Abstract

Taking benefit from recent advances in both phase retrieval and estimation of refractive indices from holographic measurements, we propose a unified framework to reconstruct them from intensity-only measurements. Our method relies on a generic and versatile formulation of the inverse problem and includes sparsity constraints. Its modularity enables the use of a variety of forward models, from simple linear ones to more sophisticated nonlinear ones, as well as various regularizers. We present reconstructions that deploy either the beam-propagation method or the iterative Lippmann-Schwinger model, combined with total-variation regularization. They suggest that our proposed (intensity-only) method can reach the same performance as reconstructions from holographic (complex) data. This is of particular interest from a practical point of view because it allows one to simplify the acquisition setup.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2017 (3)

2016 (5)

T. Zhang, C. Godavarthi, P. C. Chaumet, G. Maire, H. Giovannini, A. Talneau, M. Allain, K. Belkebir, and A. Sentenac, “Far-field diffraction microscopy at λ/10 resolution,” Optica 3, 609–612 (2016).
[Crossref]

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

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. Imaging 2, 59–70 (2016).
[Crossref]

K. Guo, S. Dong, and G. Zheng, “Fourier ptychography for brightfield, phase, darkfield, reflective, multi-slice, and fluorescence imaging,” IEEE J. Sel. Topics Quantum Electron. 22, 77–88 (2016).
[Crossref]

F. Soulez, E. Thiébaut, A. Schutz, A. Ferrari, F. Courbin, and M. Unser, “Proximity operators for phase retrieval,” Appl. Opt. 55, 7412–7421 (2016).
[Crossref] [PubMed]

2015 (7)

2014 (1)

2013 (4)

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

Y. Nesterov, “Gradient methods for minimizing composite functions,” Math. Prog. 140, 125–161 (2013).
[Crossref]

H. Attouch, J. Bolte, and B. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Math. Prog. 137, 91–129 (2013).
[Crossref]

S. Lefkimmiatis, J. Ward, and M. Unser, “Hessian Schatten-norm regularization for linear inverse problems,” IEEE Trans. Image Process. 22, 1873–1888 (2013).
[Crossref] [PubMed]

2012 (3)

W. Van den Broek and C. T. Koch, “Method for retrieval of the three-dimensional object potential by inversion of dynamical electron scattering,” Phys. Rev. Lett. 109, 245502 (2012).
[Crossref]

A. M. Maiden, M. J. Humphry, and J. M. Rodenburg, “Ptychographic transmission microscopy in three dimensions using a multi-slice approach,” J. Opt. Soc. Am. A 29, 1606–1614 (2012).
[Crossref]

E. Mudry, P. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

2011 (3)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40, 120–145 (2011).
[Crossref]

G. Zheng, C. Kolner, and C. Yang, “Microscopy refocusing and dark-field imaging by using a simple LED array,” Opt. Lett. 36, 3987–3989 (2011).
[Crossref] [PubMed]

2009 (3)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[Crossref]

Y. Sung, W. Choi, C. Fang-Yen, K. Badizadegan, R. R. Dasari, and M. S. Feld, “Optical diffraction tomography for high resolution live cell imaging,” Opt. Express 17, 266–277 (2009).
[Crossref] [PubMed]

P. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

2008 (1)

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19, 074009 (2008).
[Crossref]

2007 (2)

M. Yurkin and A. Hoekstra, “The discrete dipole approximation: An overview and recent developments,” J. Quant. Spectrosc. Radiat. Transf. 106, 558–589 (2007).
[Crossref]

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717 (2007).
[Crossref] [PubMed]

2006 (2)

2005 (3)

P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Model Simul. 4, 1168–1200 (2005).
[Crossref]

J. Geffrin, P. Sabouroux, and C. Eyraud, “Free space experimental scattering database continuation: experimental set-up and measurement precision,” Inverse Probl. 21, S117 (2005).
[Crossref]

K. Belkebir, P. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

2004 (2)

G. Gbur and E. Wolf, “The information content of the scattered intensity in diffraction tomography,” Inf. Sci. 162, 3–20 (2004).
[Crossref]

