Abstract

Fourier Ptychography is a new computational microscopy technique that achieves gigapixel images with both wide field of view and high resolution in both phase and amplitude. The hardware setup involves a simple replacement of the microscope’s illumination unit with a programmable LED array, allowing one to flexibly pattern illumination angles without any moving parts. In previous work, a series of low-resolution images was taken by sequentially turning on each single LED in the array, and the data were then combined to recover a bandwidth much higher than the one allowed by the original imaging system. Here, we demonstrate a multiplexed illumination strategy in which multiple randomly selected LEDs are turned on for each image. Since each LED corresponds to a different area of Fourier space, the total number of images can be significantly reduced, without sacrificing image quality. We demonstrate this method experimentally in a modified commercial microscope. Compared to sequential scanning, our multiplexed strategy achieves similar results with approximately an order of magnitude reduction in both acquisition time and data capture requirements.

© 2014 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2014

2013

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature494, 68–71 (2013).
[CrossRef] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics7, 739–745 (2013).
[CrossRef]

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett.38, 4845–4848 (2013).
[CrossRef] [PubMed]

2012

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods9, 1195–1197 (2012).
[CrossRef] [PubMed]

L. Tian, J. Lee, S. B. Oh, and G. Barbastathis, “Experimental compressive phase space tomography,” Opt. Express20, 8296–8308 (2012).
[CrossRef] [PubMed]

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

2011

2010

L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics1, 15 (2010).
[CrossRef]

2009

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy235, 144–162 (2009).
[CrossRef]

T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperturefourier holographic optical microscopy,” Opt. Express17, 7873–7892 (2009).
[CrossRef] [PubMed]

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

2008

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

M. Guizar-Sicairos and J. R. Fienup, “Phase retrieval with transverse translation diversity: a nonlinear optimization approach,” Opt. Express16, 7264–7278 (2008).
[CrossRef] [PubMed]

2007

C. Rydberg, J. Bengtsson, and , “Numerical algorithm for the retrieval of spatial coherence properties of partially coherent beams from transverse intensity measurements,” Opt. Express15, 13613–13623 (2007).
[CrossRef] [PubMed]

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence29, 1339–1354 (2007).
[CrossRef]

2006

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett.97, 168102 (2006).
[CrossRef] [PubMed]

2004

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

1984

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy133, 27–39 (1984).
[CrossRef]

1982

1971

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik34, 275–284 (1971).

Alexandrov, S. A.

Barbastathis, G.

Beck, A.

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing62, 928–938 (2014).
[CrossRef]

Belhumeur, P. N.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence29, 1339–1354 (2007).
[CrossRef]

Bengtsson, J.

Bernussi, A. A.

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

Chu, K. K.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods9, 1195–1197 (2012).
[CrossRef] [PubMed]

de Peralta, L. G.

Desai, D. B.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

Dominguez, D.

Eldar, Y.

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing62, 928–938 (2014).
[CrossRef]

Faulkner, H. M.

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

Fienup, J. R.

Fleischer, J.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Ford, T. N.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods9, 1195–1197 (2012).
[CrossRef] [PubMed]

Gerchberg, R.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik34, 275–284 (1971).

Guizar-Sicairos, M.

Gutzler, T.

Hamilton, D.

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy133, 27–39 (1984).
[CrossRef]

Hillman, T. R.

Horstmeyer, R.

X. Ou, R. Horstmeyer, C. Yang, and G. Zheng, “Quantitative phase imaging via fourier ptychographic microscopy,” Opt. Lett.38, 4845–4848 (2013).
[CrossRef] [PubMed]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics7, 739–745 (2013).
[CrossRef]

Johnson, I.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

Kolner, C.

Kynde, S.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

Lee, J.

Levoy, M.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy235, 144–162 (2009).
[CrossRef]

Maiden, A. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

Marti, O.

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

McDowall, I.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy235, 144–162 (2009).
[CrossRef]

Menzel, A.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature494, 68–71 (2013).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

Mertz, J.

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods9, 1195–1197 (2012).
[CrossRef] [PubMed]

Molina, L.

Nayar, S. K.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence29, 1339–1354 (2007).
[CrossRef]

O’Loughlin, T.

Oh, S. B.

Ou, X.

Pantanowitz, L.

