Abstract

Alternating projection based methods, such as ePIE and rPIE, have been used widely in ptychography. However, they only work well if there are adequate measurements (diffraction patterns); in the case of sparse data (i.e. fewer measurements) alternating projection underperforms and might not even converge. In this paper, we propose semi-implicit relaxed Douglas-Rachford (sDR), an accelerated iterative method, to solve the classical ptychography problem. Using both simulated and experimental data, we show that sDR improves the convergence speed and the reconstruction quality relative to extended ptychographic iterative engine (ePIE) and regularized ptychographic iterative engine (rPIE). Furthermore, in certain cases when sparsity is high, sDR converges while ePIE and rPIE fail or encounter slow convergence. To facilitate others to use the algorithm, we post the Matlab source code of sDR on a public website (www.physics.ucla.edu/research/imaging/sDR/index.html). We anticipate that this algorithm can be generally applied to the ptychographic reconstruction of a wide range of samples in the physical and biological sciences.

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

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References

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2018 (5)

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

J. Deng, Y. H. Lo, M. Gallagher-Jones, S. Chen, A. Pryor, Q. Jin, Y. P. Hong, Y. S. G. Nashed, S. Vogt, J. Miao, and C. Jacobsen, “Correlative 3d x-ray fluorescence and ptychographic tomography of frozen-hydrated green algae,” Sci. Adv. 4(11), eaau4548 (2018).
[Crossref]

Y. Jiang, Z. C. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
[Crossref]

Y. H. Lo, L. Zhao, M. Gallagher-Jones, A. Rana, J. J. Lodico, W. Xiao, B. C. Regan, and J. Miao, “In situ coherent diffractive imaging,” Nat. Commun. 9(1), 1826 (2018).
[Crossref]

M. Rose, T. Senkbeil, A. R. von Gundlach, S. Stuhr, C. Rumancev, D. Dzhigaev, I. Besedin, P. Skopintsev, L. Loetgering, J. Viefhaus, A. Rosenhahn, and I. A. Vartanyants, “Quantitative ptychographic bio-imaging in the water window,” Opt. Express 26(2), 1237–1254 (2018).
[Crossref]

2017 (5)

A. Maiden, D. Johnson, and P. Li, “Further improvements to the ptychographical iterative engine,” Optica 4(7), 736–745 (2017).
[Crossref]

S. Gao, P. Wang, F. Zhang, G. T. Martinez, P. D. Nellist, X. Pan, and A. I. Kirkland, “Electron ptychographic microscopy for three-dimensional imaging,” Nat. Commun. 8(1), 163 (2017).
[Crossref]

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

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]

M. Gallagher-Jones, C. S. B. Dias, A. Pryor, K. Bouchmella, L. Zhao, Y. H. Lo, M. B. Cardoso, D. Shapiro, J. Rodriguez, and J. Miao, “Correlative cellular ptychography with functionalized nanoparticles at the fe l-edge,” Sci. Rep. 7(1), 4757 (2017).
[Crossref]

2016 (3)

2015 (7)

S. Bubeck, “Convex optimization: Algorithms and complexity,” FNT in Machine Learning 8(3-4), 231–357 (2015).
[Crossref]

A. M. Maiden, M. C. Sarahan, M. D. Stagg, S. M. Schramm, and M. J. Humphry, “Quantitative electron phase imaging with high sensitivity and an unlimited field of view,” Sci. Rep. 5(1), 14690 (2015).
[Crossref]

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]

J. Miao, T. Ishikawa, I. K. Robinson, and M. M. Murnane, “Beyond crystallography: Diffractive imaging using coherent x-ray light sources,” Science 348(6234), 530–535 (2015).
[Crossref]

R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8(1), 426–457 (2015).
[Crossref]

L. Bian, J. Suo, G. Zheng, K. Guo, F. Chen, and Q. Dai, “Fourier ptychographic reconstruction using wirtinger flow optimization,” Opt. Express 23(4), 4856–4866 (2015).
[Crossref]

R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17(5), 053044 (2015).
[Crossref]

2014 (4)

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89(6), 064101 (2014).
[Crossref]

