Abstract

Fourier ptychographic microscopy (FPM) is implemented through aperture scanning by an LCOS spatial light modulator at the back focal plane of the objective lens. This FPM configuration enables the capturing of the complex scattered field for a 3D sample both in the transmissive mode and the reflective mode. We further show that by combining with the compressive sensing theory, the reconstructed 2D complex scattered field can be used to recover the 3D sample scattering density. This implementation expands the scope of application for FPM and can be beneficial for areas such as tissue imaging and wafer inspection.

© 2016 Optical Society of America

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References

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    [Crossref]
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2016 (1)

R. Horstmeyer, R. Heintzmann, G. Popescu, L. Waller, and C. Yang, “Standardizing the resolution claims for coherent microscopy,” Nat. Photonics 10, 68–71 (2016).
[Crossref]

2015 (2)

2014 (4)

2013 (2)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7, 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]

2010 (1)

2009 (1)

2007 (1)

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

2006 (4)

E. J. Candes, J .K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure and Applied Math. 59, 1207–1223 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, „ “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

2005 (1)

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

1997 (1)

1992 (1)

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

1926 (1)

M. Born, “Quantenmechanik der stoßvorgänge,” Zeitschrift für Physik 38, 803–827 (1926).
[Crossref]

Bach, F. R.

J. Mairal, J. Ponce, G. Sapiro, A. Zisserman, and F. R. Bach, “Supervised dictionary learning,” Proc. Adv. Neural Inf. Process. Syst. Conf. pp. 1033–1040 (2009).

Bian, Z.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

Blahut, R. E.

R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004).
[Crossref]

Born, M.

M. Born, “Quantenmechanik der stoßvorgänge,” Zeitschrift für Physik 38, 803–827 (1926).
[Crossref]

Brady, D. J.

Candes, E. J.

E. J. Candes, J .K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure and Applied Math. 59, 1207–1223 (2006).
[Crossref]

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

Candès, E. J.

E. J. Candès and T. Tao, „ “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Choi, K.

Chung, J.

Dong, S.

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Fatemi, E.

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

Figueiredo, M. A.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

Goodman, J. W.

Guo, K.

Heintzmann, R.

R. Horstmeyer, R. Heintzmann, G. Popescu, L. Waller, and C. Yang, “Standardizing the resolution claims for coherent microscopy,” Nat. Photonics 10, 68–71 (2016).
[Crossref]

Horisaki, R.

Horstmeyer, R.

Kubota, S.

Lim, S.

Mairal, J.

J. Mairal, J. Ponce, G. Sapiro, A. Zisserman, and F. R. Bach, “Supervised dictionary learning,” Proc. Adv. Neural Inf. Process. Syst. Conf. pp. 1033–1040 (2009).

Marks, D. L.

Osher, S.

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

Ou, X.

Ponce, J.

J. Mairal, J. Ponce, G. Sapiro, A. Zisserman, and F. R. Bach, “Supervised dictionary learning,” Proc. Adv. Neural Inf. Process. Syst. Conf. pp. 1033–1040 (2009).

Popescu, G.

R. Horstmeyer, R. Heintzmann, G. Popescu, L. Waller, and C. Yang, “Standardizing the resolution claims for coherent microscopy,” Nat. Photonics 10, 68–71 (2016).
[Crossref]

Romberg, J .K.

E. J. Candes, J .K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure and Applied Math. 59, 1207–1223 (2006).
[Crossref]

Romberg, J.

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

Rudin, L. I.

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

Sapiro, G.

J. Mairal, J. Ponce, G. Sapiro, A. Zisserman, and F. R. Bach, “Supervised dictionary learning,” Proc. Adv. Neural Inf. Process. Syst. Conf. pp. 1033–1040 (2009).

Shiradkar, R.

Tao, T.

E. J. Candes, J .K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure and Applied Math. 59, 1207–1223 (2006).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, „ “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

Tian, L.

Waller, L.

R. Horstmeyer, R. Heintzmann, G. Popescu, L. Waller, and C. Yang, “Standardizing the resolution claims for coherent microscopy,” Nat. Photonics 10, 68–71 (2016).
[Crossref]

L. Tian and L. Waller, “3D intensity and phase imaging from light field measurements in an LED array microscope,” Optica 2, 104–111 (2015).
[Crossref]

Xin, H.

Yamaguchi, I.

Yang, C.

Zhang, T.

Zheng, G.

Zisserman, A.

J. Mairal, J. Ponce, G. Sapiro, A. Zisserman, and F. R. Bach, “Supervised dictionary learning,” Proc. Adv. Neural Inf. Process. Syst. Conf. pp. 1033–1040 (2009).

