Abstract

The transport-of-intensity equation (TIE) is a well-established non-interferometric phase retrieval approach, which enables quantitative phase imaging (QPI) of transparent sample simply by measuring the intensities at multiple axially displaced planes. Nevertheless, it still suffers from two fundamentally limitations. First, it is quite susceptible to low-frequency errors (such as “cloudy” artifacts), which results from the poor contrast of the phase transfer function (PTF) near the zero frequency. Second, the reconstructed phase tends to blur under spatially low-coherent illumination, especially when the defocus distance is beyond the near Fresnel region. Recent studies have shown that the shape of the illumination aperture has a significant impact on the resolution and phase reconstruction quality, and by simply replacing the conventional circular illumination aperture with an annular one, these two limitations can be addressed, or at least significantly alleviated. However, the annular aperture was previously empirically designed based on intuitive criteria related to the shape of PTF, which does not guarantee optimality. In this work, we optimize the illumination pattern to maximize TIE’s performance based on a combined quantitative criterion for evaluating the “goodness” of an aperture. In order to make the size of the solution search space tractable, we restrict our attention to binary-coded axis-symmetric illumination patterns only, which are easier to implement and can generate isotropic TIE PTFs. We test the obtained optimal illumination by imaging both a phase resolution target and HeLa cells based on a small-pitch LED array, suggesting superior performance over other suboptimal patterns in terms of both signal-to-noise ratio (SNR) and spatial resolution.

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

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References

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2018 (2)

2017 (4)

T. Chakraborty and J. C. Petruccelli, “Source diversity for contrast transfer function imaging,” Proc. SPIE 10222, 102220R (2017).
[Crossref]

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref] [PubMed]

J. Li, Q. Chen, J. Zhang, Y. Zhang, L. Lu, and C. Zuo, “Efficient quantitative phase microscopy using programmable annular LED illumination,” Biomed. Opt. Express 8, 4687–4705 (2017).
[Crossref] [PubMed]

T. Chakraborty and J. C. Petruccelli, “Source diversity for transport of intensity phase imaging,” Opt. Express 25, 9122–9137 (2017).
[Crossref] [PubMed]

2016 (2)

2015 (5)

2014 (3)

2013 (6)

2012 (1)

2011 (2)

2010 (2)

2009 (1)

2006 (2)

2005 (1)

J. W. Lichtman and J.-A. Conchello, “Fluorescence microscopy,” Nat. Methods 2, 910–919 (2005).
[Crossref] [PubMed]

2004 (2)

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. 21, 828–831 (2004).
[Crossref]

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

2002 (1)

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

1998 (1)

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

1985 (1)

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. 2, 121–127 (1985).
[Crossref]

1983 (1)

1955 (1)

G. Nomarski and A. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall 2, 121–128 (1955).
[Crossref]

1953 (1)

H. Hopkins, “On the diffraction theory of optical images,” Proc. Phys. Soc. London A 217, 408–432 (1953).
[Crossref]

1942 (1)

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Alferi, D.

Alieva, T.

Anastasio, M. A.

Asundi, A.

Badizadegan, K.

Barbastathis, G.

Barone-Nugent, E. D.

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

Barty, A.

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Berty, A.

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

Bhaduri, Basanta

Carney, P. S.

Castello, M.

Chakraborty, T.

T. Chakraborty and J. C. Petruccelli, “Source diversity for contrast transfer function imaging,” Proc. SPIE 10222, 102220R (2017).
[Crossref]

T. Chakraborty and J. C. Petruccelli, “Source diversity for transport of intensity phase imaging,” Opt. Express 25, 9122–9137 (2017).
[Crossref] [PubMed]

Chen, M.

Chen, Q.

