Abstract

Fourier ptychography microscopy (FPM) is a recently developed microscopic imaging method that allows the recovery of a high-resolution complex image by combining a sequence of bright and darkfield images acquired under inclined illumination. The capacity of FPM for high resolution imaging at low magnification makes it particularly attractive for applications in digital pathology which require imaging of large specimens such as tissue sections and blood films. To date most applications of FPM have been limited to imaging thin samples, simplifying both image reconstruction and analysis. In this work we show that, for samples of intermediate thickness (defined here as less than the depth of field of a raw captured image), numerical propagation of the reconstructed complex field allows effective digital refocusing of FPM images. The results are validated by comparison against images obtained with an equivalent high numerical aperture objective lens. We find that post reconstruction refocusing (PRR) yields images comparable in quality to adding a defocus term to the pupil function within the reconstruction algorithm, while reducing computing time by several orders of magnitude. We apply PRR to visualize FPM images of Giemsa-stained peripheral blood films and present a novel image processing pipeline to construct an effective extended depth of field image which optimally displays the 3D sample structure in a 2D image. We also show how digital refocusing allows effective correction of the chromatic focus shifts inherent to the low magnification objective lenses used in FPM setups, improving the overall quality of color FPM images.

Published by The Optical Society under the terms of the Creative Commons Attribution 4.0 License. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI.

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References

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    [Crossref]
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    [Crossref]
  5. J. Zhang, T. Xu, Z. Shen, Y. Qiao, and Y. Zhang, “Fourier ptychographic microscopy reconstruction with multiscale deep residual network,” Opt. Express 27(6), 8612 (2019).
    [Crossref]
  6. T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
    [Crossref]
  7. K. Guo, S. Dong, and G. Zheng, “Fourier Ptychography for Brightfield, Phase, Darkfield, Reflective, Multi-Slice, and Fluorescence Imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016).
    [Crossref]
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    [Crossref]
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    [Crossref]
  22. A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
    [Crossref]
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    [Crossref]
  24. F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
    [Crossref]

2019 (3)

2018 (1)

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

2016 (2)

K. Guo, S. Dong, and G. Zheng, “Fourier Ptychography for Brightfield, Phase, Darkfield, Reflective, Multi-Slice, and Fluorescence Imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016).
[Crossref]

R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3(8), 827 (2016).
[Crossref]

2015 (5)

2014 (3)

2013 (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

2012 (1)

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
[Crossref]

2008 (1)

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

2007 (1)

J. Bioucas-Dias, “Phase Unwrapping via Graph Cuts,” IEEE Trans. on Image Process. 16(3), 698–709 (2007).
[Crossref]

2004 (2)

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

2001 (1)

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
[Crossref]

1982 (1)

Aidukas, T.

T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
[Crossref]

Amodaj, N.

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Becerra, J. M.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
[Crossref]

Berent, J.

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

Bevilacqua, A.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
[Crossref]

Bioucas-Dias, J.

J. Bioucas-Dias, “Phase Unwrapping via Graph Cuts,” IEEE Trans. on Image Process. 16(3), 698–709 (2007).
[Crossref]

Bovik, A. C.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Chen, M.

Chen, Q.

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

Chowdhury, S.

Chung, J.

R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3(8), 827 (2016).
[Crossref]

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PLoS One 10(7), e0133489 (2015).
[Crossref]

Debailleul, M.

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

Dong, J.

Dong, S.

K. Guo, S. Dong, and G. Zheng, “Fourier Ptychography for Brightfield, Phase, Darkfield, Reflective, Multi-Slice, and Fluorescence Imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016).
[Crossref]

Eckert, R.

S. Chowdhury, M. Chen, R. Eckert, D. Ren, F. Wu, N. Repina, and L. Waller, “High-resolution 3D refractive index microscopy of multiple-scattering samples from intensity images,” Optica 6(9), 1211 (2019).
[Crossref]

T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
[Crossref]

Edelstein, A. D.

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Fan, Y.

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

Fienup, J. R.

Forster, B.

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

Georges, V.

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

Goodman, J. W.

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

Guo, K.

K. Guo, S. Dong, and G. Zheng, “Fourier Ptychography for Brightfield, Phase, Darkfield, Reflective, Multi-Slice, and Fluorescence Imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016).
[Crossref]

Haeberlé, O.

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

Harvey, A.

T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
[Crossref]

Horstmeyer, R.

Inoue, S.

S. Inoue and R. Oldenbourg, Handbook of Optics, Volume II, Devices, Measurements & Properties, 2nd Edition (McGraw-Hill, 1995).

Konda, P.

T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
[Crossref]

Kulkarni, R. P.

