D N N ( f ~ ) .
ξ ( A , x , y , ν x 0 , ν y 0 ) = A F 1 { e i a δ ( ν x ν x 0 , ν y ν y 0 ) + e i a δ ( ν x + ν x 0 , ν y + ν y 0 ) + e i b δ ( ν x + ν x 0 , ν y ν y 0 ) + e i b δ ( ν x ν x 0 , ν y + ν y 0 ) } ,
f noisy , n ( A , x , y , ν x 0 , ν y 0 ) = f n + ξ ( A , x , y , ν x 0 , ν y 0 )
L ( f , f noisy ) = 1 N test n = 1 N t e s t L ( f n , f noisy , n )
MAP = argmax η { VGG ( η ) } such that η p [ 0 , 1 ]
f ^ = argmin η { L ( η , f ) } .
f ^ ( ν x 0 , ν y 0 ) = G f [ ξ ( ν x 0 , ν y 0 ) ] .

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