D. A. Shapiro, Y.-S. Yu, T. Tyliszczak, J. Cabana, R. Celestre, W. Chao, K. Kaznatcheev, A. L. D. Kilcoyne, F. Maia, S. Marchesini, Y. S. Meng, T. Warwick, L. L. Yang, and H. A. Padmore, “Chemical composition mapping with nanometre resolution by soft x-ray microscopy,” Nat. Photonics 8(10), 765–769 (2014).
[Crossref]

N. Parikh and S. Boyd, “Proximal algorithms,” FNT in Optimization 1(3), 127–239 (2014).
[Crossref]

A. Tripathi, I. McNulty, and O. G. Shpyrko, “Ptychographic overlap constraint errors and the limits of their numerical recovery using conjugate gradient descent methods,” Opt. Express 22(2), 1452–1466 (2014).
[Crossref]

2013 (4)

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(2), 312–318 (2013).
[Crossref]

J. Marrison, L. Räty, P. Marriott, and P. O’Toole, “Ptychography – a label free, high-contrast imaging technique for live cells using quantitative phase information,” Sci. Rep. 3(1), 2369 (2013).
[Crossref]

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013).
[Crossref]

F. Zhang, I. Peterson, J. Vila-Comamala, A. Diaz, F. Berenguer, R. Bean, B. Chen, A. Menzel, I. K. Robinson, and J. M. Rodenburg, “Translation position determination in ptychographic coherent diffraction imaging,” Opt. Express 21(11), 13592–13606 (2013).
[Crossref]

2012 (5)

A. M. Maiden, M. J. Humphry, M. C. Sarahan, B. Kraus, and J. M. Rodenburg, “An annealing algorithm to correct positioning errors in ptychography,” Ultramicroscopy 120, 64–72 (2012).
[Crossref]

P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14(6), 063004 (2012).
[Crossref]

P. Godard, M. Allain, V. Chamard, and J. Rodenburg, “Noise models for low counting rate coherent diffraction imaging,” Opt. Express 20(23), 25914–25934 (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(8), 1606–1614 (2012).
[Crossref]

Z. Wen, C. Yang, X. Liu, and S. Marchesini, “Alternating direction methods for classical and ptychographic phase retrieval,” Inverse Probl. 28(11), 115010 (2012).
[Crossref]

2010 (1)

M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
[Crossref]

2009 (3)

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy 109(10), 1256–1262 (2009).
[Crossref]

I. Robinson and R. Harder, “Coherent x-ray diffraction imaging of strain at the nanoscale,” Nat. Mater. 8(4), 291–298 (2009).
[Crossref]

M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. Relat. Phenom. 170(1-3), 4–12 (2009).
[Crossref]

2008 (2)

P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning x-ray diffraction microscopy,” Science 321(5887), 379–382 (2008).
[Crossref]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express 16(10), 7264–7278 (2008).
[Crossref]

2007 (3)

J. M. Rodenburg, A. C. Hurst, A. G. Cullis, B. R. Dobson, F. Pfeiffer, O. Bunk, C. David, K. Jefimovs, and I. Johnson, “Hard-x-ray lensless imaging of extended objects,” Phys. Rev. Lett. 98(3), 034801 (2007).
[Crossref]

K. J. Gaffney and H. N. Chapman, “Imaging atomic structure and dynamics with ultrafast x-ray scattering,” Science 316(5830), 1444–1448 (2007).
[Crossref]

S. Marchesini, “Invited article: A unified evaluation of iterative projection algorithms for phase retrieval,” Rev. Sci. Instrum. 78(1), 011301 (2007).
[Crossref]

2005 (2)

D. R. Luke, “Relaxed averaged alternating reflections for diffraction imaging,” Inverse Probl. 21(1), 37–50 (2005).
[Crossref]

M. Van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151(3), 250–262 (2005).
[Crossref]

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(6742), 342–344 (1999).
[Crossref]

1998 (1)

1992 (1)

J. Eckstein and D. P. Bertsekas, “On the douglas-rachford splitting method and the proximal point algorithm for maximal monotone operators,” Math. Program. 55(1-3), 293–318 (1992).
[Crossref]

1991 (1)