L. Crocco, M. D’Urso, and T. Isernia, “Inverse scattering from phaseless measurements of the total field on a closed curve,” J. Opt. Soc. Am. A 21, 622–631 (2004).
[Crossref]

2002 (1)

A. Abubakar and P. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495 (2002).
[Crossref]

2000 (1)

M. Lambert and D. Lesselier, “Binary-constrained inversion of a buried cylindrical obstacle from complete and phaseless magnetic fields,” Inverse Probl. 16, 563 (2000).
[Crossref]

1999 (1)

P. M. van den Berg, A. Van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325 (1999).
[Crossref]

1998 (1)

1997 (1)

T. Takenaka, D. J. Wall, H. Harada, and M. Tanaka, “Reconstruction algorithm of the refractive index of a cylindrical object from the intensity measurements of the total field,” Microw. Opt. Technol. Lett. 14, 182–188 (1997).
[Crossref]

1995 (1)

1993 (1)

1992 (2)

M. H. Maleki, A. J. Devaney, and A. Schatzberg, “Tomographic reconstruction from optical scattered intensities,” J. Opt. Soc. Am. A 9, 1356–1363 (1992).
[Crossref]

L. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Physica D 60, 259–268 (1992).
[Crossref]

1981 (2)

J. Van Roey, J. Van der Donk, and P. Lagasse, “Beam-propagation method: Analysis and assessment,” J. Opt. Soc. Am. A 71, 803–810 (1981).
[Crossref]

A. J. Devaney, “Inverse-scattering theory within the Rytov approximation,” Opt. Lett. 6, 374–376 (1981).
[Crossref] [PubMed]

1976 (1)

J. Fleck, J. Morris, and M. Feit, “Time-dependent propagation of high energy laser beams through the atmosphere,” Appl. Phys. A 10, 129–160 (1976).
[Crossref]

1971 (1)

R. Gerchberg and W. O. Saxton, “Phase determination from image and diffraction plane pictures in electron-microscope,” Optik 34, 275–284 (1971).

1969 (1)

E. Wolf, “Three-dimensional structure determination of semi-transparent objects from holographic data,” Opt. Commun. 1, 153–156 (1969).
[Crossref]

1962 (1)

J. J. Moreau, “Fonctions convexes duales et points proximaux dans un espace hilbertien,” C.R. Acad. Sci. Paris Ser. A Math. 255, 2897–2899 (1962).

Abubakar, A.

A. Abubakar and P. van den Berg, “The contrast source inversion method for location and shape reconstructions,” Inverse Probl. 18, 495 (2002).
[Crossref]

P. M. van den Berg, A. Van Broekhoven, and A. Abubakar, “Extended contrast source inversion,” Inverse Probl. 15, 1325 (1999).
[Crossref]

Aeppli, G.

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

Allain, M.

Attouch, H.

H. Attouch, J. Bolte, and B. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Math. Prog. 137, 91–129 (2013).
[Crossref]

Badizadegan, K.

Batey, D. J.

P. Li, D. J. Batey, T. B. Edo, and J. M. Rodenburg, “Separation of three-dimensional scattering effects in tilt-series Fourier ptychography,” Ultramicroscopy 158, 1–7 (2015).
[Crossref] [PubMed]

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Imaging Sci. 2, 183–202 (2009).
[Crossref]

Belkebir, K.

Bian, L.

Bolte, J.

H. Attouch, J. Bolte, and B. Svaiter, “Convergence of descent methods for semi-algebraic and tame problems: Proximal algorithms, forward–backward splitting, and regularized Gauss–Seidel methods,” Math. Prog. 137, 91–129 (2013).
[Crossref]

Bostan, E.

D. Ren, E. Bostan, L. Yeh, and L. Waller, “Total-variation regularized Fourier ptychographic microscopy with multiplexed coded illumination,” in “Mathematics in Imaging,” (Optical Society of America, 2017), paper MM3C–5.
[Crossref]

Boufounos, P. T.

U. S. Kamilov, D. Liu, H. Mansour, and P. T. Boufounos, “A recursive Born approach to nonlinear inverse scattering,” IEEE Signal Process. Lett. 23, 1052–1056 (2016).
[Crossref]

H. Y. Liu, D. Liu, H. Mansour, P. T. Boufounos, L. Waller, and U. S. Kamilov, “SEAGLE: Sparsity-driven image reconstruction under multiple scattering,” IEEE Trans. Comput. Imaging, https://arxiv.org/abs/1705.04281 (2017).

