Abstract

Since their inception in the 1930–1960s, the research disciplines of computational imaging and machine learning have followed parallel tracks and, during the last two decades, experienced explosive growth drawing on similar progress in mathematical optimization and computing hardware. While these developments have always been to the benefit of image interpretation and machine vision, only recently has it become evident that machine learning architectures, and deep neural networks in particular, can be effective for computational image formation, aside from interpretation. The deep learning approach has proven to be especially attractive when the measurement is noisy and the measurement operator ill posed or uncertain. Examples reviewed here are: super-resolution; lensless retrieval of phase and complex amplitude from intensity; photon-limited scenes, including ghost imaging; and imaging through scatter. In this paper, we cast these works in a common framework. We relate the deep-learning-inspired solutions to the original computational imaging formulation and use the relationship to derive design insights, principles, and caveats of more general applicability. We also explore how the machine learning process is aided by the physics of imaging when ill posedness and uncertainties become particularly severe. It is hoped that the present unifying exposition will stimulate further progress in this promising field of research.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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Y.-J. Jo, H. Cho, S. Y. Lee, G. Choi, G. Kim, H.-S. Min, and Y.-K. Park, “Quantitative phase imaging and artificial intelligence: a review,” IEEE J. Sel. Top. Quantum Electron. 25, 6800914 (2019).
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2018 (20)

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2017 (10)

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2016 (11)

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A. Kadambi, H. Zhao, B. Shi, and R. Raskar, “Occluded imaging with time-of-flight sensors,” ACM Trans. Graph. 35, 1–12 (2016).
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2015 (11)

L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
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2014 (10)

N. Srivastava, G. Hinton, A. Krizhevsky, I. Sutskever, and R. Salakhutdinov, “Dropout: a simple way to prevent neural networks from overfitting,” J. Mach. Learn. Res. 15, 1929–1958 (2014).

O. Katz, P. Heidmann, M. Fink, and S. Gigan, “Non-invasive single-shot imaging through scattering layers and around corners via speckle correlations,” Nat. Photonics 8, 784–790 (2014).
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Y. Yunhui, A. Shanker, L. Tian, L. Waller, and G. Barbastathis, “Low-noise phase imaging by hybrid uniform and structured illumination transport of intensity equation,” Opt. Express 22, 26696–26711(2014).
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A. Yevick, M. Hannel, and D. G. Grier, “Machine-learning approach to holographic particle characterization,” Opt. Express 22, 26884–26890 (2014).
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2013 (5)

2012 (9)

C. Zhao, W. Gong, M. Chen, E. Li, H. Wang, W. Xu, and S. Han, “Ghost imaging lidar via sparsity constraints,” Appl. Phys. Lett. 101, 141123 (2012).
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Y. Liu, L. Tian, J. W. Lee, H. Y. H. Huang, M. S. Triantafyllou, and G. Barbastathis, “Scanning-free compressive holography for object localization with subpixel accuracy,” Opt. Lett. 37, 3357–3359(2012).
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O. Gupta, T. Willwacher, A. Velten, A. Veeraraghavan, and R. Raskar, “Reconstruction of hidden 3d shapes using diffuse reflections,” Opt. Express 20, 19096–19108 (2012).
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L. Tian, J. C. Petruccelli, and G. Barbastathis, “Nonlinear diffusion regularization for transport of intensity phase imaging,” Opt. Lett. 37, 4131–4133 (2012).
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2010 (8)

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).
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S. Popoff, G. Lerosey, M. Fink, A. Boccara, and S. Gigan, “Image transmission through an opaque material,” Nat. Commun. 1, 81 (2010).
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2009 (9)

J. Cheng, “Ghost imaging through turbulent atmosphere,” Opt. Express 17, 7916–7921 (2009).
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2008 (6)

B. I. Erkmen and J. H. Shapiro, “Unified theory of ghost imaging with Gaussian-state light,” Phys. Rev. A 77, 043809 (2008).
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Figures (23)

