Abstract

This paper includes a tutorial on how to reconstruct in-line holograms using an inverse problems approach, starting with modeling the observations, selecting regularizations and constraints, and ending with the design of a reconstruction algorithm. A special focus is placed on the connections between the numerous alternating projections strategies derived from Fienup’s phase retrieval technique and the inverse problems framework. In particular, an interpretation of Fienup’s algorithm as iterates of a proximal gradient descent for a particular cost function is given. Reconstructions from simulated and experimental holograms of micrometric beads illustrate the theoretical developments. The results show that the transition from alternating projections techniques to the inverse problems formulation is straightforward and advantageous.

© 2019 Optical Society of America

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2019 (3)

2018 (4)

F. Jolivet, F. Momey, L. Denis, L. Méès, N. Faure, N. Grosjean, F. Pinston, J.-L. Marié, and C. Fournier, “Regularized reconstruction of absorbing and phase objects from a single in-line hologram, application to fluid mechanics and micro-biology,” Opt. Express 26, 8923–8940 (2018).
[Crossref]

Y. Wu and A. Ozcan, “Lensless digital holographic microscopy and its applications in biomedicine and environmental monitoring,” Methods 136, 4–16 (2018).
[Crossref]

O. Flasseur, F. Jolivet, F. Momey, L. Denis, and C. Fournier, “Improving color lensless microscopy reconstructions by self-calibration,” Proc. SPIE 10677, 106771A (2018).
[Crossref]

J. Bailleul, B. Simon, M. Debailleul, L. Foucault, N. Verrier, and O. Haeberlé, “Tomographic diffractive microscopy: towards high-resolution 3-D real-time data acquisition, image reconstruction and display of unlabeled samples,” Opt. Commun. 422, 28–37 (2018).
[Crossref]

2017 (3)

2016 (4)

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]

D. Noll and A. Rondepierre, “On local convergence of the method of alternating projections,” Found. Comput. Math. 16, 425–455 (2016).
[Crossref]

J. Song, C. L. Swisher, H. Im, S. Jeong, D. Pathania, Y. Iwamoto, M. Pivovarov, R. Weissleder, and H. Lee, “Sparsity-based pixel super resolution for lens-free digital in-line holography,” Sci. Rep. 6, 24681 (2016).
[Crossref]

Y. Rivenson, Y. Wu, H. Wang, Y. Zhang, A. Feizi, and A. Ozcan, “Sparsity-based multi-height phase recovery in holographic microscopy,” Sci. Rep. 6, 37862 (2016).
[Crossref]

2015 (1)

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

2014 (3)

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 127–239 (2014).
[Crossref]

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

R. Horisaki, Y. Ogura, M. Aino, and J. Tanida, “Single-shot phase imaging with a coded aperture,” Opt. Lett. 39, 6466–6469 (2014).
[Crossref]

2013 (2)

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

J. A. Rodriguez, R. Xu, C.-C. Chen, Y. Zou, and J. Miao, “Oversampling smoothness: an effective algorithm for phase retrieval of noisy diffraction intensities,” J. Appl. Crystallogr. 46, 312–318 (2013).
[Crossref]

2012 (1)

2011 (1)

2010 (4)

W. Bishara, T.-W. Su, A. F. Coskun, and A. Ozcan, “Lensfree on-chip microscopy over a wide field-of-view using pixel super-resolution,” Opt. Express 18, 11181–11191 (2010).
[Crossref]

Y. Rivenson, A. Stern, and B. Javidi, “Compressive Fresnel holography,” J. Display Technol. 6, 506–509 (2010).
[Crossref]

R. A. Dilanian, G. J. Williams, L. W. Whitehead, D. J. Vine, A. G. Peele, E. Balaur, I. McNulty, H. M. Quiney, and K. A. Nugent, “Coherent diffractive imaging: a new statistically regularized amplitude constraint,” New J. Phys. 12, 093042 (2010).
[Crossref]

H. N. Chapman and K. A. Nugent, “Coherent lensless x-ray imaging,” Nat. Photonics 4(12), 833–839 (2010).
[Crossref]

2009 (3)

