GPU-based deep convolutional neuralnetwork for tomographic phasemicroscopy with l1 fitting andregularization

Hui QiaoJiamin WuXiaoxu LiMorteza H. ShorehJingtao FanQionghai Dai

Hui Qiao, Jiamin Wu, Xiaoxu Li, Morteza H. Shoreh, Jingtao Fan, Qionghai Dai, “GPU-based deepconvolutional neural network for tomographic phase microscopy with l1 fitting and regularization,”J. Biomed. Opt. 23(6), 066003 (2018), doi: 10.1117/1.JBO.23.6.066003.

GPU-based deep convolutional neural networkfor tomographic phase microscopy with l1 fittingand regularization

Hui Qiao,a Jiamin Wu,a Xiaoxu Li,a Morteza H. Shoreh,b Jingtao Fan,a and Qionghai Daia,*aTsinghua University, Department of Automation, Beijing, ChinabÉcole Polytechnique Fédérale de Lausanne, School of Engineering, Laboratory of Optics, Lausanne, Switzerland

Abstract. Tomographic phase microscopy (TPM) is a unique imaging modality to measure the three-dimen-sional refractive index distribution of transparent and semitransparent samples. However, the requirement ofthe dense sampling in a large range of incident angles restricts its temporal resolution and prevents its appli-cation in dynamic scenes. Here, we propose a graphics processing unit-based implementation of a deep con-volutional neural network to improve the performance of phase tomography, especially with much fewer incidentangles. As a loss function for the regularized TPM, the l1-norm sparsity constraint is introduced for both data-fidelity term and gradient-domain regularizer in the multislice beam propagation model. We compare our methodwith several state-of-the-art algorithms and obtain at least 14 dB improvement in signal-to-noise ratio.Experimental results on HeLa cells are also shown with different levels of data reduction. © The Authors.

Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires

full attribution of the original publication, including its DOI. [DOI: 10.1117/1.JBO.23.6.066003]

Keywords: tomographic phase microscopy; l1 data-fidelity; GPU-based implementation; neural network; refractive index.

Paper 180094R received Feb. 13, 2018; accepted for publication May 21, 2018; published online Jun. 14, 2018.

1 IntroductionMost biological samples such as live cells have low contrastin intensity but exhibit strong phase contrast. Phase contrastmicroscopy is then widely applied in various biomedicalimaging applications.1 In the past decades, the development ofquantitative phase imaging2,3 gives rise to a label-free imagingmodality, tomographic phase microscopy (TPM), which dealswith the three-dimensional (3-D) refractive index distribution ofthe sample.4–6 The label-free and noninvasive character makesit attractive in biomedical imaging, especially for culturedcells.7,8

However, most of the current methods require around 50quantitative phase images acquired at different angles9–11 or dif-ferent depths6 for optical tomography. This speed limitationgreatly restricts its field of applications. For example, the differ-ence of the refractive index may be blurred during the angular(or axial) scanning when observing fast-evolving cell dynamicsor implementing high-throughput imaging cytometry.11 Anotherchallenge for TPM is the missing cone problem, which limits itsreconstruction performance, especially for limited axial resolu-tion compared with the subnanometer optical-path-lengthsensitivity.12

To relieve the missing cone problem, many methods havebeen developed for better signal-to-noise ratio (SNR) withfewer images. Different regularizations such as the positivityof the refractive index differences4,13 and the sparsity in sometransform domain14,15 are added to an iterative reconstructionframework based on the theory of diffraction tomography,16,17

for reducing the artifacts induced by the missing cone problem

and the limited sampling rates in Fourier domain. Bothintensity-coded and phase-coded structured illuminationmethods further promote the performance by their better multi-plexing ability compared with conventional plane-waveillumination.18,19 However, these methods suffer from thegreat degradation when the scattering effects become significantin the sample. The beam propagation method (BPM)20 is thenapplied in phase tomography to provide a more accuratemodel by considering the nonlinear light propagation withscattering.21,22 And the multislice propagation modeling is def-initely similar to the neural network in the field of machinelearning.23,24 By combining the nonlinear modeling and thesparse constraint in the gradient domain, the Psaltis grouphas validated the competitive capability of this learningapproach over conventional methods.21,23 Despite its successin modeling with l2-norm constraint, the current method isstill a preliminary network, especially compared with thestate-of-the-art deep learning frameworks,25 and the iterativereconstruction is challenging to deploy in practice due to thehigh computational cost and the difficulty of the hyperparameterselection. More potential can be exploited in both optimizationalgorithms and better network architectures.