L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics1, 15 (2010).
[CrossRef]

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

Ratner, N.

N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.

Rodenburg, J. M.

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

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

Rydberg, C.

Sampson, D. D.

Saxton, W.

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik34, 275–284 (1971).

Schechner, Y. Y.

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence29, 1339–1354 (2007).
[CrossRef]

N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.

Shechtman, Y.

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing62, 928–938 (2014).
[CrossRef]

Sheppard, C.

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy133, 27–39 (1984).
[CrossRef]

Situ, G.

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Thibault, P.

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature494, 68–71 (2013).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

Tian, L.

Waller, L.

L. Tian, J. Wang, and L. Waller, “3D differential phase-contrast microscopy with computational illumination using an LED array,” Opt. Lett.39, 1326–1329 (2014).
[CrossRef] [PubMed]

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

Wang, J.

Yang, C.

Zhang, Z.

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy235, 144–162 (2009).
[CrossRef]

Zheng, G.

Appl. Opt.

Appl. Phys. Lett.

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

IEEE Trans. Pattern Analysis Machine Intelligence

Y. Y. Schechner, S. K. Nayar, and P. N. Belhumeur, “Multiplexing for optimal lighting,” IEEE Trans. Pattern Analysis Machine Intelligence29, 1339–1354 (2007).
[CrossRef]

IEEE Trans. Sig. Processing

Y. Shechtman, A. Beck, and Y. Eldar, “GESPAR: Efficient phase retrieval of sparse signals,” IEEE Trans. Sig. Processing62, 928–938 (2014).
[CrossRef]

J. Microscopy

D. Hamilton and C. Sheppard, “Differential phase contrast in scanning optical microscopy,” J. Microscopy133, 27–39 (1984).
[CrossRef]

M. Levoy, Z. Zhang, and I. McDowall, “Recording and controlling the 4D light field in a microscope using microlens arrays,” J. Microscopy235, 144–162 (2009).
[CrossRef]

J. Pathology Informatics

L. Pantanowitz, “Digital images and the future of digital pathology,” J. Pathology Informatics1, 15 (2010).
[CrossRef]

Nat. Photonics

L. Waller, G. Situ, and J. Fleischer, “Phase-space measurement and coherence synthesis of optical beams,” Nat. Photonics6, 474–479 (2012).
[CrossRef]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier Ptychographic microscopy,” Nat. Photonics7, 739–745 (2013).
[CrossRef]

Nature

P. Thibault and A. Menzel, “Reconstructing state mixtures from diffraction measurements,” Nature494, 68–71 (2013).
[CrossRef] [PubMed]

Nature Methods

T. N. Ford, K. K. Chu, and J. Mertz, “Phase-gradient microscopy in thick tissue with oblique back-illumination,” Nature Methods9, 1195–1197 (2012).
[CrossRef] [PubMed]

Opt. Express

Opt. Lett.

Optik

R. Gerchberg and W. Saxton, “Phase determination for image and diffraction plane pictures in the electron microscope,” Optik34, 275–284 (1971).

Phys. Rev. Lett.

S. A. Alexandrov, T. R. Hillman, T. Gutzler, and D. D. Sampson, “Synthetic aperture fourier holographic optical microscopy,” Phys. Rev. Lett.97, 168102 (2006).
[CrossRef] [PubMed]

Ultramicroscopy

A. M. Maiden and J. M. Rodenburg, “An improved ptychographical phase retrieval algorithm for diffractive imaging,” Ultramicroscopy109, 1256–1262 (2009).
[CrossRef] [PubMed]

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy109, 338–343 (2009).
[CrossRef] [PubMed]

O. Bunk, M. Dierolf, S. Kynde, I. Johnson, O. Marti, and F. Pfeiffer, “Influence of the overlap parameter on the convergence of the ptychographical iterative engine,” Ultramicroscopy108, 481–487 (2008).
[CrossRef]

Other

N. Ratner and Y. Y. Schechner, “Illumination multiplexing within fundamental limits,” in “Computer Vision and Pattern Recognition” (IEEE, 2007), pp. 1–8.

