Abstract

We introduce a Bayesian framework for the ptychotomographic imaging of 3D objects under photon-limited conditions. This approach is significantly more robust to measurement noise by incorporating prior information on the probabilities of the object features and the measurement process into the reconstruction problem, and it can improve both the temporal and spatial resolution of current and future ptychography instruments. We use the alternating direction method of multipliers to solve the proposed optimization problem, where the ptychography and tomography subproblems are both solved using the conjugate-gradient method. The effectiveness of the framework is demonstrated by reconstructing a synthetic 3D integrated chip with a significantly reduced photon-budget for the simulated experiment.

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2019 (4)

S. Aslan, V. Nikitin, D. J. Ching, T. Bicer, S. Leyffer, and D. Gürsoy, “Joint ptycho-tomography reconstruction through alternating direction method of multipliers,” Opt. Express 27(6), 9128–9143 (2019).
[Crossref]

V. Nikitin, M. Carlsson, F. Andersson, and R. Mokso, “Four-dimensional tomographic reconstruction by time domain decomposition,” IEEE Transactions on Comput. Imaging 5(3), 409–419 (2019).
[Crossref]

S. Sala, D. J. Batey, A. Prakash, S. Ahmed, C. Rau, and P. Thibault, “Ptychographic x-ray computed tomography at a high-brilliance x-ray source,” Opt. Express 27(2), 533–542 (2019).
[Crossref]

H. Chang, P. Enfedaque, and S. Marchesini, “Blind ptychographic phase retrieval via convergent alternating direction method of multipliers,” SIAM J. on Imaging Sci. 12(1), 153–185 (2019).
[Crossref]

2018 (5)

S. Jiang, K. Guo, J. Liao, and G. Zheng, “Solving Fourier ptychographic imaging problems via neural network modeling and tensorflow,” Biomed. Opt. Express 9(7), 3306–3319 (2018).
[Crossref]

T. Nguyen, Y. Xue, Y. Li, L. Tian, and G. Nehmetallah, “Deep learning approach for Fourier ptychography microscopy,” Opt. Express 26(20), 26470–26484 (2018).
[Crossref]

H. Chang, P. Enfedaque, Y. Lou, and S. Marchesini, “Partially coherent ptychography by gradient decomposition of the probe,” Acta Crystallogr., Sect. A: Found. Adv. 74(3), 157–169 (2018).
[Crossref]

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast Laplace transforms for the exponential Radon transform,” J. Fourier Analysis Appl. 24(2), 431–450 (2018).
[Crossref]

F. Pfeiffer, “X-ray ptychography,” Nat. Photonics 12(1), 9–17 (2018).
[Crossref]

2017 (4)

M. Holler, M. Guizar-Sicairos, E. H. R. Tsai, R. Dinapoli, E. Müller, O. Bunk, J. Raabe, and G. Aeppli, “High-resolution non-destructive three-dimensional imaging of integrated circuits,” Nature 543(7645), 402–406 (2017).
[Crossref]

D. Gürsoy, “Direct coupling of tomography and ptychography,” Opt. Lett. 42(16), 3169–3172 (2017).
[Crossref]

T. Bicer, D. Gürsoy, V. D. Andrade, R. Kettimuthu, W. Scullin, F. D. Carlo, and I. T. Foster, “Trace: a high-throughput tomographic reconstruction engine for large-scale datasets,” Adv. Struct. Chem. Imaging 3(1), 6–0 (2017).
[Crossref]

D. Gursoy, Y. P. Hong, K. He, K. Hujsak, S. Yoo, S. Chen, and Y. L. et al., “Rapid alignment of nanotomography data using joint iterative reconstruction and reprojection,” Sci. Rep. 7(1), 11818 (2017).
[Crossref]

2016 (4)

D. M. Pelt, D. Gürsoy, W. J. Palenstijn, J. Sijbers, F. De Carlo, and K. J. Batenburg, “Integration of TomoPy and the ASTRA toolbox for advanced processing and reconstruction of tomographic synchrotron data,” J. Synchrotron Radiat. 23(3), 842–849 (2016).
[Crossref]

S. Marchesini, H. Krishnan, B. J. Daurer, D. A. Shapiro, T. Perciano, J. A. Sethian, and F. R. N. C. Maia, “SHARP: a distributed GPU-based ptychographic solver,” J. Appl. Crystallogr. 49(4), 1245–1252 (2016).
[Crossref]