Fig. 1.
Fig. 1. General computational imaging system. The illumination source or source array is shaped by the condenser optics according to the operator Hi before reaching the object f. The radiation is subsequently shaped again by the imaging optics with collection operator Hc, and the intensity g is sampled by the digital camera. The signal g is then processed by the CI algorithm, which takes into account prior knowledge Φ about the class of objects being imaged (regularizer) in addition to the physical models Hi, Hc. The result of the computation is the image f^. For detailed notation and description, please see Section 2.
Fig. 2.
Fig. 2. Simplified schematic of a generic deep neural network. The input layer consists of the components of the input vector u. The dark circles denote the activation (nonlinear) elements, whereas wkj(l) is the weight connecting the j-th activation element in layer (l) with the k-th activation element in layer (l+1). The last (L)-th layer produces an estimate of the output vector v. The quality of the approximation relating input to output depends on the training; please see Section 3.
Fig. 3.
Fig. 3. CI architecture of Fig. 1 with an ML engine producing the image. As we see in Section 4, the ML engine generally includes a multilayered architecture such as the one shown in Fig. 2 and is informed on the physics of the illumination and collection optics, Hi, Hc, respectively, and the prior Φ. The three components Hi, Hc, Φ or their combinations are incorporated in the ML engine either explicitly as approximants (Section 3.F) or implicitly through training with examples.
Fig. 4.
Fig. 4. Simplified schematics of layer width progression strategies for DNNs. (a) Contracting architecture. (b) Expanding architecture. (c) Encoder–decoder architecture.
Fig. 5.
Fig. 5. Behavior of the training, test, and validation error as function of training epoch t.
Fig. 6.
Fig. 6. Convolutional encoding principle between layers (l1) and l of decreasing width. Shift-invariant weights of limited range feed into activation units, e.g., ReLU, followed by a pooling layer.
Fig. 7.
Fig. 7. Detailed implementation of the encoder–decoder architecture, Fig. 4(c), and the convolutional principle, Fig. 6, in the original U-net that was built for image segmentation (after [107], reprinted with permission from Springer).
Fig. 8.
Fig. 8. Simplified schematic of residual learning through bypass connections with unit weight (dashed lines) [149].
Fig. 9.
Fig. 9. Training an image-forming DNN1 with a perceptual loss function generated by a content- and style-sensitive DNN2 (after [170]).
Fig. 10.
Fig. 10. Ways to implement the ML engine in Fig. 3, with or without physical priors. (a) End-to-end ML engine. (b) Recurrent physics-informed ML engine [176]. (c) Cascaded physics-informed ML engine [176]. (d) Single-pass physics-informed ML engine. Here, H*=HT if the forward operator H is linear; otherwise, even a crude approximation to the nonlinear inverse has often been sufficient.
Fig. 11.
Fig. 11. Evolution of the image estimate along the cascaded ML engine in Fig. 10(c) (reprinted with permission from [177], Fig. 10 ). Top row, left to right, and in the notation used in the present review: undersampled raw intensity image g, f^[0], f^[1], f^[2], and f^f^[3]. Bottom row: spatial Fourier transform magnitude of the corresponding images from the top row.
Fig. 12.
Fig. 12. Deep learning microscopy (adapted from [199], Figs. 1 and 5, with permission). (a) Training of the end-to-end ML engine with low-resolution (downsampled and blurred) inputs to the DNN produced by a 40×0.95NA objective lens and high-resolution outputs produced by a 100×1.4NA objective lens. (b) Operation of the trained DNN engine, with the test outputs successfully upsampled and deblurred. (c) Resolution test demonstrating the DNN with inputs from the 40×0.95NA objective lens achieving in testing performance comparable to the raw performance of the 100×1.4NA objective lens the DNN was trained with.
Fig. 13.
Fig. 13. Phase extraction neural network (PhENN) (reprinted from [234], Fig. 5, with permission). Columns I and II are the ground truth pixel values driving the SLM and corresponding phase images f produced by the SLM, according to calibration. Groups of columns III-V, VI-VIII, and IX-XI are different propagation distances z=37.5cm, 67.5 cm, and 97.5 cm, respectively. Columns III, VI, IX are the raw images g captured by the camera. Columns IV, VII, X are reconstructions f^ when PhENN was trained with Faces-LFW [235], whereas columns V, VIII, XI are estimates f^ when PhENN was trained with ImageNet [173]. The rows represent different databases the PhENN was tested on as a, Faces-LFW (disjoint from training); b, ImageNet (disjoint from training); c, Characters [236]; d, MNIST [237]; e, Faces-ATT [238]; f, CIFAR [239].
Fig. 14.