2008 (1)

A. Ribes and F. Schmitt, “Linear inverse problems in imaging,” IEEE Signal Process. Mag. 25(4), 84–99 (2008).
[Crossref]

2007 (5)

T. Latychevskaia and H.-W. Fink, “Solution to the twin image problem in holography,” Phys. Rev. Lett. 98, 233901 (2007).
[Crossref]

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

S. Marchesini, “Invited article: a unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78, 011301 (2007).
[Crossref]

F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse-problem approach for particle digital holography: accurate location based on local optimization,” J. Opt. Soc. Am. A 24, 1164–1171 (2007).
[Crossref]

F. Soulez, L. Denis, E. Thiébaut, C. Fournier, and C. Goepfert, “Inverse problem approach in particle digital holography: out-of-field particle detection made possible,” J. Opt. Soc. Am. A 24, 3708–3716 (2007).
[Crossref]

2006 (1)

M. Defrise, F. Noo, R. Clackdoyle, and H. Kudo, “Truncated Hilbert transform and image reconstruction from limited tomographic data,” Inverse Probl. 22, 1037 (2006).
[Crossref]

2005 (1)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21, 37–50 (2005).
[Crossref]

2004 (3)

S. P. Hau-Riege, H. Szoke, H. N. Chapman, A. Szoke, S. Marchesini, A. Noy, H. He, M. Howells, U. Weierstall, and J. C. H. Spence, “SPEDEN: reconstructing single particles from their diffraction patterns,” Acta Crystallogr. Sec. A 60, 294–305 (2004).
[Crossref]

I. Daubechies, M. Defrise, and C. De Mol, “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math. 57, 1413–1457 (2004).
[Crossref]

S. Sotthivirat and J. A. Fessler, “Penalized-likelihood image reconstruction for digital holography,” J. Opt. Soc. Am. A 21, 737–750 (2004).
[Crossref]

2003 (2)

H. H. Bauschke, P. L. Combettes, and D. R. Luke, “Hybrid projection-reflection method for phase retrieval,” J. Opt. Soc. Am. A 20, 1025–1034 (2003).
[Crossref]

V. Elser, “Solution of the crystallographic phase problem by iterated projections,” Acta Crystallogr. Sec. A 59, 201–209 (2003).
[Crossref]

2002 (2)

1999 (1)

J. Miao, P. Charalambous, J. Kirz, and D. Sayre, “Extending the methodology of x-ray crystallography to allow imaging of micrometre-sized non-crystalline specimens,” Nature 400, 342–344 (1999).
[Crossref]

1998 (1)

1997 (1)

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[Crossref]

1994 (1)

J. A. Fessler, “Penalized weighted least-squares image reconstruction for positron emission tomography,” IEEE Trans. Med. Imag. 13, 290–300 (1994).
[Crossref]

1992 (1)

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

1984 (1)

1982 (1)

1980 (1)

J. Nocedal, “Updating quasi-Newton matrices with limited storage,” Math. Comp. 35, 773–782 (1980).
[Crossref]

1978 (1)

1974 (1)

1972 (1)

R. Gerchberg and W. Saxton, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

1969 (1)

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

1948 (1)

D. Gabor, “A new microscopic principle,” Nature 161, 777–778 (1948).
[Crossref]

1908 (1)

G. Mie, “Beiträge zur Optik trüber Medien, speziell kolloidaler Metallösungen,” Ann. Phys. 330, 377–445 (1908).
[Crossref]

Aino, M.

Allier, C.

Aubert, G.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Bailleul, J.

J. Bailleul, B. Simon, M. Debailleul, L. Foucault, N. Verrier, and O. Haeberlé, “Tomographic diffractive microscopy: towards high-resolution 3-D real-time data acquisition, image reconstruction and display of unlabeled samples,” Opt. Commun. 422, 28–37 (2018).
[Crossref]

Balaur, E.