J. Sun, Q. Chen, J. Zhang, Y. Fan, and C. Zuo, “Single-shot quantitative phase microscopy based on color-multiplexed Fourier ptychography,” Opt. Lett. 43, 3365–3368 (2018).
[Crossref] [PubMed]

J. Li, Q. Chen, J. Sun, J. Zhang, J. Ding, and C. Zuo, “Three-dimensional tomographic microscopy technique with multi-frequency combination with partially coherent illuminations,” Biomed. Opt. Express 9, 2526–2542 (2018).
[Crossref]

J. Li, Q. Chen, J. Zhang, Y. Zhang, L. Lu, and C. Zuo, “Efficient quantitative phase microscopy using programmable annular LED illumination,” Biomed. Opt. Express 8, 4687–4705 (2017).
[Crossref] [PubMed]

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref] [PubMed]

J. Li, Q. Chen, J. Sun, J. Zhang, and C. Zuo, “Multimodal computational microscopy based on transport of intensity equation,” J. Biomed. Opt. 21, 126003 (2016).
[Crossref] [PubMed]

C. Zuo, J. Sun, J. Zhang, Y. Hu, and Q. Chen, “Lensless phase microscopy and diffraction tomography with multi-angle and multi-wavelength illuminations using a LED matrix,” Opt. Express 23, 14314–14328 (2015).
[Crossref] [PubMed]

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective,” Opt. Laser Eng. 71, 20–32 (2015).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” Opt. Lett. 38, 3538–3541 (2013).
[Crossref] [PubMed]

C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter-theory and applications,” Opt. Express 21, 5346–5362 (2013).
[Crossref] [PubMed]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Phase aberration compensation in digital holographic microscopy based on principal component analysis,” Opt. Lett. 38, 1724–1726 (2013).
[Crossref] [PubMed]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21, 24060–24075 (2013).
[Crossref] [PubMed]

Chen, R.

Chen, X.

Conchello, J.-A.

J. W. Lichtman and J.-A. Conchello, “Fluorescence microscopy,” Nat. Methods 2, 910–919 (2005).
[Crossref] [PubMed]

Dasari, R. R.

De Nicola, S.

De Petrocellis, L.

Depeursinge, C.

Diaspro, A.

Ding, H.

Ding, J.

Falaggis, K.

Fan, Y.

Feld, M. S.

Ferraro, P.

Finizio, A.

Gaylord, T. K.

Gillette, M. U.

Heintzmann, R.

Hopkins, H.

H. Hopkins, “On the diffraction theory of optical images,” Proc. Phys. Soc. London A 217, 408–432 (1953).
[Crossref]

Horstmeyer, R.

Hu, Y.

Jenkins, M. H.

Kim, M. K.

M. K. Kim, Digital Holographic Microscopy: Principles, Techniques, and Applications (Springer, 2011).
[Crossref]

Kou, S. S.

Kozacki, T.

Li, J.

J. Li, Q. Chen, J. Sun, J. Zhang, J. Ding, and C. Zuo, “Three-dimensional tomographic microscopy technique with multi-frequency combination with partially coherent illuminations,” Biomed. Opt. Express 9, 2526–2542 (2018).
[Crossref]

J. Li, Q. Chen, J. Zhang, Y. Zhang, L. Lu, and C. Zuo, “Efficient quantitative phase microscopy using programmable annular LED illumination,” Biomed. Opt. Express 8, 4687–4705 (2017).
[Crossref] [PubMed]

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref] [PubMed]

J. Li, Q. Chen, J. Sun, J. Zhang, and C. Zuo, “Multimodal computational microscopy based on transport of intensity equation,” J. Biomed. Opt. 21, 126003 (2016).
[Crossref] [PubMed]

Lichtman, J. W.

J. W. Lichtman and J.-A. Conchello, “Fluorescence microscopy,” Nat. Methods 2, 910–919 (2005).
[Crossref] [PubMed]

Liu, Z.

Long, J. M.

Lu, L.

Marquet, P.

Martinez-Carranza, J.

McMahon, P.

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Mehta, S. B.

Millet, L.

Mir, M.

Mir, Mustafa

Nomarski, G.

G. Nomarski and A. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall 2, 121–128 (1955).
[Crossref]

Nugent, K.

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

Nugent, K. A.

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

Ou, X.

Paganin, D.