P. Tseng, “Applications of a splitting algorithm to decomposition in convex programming and variational inequalities,” SIAM J. Control Optim. 29(1), 119–138 (1991).
[Crossref]

1982 (1)

1979 (1)

P.-L. Lions and B. Mercier, “Splitting algorithms for the sum of two nonlinear operators,” SIAM J. Numer. Anal. 16(6), 964–979 (1979).
[Crossref]

1976 (1)

R. Hegerl and W. Hoppe, “Influence of electron noise on three-dimensional image reconstruction,” Zeitschrift für Naturforschung A 31(12), 1717–1721 (1976).
[Crossref]

1956 (1)

J. Douglas and H. H. Rachford, “On the numerical solution of heat conduction problems in two and three space variables,” Trans. Amer. Math. Soc. 82(2), 421 (1956).
[Crossref]

Adams, D. E.

D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
[Crossref]

Aeppli, G.

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]

Allain, M.

Allen, L. J.

A. J. D’Alfonso, A. J. Morgan, A. W. C. Yan, P. Wang, H. Sawada, A. I. Kirkland, and L. J. Allen, “Deterministic electron ptychography at atomic resolution,” Phys. Rev. B 89(6), 064101 (2014).
[Crossref]

Ames, B.

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M. Gallagher-Jones, C. S. B. Dias, A. Pryor, K. Bouchmella, L. Zhao, Y. H. Lo, M. B. Cardoso, D. Shapiro, J. Rodriguez, and J. Miao, “Correlative cellular ptychography with functionalized nanoparticles at the fe l-edge,” Sci. Rep. 7(1), 4757 (2017).
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M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. Relat. Phenom. 170(1-3), 4–12 (2009).
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A. M. Maiden, M. C. Sarahan, M. D. Stagg, S. M. Schramm, and M. J. Humphry, “Quantitative electron phase imaging with high sensitivity and an unlimited field of view,” Sci. Rep. 5(1), 14690 (2015).
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M. Howells, T. Beetz, H. Chapman, C. Cui, J. Holton, C. Jacobsen, J. Kirz, E. Lima, S. Marchesini, H. Miao, D. Sayre, D. Shapiro, J. Spence, and D. Starodub, “An assessment of the resolution limitation due to radiation-damage in x-ray diffraction microscopy,” J. Electron Spectrosc. Relat. Phenom. 170(1-3), 4–12 (2009).
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R. Hesse, D. R. Luke, S. Sabach, and M. K. Tam, “Proximal heterogeneous block implicit-explicit method and application to blind ptychographic diffraction imaging,” SIAM J. Imaging Sci. 8(1), 426–457 (2015).
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D. F. Gardner, M. Tanksalvala, E. R. Shanblatt, X. Zhang, B. R. Galloway, C. L. Porter, R. Karl Jr, C. Bevis, D. E. Adams, H. C. Kapteyn, M. M. Murnane, and G. F. Mancini, “Subwavelength coherent imaging of periodic samples using a 13.5 nm tabletop high-harmonic light source,” Nat. Photonics 11(4), 259–263 (2017).
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Y. Jiang, Z. C. Chen, Y. Han, P. Deb, H. Gao, S. Xie, P. Purohit, M. W. Tate, J. Park, S. M. Gruner, V. Elser, and D. A. Muller, “Electron ptychography of 2d materials to deep sub-ångström resolution,” Nature 559(7714), 343–349 (2018).
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P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature 494(7435), 68–71 (2013).
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P. Thibault and M. Guizar-Sicairos, “Maximum-likelihood refinement for coherent diffractive imaging,” New J. Phys. 14(6), 063004 (2012).
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M. Dierolf, A. Menzel, P. Thibault, P. Schneider, C. M. Kewish, R. Wepf, O. Bunk, and F. Pfeiffer, “Ptychographic x-ray computed tomography at the nanoscale,” Nature 467(7314), 436–439 (2010).
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R. Horstmeyer, R. Y. Chen, X. Ou, B. Ames, J. A. Tropp, and C. Yang, “Solving ptychography with a convex relaxation,” New J. Phys. 17(5), 053044 (2015).
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Figures (8)