R. A. Dilanian, G. J. Williams, L. W. Whitehead, D. J. Vine, A. G. Peele, E. Balaur, I. McNulty, H. M. Quiney, and K. A. Nugent, “Coherent diffractive imaging: a new statistically regularized amplitude constraint,” New J. Phys. 12, 093042 (2010).
[Crossref]

Baraniuk, R. G.

M. L. Moravec, J. K. Romberg, and R. G. Baraniuk, “Compressive phase retrieval,” Proc. SPIE 6701, 670120 (2007).
[Crossref]

Barlaud, M.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Bauschke, H. H.

Beck, A.

Y. Shechtman, A. Beck, and Y. C. Eldar, “GESPAR: efficient phase retrieval of sparse signals,” IEEE Trans. Signal Process. 62, 928–938 (2014).
[Crossref]

A. Beck and M. Teboulle, “Fast gradient-based algorithms for constrained total variation image denoising and deblurring problems,” IEEE Trans. Image Process. 18, 2419–2434 (2009).
[Crossref]

Berdeu, A.

Bishara, W.

Blanc-Féraud, L.

P. Charbonnier, L. Blanc-Féraud, G. Aubert, and M. Barlaud, “Deterministic edge-preserving regularization in computed imaging,” IEEE Trans. Image Process. 6, 298–311 (1997).
[Crossref]

Bordy, T.

Boss, D.

Y. Cotte, F. Toy, P. Jourdain, N. Pavillon, D. Boss, P. Magistretti, P. Marquet, and C. Depeursinge, “Marker-free phase nanoscopy,” Nat. Photonics 7, 113–117 (2013).
[Crossref]

Boyd, S.

N. Parikh and S. Boyd, “Proximal algorithms,” Found. Trends Optim. 1, 127–239 (2014).
[Crossref]

Brady, D. J.

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

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A. Drémeau and F. Krzakala, “Phase recovery from a Bayesian point of view: the variational approach,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2015), pp. 3661–3665.

A. M. Tillmann, Y. C. Eldar, and J. Mairal, “Dictionary learning from phaseless measurements,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2016), pp. 4702–4706.

S. Mukherjee and C. S. Seelamantula, “An iterative algorithm for phase retrieval with sparsity constraints: application to frequency domain optical coherence tomography,” in IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP) (2012), pp. 553–556.

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Figures (9)

