Abstract

Although ptychography does not require a precise knowledge of the illumination wavefront, common implementations rely upon assumptions such as accurate knowledge of the scan positions and constant illumination. Limited validity of these assumptions results in deterioration of the reconstruction quality. We present a generalized ptychography method that optimizes the reconstruction along multiple directions. In our manuscript, we demonstrate that the additional flexibility not only helps to amend imprecisions of the ptychography model in a self-consistent way but additionally leads to faster convergence without a significant increase of the computational cost per iteration.

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

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References

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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  34. P. Thibault, M. Dierolf, A. Menzel, O. Bunk, C. David, and F. Pfeiffer, “High-resolution scanning X-ray diffraction microscopy,” Science 321, 379–382 (2008).
    [Crossref] [PubMed]
  35. M. Van Heel and M. Schatz, “Fourier shell correlation threshold criteria,” J. Struct. Biol. 151, 250–262 (2005).
    [Crossref] [PubMed]
  36. M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  41. R. Dinapoli, A. Bergamaschi, B. Henrich, R. Horisberger, I. Johnson, A. Mozzanica, E. Schmid, B. Schmitt, A. Schreiber, X. Shi, and et al., “Eiger: Next generation single photon counting detector for x-ray applications,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 650, 79–83 (2011).
    [Crossref]

2017 (1)

2016 (2)

2015 (8)

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 processing magazine 32, 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, 530–535 (2015).
[Crossref] [PubMed]

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, 053044 (2015).
[Crossref] [PubMed]

J. Deng, Y. S. Nashed, S. Chen, N. W. Phillips, T. Peterka, R. Ross, S. Vogt, C. Jacobsen, and D. J. Vine, “Continuous motion scan ptychography: characterization for increased speed in coherent X-ray imaging,” Opt. Express 23, 5438–5451 (2015).
[Crossref] [PubMed]

N. Burdet, X. Shi, D. Parks, J. N. Clark, X. Huang, S. D. Kevan, and I. K. Robinson, “Evaluation of partial coherence correction in X-ray ptychography,” Opt. express 23, 5452–5467 (2015).
[Crossref] [PubMed]

M. Odstrcil, J. Bussmann, D. Rudolf, R. Bresenitz, J. Miao, W. Brocklesby, and L. Juschkin, “Ptychographic imaging with a compact gas-discharge plasma extreme ultraviolet light source,” Opt. Lett. 40, 5574–5577 (2015).
[Crossref] [PubMed]

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, 33214–33240 (2015).
[Crossref]

J. C. da Silva and A. Menzel, “Elementary signals in ptychography,” Opt. Express 23, 33812–33821 (2015).
[Crossref]

2014 (7)

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22, 4960–4972 (2014).
[Crossref] [PubMed]

X. Huang, H. Yan, R. Harder, Y. Hwu, I. K. Robinson, and Y. S. Chu, “Optimization of overlap uniformness for ptychography,” Opt. Express 22, 12634–12644 (2014).
[Crossref] [PubMed]

M. Guizar-Sicairos, I. Johnson, A. Diaz, M. Holler, P. Karvinen, H.-C. Stadler, R. Dinapoli, O. Bunk, and A. Menzel, “High-throughput ptychography using Eiger: scanning X-ray nano-imaging of extended regions,” Opt. Express 22, 14859–14870 (2014).
[Crossref] [PubMed]

J. N. Clark, X. Huang, R. J. Harder, and I. K. Robinson, “Continuous scanning mode for ptychography,” Opt. Lett. 39, 6066–6069 (2014).
[Crossref] [PubMed]

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, 3857 (2014).
[Crossref] [PubMed]

P. M. Pelz, M. Guizar-Sicairos, P. Thibault, I. Johnson, M. Holler, and A. Menzel, “On-the-fly scans for X-ray ptychography,” Appl. Phys. Lett. 105, 251101 (2014).
[Crossref]

J. Qian, C. Yang, A. Schirotzek, F. Maia, and S. Marchesini, “Efficient algorithms for ptychographic phase retrieval,” Inverse Problems and Applications, Contemp. Math 615, 261–280 (2014).