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PLoS One 10(7), e0133489 (2015).
[Crossref]

Lauer, V.

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

Li, X.

Liu, Z.

Marshall, D.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
[Crossref]

Oldenbourg, R.

S. Inoue and R. Oldenbourg, Handbook of Optics, Volume II, Devices, Measurements & Properties, 2nd Edition (McGraw-Hill, 1995).

Ou, X.

Piccinini, F.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
[Crossref]

Pinkard, H.

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Qiao, Y.

Ramchandran, K.

Ren, D.

Repina, N.

Sage, D.

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

Sheikh, H. R.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Shen, Z.

Simon, B.

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

Simoncelli, E. P.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Soltanolkotabi, M.

Stuurman, N.

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Sun, J.

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

Tang, G.

Terrero, J. J.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
[Crossref]

Tesei, A.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
[Crossref]

Tian, L.

Tsuchida, M. A.

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Unser, M.

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

Valdecasas, A. G.

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
[Crossref]

Vale, R. D.

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Van De Ville, D.

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

Waller, L.

Wang, Z.

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

Wu, F.

Xu, T.

Yang, C.

Yeh, L.-H.

Zhang, J.

J. Zhang, T. Xu, Z. Shen, Y. Qiao, and Y. Zhang, “Fourier ptychographic microscopy reconstruction with multiscale deep residual network,” Opt. Express 27(6), 8612 (2019).
[Crossref]

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

Zhang, Y.

Zheng, G.

K. Guo, S. Dong, and G. Zheng, “Fourier Ptychography for Brightfield, Phase, Darkfield, Reflective, Multi-Slice, and Fluorescence Imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016).
[Crossref]

R. Horstmeyer, J. Chung, X. Ou, G. Zheng, and C. Yang, “Diffraction tomography with Fourier ptychography,” Optica 3(8), 827 (2016).
[Crossref]

X. Ou, R. Horstmeyer, G. Zheng, and C. Yang, “High numerical aperture Fourier ptychography: principle, implementation and characterization,” Opt. Express 23(3), 3472–3491 (2015).
[Crossref]

X. Ou, G. Zheng, and C. Yang, “Embedded pupil function recovery for Fourier ptychographic microscopy,” Opt. Express 22(5), 4960–4972 (2014).
[Crossref]

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Zhong, J.

Zoli, W.

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
[Crossref]

Zuo, C.

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

Appl. Opt. (1)

Biomed. Opt. Express (1)

IEEE J. Sel. Top. Quantum Electron. (1)

K. Guo, S. Dong, and G. Zheng, “Fourier Ptychography for Brightfield, Phase, Darkfield, Reflective, Multi-Slice, and Fluorescence Imaging,” IEEE J. Sel. Top. Quantum Electron. 22(4), 77–88 (2016).
[Crossref]

IEEE Trans. on Image Process. (2)

Z. Wang, A. C. Bovik, H. R. Sheikh, and E. P. Simoncelli, “Image Quality Assessment: From Error Visibility to Structural Similarity,” IEEE Trans. on Image Process. 13(4), 600–612 (2004).
[Crossref]

J. Bioucas-Dias, “Phase Unwrapping via Graph Cuts,” IEEE Trans. on Image Process. 16(3), 698–709 (2007).
[Crossref]

J. Biol. Methods (1)

A. D. Edelstein, M. A. Tsuchida, N. Amodaj, H. Pinkard, R. D. Vale, and N. Stuurman, “Advanced methods of microscope control using µManager software,” J. Biol. Methods 1(2), 10 (2014).
[Crossref]

Meas. Sci. Technol. (1)

M. Debailleul, B. Simon, V. Georges, O. Haeberlé, and V. Lauer, “Holographic microscopy and diffractive microtomography of transparent samples,” Meas. Sci. Technol. 19(7), 074009 (2008).
[Crossref]

Micron (1)

A. G. Valdecasas, D. Marshall, J. M. Becerra, and J. J. Terrero, “On the extended depth of focus algorithms for bright field microscopy,” Micron 32(6), 559–569 (2001).
[Crossref]

Microsc. Res. Tech. (2)

B. Forster, D. Van De Ville, J. Berent, D. Sage, and M. Unser, “Complex wavelets for extended depth-of-field: A new method for the fusion of multichannel microscopy images,” Microsc. Res. Tech. 65(1-2), 33–42 (2004).
[Crossref]

F. Piccinini, A. Tesei, W. Zoli, and A. Bevilacqua, “Extended depth of focus in optical microscopy: Assessment of existing methods and a new proposal,” Microsc. Res. Tech. 75(11), 1582–1592 (2012).
[Crossref]

Nat. Photonics (1)