Fig. 1.
Fig. 1. Hologram reconstruction based on inverse problems approaches: the direct model connects the sample of interest to the measurements (the holograms); inverting this model leads to a reconstruction and/or estimation of the optical parameters of the sample. The case chosen as an illustration in this paper is highlighted in yellow.
Fig. 2.
Fig. 2. In-line holography principle: a plane wave illuminates the sample plane with normal incidence. The presence of inhomogeneity in the spatial distribution of the complex-valued refractive index, in the sample plane, distorts the illumination wave. A diffraction pattern is recorded on the sensor plane.
Fig. 3.
Fig. 3. Illustration of the behavior of several regularization terms on reconstructions (in a region of interest) of a simulated hologram (a) using a regularized inverse problems approach. Details on the method, the simulation, and reconstruction parameters are given in Section 5 and Tables 1 and 2. (a) Data (in-line hologram). The yellow frame indicates the region of interest that is extracted from the field of view for visualization. (b) Ground truth phase. (c)–(f) Reconstructed images using the following regularizations: (c) squared $ {l_2} $ norm of the gradient $ {{\cal J}_{{{l}_{2}}\nabla }} $ [Eq. (15)], (d) edge-preserving smoothness $ {{\cal J}_{{{\text{TV}}_ \epsilon }}} $ [Eq. (16)], (e) $ {l_1} $ sparsity constraint $ {{\cal J}_{{l_1}}} $ [Eq. (17)], and (f) composition of the $ {l_1} $ sparsity constraint and edge-preserving smoothness $ {{\cal J}_{\textsf{reg}}} = {{\cal J}_{{l_1}}} + {{\cal J}_{{{\text{TV}}_ \epsilon }}} $ [Eq. (18)]. Red curves show a line profile passing through two particles of the image.
Fig. 4.
Fig. 4. Principle that underlies proximal gradient methods is the iterative minimization of a local approximation of the original cost function. If the parameter $ t $ is chosen small enough, the approximation is majorant, and minimizing the approximation improves the current solution until convergence. When the non-smooth component $ {\cal H} $ is separable (as is the case of the $ {\ell _1} $ norm), minimizing the local approximation is easily done [see Fig. (5)].
Fig. 5.
Fig. 5. Computation of the proximal operator: an illustration with the case of the $ {l_1} $ norm under positivity constraints. Values that are smaller than $ \mu t $ are mapped to zero by the proximal operator. Larger values of the regularization weight $ \mu $ therefore lead to more values being set to zero, i.e., a sparser solution.
Fig. 6.
Fig. 6. Phase image reconstructed from simulated data with different methods and regularization terms. (a) Data (in-line hologram). The yellow frames indicate the regions of interest that are extracted from the field of view for visualization. (b) Ground truth phase. (c) Evolution of the normalized cost criterion (minimum and maximum values ranged between zero and one) for each reconstruction. (d)–(i) Reconstructed images using (d) Fienup method, (e) Fienup method + soft-thresholding, (f) Fienup method + soft-thresholding, stopped at 10 iterations, (g) RI method using FISTA with soft-thresholding, (h) RI method using FISTA with soft-thresholding + edge-preserving, (i) RI method using FISTA with soft-thresholding + edge-preserving, stopped at 10 iterations. For each reconstruction, a positivity constraint is imposed on the solution. The values of the hyperparameters are indicated in Table 2. Black dots delimit the contour of the ground truth image. Red curves show line profiles passing through particles of the image.
Fig. 7.
Fig. 7. Phase image reconstructed from experimental data with different methods and regularization terms. (a) Data (in-line hologram). The yellow frame indicates the region of interest that is extracted from the field of view for visualization. (b) Evolution of the normalized cost criterion (minimum and maximum values ranged between zero and one) for each reconstruction. (c)–(h) Reconstructed images using (c) Fienup method, (d) Fienup method + soft-thresholding, (e) Fienup method + soft-thresholding, stopped at 10 iterations, (f) RI method using FISTA with soft-thresholding, (g) RI method using FISTA with soft-thresholding + edge-preserving, (h) RI method using FISTA with soft-thresholding + edge-preserving, stopped at 10 iterations. For each reconstruction, a positivity constraint is imposed on the solution. The values of the hyperparameters are indicated in Table 2. Red curves show a line profile passing through two particles of the image.
Fig. 8.
Fig. 8. Reconstruction (detection) of an out-of-field particle using RI method and field-of-view extension. (a) Data in-line hologram highlighting (in yellow) a part of the out-of-field hologram. (b) Ground truth image. (c), (d) Reconstructions with two different sets of regularization weights values [cf. Table 2]. The red frames show the initial data field of view (pixels seen by the detector). The green frames are zooms on the out-of-field particle.
Fig. 9.
Fig. 9. Derivation of a reconstruction algorithm from a cost function.

Tables (6)

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Algorithm 1. Fienup’s ER algorithm [4]

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Algorithm 2. FISTA algorithm [61] for solving problem Eq. (19)

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Table 1. Experimental and Simulation Parameters of In-Line Holograms Data

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Table 2. Reconstruction Parameters for Simulations and Experiments in Figs. 3,68

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Algorithm 3. Detailed implementation of Algorithm 1

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Algorithm 4. Detailed implementation of Algorithm 2

Equations (29)