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

D. Paganin and K. A. Nugent, “Noninterferometric phase imaging with partially coherent light,” Phys. Rev. Lett. 80, 2586 (1998).
[Crossref]

Park, Y.

Petruccelli, J. C.

Pham, Hoa

Pierattini, G.

Popescu, G.

Popescu, Gabriel

Qu, W.

Rodrigo, J. A.

Rogers, J.

Roth, S.

Schoonover, R. W.

Sheppard, C. J.

Sheppard, C. J. R.

S. S. Kou, L. Waller, G. Barbastathis, P. Marquet, C. Depeursinge, and C. J. R. Sheppard, “Quantitative phase restoration by direct inversion using the optical transfer function,” Opt. Lett. 36, 2671–2673 (2011).
[Crossref] [PubMed]

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. 21, 828–831 (2004).
[Crossref]

Streibl, N.

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. 2, 121–127 (1985).
[Crossref]

Sun, J.

Teague, M. R.

Tian, L.

Unarunotai, S.

Vicidomini, G.

Waller, L.

Wang, J.

Wang, Z.

Weill, A.

G. Nomarski and A. Weill, “Application à la métallographie des méthodes interférentielles à deux ondes polarisées,” Rev. Metall 2, 121–128 (1955).
[Crossref]

Yang, C.

Yeh, L.-H.

Yu, Y.

Zernike, F.

F. Zernike, “Phase contrast, a new method for the microscopic observation of transparent objects,” Physica 9(7), 686–698 (1942).
[Crossref]

Zhang, J.

Zhang, Y.

Zheng, G.

Zhong, J.

Zuo, C.

J. Li, Q. Chen, J. Sun, J. Zhang, J. Ding, and C. Zuo, “Three-dimensional tomographic microscopy technique with multi-frequency combination with partially coherent illuminations,” Biomed. Opt. Express 9, 2526–2542 (2018).
[Crossref]

J. Sun, Q. Chen, J. Zhang, Y. Fan, and C. Zuo, “Single-shot quantitative phase microscopy based on color-multiplexed Fourier ptychography,” Opt. Lett. 43, 3365–3368 (2018).
[Crossref] [PubMed]

J. Li, Q. Chen, J. Zhang, Y. Zhang, L. Lu, and C. Zuo, “Efficient quantitative phase microscopy using programmable annular LED illumination,” Biomed. Opt. Express 8, 4687–4705 (2017).
[Crossref] [PubMed]

C. Zuo, J. Sun, J. Li, J. Zhang, A. Asundi, and Q. Chen, “High-resolution transport-of-intensity quantitative phase microscopy with annular illumination,” Sci. Rep. 7(1), 7654 (2017).
[Crossref] [PubMed]

J. Li, Q. Chen, J. Sun, J. Zhang, and C. Zuo, “Multimodal computational microscopy based on transport of intensity equation,” J. Biomed. Opt. 21, 126003 (2016).
[Crossref] [PubMed]

C. Zuo, J. Sun, J. Zhang, Y. Hu, and Q. Chen, “Lensless phase microscopy and diffraction tomography with multi-angle and multi-wavelength illuminations using a LED matrix,” Opt. Express 23, 14314–14328 (2015).
[Crossref] [PubMed]

C. Zuo, Q. Chen, L. Tian, L. Waller, and A. Asundi, “Transport of intensity phase retrieval and computational imaging for partially coherent fields: The phase space perspective,” Opt. Laser Eng. 71, 20–32 (2015).
[Crossref]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Noninterferometric single-shot quantitative phase microscopy,” Opt. Lett. 38, 3538–3541 (2013).
[Crossref] [PubMed]

C. Zuo, Q. Chen, Y. Yu, and A. Asundi, “Transport-of-intensity phase imaging using Savitzky-Golay differentiation filter-theory and applications,” Opt. Express 21, 5346–5362 (2013).
[Crossref] [PubMed]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “Phase aberration compensation in digital holographic microscopy based on principal component analysis,” Opt. Lett. 38, 1724–1726 (2013).
[Crossref] [PubMed]

C. Zuo, Q. Chen, W. Qu, and A. Asundi, “High-speed transport-of-intensity phase microscopy with an electrically tunable lens,” Opt. Express 21, 24060–24075 (2013).
[Crossref] [PubMed]

Zysk, A. M.