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Figures (5)

Fig. 1
Fig. 1

Summary of Fourier Ptychography (FP) in an LED array microscope. (a) A sample is illuminated from different angles by turning on different LEDs of an array. (b) Our experimental setup on a Nikon TE300 microscope. (c) Images taken with different LEDs contain information from different spatial frequency areas of the sample. The central (brightfield) LED fills an area defined by the NA (0.1) of the objective. Images taken with top and left (dark field) LEDs result in accentuated edges along the corresponding orientations. (d) FP reconstructs a high resolution image from many LEDs, while simultaneously estimating aberrations.

Fig. 2
Fig. 2

Sample datasets for multiplexed illumination coding in Fourier Ptychography. (Top) Four randomly chosen LEDs are turned on for each measurement. (Middle) The captured images corresponding to each LED pattern. (Bottom) Fourier coverage of the sample’s Fourier space for each of the LED patterns (drawn to scale). Turning on multiple well-separated LEDs allows information from multiple areas of Fourier space to pass through the system simultaneously. The center unshaded circle represents the NA of the objective lens.

Fig. 3
Fig. 3

Flow chart of the reconstruction algorithm.

Fig. 4
Fig. 4

Experimental results for multiplexed illumination of a resolution target. (a) The original low resolution image from a 4 × 0.1 NA objective taken with only the central LED on. (b) A zoom-in on the smallest features. (c) Reconstruction result from sequential FP with Nimg = 293 single LED images having a total acquisition time of T=586s. (d) Multiplexing 4 LEDs for each image while preserving the number of measurements Nimg = 293 reduces the total acquisition time to T=293s. (e) The multiplexed illumination also allows reduction of the number of measurements to Nimg = 74 with T=74s.

Fig. 5
Fig. 5

Experimental results for multiplexed illumination of a stained dog stomach cardiac region sample. (a) The original low resolution image from a 4× 0.1NA objective taken with only the central LED on. (b1, c1) Zoom-in of the regions denoted by red and green squares. (b2, c2) Amplitude and (b5, c5) phase reconstructions from sequential FP with Nimg = 293 single LED images and a total acquisition time of T=586s. (b3, c3) Amplitude and (b6, c6) phase reconstructions from multiplexing 8 LEDs with Nimg = 293 and T=293s. Amplitude (b4, c7) and phase (b4, c7) reconstructions for multiplexing 8 LEDs with Nimg = 40 and T=40s.