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t _ ( r ) = 1 + o _ ( r ) = ρ ( r ) e i φ ( r ) .
I ( r ) = | R 2 h _ z ( r r ) a _ 0 ( r ) t _ ( r ) d r | 2 = | h _ z r ( a _ 0 t _ ) | 2 ,
h _ z ( r ) = 1 i λ z exp ( i π λ z r 2 ) .
I ( r ) = | a _ 0 | 2 | 1 + h _ z r o _ | 2 = | a _ 0 | 2 ( 1 + 2 R e ( h _ z r o _ ) + | h _ z r o _ | 2 ) model m ( o _ ) .
d = c m ( o _ ) + η = c | 1 + H _ z o _ | 2 + η ,
t _ ( r ) = 1 + o _ ( r ) = e i φ ( r ) 1 + i φ ( r ) = 1 + i o ( r ) ;
m ~ ( o ) = 1 2 J m ( h _ z ) o .
d = c m ~ ( o ) + η with m ~ ( o ) = ( 1 + G z o ) ,
J fid ( c , o , d ) = c m ~ ( o ) d W 2 = ( c m ~ ( o ) d ) T W ( c m ~ ( o ) d ) = q w q ( m ~ q ( o ) d q ) 2 ( if errors are uncorrelated,i.e. , W is diagonal ) = q w q ( 1 + [ g z o ] q d q ) 2 ,
{ o , c } = argmin o , c J fid ( c , o , d )
c ( o ) = m ~ ( o ) T W d m ~ ( o ) T W m ~ ( o ) .
c ( o ) = q w q m ~ q ( o ) d q q w q m ~ q ( o ) 2 .
o = arg min o J fid ( c ( o ) , o , d ) .
o = arg min o O J fid ( c ( o ) , o , d ) + J reg ( o , θ ) ,
J l 2 ( o , μ ) = μ q | | q o | | 2 2 ,
J TV ϵ ( o , μ , ϵ ) = μ q | | q o | | 2 2 + ϵ 2 ,
J l 1 ( o , μ ) = μ o 1 = μ q | o q | .
J reg ( o , θ ) = J l 1 ( o , μ l 1 ) + J TV ϵ ( o , μ TV , ϵ TV ) ,
o = arg min o 0 J fid ( c ( o ) , o , d ) smooth part G + J l 1 ( o , μ ) non smooth part H = arg min o 0 c ( o ) m ~ ( o ) d W 2 + μ o 1 .
o ( i + 1 ) = T μ t ( o ( i ) t G ( o ( i ) ) ) = T μ t ( o ( i ) t J fid ( c ( o ( i ) ) , o ( i ) , d ) ) = T μ t ( o ( i ) 2 t c ( o ( i ) ) G z T W ( c ( o ( i ) ) m ~ ( o ( i ) ) d ) ) ,
T α ( o ) q = max ( 0 , o q α ) .
o _ ( i + 1 ) = P O ( H _ z ( d ¯ a _ z ( i + 1 / 2 ) | a _ z ( i + 1 / 2 ) | 1 ) ) ,
o _ ( i + 1 ) P O [ o _ ( i ) H _ z ( a _ z ( i + 1 / 2 ) | a _ z ( i + 1 / 2 ) | ( | a _ z ( i + 1 / 2 ) | d ¯ ) ) ] .
J fid ( o , d ¯ ) = | a z _ ( o _ ) | d ¯ 2 2 = | 1 + H _ z o _ | d ¯ 2 2 .
ι O ( o _ ) = { 0 if o _ O + otherwise .
o _ = a r g m i n o _ | 1 + H _ z o _ | d ¯ 2 2 s m o o t h p a r t G + ι O ( o _ ) n o n s m o o t h p a r t H ,
o = a r g m i n o 0 J fid ( c ( o ) , o , d ) + J T V ϵ ( o , μ T V ) s m o o t h p a r t G + J l 1 ( o , μ l 1 ) n o n s m o o t h p a r t H = a r g m i n o 0 c ( o ) m ~ ( o ) d W 2 + μ T V q | | q o | | 2 2 + ϵ 2 + μ l 1 o 1 .
o ( i + 1 ) = T μ l 1 t ( o ( i ) t ( J fid ( c ( o ( i ) ) , o ( i ) , d ) + J T V ϵ ( o ( i ) , μ TV ) ) ) = T μ l 1 t ( ( q o ( i ) | | q o ( i ) | | 2 2 + ϵ 2 ) o ( i ) 2 t c ( o ( i ) ) G z T W ( c m ~ ( o ( i ) ) d ) t μ TV q q T ( q o ( i ) | | q o ( i ) | | 2 2 + ϵ 2 ) ) .
m ~ ( o ) = T ( 1 + G z o ) .