Appl. Opt. (1)

Comm. Pure and Applied Math. (1)

E. J. Candes, J .K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Comm. Pure and Applied Math. 59, 1207–1223 (2006).
[Crossref]

IEEE Trans. Image Process. (1)

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16, 2992–3004 (2007).
[Crossref] [PubMed]

IEEE Trans. Inf. Theory (4)

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Inf. Theory 51, 4203–4215 (2005).
[Crossref]

E. J. Candès, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52, 489–509 (2006).
[Crossref]

E. J. Candès and T. Tao, „ “Near-optimal signal recovery from random projections: Universal encoding strategies?” IEEE Trans. Inf. Theory 52, 5406–5425 (2006).
[Crossref]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
[Crossref]

Nat. Photonics (2)

R. Horstmeyer, R. Heintzmann, G. Popescu, L. Waller, and C. Yang, “Standardizing the resolution claims for coherent microscopy,” Nat. Photonics 10, 68–71 (2016).
[Crossref]

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

Opt. Express (6)

Opt. Lett. (2)

Optica (1)

Physica D. (1)

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

Zeitschrift für Physik (1)

M. Born, “Quantenmechanik der stoßvorgänge,” Zeitschrift für Physik 38, 803–827 (1926).
[Crossref]

Other (5)

R. E. Blahut, Theory of Remote Image Formation (Cambridge University Press, 2004).
[Crossref]

J. M. Bioucas-Dias and M. A. Figueiredo, “Two-step Iterative Shrinkage/Thresholding Algorithm for Linear Inverse Problems,” http://www.lx.it.pt/bioucas/TwIST/TwIST.htm

R. Horstmeyer and C. Yang, “Diffraction tomography with Fourier ptychography,” arXiv:1510.08756(2015).

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005), 1st ed.

J. Mairal, J. Ponce, G. Sapiro, A. Zisserman, and F. R. Bach, “Supervised dictionary learning,” Proc. Adv. Neural Inf. Process. Syst. Conf. pp. 1033–1040 (2009).

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Figures (8)

Fig. 1
Fig. 1

Principle of aperture scanning Fourier ptychographic microscopy (a), and schematic for transmissive (b) as well as reflective (c) mode microscope system. f: focal length of the lens; L, lens; LP, linear polarizer; Obj., objective lens; BS, beam splitter

Fig. 2
Fig. 2

(a) The arrangement of aperture scanning sequence on the SLM plane. A circular aperture is scanned following the blue trajectory. The covered area provides a synthesized NA of 0.36, which is within the objective NA of 0.4. (b–c) Siemens star target intensity image in transmissive mode ASFPM. (d)intensity distribution of the red circle in (c), showing a resolution of 1.8 μm which matches the theoretical resolution of the synthesized NA.

Fig. 3
Fig. 3

Transmissive mode ASFPM image of a spirogyra slide. (a1–a2) Amplitude and phase distribution of the reconstructed hologram. (b1–b3) Refocused amplitude image of the hologram at different focal planes, in which different filaments come into focus as indicated by red arrows.

Fig. 4
Fig. 4

Reflective mode ASFPM image of a microprocessor chip. (a) Picture of the microprocessor. (b1–b2) Amplitude and phase of the reconstructed hologram. The hologram is digitally propagated to −3μm (c1–e1) and 2μm (c2–e2) away from the objective lens’ focal plane and three sub-regions are zoomed in. Intensity image captured with the aperture opened at NA=0.36 and microprocessor chip physically moved −3μm (c3–e3) and 2μm (c4–e4) are shown as a comparison to the digitally propagated results.

Fig. 5
Fig. 5

ASFPM image of 10μm microspheres on silicon wafer, immersed in mineral oil. Reconstructed amplitude (a) and phase (b) distribution of the hologram. (c) The phase distribution through the dashed line in (b) is converted to the microsphere’s thickness and plotted with a blue line. The theoretical thickness distribution is plot with a black line, showing a good match with the measurement.

Fig. 6
Fig. 6

Matrix representation of the compressive sampling model

Fig. 7
Fig. 7

Decompressive recovery of a microsphere sample. (a1–a2) Reconstructed amplitude and phase information of the sample using ASFPM. (a3) Two-layer sample configuration. (b–c) Amplitude and phase from decompressive recovery at (b1–b2) 2μm and (c1–c2) 45μm focal planes. (b3) and (c3) are the thickness profiles of microsphere calculated from the recovered phase through the two lines in (b2) and (c2), shown in blue. The red lines show the theoretical thickness value of the 4.3μm microsphere.

Fig. 8
Fig. 8

Decompressive recovery of spirogyra sample from Fig. 3. (a1–a2) Amplitude and phase reconstructed from ASFPM. (b1–d1) Amplitude and (b2–d2) phase recovered from decompressive recovery at −9μm, 6 μm and 17 μm plane.

Equations (6)

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E ( x , y , z ) = η ( x - y , z ) h ( x x , y y , z z ) d x d y d z
E n 1 n 2 = E ( n 1 Δ , n 2 Δ ) = 1 ( 2 π ) 2 d z d x d y d k x d k y d x d y η ( x , y , z ) e i k z e i ( k x x + k y y ) m 1 m 2 δ ( x m 1 Δ ) δ ( y m 2 Δ ) e i z k 2 k x 2 k y 2 δ ( z z ) e i ( k x x + k y y ) δ ( x n 1 Δ ) δ ( y n 2 Δ ) m 1 m 2 δ ( k x m 1 Δ k ) δ ( k y m 2 Δ k ) l δ ( z l Δ z ) = 1 N 2 l m 1 m 2 [ m 1 m 2 η m 1 , m 2 l e i 2 π m 1 m 1 + m 2 m 2 N ] e i k l Δ z e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 e i 2 π n 1 m 1 + n 2 m 2 N ,
E n 1 n 2 = 2 D 1 { l η ^ m 1 m 2 l e i k l Δ z e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 } = l 2 D 1 { η ^ m 1 m 2 l e i k l Δ z e i l Δ z k 2 m 1 2 Δ k 2 m 2 2 Δ k 2 } ,
g = G 2 D QBf ,
f ^ = arg min f f TV such that g = Hf ,
f TV = l n 1 n 2 | ( f l ) n 1 , n 2 | ,

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