In this paper, we propose a graphics processing unit (GPU)-based implementation of a deep convolutional neural network(CNN) to simulate the multislice beam propagation for TPM.A loss function consisting of an l1-norm data-fidelity termand an l1-norm gradient-domain regularizer is devised toachieve higher reconstruction quality even with fewer trainingdata. To deal with the vast quantities of parameters and regular-izers, we apply the adaptive moment estimation (Adam)algorithm26 for optimization, which can also be regarded asthe training process of the CNN. Compared with previousworks using stochastic gradient descent,23,24 our method ensures*Address all correspondence to: Qionghai Dai, E-mail: qhdai@tsinghua.edu.cn

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Journal of Biomedical Optics 23(6), 066003 (June 2018)

a faster convergence and a better robustness to the initial value.Both simulation and experimental results on polystyrene beads,and HeLa cells are shown to validate its reconstruction perfor-mance. We anticipate that our work can not only boost the per-formance of optical tomography, but also guide moreapplications of deep learning in the optics field.

2 Materials and Methods

2.1 Experimental Setup

Figure 1 shows the schematic diagram of the experimentalsetup. In our system,23 the sample placed between two coverglasses is illuminated sequentially at multiple angles and thescattered light is holographically recorded. A laser beam(λ ¼ 561 nm) is split into sample and reference arms by thefirst beam splitter. In the sample arm, a galvo mirror variesthe angle of illumination on the sample using the 4F system cre-ated by L1 and OB1. The light transmitted through the sample isimaged onto the CMOS camera via the 4F system created byOB2 and L2. The beam splitter (BS2) recombines the sampleand reference laser beams, forming a hologram at the imageplane. The numerical apertures (NAs) of OB1 and OB2 are1.45 and 1.4, respectively. For data acquisition, we capturemultiple tomographic phase images by near-plane-wave illumi-nation (Gaussian beam) with equally spaced incident angles. Weuse a differential measurement between the phase on a portion ofthe field of view on the detector that does not include the cell andthe cell itself to maintain phase stability. Accordingly, complexamplitudes extracted from the measurements constitute thetraining set of our proposed CNN.

2.2 Beam Propagation Method

We build the CNN, based on the forward model of light propa-gation,21,23 to model the diffraction and propagation effects oflight-waves. It is known that the scalar inhomogeneousHelmholtz equation completely characterizes the light field atall spatial positions in a time-independent form

EQ-TARGET;temp:intralink-;e001;63;334½∇2 þ k2ðrÞ�uðrÞ ¼ 0; (1)

where r ¼ ðx; y; zÞ denotes a spatial position, u is the total light-field at r, ∇2 ¼ ð ∂2

∂x2 þ ∂2∂y2 þ ∂2

∂z2Þ is the Laplacian, and kðrÞ is thewave number of the light field at r. The wave number dependson the local refractive index distribution nðrÞ asEQ-TARGET;temp:intralink-;e002;326;704kðrÞ ¼ k0nðrÞ ¼ k0½n0 þ δnðrÞ�; (2)

where k0 ¼ 2π∕λ is the wave number in vacuum, n0 is therefractive index of the medium, and the local variation δnðrÞis caused by the sample inhomogeneities. By introducingthe complex envelope aðrÞ of the paraxial wave uðrÞ ¼aðrÞ expðjk0n0zÞ for BPM, we can obtain an evolutionequation 21 in which z plays the role of evolution parameterEQ-TARGET;temp:intralink-;e003;326;607

aðx; y; zþ δzÞ ¼ ejk0δnðrÞδz

×

264F−1

8><>:e

−j

�ω2xþω2y

k0n0þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik20n20−ω2x−ω

2y

p�

δz

9>=>; � að·; ·; zÞ

375; (3)

where δz is a sufficiently small but a finite z step, ωx and ωy

represent angular frequency coordinates in the Fourier domain,að·; ·; zÞ expresses the two-dimensional (2-D) complex envelopeat z depth, � refers to a convolution operator, and F−1f·gmeansthe 2-D inverse Fourier transform.