A. Chambolle and T. Pock, “An introduction to continuous optimization for imaging,” Acta Numer. 25, 161–319 (2016).
[Crossref]

F. Andersson, M. Carlsson, and V. V. Nikitin, “Fast algorithms and efficient GPU implementations for the Radon transform and the back-projection operator represented as convolution operators,” SIAM J. on Imaging Sci. 9(2), 637–664 (2016).
[Crossref]

2015 (9)

L.-H. Yeh, J. Dong, J. Zhong, L. Tian, M. Chen, G. Tang, M. Soltanolkotabi, and L. Waller, “Experimental robustness of Fourier ptychography phase retrieval algorithms,” Opt. Express 23(26), 33214–33240 (2015).
[Crossref]

L. Li, X. Wang, and G. Wang, “Alternating direction method of multipliers for separable convex optimization of real functions in complex variables,” Math. Probl. Eng. 2015, 1–14 (2015).
[Crossref]

M. Holler, A. Diaz, M. Guizar-Sicairos, P. Karvinen, E. Färm, E. Härkönen, M. Ritala, A. Menzel, J. Raabe, and O. Bunk, “X-ray ptychographic computed tomography at 16 nm isotropic 3D resolution,” Sci. Rep. 4(1), 3857 (2015).
[Crossref]

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Figures (7)

Fig. 1.
Fig. 1. Schematic of a generic ptychotomography setup for imaging a 3D sample. A coherent x-ray beam spot propagates through the sample, and the diffraction pattern of the transmitted radiation is measured by a pixelated photon counting detector. In order to acquire ptychography data $d$, the sample $u$ is raster-scanned at different positions and views $\theta$ by using the rotation and translation stages. Detected photons at large scattering angles contain high-frequency information about the object, and they often follow a Poisson process at low doses of radiation exposure.
Fig. 2.
Fig. 2. Synthetic model of the chip with features representing different resolution levels. Pixel size is ${10} \,{\textrm {nm}}$, number of pixels $(512\times \!512\times \!512)$. Right panel shows the amplitude difference between real and imaginary parts of the complex refractive index.
Fig. 3.
Fig. 3. Illumination probe function and randomly shifted scan patterns used in simulations (left); photon counts on the detector when scanning the chip model for different probe intensity levels $I$ (middle, right).
Fig. 4.
Fig. 4. Comparison of reconstructions of a slice of the 3D synthetic chip with the maximum likelihood $u_\textrm {ML}=\delta _\textrm {ML}+i\beta _\textrm {ML}$ and least-squares $u_\textrm {LS}=\delta _\textrm {LS}+i\beta _\textrm {LS}$ estimates by using different probe intensity levels $I$. Parameters: object size $(512\!\times \!512\!\times \!512)$, detector size $(128\!\times \!128)$, probe overlap 50%, number of projection angles 768. The ADMM scheme was run for 300 outer iterations with 4 inner CG iterations for each ptychography and tomography subproblem.
Fig. 5.
Fig. 5. L-curve analysis in logarithmic scale for choosing optimal regularization parameter $\alpha$. The panels have inscribed small reconstructed regions for $\alpha \!=\!1\mathrm {e}\!-\!09,1\mathrm {e}\!-\!08,$ and $1\mathrm {e}\!-\!07$, as well as horizontal profiles marked with different colors.
Fig. 6.
Fig. 6. MAP reconstructions ($u_\textrm {MAP}=\delta _\textrm {MAP}+i\beta _\textrm {MAP}$) of a slice of the 3D synthetic chip by using different probe intensity levels $I$. The TV penalty coefficient $\alpha$ is chosen with respect to the L-curve criterion.
Fig. 7.
Fig. 7. Reconstructions of a slice of the 3D synthetic chip from the incomplete number of measurements with $u_\textrm {MAP}$ estimate and TV regularization by using different probe intensity levels $I$. Parameters: object size $(512\!\times \!512\!\times \!512)$, detector size $(128\!\times \!128)$, probe overlap 25%, number of projection angles 192. The ADMM scheme was run for 300 outer iterations with 4 inner CG iterations for each ptychography and tomography subproblem.