2013 (4)

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 31927 (2013).
[Crossref] [PubMed]

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

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. Hastings, D. Nilsson, F. Uhlen, U. Vogt, H. M. Hertz, and S. Christian, “Full spatial characterization of a nanofocused X-ray free-electron laser beam by ptychographic imaging,” Sci. Rep. 3, 1633 (2013).
[Crossref] [PubMed]

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, 13592–13606 (2013).
[Crossref] [PubMed]

2012 (2)

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

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

2011 (1)

R. Dinapoli, A. Bergamaschi, B. Henrich, R. Horisberger, I. Johnson, A. Mozzanica, E. Schmid, B. Schmitt, A. Schreiber, X. Shi, and et al., “Eiger: Next generation single photon counting detector for x-ray applications,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 650, 79–83 (2011).
[Crossref]

2010 (1)

P. A. Penczek, “Chapter three-resolution measures in molecular electron microscopy,” Methods Enzymol. 482, 73–100 (2010).
[Crossref]

2009 (2)

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

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

2008 (4)

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

J. Rodenburg, “Ptychography and related diffractive imaging methods,” Adv. Imag. Elect. Phys. 150, 87–184 (2008).
[Crossref]

M. Guizar-Sicairos, S. T. Thurman, and J. R. Fienup, “Efficient subpixel image registration algorithms,” Opt. Lett. 33, 156–158 (2008).
[Crossref] [PubMed]

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

2005 (1)

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

2004 (1)

J. M. Rodenburg and H. M. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85, 4795–4797 (2004).
[Crossref]

1996 (1)

H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[Crossref]

1992 (1)

J. Rodenburg and R. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. Roy. Soc. London A 339, 521–553 (1992).
[Crossref]

1984 (1)

E. H. Adelson, C. H. Anderson, J. R. Bergen, P. J. Burt, and J. M. Ogden, “Pyramid methods in image processing,” RCA engineer 29, 33–41 (1984).

1978 (1)

1969 (1)

E. Polak and G. Ribiere, “Note sur la convergence de méthodes de directions conjuguées,” Revue française d’informatique et de recherche opérationnelle. Série rouge 3, 35–43 (1969).
[Crossref]

Adelson, E. H.

E. H. Adelson, C. H. Anderson, J. R. Bergen, P. J. Burt, and J. M. Ogden, “Pyramid methods in image processing,” RCA engineer 29, 33–41 (1984).

Allain, M.

Ames, B.

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, 053044 (2015).
[Crossref] [PubMed]

Anderson, C. H.

E. H. Adelson, C. H. Anderson, J. R. Bergen, P. J. Burt, and J. M. Ogden, “Pyramid methods in image processing,” RCA engineer 29, 33–41 (1984).

Arnold, B.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. Hastings, D. Nilsson, F. Uhlen, U. Vogt, H. M. Hertz, and S. Christian, “Full spatial characterization of a nanofocused X-ray free-electron laser beam by ptychographic imaging,” Sci. Rep. 3, 1633 (2013).
[Crossref] [PubMed]

Baksh, P.

Baksh, P. D.

Bates, R.

J. Rodenburg and R. Bates, “The theory of super-resolution electron microscopy via Wigner-distribution deconvolution,” Phil. Trans. Roy. Soc. London A 339, 521–553 (1992).
[Crossref]

Bean, R.

Berenguer, F.

Bergamaschi, A.

R. Dinapoli, A. Bergamaschi, B. Henrich, R. Horisberger, I. Johnson, A. Mozzanica, E. Schmid, B. Schmitt, A. Schreiber, X. Shi, and et al., “Eiger: Next generation single photon counting detector for x-ray applications,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 650, 79–83 (2011).
[Crossref]

Bergen, J. R.

E. H. Adelson, C. H. Anderson, J. R. Bergen, P. J. Burt, and J. M. Ogden, “Pyramid methods in image processing,” RCA engineer 29, 33–41 (1984).

Boden, S.

Boden, S. A.

Bresenitz, R.

Brocklesby, W.

Brocklesby, W. S.

Bunk, O.

M. Guizar-Sicairos, I. Johnson, A. Diaz, M. Holler, P. Karvinen, H.-C. Stadler, R. Dinapoli, O. Bunk, and A. Menzel, “High-throughput ptychography using Eiger: scanning X-ray nano-imaging of extended regions,” Opt. Express 22, 14859–14870 (2014).
[Crossref] [PubMed]

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, 3857 (2014).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

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

Burdet, N.

Burt, P. J.

E. H. Adelson, C. H. Anderson, J. R. Bergen, P. J. Burt, and J. M. Ogden, “Pyramid methods in image processing,” RCA engineer 29, 33–41 (1984).

Bussmann, J.

Card, R.

Chad, J.

Chamard, V.

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 processing magazine 32, 87–109 (2015).
[Crossref]

H. N. Chapman, “Phase-retrieval X-ray microscopy by Wigner-distribution deconvolution,” Ultramicroscopy 66, 153–172 (1996).
[Crossref]

Chen, B.

Chen, M.

Chen, R. Y.

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, 053044 (2015).
[Crossref] [PubMed]

Chen, S.

Christian, S.

