Abstract

Fourier ptychographic microscopy is a computational microscopy technique to achieve wide-field and super-resolution complex imaging which has been developed in recent years. The method is based on illuminating the sample by a light source array, and then computationally integrating different images correspondent to each of the sources, in the Fourier domain. Knowledge of the exact relative position of the light sources and the sample is critical for the quality of the final recovered image. In this paper, we present an iterative approach towards correcting the position in the Fourier domain based on Newton's method. Also, an analysis is presented which shows the relation between the position error and the deterioration of the final recovery quality. The effectiveness of the presented method in improving the quality of the final recovered image is demonstrated using simulation and experimental results. Moreover, the method is shown to be more stable and robust to noises in comparison with the state-of-the-art algorithm.

© 2017 Optical Society of America

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2017 (1)

2016 (4)

2015 (6)

2014 (7)

2013 (4)

2012 (2)

D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. J. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express 20(9), 9911–9918 (2012).
[PubMed]

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

2011 (2)

2010 (3)

2009 (3)

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

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[PubMed]

T. R. Hillman, T. Gutzler, S. A. Alexandrov, and D. D. Sampson, “High-resolution, wide-field object reconstruction with synthetic aperture Fourier holographic optical microscopy,” Opt. Express 17(10), 7873–7892 (2009).
[PubMed]

2008 (1)

2007 (4)

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[PubMed]

V. Mico, Z. Zalevsky, and J. García, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276(2), 209–217 (2007).

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).

S. Marchesini, “Phase retrieval and saddle-point optimization,” J. Opt. Soc. Am. A 24(10), 3289–3296 (2007).
[PubMed]

2006 (1)

2004 (2)

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).

2002 (1)

1995 (1)

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).

1982 (1)

Alexandrov, S. A.

Allen, L. J.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Barbastathis, G.

Barsi, C.

Bauschke, H. H.

Bian, L.

Bian, Z.

Boppart, S. A.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[PubMed]

Bourgeois, L.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Bunk, O.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[PubMed]

Carney, P. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[PubMed]

Chen, F.

Chen, M.

Chen, Q.

Chen, W.

Chen, X.

Chien, W.-C.

Choi, W.

Choi, Y.

Chung, J.

Claus, D.

Combettes, P. L.

D’Alfonso, A. J.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Dai, Q.

Dasari, R. R.

Dierolf, M.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[PubMed]

Dilworth, D. S.

Dong, J.

Dong, S.

Dwyer, C.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Etheridge, J.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Fang-Yen, C.

Faulkner, H. M. L.

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).

Feld, M. S.

Fienup, J. R.

Fleischer, J. W.

García, J.

L. Granero, V. Micó, Z. Zalevsky, and J. García, “Synthetic aperture superresolved microscopy in digital lensless Fourier holography by time and angular multiplexing of the object information,” Appl. Opt. 49(5), 845–857 (2010).
[PubMed]

V. Mico, Z. Zalevsky, and J. García, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276(2), 209–217 (2007).

Gesell, L. H.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).

Granero, L.

Guizar-Sicairos, M.

Guo, K.

Gutzler, T.

Hillman, T. R.

Horstmeyer, R.

Hue, F.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).

Humphry, M. J.

Kim, M.

Kuang, C.

Kutz, J. N.

Lapides, J.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).

Lee, J.

Lee, K. J.

Leith, E. N.

Li, X.

Liang, R.

S. Pacheco, G. Zheng, and R. Liang, “Reflective Fourier ptychography,” J. Biomed. Opt. 21(2), 26010 (2016).
[PubMed]

Liu, E.

Liu, Z.

Lu, C.-H.

Lu, H.

Luke, D. R.

Ma, Y.

Maiden, A. M.

D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. J. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express 20(9), 9911–9918 (2012).
[PubMed]

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
[PubMed]

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

Marchesini, S.

Marks, D. L.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[PubMed]

McNulty, I.

Menzel, A.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[PubMed]

Mico, V.

V. Mico, Z. Zalevsky, and J. García, “Synthetic aperture microscopy using off-axis illumination and polarization coding,” Opt. Commun. 276(2), 209–217 (2007).

Micó, V.

Midgley, P. A.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).

Morgan, A. J.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Nanda, P.

Nugent, K. A.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Osten, W.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).

Ou, X.

Pacheco, S.

S. Pacheco, G. Zheng, and R. Liang, “Reflective Fourier ptychography,” J. Biomed. Opt. 21(2), 26010 (2016).
[PubMed]

Pedrini, G.

F. Zhang, G. Pedrini, and W. Osten, “Phase retrieval of arbitrary complex-valued fields through aperture-plane modulation,” Phys. Rev. A 75, 043805 (2007).

Pfeiffer, F.