Fig. 14. Retrieval of phase and amplitude with a physics-informed ML engine (reprinted with permission from Springer [240], Fig. 2 ). (a)–(h) Pap smear and (j)–(p) breast tissue section reconstructions. (a), (i) Zoom-ins to the (optical) backpropagation results from a single intensity image; (h), (p) corresponding bright-field microscopy images, shown for comparison. (b), (c), (j), (k) (Optical) backpropagation results showing artifacts due to twin image and self-interference effects; (d), (e), (l), (m) corresponding ML engine reconstructions from a single hologram, showing quality comparable to (f), (g), (n), (o) traditional reconstructions from eight holograms.
Fig. 15.
Fig. 15. Physics-informed DNN trained with samples at different propagation distances of up to ±100μm from the focal plane, exhibiting improved depth invariance compared to (optical) backpropagation and multi-height phase retrieval (MH-PR) (reprinted with permission [244], Fig. 3). The results are from a human breast tissue sample, captured at different propagation distances dz. Distances marked in red exceed the training range of the DNN. Scale bar=20μm.
Fig. 16.
Fig. 16. Resolution improvement by spectral premodulation of PhENN (from [156], Figs. 5, 8, and 9; reprinted with permission). (a) Two-phase point-object resolution target, implemented on the SLM; (b) PhENN image, not resolving phase point-objects separated by three pixels; (c) with spectral pre-modulation, the three pixel-separated phase-point objects are resolved; (d) sample test image from the ImageNet database; (e) PheNN reconstruction; (f) PhENN reconstruction with spectral pre-modulation, showing sharper features but also edge-enhancement artifacts.
Fig. 17.
Fig. 17. ML applied to reconstruct a severely photon-limited scene (after [169], Fig. 5; reprinted with permission). (a) Signal captured by a Fujifilm X-T2 camera with ISO 800, aperture f/7.1, and exposure of 1/30 s, with illuminance of approximately 1 lux; (b) reconstruction using denoising, deblurring, and enhancement, which the authors refer to as “traditional image processing pipeline”; (d) reconstruction obtained using the radiational image processing pipeline on a Sony α7SII camera; (e) reconstruction obtained using BM3D [264], considered as benchmark for denoising; (c), (f) ML engine reconstructions of the respective dark images.
Fig. 18.
Fig. 18. ML applied to quantitative phase retrieval on a severely photon-limited signal (after [265], Fig. 2; reprinted with permission). (a) Ground truth f for a test example; (b), (f) raw intensity signal at photon flux of 1,050 photons/pixel; (c), (g) corresponding outputs f^[0]of the approximant H* of Fig. 10(d), implemented as a single-iteration Gerchberg–Saxton algorithm; (d), (h) corresponding reconstructions by the end-to-end ML engine of Fig. 10(a); (e), (i) corresponding reconstructions by physics-informed single-pass ML engine according to Fig. 10(d). In (a) and reconstructions (c)–(e, (g)–(i), grayscale tone represents phase delay [0,2π).
Fig. 19.
Fig. 19. Experimental computational ghost image reconstructions with sampling ratio β=0.1 (left-hand side group) and β=0.05 (right-hand side group) (after [279]; reprinted with permission from Springer Nature). Top row: ground truth; second-row: basic ghost reconstruction according to [273]; third row: deep learning ghost reconstructions using the images in the second row as approximants f^[0] to the physics-informed ML engine of Fig. 10(d); last row: compressive ghost reconstructions according to [278].
Fig. 20.
Fig. 20. Learning-based imaging through diffusers using support vector regression (SVR) (after [18], Fig. 3; reprinted with permission). (a) Training examples from a database of faces and (b) their corresponding speckle patterns; (c) test face examples and their corresponding (d) speckle patterns, (e) test reconstructions using SVR, and (f) test reconstructions using pattern matching.
Fig. 21.
Fig. 21. Generalization study of the SVR method (after [18], Fig. 5; reprinted with permission). (a) Test non-face examples and their corresponding (b) speckle patterns, (c) test reconstructions using SVR, and (d) test reconstructions using pattern matching.
Fig. 22.
Fig. 22. Learning-based imaging through optically thick diffusers using a fully connected DNN with MAE TLF (after [299], Fig. 3; reprinted with permission). First row: speckle patterns input to the DNN; second row: corresponding ground-truth images selected among a database of English letters; third row: corresponding reconstructions by the DNN.
Fig. 23.
Fig. 23. Learning-based imaging through diffusers using IDiffNet: a residual-convolutional DNN with NPCC TLF (after [155], Fig. 6; reprinted with permission). Columns I, II; ground truth pixel values and fields modulated by the SLM, after calibration; III-VI: results with 600-grit diffuser. III are the raw images, IV-VI test reconstructions from IDiffNet trained with Faces-LFW [235], ImageNet [173], and MNIST [237] datasets, respectively. Rows correspond to the dataset the test image is drawn from, as (a) Faces-LFW, (b) ImageNet, (c) Characters [236], (d) MNIST, (e) Faces-ATT [238], (f) CIFAR [239].