2.3 GPU-Based Implementation of CNN

A schematic architecture of our CNN is shown in Fig. 2. Forconstructing our neural network, we divide the computationalsample space into thin slices with the sampling interval δzalong the propagation direction z. One slice corresponds toone layer in CNN. Within each layer, neurons specify the dis-cretized light-field with transverse sampling intervals δx and δy,respectively. The input layer is the incident field upon the sam-ple. In terms of the Eq. (3), inputs are then passed from nodes ofeach layer to the next, with adjacent layers connected by alter-nating operations of convolution and multiplication. At the verylast layer of our CNN, the output complex field amplitude isthen bandlimited by the NA of the imaging system composed

BS1

BS2

GM

M

L1

L2

OB1

OB2

Sample

L3

L4

CMOS

Laser

0

127

255

(a.u.)

Fig. 1 Experimental setup (BS, beam splitter; GM, galva mirror; L, lens; M, mirror; and OB, objective) andmeasured hologram by CMOS. Scale bar, 10 μm.

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Qiao et al.: GPU-based deep convolutional neural network for tomographic phase microscopy. . .

of lenses OB2 and L2 in Fig. 1. We implement the proposednetwork on the basis of TensorFlow framework. The connectionweight δnðrÞ can be trained using the Adam algorithm for opti-mization on the following minimization problem:

EQ-TARGET;temp:intralink-;e004;63;479

minδn

1

M

XMm¼1

kYmðδnÞ − GmðδnÞk1 þ τRðδnÞ

s:t: RðδnÞ ¼Xr

k∇δnðrÞk1 and δn ≥ 0; (4)

whereM denotes the number of measured views, k ·k1 indicatesthe l1-norm, and ∇ ¼ ð ∂

∂x ;∂∂y ;

∂∂zÞ is the differential operator. For

a given view m, Ym, and Gm are the output of the last layer andthe actual measurement acquired by the optical system, respec-tively. The design of our loss function will be specifically dis-cussed in Sec. 4.1. Compared with the l2-norm, the l1 data-fidelity term relaxes the intrinsic assumptions on the distributionof noise (symmetry and no heavy tails) and suits better for themeasurements containing outliers. Hence, it can be effectivelyapplied to the biomedical imaging especially when the noisemodel is heavy-tailed and undetermined.27 As a regularizationterm, RðδnÞ imposes the l1-norm sparsity constraint on a gra-dient domain according to its better characteristic for thereconstruction from higher incomplete frequency informationthan l2-norm,28,29 whereas τ is the positive parameter control-ling the influence of regularization. The positivity constrainttakes advantage of the assumption that the index perturbationis real and positive when imaging weakly absorbing samplessuch as biological cells. The subgradient method30 plays animportant role in machine learning for solving the optimizationframework under l1-norm and the Ref. 26 has verified the theo-retical convergence properties of the Adam algorithm, whichwill be specifically discussed in Sec. 4.3. We perform the neuralnetwork computations on 4 NVIDIATITAN Xp graphics cardsand the processing time to run the learning algorithm (100 iter-ations) on 256 × 256 × 160 nodes is nearly 9 min. Obviously,it is possible to make the optimization of hyperparameters,which have an important effect on results, a more reproducibleand automated process and thus is beneficial for trainingthe large-scale and often deep multilayer neural networks

successfully and efficiently.31 The full implementation andthe trained networks are available at https://github.com/HuiQiaoLightning/CNNforTPM.

3 ResultsFor demonstration, we evaluate the designed network by bothsimulation and experimental results of the TPM as describedbefore. To make a reasonable comparison, selected hyperpara-meters have been declared for all the other reconstructionmethods. The selection of hyperparameters will be specificallydiscussed in Sec. 4.2.

3.1 Tomographic Reconstruction of Simulated Data

In simulation, we consider a situation of three 5 μm beads ofrefractive index n ¼ 1.548 immersed into oil of refractiveindex n0 ¼ 1.518 shown in Fig. 3. The centers of the beadsare placed at ð0; 0;−3Þ, (0, 0, 3), and (0, 5, 0), respectively,with the unit of micron. The training set of the framework issimulated as 81 complex amplitudes extracted from the digi-tal-holography measurements with different angles of incidenceevenly distributed in ½−π∕4; π∕4� by BPM, whereas the illumi-nation is tilted perpendicular to the x-axis and the angle is speci-fied with respect to the optical axis z. The size of thereconstructed volume is 23.04 μm × 23.04 μm × 23.04 μm,

Nx

Ny

Incident field

Back propagation

Output field

j ( , )1{e }x yker z

( , )x yker

0j ( )e k n zr

n r

Conv term Mult term

Nz layersConvolution Multiplication Lowpass filtering

Measured field

Loss function

Fig. 2 Detailed schematic of our CNN architecture, indicating the number of layers (Nz), nodes(Nx × Ny ) in each layer and operations between adjacent layers. Here, kerðωx ; ωy Þ signifies

ðω2x þ ω2

y Þ∕ðk0n0 þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik20n

20 − ω2

x − ω2y

qÞ and we take the δnðrÞ of a polystyrene bead as example.

x

y z

000

Cross-sectionalviews

Fig. 3 Simulation geometry comprising three spherical beads with arefractive index difference of 0.03 compared with the background.