P. Thibault, M. Dierolf, O. Bunk, A. Menzel, and F. Pfeiffer, “Probe retrieval in ptychographic coherent diffractive imaging,” Ultramicroscopy 109(4), 338–343 (2009).
[PubMed]

Price, C. H.

T. M. Turpin, L. H. Gesell, J. Lapides, and C. H. Price, “Theory of the synthetic aperture microscope,” Proc. SPIE 2566, 230–240 (1995).

Putkunz, C. T.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Ralston, T. S.

T. S. Ralston, D. L. Marks, P. S. Carney, and S. A. Boppart, “Interferometric synthetic aperture microscopy,” Nat. Phys. 3(2), 129–134 (2007).
[PubMed]

Ramchandran, K.

Roberts, A.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Rodenburg, J. M.

D. Claus, A. M. Maiden, F. Zhang, F. G. R. Sweeney, M. J. Humphry, H. Schluesener, and J. M. Rodenburg, “Quantitative phase contrast optimised cancerous cell differentiation via ptychography,” Opt. Express 20(9), 9911–9918 (2012).
[PubMed]

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).

A. M. Maiden, J. M. Rodenburg, and M. J. Humphry, “Optical ptychography: a practical implementation with useful resolution,” Opt. Lett. 35(15), 2585–2587 (2010).
[PubMed]

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

H. M. L. Faulkner and J. M. Rodenburg, “Movable aperture lensless transmission microscopy: a novel phase retrieval algorithm,” Phys. Rev. Lett. 93(2), 023903 (2004).
[PubMed]

J. M. Rodenburg and H. M. L. Faulkner, “A phase retrieval algorithm for shifting illumination,” Appl. Phys. Lett. 85(20), 4795–4797 (2004).

Sampson, D. D.

Schluesener, H.

Scholten, R. E.

C. T. Putkunz, A. J. D’Alfonso, A. J. Morgan, M. Weyland, C. Dwyer, L. Bourgeois, J. Etheridge, A. Roberts, R. E. Scholten, K. A. Nugent, and L. J. Allen, “Atom-scale ptychographic electron diffractive imaging of boron nitride cones,” Phys. Rev. Lett. 108(7), 073901 (2012).
[PubMed]

Shiradkar, R.

Shpyrko, O. G.

Situ, G.

So, P. T. C.

Soltanolkotabi, M.

Song, P.

Sun, J.

Sung, Y.

Suo, J.

Sweeney, F.

F. Hue, J. M. Rodenburg, A. M. Maiden, F. Sweeney, and P. A. Midgley, “Wave-front phase retrieval in transmission electron microscopy via ptychography,” Phys. Rev. B 82, 121415 (2010).

Sweeney, F. G. R.

Tang, G.

Thibault, P.

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Figures (5)

Fig. 1
Fig. 1 A schematic diagram of a classic FPM setup.
Fig. 2
Fig. 2 Recovery experiment using the traditional FP with absolute positional error and without absolute positional error. (a) The input high-resolution intensity and phase images. (b) The recovered high-resolution intensity and phase images for the conditional FP algorithm without absolute positional errors. (c) The recovered high-resolution intensity and phase images for the conditional FP algorithm with absolute positional errors. (d) The recovered high-resolution intensity and phase images from (c). (e) The spectrogram for the input high-resolution image and location map with and without errors. (f) The MSE of the recovery result by the traditional FP algorithm with and without absolution position errors.
Fig. 3
Fig. 3 The recovered results for the traditional FP, pc-FP, and sc-FP algorithms. (a) The recovered high-resolution intensity and phase images. (b) The MSE of the recovered high-resolution intensity image for running the pc-FP algorithm 10 times (black) and running the sc-FP algorithm 10 times (red). (g) The frequency domain position distribution map containing the ideal, actual, and corrected positions for the pc-FP and sc-FP algorithms.
Fig. 4
Fig. 4 The recovered results for the traditional FP, pc-FP, and sc-FP algorithms with different types of noise added. The value of the upper left corner of the image is MSE value.
Fig. 5
Fig. 5 Experimental results for one region in a classic USAF sample obtained from the traditional FP, pc-FP, and sc-FP algorithms.