Equations (40)

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g=Hf.
g=P{Hf}+T.
f^=argminf{Hfg22+αΦ(f)},
Φ(f)=f22.
f^=(HTH+α1)1HTg,
TVx,yf2,
f˜=Df
v=NN(u)
sk(l)=r(jwkj(l1)sj(l1)+wk0(l1)),
sk(1)=uk,
sk(L)=vk.
r(ξ)=max(0,ξ)={0,ifξ0,ξ,otherwise.
L{u(n),v˜(n)}n=1,,NnE(NN(u(n)),v˜(n)),
W˜=argminWL{u(n),v˜(n)}n=1,,N,
w(t+1)=w(t)+η[Lw].
Ltest{u(m),v˜(m)}m=1,,MmEtest(NN(u(m)),v˜(m)).
W˜=argminWL{u(n),v˜(n)}n=1,,N+αW22.
w[t]w[0]eηαt;
ζj(l1)=r(m=MMwm(l1)sjm(l1)),
stride: sk(l)=ζ2k1(l1),
max pooling: sk(l)=maxk(ζ2k1(l1),ζ2k(l1)),
average pooling:sk(l)=ζ2k1(l1)+ζ2k(l1)2.
EMSE(v,v˜)=p(v(p)v˜(p))2,and
EMAE(v,v˜)=p|v(p)v˜(p)|.
a=1Ppa(p)
Cab=1P1p(aa)(bb).
ENPCC(v,v˜)=Cvv˜CvvCv˜v˜.
ESSIM(v,v˜)=(2vv˜+c1)(2Cvv˜+c2)(v2+v˜2+c1)(Cvv2+Cv˜v˜2+c2).
ECE=p[σ(f)logχ+(1σ(f))log(1χ)],where
σ(f)=efkjefjavailable instanceskof the datafk.
EDiscP=EVvpdatalogd(v)+EVvpmodel(1logd(v)).
EPFL(f^,f)=lμls(l)(f^)s(l)(f)22,
E=μ1E1+μ2E2+
f^=NN(g).
f^[m+1]=PΦ{f^[m]+αHT(gHf^[m])}
=PΦ{αHTg+(1αHTH)f^[m]}.
N=1αHTH
β=#displayedSLMpatterns#reconstructed pixels
f^=n=1NγnK(g(n),g)+f0,
Eεins(v,v˜)={0,if|vv˜|ε,|vv˜|ε,otherwise.