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

i m ( r ) = | [ O ( k k m ) P ( k ) ] ( r ) | 2 ,
I p ( r ) = m p i m ( r ) = m p | [ O ( k k m ) P ( k ) ] ( r ) | 2 ,
A p , m = { 1 , m p 0 , otherwise , m = 1 , , N LED , p = 1 , , N img .
min O ( k ) , P ( k ) , { b p } p = 1 N img p = 1 N img r | I p ( r ) ( m p | [ O ( k k m ) P ( k ) ] ( r ) | 2 + b p ) | 2 .
I ^ p ( r ) = I p ( r ) b ^ p .
Ψ m ( k ) = O ( k k m ) P ( k ) ,
ψ m ( i ) ( r ) = [ Ψ m ( i ) ( k ) ] ( r ) , m p .
Φ m ( i ) ( k ) = [ ϕ m ( i ) ( r ) ] 1 ( k ) with ϕ m ( i ) ( r ) = I ^ p ( r ) m p | ψ m ( i ) ( r ) | 2 ψ m ( i ) ( r ) , m p .
O ( i + 1 ) ( k ) = O ( i ) ( k ) + m p | P ( i ) ( k + k m ) | [ P ( i ) ( k + k m ) ] * [ Φ m ( i ) ( k + k m ) O ( i ) ( k ) P ( i ) ( k + k m ) ] | P ( i ) ( k ) | max ( m p | P ( i ) ( k + k m ) | 2 + δ 1 )
P ( i + 1 ) ( k ) = P ( i ) ( k ) + m p | O ( i ) ( k k m ) | [ O ( i ) ( k k m ) ] * [ Φ m ( i ) ( k ) O ( i ) ( k k m ) P ( i ) ( k ) ] | O ( i ) ( k ) | max ( m p | O ( i ) ( k k m ) | 2 + δ 2 ) ,
𝒞 p , 1 { { Ψ m ( k ) } m p : I ^ p ( r ) = m p | [ Ψ m ( k ) ] ( r ) | 2 , r }
𝒞 p , 2 { { Ψ m ( k ) } m p : Ψ m ( k ) = O ( k k m ) P ( k ) , m p , O ( k ) , P ( k ) , k } .
i p mod N img .
{ Φ m ( i ) ( k ) } m p = arg min Φ m ( k ) m p k | Φ m ( k ) Ψ m ( i ) ( k ) | 2 , s . t . { Φ m ( k ) } m p 𝒞 p , 1 .
{ ϕ m ( i ) ( r ) } m p = arg min ϕ m ( r ) m p r | ϕ m ( r ) ψ m ( i ) ( r ) | 2 , s . t . I ^ p ( r ) = m p | ϕ m ( r ) | 2 ,
{ ϕ m ( i ) ( r ) } m p = arg min ϕ m ( r ) m p | ϕ m ( r ) ψ m ( i ) ( r ) | 2 , s . t . I ^ p ( r ) = m p | ϕ m ( r ) | 2 ,
{ ϕ m ( i ) ( r ) , λ } m p = arg min ϕ m ( r ) , λ m p | ϕ m ( r ) ψ m ( i ) ( r ) | 2 + λ ( I ^ p ( r ) m p | ϕ m ( r ) | 2 ) .
ϕ m ( i ) ( r ) = I ^ p ( r ) m p | ψ m ( i ) ( r ) | 2 ψ m ( i ) ( r ) , m p .
Φ m ( i ) ( k ) = [ ϕ m ( i ) ( r ) ] 1 ( k ) ,
Ψ m ( i + 1 ) ( k ) = O ( i + 1 ) ( k k m ) P ( i + 1 ) ( k ) , m p
{ O ( i + 1 ) ( k ) , P ( i + 1 ) ( k ) } = arg min O ( k ) , P ( k ) m p k | O ( k k m ) P ( k ) Φ m ( i ) ( k ) | 2 .
{ O ( i + 1 ) ( k ) , P ( i + 1 ) ( k ) } = arg min O ( k ) , P ( k ) m p | O ( k k m ) P ( k ) Φ m ( i ) ( k ) | 2 .
f ( O ( k k m ) , P ( k ) ) O ( k k m ) = 2 m p [ O ( k k m ) P ( k ) Φ m ( i ) ( k ) ] [ P ( k ) ] *
f ( O ( k k m ) , P ( k ) ) P ( k ) = 2 m p [ O ( k k m ) P ( k ) Φ m ( i ) ( k ) ] [ O ( k k m ) ] *
O ( i + 1 ) ( k ) = m p [ P ( i + 1 ) ( k + k m ) ] * Φ m ( i ) ( k + k m ) m p | P ( i + 1 ) ( k + k m ) | 2
P ( i + 1 ) ( k ) = m p [ O ( i + 1 ) ( k k m ) ] * Φ m ( i ) ( k ) m p | O ( i + 1 ) ( k k m ) | 2 .
2 f ( O ( k k m ) , P ( k ) ) O ( k k m ) 2 = 2 m p | P ( k ) | 2
2 f ( O ( k k m ) , P ( k ) ) P ( k ) 2 = 2 m p | O ( k k m ) | 2 .
O ( i , + 1 ) ( k ) = O ( i , ) ( k ) + step-size [ 2 f ( O ( k k m ) , P ( k ) ) O ( k k m ) 2 ] 1 f ( O ( k k m ) , P ( k ) ) O ( k )
P ( i , + 1 ) ( k ) = P ( i , ) ( k ) + step-size [ 2 f ( O ( k k m ) , P ( k ) ) P ( k ) 2 ] 1 f ( O ( k k m ) , P ( k ) ) P ( k ) .
O ( i , + 1 ) ( k ) = O ( i , ) ( k ) + α ( i , ) ( k + k m ) m p | P ( i , ) ( k + k m ) | 2 + δ 1 × m p [ P ( i , ) ( k + k m ) ] * [ Φ m ( i ) ( k + k m ) O ( i , ) ( k ) P ( i , ) ( k + k m ) ]
P ( i , + 1 ) ( k ) = P ( i , ) ( k ) + β ( i , ) ( k ) m p | O ( i , ) ( k k m ) | 2 + δ 2 × m p [ O ( i , ) ( k k m ) ] * [ Φ m ( i ) ( k ) O ( i , ) ( k k m ) P ( i , ) ( k ) ] .
{ O ( i + 1 ) ( k ) , P ( i + 1 ) ( k ) } = { O ( i , L ) ( k ) , P ( i , L ) ( k ) } .

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