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with the sampling steps of δx ¼ δy ¼ δz ¼ 144 nm. For the net-work hyperparameters, we choose 600 training iterations in ourGPU-based implementation with the batch size of 20, the initiallearning rate of 0.001, and the regularization coefficient ofτ ¼ 1.5. The reconstructed results by our method and otherreconstruction methods are shown in Fig. 4. The SNR definedin Ref. 21 of our result is 25.56 dB, 14 dB higher than the pre-vious works. We can also observe much sharper edges of thereconstructed beads at the interface with less noise in the back-ground from Fig. 5. The comparison between the proposed lossfunction and other regularized loss functions proves the higherreconstruction quality of the l1-norm constraint than the l2-casedirectly.

In addition, we analyze the performance of our method underdifferent noise levels and reduced sampling angles. For the noisetest, we add Gaussian noise of different power levels to the 81simulated measured complex amplitudes, which are representedas different SNRs of the training data. From the curve of thereconstructed SNR versus the noise level, as shown in Fig. 6(a),we can find our method maintains more robustness to the noisethan other methods. This is especially useful in the case ofshorter exposure time for higher scanning speed, where the dataare always readout-noise limited. For the test of reducedsampling angles, we keep the range of incident angles fixedfrom −π∕4 to π∕4. The total number of the incident anglesfor the network training decreases from 81. The curve of thereconstructed SNR versus the number of the incident anglesis shown in Fig. 6(b). Even with as few as 11 incident angles,we can still achieve comparable performance as the previousmethods with 81 angles. This nearly eight-time improvementfacilitates the development of high-speed 3-D refractive indeximaging.

3.2 Tomographic Reconstruction of a BiologicalSample

To further validate the capability of the network, we displaythe experimental results on HeLa cells performed by ourtomographic phase microscope as shown in Fig. 1. In detail,we illuminate the HeLa cells in culture medium of refractiveindex n0 ¼ 1.33 from 41 incident angles evenly distributedfrom −35 deg to 35 deg. The measured hologram with an inci-dent angle of 0 deg is shown in Fig. 1. The reconstructed volumeis 36.86 μm × 36.86 μm × 23.04 μm, composed of 256 × 256 ×160 voxels (with the voxel size of 144 nm × 144 nm ×144 nm). After the selection of hyperparameters, we set theregularization coefficient to τ ¼ 5, with training iterations of100, the batch size of 20, and the initial learning rate of 0.002.The performance comparison of different methods under differ-ent levels of data reduction is shown in Fig. 7. More details canbe observed by our method even with fewer incident angles.Moreover, many fewer artifacts and noises exist in our results

(a)

(d)

(b)

(e)

(c)

(f)

Fig. 4 Reconstruction results of three 5 μm beads. Comparison of the cross-sectional slices of the 3-Drefractive index distribution of the sample along the x − y , x − z, and y − z planes reconstructed by(a) proposed CNN, (b) CNN with l1 fitting, l2 regularization (L1 L2) and the regularization coefficientof 5, (c) CNN with l2 fitting, l1 regularization (L2 L1) and the regularization coefficient of 0.1, (d) learningapproach23 implemented on the same CNN settings (LA) with the regularization coefficient of 0.6, (e) opti-cal diffraction tomography based on the Rytov approximation (ODT)13 with the positivity constraint and100 iterations, and (f) iterative reconstruction based on the filtered backprojection method4 with the pos-itivity constraint and 400 iterations. Scale bar, 5 μm.

Ref

ract

ive

inde

x

Z axis ( )

1.52

1.53

1.54

1.55

1.56

-10 -5 0 5 10

IFBPODTLAL2 L1L1 L2CNNGT

12

3

45

6

0

0123456

Fig. 5 Comparison of the refractive index profiles along the z-axisreconstructed by different algorithms and the ground truth.

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Qiao et al.: GPU-based deep convolutional neural network for tomographic phase microscopy. . .

with large data reduction than other methods, which can be seenapparently in Fig. 8.