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Bunk, O.

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

Chambolle, A.

A. Chambolle and T. Pock, “A first-order primal-dual algorithm for convex problems with applications to imaging,” J. Math. Imaging Vis. 40, 120–145 (2011).
[Crossref]

Charrière, F.

Chaumet, P.

E. Mudry, P. Chaumet, K. Belkebir, and A. Sentenac, “Electromagnetic wave imaging of three-dimensional targets using a hybrid iterative inversion method,” Inverse Probl. 28, 065007 (2012).
[Crossref]

P. Chaumet and K. Belkebir, “Three-dimensional reconstruction from real data using a conjugate gradient-coupled dipole method,” Inverse Probl. 25, 024003 (2009).
[Crossref]

K. Belkebir, P. Chaumet, and A. Sentenac, “Superresolution in total internal reflection tomography,” J. Opt. Soc. Am. A 22, 1889–1897 (2005).
[Crossref]

Chaumet, P. C.

Chen, B.

Chen, F.

Chen, M.

Choi, W.

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Found. Trends Mach. Learn. 3, 1–122 (2011).
[Crossref]

Colomb, T.

Colton, D.

D. Colton and R. Kress, Inverse Acoustic and Electromagnetic Scattering Theory (Springer Science & Business Media, 2012), Vol. 93.

Combettes, P. L.

P. L. Combettes and V. R. Wajs, “Signal recovery by proximal forward-backward splitting,” Multiscale Model Simul. 4, 1168–1200 (2005).
[Crossref]

Courbin, F.

Crocco, L.

Cuche, E.

D’Urso, M.

Dai, Q.

Dasari, R.

W. Choi, C. Fang-Yen, K. Badizadegan, S. Oh, N. Lue, R. Dasari, and M. Feld, “Tomographic phase microscopy,” Nat. Methods 4, 717 (2007).
[Crossref] [PubMed]

Dasari, R. R.

Debailleul, M.

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19, 074009 (2008).
[Crossref]

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Figures (5)

Fig. 1
Fig. 1 Optical diffraction tomography setup (intensity-only). A sample with the refractive index n ∈ ℝM is immersed in a background medium of index ñ and impinged by an incident plane wave with a given orientation (wave vector kb). The interaction of the wave with the object produces a scattered wave (forward and backward). The squared magnitude of the total field, which corresponds to the sum of the the incident and scattered waves, is recorded by the detector.
Fig. 2
Fig. 2 RI distribution for three fibers, a simulated cell, and the Shepp-Logan in the first, second, and third column, respectively. The ground truth and the reconstructions from the LFR, BPM, and LSm proposed methods are shown in Row 1 to 4, respectively. The samples are immersed in water ( n ˜ = 1.33 ). Thirty-one views were acquired with a tilted plane-wave illumination. The angles ranged from −45° to 45°. The sample is illuminated from below. The 1024 sensors are evenly placed on a straight line of length 33λ above the sample at 16.5λ from the center. The measurements were reduced to 512 using averaging.
Fig. 3
Fig. 3 Acquisition setup for the Fresnel dataset. The sensors (dots in the inner circle) correspond to the illumination angle of 0° (i.e., E1). The measurements are restricted either by reducing the number of sensors (sensors sets S241, …, S91) or the number of acquired views (emitters E1, …, E8).
Fig. 4
Fig. 4 Permittivity reconstruction of the Fresnel datasets by LSm. From left to right: FoamDielExt, FoamDielInt, and FoamTwinDiel. From top to bottom: ground truth, reconstructions from complex (using [5]) and intensity-only (proposed method) measurements, respectively, and magnitude and phase of the predicted (solid curve) vs true (dashed curve) measurements (0° illumination angle). The two curves often overlap. For the solutions from complex measurements, the regularization parameters were set at 1.6 · 10−2, 3 · 10−3, and 9 · 10−3 for FoamDielExt, FoamDielInt, and FoamTwinDiel, respectively. For the solutions from intensity-only measurements, the regularization parameters were set at 7 · 10−2, 9 · 10−3, and 4 · 10−2 for FoamDielExt, FoamDielInt, and FoamTwinDiel, respectively.
Fig. 5
Fig. 5 Permittivity reconstructions of the Fresnel dataset with a limited number of measurements. From left to right: P = 3, 5, 7, and 8 views were used to reconstruct the sample FoamDielExt. From top to bottom: The sensors were included in the sets S241, S181, S151, S121, and S91, respectively. The reconstruction error with respect to the best solution (i.e., E8, S241) is shown at the top left of each image.