Fig. 1.
Fig. 1. Flow chart of the sDR algorithm.
Fig. 2.
Fig. 2. A simulated complex object with the amplitude being a camera man image shown in (a) and the phase being a pepper image shown in (b).
Fig. 3.
Fig. 3. The reconstructions of ePIE, rPIE and sDR of a complex object consisting of $128\times 128$ pixels, a scan step size of 35 pixels and $4\times 4$ diffraction patterns. Poisson noise was added to the diffraction patterns with $R_{noise} = 3.73\%$. (a-c) The amplitude and (d-f) the phase of ePIE, rPIE and sDR reconstructions, respectively.
Fig. 4.
Fig. 4. Ptychographic reconstructions of sparse data by ePIE, rPIE and sDR. The data consist of $3\times 3$ diffraction patterns with a scan step of 50 pixels. Poisson noise was added to the diffraction patterns with $R_{noise} = 3.73\%$. (a-c) The amplitude and (d-f) the phase of ePIE, rPIE and sDR reconstructions, respectively. For this sparse data, ePIE and rPIE fail to converge in the reconstructions no matter how many iterations are used, but sDR converges to a high quality image. Furthermore, amplitudes and phases in the ePIE and rPIE reconstructions are mixed and could not be separated.
Fig. 5.
Fig. 5. The ePIE (a), rPIE (b) and sDR (c) reconstructions respectively of a sparse data with 300 iterations, where sDR obtains a better quality reconstruction than ePIE and rPIE. Both sDR and rPIE produce a larger FOV than ePIE. Scale bar $200 \mu m$.
Fig. 6.
Fig. 6. The ePIE (a), rPIE (b) and sDR (c) reconstructions of a $3.70 \times 3.70 \mu m$ region of the HeLa cell after 10000 iterations with $R_F = 15.80\%, \, 15.84\%$ and $14.40\%$, respectively. (d-f) Magnified regions ($1.66 \times 1.66 \mu m$) in (a-c), respectively. (g-l) The ePIE, rPIE, and sDR reconstructions and their magnified regions after 100 iterations with $R_F = 18.69\%, \, 16.60\%$ and $14.92\%$, respectively. sDR converges fastest among the three algorithms. Scale bar $500 nm$ and $200nm$ respectively.
Fig. 7.
Fig. 7. (a-c) The ePIE, rPIE, and sDR reconstructions of a sparse dataset and (d-f) their magnified regions after 10000 iterations with $R_F = 15.62\%, \,15.72\%$ and $13.89\%$, respectively. (g-l) The ePIE, rPIE and sDR reconstructions and their magnified regions after 300 iterations with $R_F = 19.52\%, \,16.58\%$ and $14.26\%$, respectively. To create the sparse data, we randomly pick 980 out of 2,450 diffraction patterns from the HeLa cell dataset. sDR reproduces a more distinguishable features than ePIE and rPIE. In particular, ePIE and rPIE encounter slow convergence while sDR converges faster with fewer iterations.
Fig. 8.
Fig. 8. Fourier ring correlation (FRC) of ePIE, rPIE, sDR reconstructions of the 40% dataset. FRC shows that the sDR reconstruction is more highly correlated to the ground truth(reconstruction using full dataset) than the ePIE and rPIE reconstructions.

Equations (29)