A. Schropp, R. Hoppe, V. Meier, J. Patommel, F. Seiboth, H. J. Lee, B. Nagler, E. C. Galtier, B. Arnold, U. Zastrau, J. Hastings, D. Nilsson, F. Uhlen, U. Vogt, H. M. Hertz, and S. Christian, “Full spatial characterization of a nanofocused X-ray free-electron laser beam by ptychographic imaging,” Sci. Rep. 3, 1633 (2013).
[Crossref] [PubMed]

Chu, Y. S.

Clark, J. N.

Cloetens, P.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 31927 (2013).
[Crossref] [PubMed]

Cohen, O.

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 processing magazine 32, 87–109 (2015).
[Crossref]

da Silva, J. C.

David, C.

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

Deng, J.

Diaz, A.

Dierolf, M.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 31927 (2013).
[Crossref] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109, 338–343 (2009).
[Crossref] [PubMed]

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

Dinapoli, R.

M. Guizar-Sicairos, I. Johnson, A. Diaz, M. Holler, P. Karvinen, H.-C. Stadler, R. Dinapoli, O. Bunk, and A. Menzel, “High-throughput ptychography using Eiger: scanning X-ray nano-imaging of extended regions,” Opt. Express 22, 14859–14870 (2014).
[Crossref] [PubMed]

R. Dinapoli, A. Bergamaschi, B. Henrich, R. Horisberger, I. Johnson, A. Mozzanica, E. Schmid, B. Schmitt, A. Schreiber, X. Shi, and et al., “Eiger: Next generation single photon counting detector for x-ray applications,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment 650, 79–83 (2011).
[Crossref]

Dong, J.

Eldar, Y. C.

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 processing magazine 32, 87–109 (2015).
[Crossref]

Enders, B.

M. Stockmar, P. Cloetens, I. Zanette, B. Enders, M. Dierolf, F. Pfeiffer, and P. Thibault, “Near-field ptychography: phase retrieval for inline holography using a structured illumination,” Sci. Rep. 31927 (2013).
[Crossref] [PubMed]

Färm, E.

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Figures (7)

Fig. 1
Fig. 1 Two examples of partitioning the full set of scan positions ��0 into 5 subsets ��k. (a) Compact partitions estimated by the K-means algorithm [22], (b) sparse partitions realized by a random permutation.
Fig. 2
Fig. 2 (a) Evolution of the reconstruction quality during convergence of different methods based on the simulated dataset, shown in semilogarithmic scale. The dashed lines denote the least-squares maximum-likelihood (LSQ-ML) method with different block size and the full lines correspond to other common ptychography methods: ML, DM, ePIE. LSQ-ML-s B:50 denotes a sparse set selection with the block size 50 and LSQ-ML-c B:all is the compact option using one block of full dataset size. (b) Model of the object and the illumination probe used for our simulations in the complex colorscale. Scale bar is 20 μm.
Fig. 3
Fig. 3 Spectral signal to noise ratio (SSNR) estimated from the Fourier ring correlation (FRC) [37]. Plot (a) shows the SSNR for each of the tested methods after 30 iterations and (b) is the SSNR after 1000 iterations.
Fig. 4
Fig. 4 (a) Evolution of the mean square error (MSE) of the corrected positions during convergence of the LSQ-ML algorithm with position refinement. Blue line denotes convergence with random errors only, red line is convergence with global parameters errors: scale, rotation and skewness. (b,c) Evolution of the residuum in the global geometry parameters: pixel scale (b) skewness and rotation (c).
Fig. 5
Fig. 5 (a) Evolution of the mean square error (MSE) between the refined and the optimal values for the wavefront correction for both simulated datasets. The blue line denotes the residuum between the wavefront parameters, and the red line corresponds to the intensity error. An example of the initial (b) and the final (c) displacement of the wavefront in the far field along the vertical axis. Displacement is indicated by shade of a spot at each scan position.
Fig. 6
Fig. 6 (a) Evolution of the reconstruction resolution for different algorithms on the Eiger readout chip dataset [31]. (b) SSNR in the final iteration for each reconstruction method.
Fig. 7
Fig. 7 (a) Evolution of the reconstruction quality for the LSQ-ML method: with position refinement (green line), without position refinement (red line) and starting from the already refined positions (blue line). (b) Residues between the refined and original positions after tilt and rotation correction for both scans.