4 Discussion

4.1 Comparison between L1-Norm and L2-Norm forLoss Function Design

Loss function design is a crucial element of learning algorithms,which determines the training process of the neural network.Regularized loss function comprises one data-fidelity termand one regularization term.

To the best of our knowledge, the presented study is the firstto employ l1 fitting for the regularized TPM. Generally, thechoice of the data-fidelity term depends on the specified noisedistribution. However, it is particularly common for solving

Rec

onst

ruct

ed S

NR

(dB

)

Number of incident angles0 10 20 30 40 50 60 70 80 90

0

5

10

15

20

25

IFBPODTLACNN

30

R

NS detcurtsnoceR

(dB

)

Noise level (dB)0 5 10 15 20 25 30

-15

0

5

10

15

20(a) (b)

IFBPODTLACNN

Fig. 6 Performance analysis for proposed approach with the same hyperparameter selection. (a) Thecurve of the reconstructed SNR versus the noise level and (b) the curve of the reconstructed SNR versusthe number of the incident angles.

1121

(a)

(e)

(i)

(b)

(f)

(j)

(c)

(g)

(k)

1.33

1.43n

41x

y

y

z

x

z

6

(d)

(h)

(i)

Fig. 7 Comparison of three reconstruction algorithms for various levels of data reduction on a HeLa cell.(a–d) Proposed CNN, (e–h) LA with the regularization coefficient of 1.5, (i–l) ODT with the positivity con-straint and 20 iterations, (a, e, and i) 41 training data, (b, f, and j) 21 training data, (c, g, and k) 11 trainingdata, and (d, h, and l) 6 training data. Scale bar, 10 μm.

0

Z axis ( )

xedni evitcarfeR

1.43

1.41

1.37

1.335 10 15 20

LAODT

CNN

1.39

1.35

1

23

123

Fig. 8 Comparison of the reconstructed HeLa cell refractive indexprofiles along the z-axis with 41 training data.

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Qiao et al.: GPU-based deep convolutional neural network for tomographic phase microscopy. . .

normal image restoration problems, as under various con-straints, images are always degraded with mixed noise and itis impossible to identify what type of noise is involved. Thel2-norm fitting relies on strong assumptions on the distributionof noise: there are no heavy tails and the distribution is symmet-ric. If either of these assumptions fails, then the use of l2-normis not an optimal choice. On the other hand, for the so-calledrobust formulation based on l1-norm fitting, it has beenshown that the corresponding statistics can tolerate up to50% false observations and other inconsistencies.27 Hence,l1-norm data-fidelity term relaxes the underlying requirementsfor the l2-case and is well suited to biomedical imaging espe-cially when the noise model is undetermined [as shown inFigs. 4(a)–4(d)] and mixed [as shown in Fig. 6(a)].

As for the regularization term, we finally choose the aniso-tropic total variation (TV) regularizer in our method, which is anl1 penalty directly on the image gradient. It is a very strongregularizer, which offers improvements on reconstruction qual-ity to a great extent compared with the isotropic counterpart (l2

penalty).29 Therefore, the edges are better preserved, whichcan be seen apparently from the comparison between Figs. 4(a)and 4(b).

4.2 Selection of Hyperparameters

Selection of hyperparameters has an important effect on tomo-graphic reconstruction results. In practice, many learningalgorithms involve hyperparameters (10 or more), such as initiallearning rate, minibatch size, and regularization coefficient.Reference 31 introduces a large number of recommendations fortraining feed-forward neural networks and choosing the multiplehyperparameters, which can make a substantial difference (interms of speed, ease of implementation, and accuracy) whenit comes to putting algorithms to work on real problems.Unfortunately, optimal selection of hyperparameters is challeng-ing due to the high computational cost when using traditionalregularized iterative algorithms.21,23

In this study, our GPU-based implementation of CNN runscomputation-intensive simulations at low cost and is possible tomake the optimization of hyperparameters a more reproducibleand automated process with modern computing facilities. Thus,we can gain better and more robust reconstruction performancewith the GPU-based learning method. During the simulation andexperiment, selection of hyperparameters varies with the bio-logical sample and the range of incident angles. To make a con-vincing comparison, optimal hyperparameters have beenselected for all the other reconstruction methods. The refractiveindex difference δnðrÞ is initialized with a constant value of 0 forall the methods, and different optimal regularization coefficientsare chosen for different regularized loss functions due to the dif-ferent combinations of data-fidelity term and regularizationterm. The number of iterations is set to guarantee the conver-gence of each method, as shown in Fig. 9.