Equations (19)

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sin θ x m = x r,c m ( x r,c m ) 2 + ( y r,c m ) 2 + h 2 ,sin θ y m = y r,c m ( x r,c m ) 2 + ( y r,c m ) 2 + h 2 ,
( d kx m , d ky m )=( k 0 (sin θ x m+1 sin θ x m ), k 0 (sin θ y m+1 sin θ y m )),
F -1 {F(u u 0 ,v v 0 )}=f(x,y) e i u 0 x+i v 0 y
O j+1 ( k x , k y )= O j ( k x , k y )+α | P j ( k x + k 0 sin θ x m , k y + k 0 sin θ y m ) | P j * ( k x + k 0 sin θ x m , k y + k 0 sin θ y m ) | P j ( k x , k y ) | max ( | P j ( k x + k 0 sin θ x m , k y + k 0 sin θ y m ) | 2 + δ 1 ) [ Φ j m ( k x + k 0 sin θ x m , k y + k 0 sin θ y m ) O j ( k x , k y ) P j ( k x + k 0 sin θ x m , k y + k 0 sin θ y m ) ]
P j+1 ( k x , k y )= P j ( k x , k y )+β | O j ( k x k 0 sin θ x m , k y k 0 sin θ y m ) | O j * ( k x k 0 sin θ x m , k y k 0 sin θ y m ) | O j ( k x , k y ) | max ( | O j ( k x k 0 sin θ x m , k y k 0 sin θ y m ) | 2 + δ 2 ) [ Φ j m ( k x , k y ) O j ( k x k 0 sin θ x m , k y k 0 sin θ y m ) P j ( k x , k y ) ]
Φ j m ( k x , k y )=F{ I m c (x,y) ϕ (x,y) j m | ϕ (x,y) j m | },
E= m x,y | | ϕ m (x,y) | 2 I m (x,y) | 2 ,
ϕ m (x,y)= F -1 {O( k x k 0 sin θ x m , k y k 0 sin θ y m )P( k x , k y )}= F -1 {O( k x k x m , k y k y m )P( k x , k y )}
ϕ m (x,y)= F -1 {O( k x k x m , k y k y m )P( k x , k y )}=IDFT{ O( k x k x m , k y k y m )P( k x , k y ) } = k x , k y O( k x k x m , k y k y m )P( k x , k y )exp(i2π( k x x N + k y y N )) ,
( k x m , k y m ) j+1 = ( k x m , k y m ) j α ( k x m , k y m ) j E ( k x m , k y m ) j 2 E ,
( k x m , k y m ) 2 E=[ 2 E ( k x m ) 2 2 E k x m k y m 2 E k y m k x m 2 E ( k y m ) 2 ]
E k x m =2 x,y [ | ϕ m (x,y) | 2 I m (x,y)] | ϕ m (x,y) | 2 k x m E k y m =2 x,y [ | ϕ m (x,y) | 2 I m (x,y)] | ϕ m (x,y) | 2 k y m
| ϕ m (x,y) | 2 k x m = ( ϕ m (x,y)) * ϕ m (x,y) k x m + ϕ m (x,y) ( ϕ m (x,y)) * k x m
( ϕ m (x,y)) * ϕ m (x,y) k x m = i2π N ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(xo(x,y)exp(i2π( k x m x N + k y m y N )))]
ϕ m (x,y) ( ϕ m (x,y)) * k x m = i2π N ϕ m (x,y)IDFT[ P * ( k x , k y )DFT(x o * (x,y)exp(i2π( k x m x N + k y m y N )))],
E k x m = 8π N x,y [ | ϕ m (x,y) | 2 I m (x,y)] Im{ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(xo(x,y)exp(i2π( k x m x N + k y m y N )))]}
E k y m = 8π N x,y [ | ϕ m (x,y) | 2 I m (x,y)] Im{ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(yo(x,y)exp(i2π( k x m x N + k y m y N )))]}
2 E ( k x m ) 2 =2( x,y g x m g x m + x,y [ | ϕ m (x,y) | 2 I m (x,y) ] g x m k x m ) 2 E ( k y m ) 2 =2( x,y g y m g y m + x,y [ | ϕ m (x,y) | 2 I m (x,y) ] g y m k y m ) 2 E k x m k y m =2( x,y g x m g y m + x,y [ | ϕ m (x,y) | 2 I m (x,y) ] g x m k y m ) 2 E k y m k x m =2( x,y g y m g x m + x,y [ | ϕ m (x,y) | 2 I m (x,y) ] g y m k x m ),
g x m k x m = 8ππ NN { h x m ( h x m ) * Re[ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(xxoe)]]} g x m k y m = 8ππ NN {Re[ h x m ( h y m ) * ]Re[ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(xyoe)]]} g y m k y m = 8ππ NN { h y m ( h y m ) * Re[ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(yyoe)]]} g y m k x m = 8ππ NN {Re[ h y m ( h x m ) * ]Re[ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(xyoe)]]} g x m = 4π N Im{ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(xoe)]} h x m =IDFT[P( k x , k y )DFT(xoe)] g y m = 4π N Im{ ( ϕ m (x,y)) * IDFT[P( k x , k y )DFT(yoe)]} h y m =IDFT[P( k x , k y )DFT(yoe)] oe=o(x,y)exp(i2π( k x m x N + k y m y N ))

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