Equations (42)

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ψ i , r = P r O r + r i r i 𝒩 0 ,
I i , q e : = q ( 𝒜 q , q | Ψ i , q | 2 ) + b i , q ,
𝒜 | Ψ i | 2 = | Ψ i | 2 * k ,
Ψ i opt = arg min d ( I i m , I i e ) r i 𝒩 0 ,
P i , r upd = P r + j α P , i ( j ) Δ P i , r ( j ) ,
O i , r upd = O r + r i + j α O , i ( j ) Δ O i , r ( j ) ,
min α P , i , α O , i ψ i opt ψ i upd r i 𝒩 0 ,
ψ i , r upd = P i , r upd O i , r upd .
p g ( I i | Ψ i ) = q { 1 2 π σ i , q 2 exp [ 1 2 ( I i , q m I i , q e σ i , q ) 2 ] } .
p p ( I i | Ψ i ) = q ( I i , q e ) I i , q m I i , q m ! exp ( I i , q e ) .
p a ( I i | Ψ i ) = q { 1 2 π ( σ i , q a ) 2 exp [ 1 2 ( I i , q m I i , q e σ i , q a ) 2 ] } .
p ( Ψ i ) = q [ I i , q m log ( I i , q e ) I i , q e ] Poisson log-likelihood
a ( Ψ i ) = q 1 2 ( I i , q m I i , q e σ i , q a ) 2 Amplitude log-likelihood
Ψ i , q p ( Ψ i ) = q [ 𝒜 q , q ( 1 I i , q m I i , q e ) ] Ψ i , q Gradient of Poisson log-likelihood
Ψ i , q a ( Ψ i ) = 2 W i , q q [ 𝒜 q , q ( 1 I i , q m I i , q e ) ] Ψ i , q Gradient of amplitude log-likelihood
W i , q : = 1 / ( 2 σ i , q a ) 2 [ 0 , 1 ] ,
Ψ i , q opt : = Ψ i , q α i Ψ i , q ( Ψ i ) .
{ Ψ i opt } α i = 0 .
α i , q a = 1 2 q 𝒜 q , q ,
α i , q p = 1 q 𝒜 q , q q ( I i , q e ξ i , q [ I i , q m 1 α i , q p ξ i , q ] ) q ξ i , q 2 I i , q e ,
ψ i , r opt = 𝒫 1 { Ψ i , q opt } .
χ i , r : = ψ i , r opt ψ i , r = ψ i , r opt P r O r + r i .
r : = r | ψ i , r opt ψ i , r upd | 2 ,
r r | χ i , r j α P , i ( j ) Δ P i , r ( j ) O r + r i j α O , i ( j ) Δ O i , r ( j ) P r 2 | .
( r | Δ O i , r ( j ) P r | 2 + γ r Δ O i , r ( j ) P r ( Δ P r ( j ) O r + r i ) * r [ Δ O i , r ( j ) P r ( Δ P r ( j ) O r + r i ) * ] * r | Δ P r ( j ) O r + r i | 2 + γ ) ( α O , i ( j ) α P , i ( j ) ) = ( r [ χ i , r ( Δ O i , r ( j ) P r ) * ] r [ χ i , r ( Δ P r ( j ) O r + r i ) * ] ) ,
α P , i ( j ) = [ χ i , r ( Δ P i , r ( j ) O r + r i ) * ] | Δ P i , r ( j ) O r + r i | 2 + γ .
α O , i ( j ) = [ χ i , r ( Δ O i , r ( j ) P r ) * ] | Δ O i , r ( j ) P r | 2 + γ .
Δ P i , r : = P r = χ i , r O r + r i * ,
Δ O i , r : = O r = χ i , r P r * .
Δ P ^ r : = i 𝒩 Δ P i , r i 𝒩 0 | O r + r i | 2 + δ P ,
Δ O ^ r : = i 𝒩 Δ O i , r r i i 𝒩 0 | P r r i | 2 + δ O .
χ i ψ i ,
P ^ r = P r + i 𝒩 α P , i Δ P ^ r | O r + r i | 2 i 𝒩 | O r + r i | 2 ,
O ^ r = O r + i 𝒩 α O , i Δ O ^ r | P r r i | 2 i 𝒩 | P r r i | 2 .
Δ P ( x ) : = x P r = 1 { 2 π i x ( P r ) } ,
Δ P Z j , r : = i Z j , r P r for j > 0 ,
Δ P int : = P r ,
R i : = Δ P i Δ P ,
Δ P ^ eig = Δ eig + β R ( R T Δ P eig + α ˜ eig ) / α ˜ eig 2 ,
P i , r = ( 1 + α ˜ i int ) exp ( 2 π i j α ˜ i Z j Z j , r ) 1 { exp [ 2 π i ( α ˜ i ( x ) x + α ˜ i ( y ) y ) ] { P r + α ˜ i eig Δ P eig , r } } ,
i α i ( j ) = 0 j [ 1 , N dir ] ,
SSNR ( q ) = 2 FRC ( q ) 1 FRC ( q ) .

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