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| F ( P O n ) | = I n for n = 1 , . . , N .
O ( x + r n ) = O n ( x ) if r n + x Ω n for n = 1 , , N
T := { Ψ = { Ψ n } n = 1 N : | F Ψ n | = I n  for  n = 1 , , N } S := { Ψ = { Ψ n } n = 1 N : P , O  s.t.  Ψ n = P O n  for  n = 1 , , N } } .
min Ψ i S ( Ψ ) + i T ( Ψ )
i S ( Ψ ) = { 0 Ψ S otherwise
Ψ n = Π T ( Ψ n k ) = F 1 ( I n arg ( F Ψ n k ) )
{ P k + 1 , O n k + 1 } = a r g m i n P , O n 1 2 P O n Ψ n 2
Ψ n k + 1 = P k + 1 O n k + 1
O n k + 1 = a r g m i n O n 1 2 P k O n Ψ n 2 = Ψ n P k P k + 1 = a r g m i n P 1 2 P O n k + 1 Ψ n 2 = Ψ n O n k + 1
{ P k + 1 , O n k + 1 } = a r g m i n P , O n 1 2 P O n Ψ n 2 + 1 2 s P P k 2 + 1 2 t O n O n k 2
O n k + 1 = O n k t P k + 1 ¯ ( P k + 1 O n k + 1 Ψ n ) P k + 1 = P k s O n k + 1 ¯ ( P k + 1 O n k + 1 Ψ n )
O n k + 1 = O n k t P k ¯ ( P k O n k Ψ n ) P k + 1 = P k s O n k + 1 ¯ ( P k O n k + 1 Ψ n )
O n k + 1 = O n k β O P k ¯ ( P k O n k Ψ n ) / P k max 2 P k + 1 = P k β P O n k + 1 ¯ ( P k O n k + 1 Ψ n ) / O n k + 1 max 2
Step 1: O n k + 1 = a r g m i n O n 1 2 P k O n Ψ n 2 + 1 2 t O n O n k 2 Step 2: P k + 1 = a r g m i n P 1 2 P O n k + 1 Ψ n 2 + 1 2 s P P k 2
O n k + 1 = O n k t P k ¯ ( P k O n k + 1 Ψ n ) P k + 1 = P k s O n k + 1 ¯ ( P k + 1 O n k + 1 Ψ n )
O n k + 1 = ( O n k + t P k ¯ Ψ n ) / ( 1 + t | P k | 2 ) P k + 1 = ( P k + s O n k + 1 ¯ Ψ n ) / ( 1 + s | O n k + 1 | 2 )
O n k + 1 = ( 1 β O ) P k max 2 O n k + β O P k ¯ Ψ n ( 1 β O ) P k max 2 + β O | P k | 2 P k + 1 = ( 1 β P ) O n k + 1 max 2 P k + β P O n k + 1 ¯ Ψ n ( 1 β P ) O n k + 1 max 2 + β P | O n k + 1 | 2
O n k + 1 = O k + β O P k ¯ ( Ψ n Ψ n k ) ( 1 β O ) P k max 2 + β O | P k | 2
O n k + 1 = O k + P k ¯ ( Ψ n Ψ n k ) α P k max 2 + ( 1 α ) | P k | 2
min Ψ f ( Ψ ) + g ( Ψ )
Ψ k + 1 = Ψ k + p r o x t f ( 2 p r o x t g ( Ψ k ) Ψ k ) p r o x t g ( Ψ k )
Ψ k + 1 = Ψ k + Π T ( 2 Π S ( Ψ k ) Ψ k ) Π S ( Ψ k )
Ψ k + 1 = p r o x t f ( ( 1 + σ ) p r o x t g ( Ψ k ) σ Ψ k ) + σ ( Ψ k p r o x t g ( Ψ k ) )
min Ψ n n = 1 N | F Ψ n | I n 2 + i S ( Ψ n )
p r o x t f ( Ψ k ) = a r g m i n Ψ 1 2 | F Ψ | I n 2 + 1 2 t Ψ Ψ k 2 = Ψ k + t F 1 [ I n arg ( F Ψ ) ] / ( 1 + t ) = ( 1 τ ) Ψ k + τ F 1 [ I n arg ( F Ψ ) ]
Ψ k + 1 = ( 1 τ ) ( ( 1 + σ ) Π S ( Ψ k ) σ Ψ k ) + τ Π T ( ( 1 + σ ) Π S ( Ψ k ) σ Ψ k ) + σ ( Ψ k Π S ( Ψ k ) ) = τ ( σ Ψ k + Π T ( ( 1 + σ ) Π S ( Ψ k ) σ Ψ k ) ) + ( 1 τ ( 1 + σ ) ) Π S ( Ψ k )
β O k = β O 0 k / ( K k )
R n o i s e = 1 N n = 1 N | F ( P 0 O n 0 ) | I n 1 , 1 / I n 1 , 1
R F = 1 N n = 1 N | F ( P O n ) | I n 1 , 1 / I n 1 , 1

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