Tables (1)

Tables Icon

Table 1. Computational times on NVidia Tesla P100 for recovering objects of the size $N\!\times \! N\!\times \!N$ from the ptychography data generated for $N_\theta =3N/8$ angles and with 25% probe overlap. Each of 300 outer ADMM iterations involves 4 inner ptychography and 4 inner tomography CG iterations.

Equations (29)

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| G H u | 2 = d ,
H u = exp ( i ν R u ) ,
G ψ = F Q ψ ,
p ( d | u ) = j = 1 n e | G H u | j 2 | G H u | j 2 d j d j ! .
u ML = arg min u j = 1 n { | G H u | j 2 2 d j log | G H u | j } .
u MAP = arg min u j = 1 n { | G H u | j 2 2 d j log | G H u | j } + α q ( J u ) .
u LS = arg min u j = 1 n ( | G H u | j d j ) 2 ,
min ψ , φ j = 1 n { | G ψ | j 2 2 d j log | G ψ | j } + α q ( φ ) subject to H u = ψ , J u = φ ,
L ρ , τ ( ψ , u , φ , λ , μ ) = j = 1 n { | G ψ | j 2 2 d j log | G ψ | j } + α q ( φ ) + 2 Re { λ H ( H u ψ ) } + ρ H u ψ 2 2 + 2 Re { μ H ( J u φ ) } + τ J u φ 2 2 ,
ψ k + 1 = arg min ψ L ρ , τ ( ψ , u k , φ k , λ k , μ k ) ,
u k + 1 = arg min u L ρ , τ ( ψ k + 1 , u , φ k , λ k , μ k ) ,
φ k + 1 = arg min φ L ρ , τ ( ψ k + 1 , u k + 1 , φ , λ k , μ k ) ,
λ k + 1 = λ k + ρ ( H u k + 1 ψ k + 1 ) ,
μ k + 1 = μ k + τ ( J u k + 1 φ k + 1 ) .
F ( ψ ) = j = 1 n { | G ψ | j 2 2 d j log | G ψ | j } + ρ H u k ψ + λ k / ρ 2 2 .
ψ F ( ψ ) = G H ( G ψ d ( G ψ ) ) ρ ( H u k ψ + λ k / ρ ) .
η m + 1 = ψ F ( ψ m + 1 ) + ψ F ( ψ m + 1 ) 2 2 ( ψ F ( ψ m + 1 ) ψ F ( ψ m ) ) H η m η m
F ~ ( ψ ) = j = 1 n ( | G ψ | j d ) j 2 + ρ H u k ψ + λ k / ρ 2 2 ,
ψ F ~ ( ψ ) = G H ( G ψ d G ψ | G ψ | ) ρ ( H u k ψ + λ k / ρ ) .
F ( u ) = ρ H u ψ k + 1 + λ k / ρ 2 2 + τ J u φ k + μ k / τ 2 2
H u ψ k + 1 + λ k / ρ 2 2 = e 2 π i ν R u ψ k + 1 + λ k / ρ 2 2 2 π i ν ( ψ k + 1 λ k / ρ ) ( R u + i ν 2 π log ( ψ k + 1 λ k / ρ ) ) 2 2 ,
F ( u ) ρ K R u ξ 1 2 2 + τ J u ξ 2 2 2 ,
K R u = 2 π i ν ( ψ k + 1 λ k / ρ ) R u
ξ 1 = ( ψ k + 1 λ k / ρ ) log ( ψ k + 1 λ k / ρ ) , ξ 2 = φ k μ k / τ .
u F ( u ) = ρ R T K H ( K R u ξ 1 ) + τ J T ( J u ξ 2 ) .
F ( φ ) = α q ( φ ) + τ J u k + 1 φ + μ k / τ 2 2 .
q ( φ ) = | u | 1 ,
φ ~ = u k + 1 + μ k / τ | u k + 1 + μ k / τ | max ( 0 , | u k + 1 + μ k / τ | α / τ ) ,
I = 3 J / ( m 2 s ) : α = 8 e 09 , I = 0.3 J / ( m 2 s ) : α = 1 e 08 , I = 0.03 J / ( m 2 s ) : α = 2 e 08 , I = 0.003 J / ( m 2 s ) : α = 4 e 08 .