Tables (3)

Tables Icon

Table 1 Some RI reconstruction algorithms from holographic (i.e., complex) or intensity-only measurements. Ref.: Reference. Algo.: Algorithm. Reg.: Regularization. rec. Born: Recursive Born. BPM: Beam-propagation method. LSm: Lippmann-Schwinger model. GS: Gerchberg-Saxton projection operator. E.: Embedded. TV: Total-variation constraint. a.h.: ad hoc. 3PIE: Ptychographical iterative engine. GD: Gradient descent. FBS: Forward-backward splitting. iter: Iterative.

Tables Icon

Algorithm 1 ADMM for solving Eq. (5)

Tables Icon

Table 2 Reconstruction performance. The relative error ϵ = x ^ x true 2 x true 2 is shown. The proposed method with BPM was 3 to 6 times faster than with LSm.

Equations (21)

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y p = | S ( f , u p in ) | 2 + η p , p [ 1 P ] ,
f ^ { arg min f B ( p = 1 P D ( | S p ( f ) | 2 , y p ) + τ ( Lf ) ) } .
( f ^ , v 1 ^ , , v P ^ ) { arg min f , v 1 , , v P ) X ( p = 1 P D ( | v p | 2 , y p ) + τ ( Lf ) ) } ,
X = { ( f , v 1 , , v P ) B × N × P s . t . v p = X p ( f ) p [ 1 P ] } .
( f , v 1 , , v P , w 1 , , w P , ) = p = 1 P D ( | v p | 2 , y p ) + ρ 2 S p ( f ) v p + w p / ρ 2 2 + τ ( Lf ) ,
prox 1 ρ D ( | | 2 , y p ) ( x ) = arg min v N ( 1 2 v x 2 2 1 ρ D ( | v | 2 , y p ) ) .
D ( | v | 2 , y p ) = 1 2 | v | 2 y p w p 2 ,
x N , [ prox 1 ρ D ( | | 2 , y p ) ( x ) ] n = ϱ n e i arg ( x n ) ,
q G ( ϱ ) = 4 w n p ρ ϱ 3 + ϱ ( 1 4 w n p ρ y p , n ) | x n |
prox ( τ ρ ) ( L ) ( f ) = arg min g 0 M ( 1 2 g f 2 2 + ( τ ρ ) ( Lg ) ) .
( f ) = ρ p = 1 P Re ( J S p H ( f ) ( S p ( f ) z p ( k + 1 ) ) ) ,
{ u 0 ( f ) = u 0 in u z ( f ) = ( u z δ z ( f ) * h prop δ z ) p z ( f )
h prop δ z = 1 { exp ( i δ z ( k ˜ 2 1 M x k x 2 ) ) }   ( diffraction step ) ,
p z ( f ) = exp ( i k ˜ 0 δ z δ n z ( f ) ) ( refraction step ) .
δ n ( f ) = n ˜ ( 1 + f k ˜ 1 ) .
S ( f ) = G ˜ ( u L z ( f ) ) .
u = u in + G d i a g ( f ) u ,
G ( r ) = { i 4 H 0 ( 1 ) ( k ˜ r 2 ) , in 2 D 1 4 π e i k ˜ r 2 r 2 in 3 D .
u ( f ) = arg min u M ( 1 2 ( I G d i a g ( f ) ) u u in 2 2 )
S ( f ) = G ˜ d i a g ( f ) u ( f ) + u in | Γ ,
J S ( f ) = G ˜ ( I + d i a g ( f ) ( I G d i a g ( f ) ) 1 G ) d i a g ( u ( f ) )

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