G. Zheng, R. Horstmeyer, and C. Yang, “Wide-field, high-resolution Fourier ptychographic microscopy,” Nat. Photonics 7(9), 739–745 (2013).
[Crossref]

Opt. Express (4)

Optica (4)

PLoS One (1)

J. Chung, X. Ou, R. P. Kulkarni, and C. Yang, “Counting White Blood Cells from a Blood Smear Using Fourier Ptychographic Microscopy,” PLoS One 10(7), e0133489 (2015).
[Crossref]

Sci. Rep. (2)

J. Sun, C. Zuo, J. Zhang, Y. Fan, and Q. Chen, “High-speed Fourier ptychographic microscopy based on programmable annular illuminations,” Sci. Rep. 8(1), 7669 (2018).
[Crossref]

T. Aidukas, R. Eckert, A. Harvey, L. Waller, and P. Konda, “Low-cost, sub-micron resolution, wide-field computational microscopy using opensource hardware,” Sci. Rep. 9(1), 7457 (2019).
[Crossref]

Other (2)

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

S. Inoue and R. Oldenbourg, Handbook of Optics, Volume II, Devices, Measurements & Properties, 2nd Edition (McGraw-Hill, 1995).

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Figures (8)

Fig. 1.
Fig. 1. (a) Schematic diagram illustrating the principle of FPM in which images of the sample are captured at different illumination angles using an LED array. (b) Experimental FPM setup using a low-cost off-the-shelf LED matrix.
Fig. 2.
Fig. 2. Observation of the decrease in the DoF after reconstruction. FPM amplitude images of a blood smear containing infected RBCs (red arrow) for raw images captured with the sample at two different axial positions within the DoF of the objective lens. (a)-(c) z = -2 µm; (b)-(d) z = 1 µm. (c) and (d) are aberration corrected. (e)-(f) Phase in radians of pupil functions corresponding to the system for cases (c) and (d).
Fig. 3.
Fig. 3. (a) Reference image captured from a 100x/0.8 objective manually focused at the plane for which MPs (indicated by red arrows) are optimally in focus. (b) FPM amplitude image generated using the autofocus EPRY method. (c-d) Optimal (highest SSIM) refocused images, using (c) the AS propagation term and (d) a linear weighting of frequency components. (e) SSIM of the different focal planes for both propagation kernels. Image (a) is used as the reference.
Fig. 4.
Fig. 4. (a) Flow chart depicting the pipeline developed for generating an EDoF image from a reconstructed FPM image. Size filter 1 and size filter 2 refer to size exclusion filters to remove very small objects (less than fifty pixels) and objects smaller than a cell with a diameter of 4 µm respectively.
Fig. 5.
Fig. 5. (a) Reference image captured using a 100x/0.8 objective lens (b-d) EDoF images computed using (b) the variance method (c) the complex wavelet transform method (d) our RBC segmentation method. The same numerically computed focal series was used for (b), (c) and (d).
Fig. 6.
Fig. 6. Performance of digital refocusing methods as a function of focus shift. The bottom right panel shows three reconstructed features without refocusing (black), using the IPM method (blue) and the PRR method (red) for a defocus distance of 215 µm.
Fig. 7.
Fig. 7. EDoF image of a Giemsa stained blood film captured in air. For each highlighted boxed region (red) the corresponding raw captured image (blue), the reconstructed (unfocused) high resolution image (green) and a reference image captured using a 100x/0.8 objective lens (purple) are also shown.
Fig. 8.
Fig. 8. Application of PRR and EDoF methods to correct chromatic focus shifts in FPM. (a) (left) A blood smear sample coated with oil is located outside the DoF of the objective lens for two of the three color channels. (centre) Individual color channel images can be refocused using PRR (where zR, zG, zB are different planes within the numerically generated focal series), before being combined into a final corrected RGB image (right). (b) (left) Reconstructed FPM image of the same sample positioned within the DoF of the objective lens for all illumination wavelengths. (centre) 2D EDoF images are reconstructed separately for each color channel, to optimally display 3D image information. These single color EDoF images can be combined to yield a final color EDoF image free from artefacts associated with chromatic focal shifts (right).

Equations (4)

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D o F = λ n N A 2 + n M N A e ,
I i = | iFT [ T ( u α i , v β i ) P ( u , v ) ] | 2
U ( x , y , z ) = iFT ( FT [ U ( x , y , z 0 ) ] H ( u , v ) ) with  H ( u , v ) = exp ( j 2 π λ z 1 ( λ u ) 2 ( λ v ) 2 )
H ( u , v ) = exp ( j 2 π λ z ) exp ( j π λ z ( λ u ) 2 + ( λ v ) 2 )

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