4.3 Subgradient Method and Adam Algorithm

In convex analysis,30 the subgradient generalizes the derivativeto functions that are not differentiable. A vector g ∈ Rn is a sub-gradient of a convex function at x if

EQ-TARGET;temp:intralink-;e005;63;110fðyÞ ≥ fðxÞ þ gTðy − xÞ ∀y: (5)

If f is convex and differentiable, then its gradient at x is asubgradient. But a subgradient can exist even when f is not

differentiable at x. There can be more than one subgradient ofa function f at a point x. The set of all subgradients at x is calledthe subdifferential, and is denoted by ∂fðxÞ. Considering theabsolute value function jxj, the subdifferential is

EQ-TARGET;temp:intralink-;e006;326;528∂jxj ¼8<:

1; x > 0

−1; x < 0

½−1; 1�; x ¼ 0

: (6)

Subgradient methods are subgradient-based iterativemethods for solving nondifferentiable convex minimizationproblems.

Adam is an algorithm for first-order (sub)gradient-basedoptimization of stochastic objective functions, based on adaptiveestimates of lower-order moments. The method is aimed towardmachine learning problems with large datasets and/or high-dimensional parameter spaces. The method is also appropriatefor nonstationary objectives and problems with very noisy and/or sparse gradients. Adam works well in practice and comparesfavorably with other stochastic optimization methods regardingthe computational performance and convergence rate.26 It isstraightforward to implement, is computationally efficient,and has little memory requirements, which is robust and wellsuited to TPM. Compared with the stochastic proximal gradientdescent (SPGD) algorithm reported in Ref. 21, our GPU-basedCNN trained with the Adam algorithm for optimization con-verges to the same SNR level and achieves twice the rate of con-vergence as shown in Fig. 10. To show the higher convergencerate of Adam fairly, we use the proposed l1-norm loss function

0 200 400 600 8000

5

10

15

20

25

30

R

NS de tcur tsnoceR

(dB

)

IFBPODTLAL2 L1L1 L2CNN

Iterations

123456

1

2

3

4

5

6

Fig. 9 Reconstructed SNR plotted as a function of the number of iter-ations for different reconstruction methods on simulated data.Hyperparameters are declared in Sec. 3.1.

AdamSPGD

30

25

20

15

10

5

00 400 800 1200 1600 2000

Iterations

R

NS detcurtsnoceR

(dB

)

Fig. 10 Reconstructed SNR of proposed approach plotted as a func-tion of the number of iterations for two different training optimizationalgorithms on simulated data with the same hyperparametersdeclared in Sec. 3.1.

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Qiao et al.: GPU-based deep convolutional neural network for tomographic phase microscopy. . .

for both the Adam and SPGD training processes here, thus pro-ducing the same reconstructed SNR after convergence.

5 ConclusionWe have demonstrated a GPU-based implementation of deepCNN to model the propagation of light in inhomogeneous samplefor TPM and have applied it to both synthetic and biological sam-ples. The experimental results verify its superior reconstructionperformance over other tomographic reconstruction methods,especially when we take fewer measurements. Furthermore,our CNN is much more general under different optical systemsand arbitrary illumination patterns as its design is illumination-independent. Importantly, this approach can not only enlargethe applications of optical tomography in biomedical imaging,but also open rich perspectives for the potential of deep neuralnetworks in the optical society.

DisclosuresThe authors have no relevant financial interests in the article andno other potential conflicts of interest to disclose.

AcknowledgmentsThis work was supported by the National Natural ScienceFoundation of China (Nos. 61327902 and 61671265). Theauthors gratefully acknowledge Ulugbek S. Kamilov, AlexandreGoy, and Demetri Psaltis for providing the code and their helpfulsuggestions.

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Hui Qiao is a PhD candidate at the Department of Automation,Tsinghua University, Beijing, China. His research interests includebiomedical imaging and deep learning.

Qionghai Dai is currently a professor at the Department ofAutomation, Tsinghua University, Beijing, China. He received his PhDfrom Northeastern University, Liaoning province, China, in 1996.His research interests include microscopy imaging for life science,computational photography, computer vision, and 3-D video.

Biographies for the other authors are not available.

Journal of Biomedical Optics 066003-7 June 2018 • Vol. 23(6)

Qiao et al.: GPU-based deep convolutional neural network for tomographic phase microscopy. . .