Appl. Opt. (1)

Biomed. Opt. Express (2)

J. Biomed. Opt. (1)

J. Li, Q. Chen, J. Sun, J. Zhang, and C. Zuo, “Multimodal computational microscopy based on transport of intensity equation,” J. Biomed. Opt. 21, 126003 (2016).
[Crossref] [PubMed]

J. Microsc. (2)

D. Paganin, A. Barty, P. McMahon, and K. Nugent, “Quantitative phase-amplitude microscopy. III. The effects of noise,” J. Microsc. 214, 51–61 (2004).
[Crossref] [PubMed]

E. D. Barone-Nugent, A. Berty, and K. A. Nugent, “Quantitative phase-amplitude microscopy I: optical microscopy,” J. Microsc. 206, 194–203 (2002).
[Crossref] [PubMed]

J. Opt. Soc. Am. (3)

M. R. Teague, “Deterministic phase retrieval: a Greens function solution,” J. Opt. Soc. Am. 73(11), 1434–1441 (1983).
[Crossref]

C. J. R. Sheppard, “Defocused transfer function for a partially coherent microscope and application to phase retrieval,” J. Opt. Soc. Am. 21, 828–831 (2004).
[Crossref]

N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. 2, 121–127 (1985).
[Crossref]

Nat. Methods (1)

J. W. Lichtman and J.-A. Conchello, “Fluorescence microscopy,” Nat. Methods 2, 910–919 (2005).
[Crossref] [PubMed]

Opt. Express (12)

J. C. Petruccelli, L. Tian, and G. Barbastathis, “The transport of intensity equation for optical path length recovery using partially coherent illumination,” Opt. Express 21, 14430–14441 (2013).
[Crossref] [PubMed]

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Supplementary Material (1)

NameDescription
» Visualization 1       the fast switching of binary coded LED illumination patterns captured from LED array and objective pupil

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Figures (8)

Fig. 1
Fig. 1 Segmentation of illumination source and labeling of all combinatorial illumination patterns. The incoherent illumination source is evenly divided into a lot of discrete annular patterns with same center point, and these annuli are labeled from the LSB R0 to the MSB R11 with 12-bit. The radius cut-profile of each illumination pattern is illustrated as a PWM wave or binary number, and the corresponding illumination pattern can be numbered from 1 to 4095 in decimal.
Fig. 2
Fig. 2 2D images and line profiles of PTF of five selected illumination patterns under weak defocusing assumptions.
Fig. 3
Fig. 3 The imaging performance evaluation of PTF of all illumination source patterns using three crucial criteria, including cutoff frequency, zero crossings, and mean value of absolute of PTF. (a-c) Example of binary-coded illumination pattern, corresponding 2D PTF image in color map, and area plot of radius cut-line of PTF. (d1-d2) The cutoff frequency value of PTF of all illumination patterns is calculated and the illumination patterns whose cutoff frequency equals the normalized spatial threshold value 1 + N 1 N are retained. (e1-e2) Statistics of zero crossings of source patterns with the maximum cutoff frequency value, and the preserved source patterns with no zero crossings. (f1-f2) Mean value of absolute PTF curves with both the maximum cutoff frequency and no zero crossings are calculated, and the PTFs of top three mean values are selected. (g1-g3) Enlarged PTFs of top three mean values. (h1-h3) Area plots of radius cut-line of the top three best PTF with the same cutoff frequency value.
Fig. 4
Fig. 4 Comparative line profiles of PTFs of illumination patterns with different index. (a) The cut-lines of PTF of source patterns with special index from 128 (12’b0000 1000 0000) to 2048 (12’b1000 0000 0000). (b) The cut-lines of PTF of source patterns with the same maximum outer radius and different annular width from 2048 (12’b1000 0000 0000) to 3840 (12’b1111 0000 0000). (c) The cut-lines of PTF of combined source patterns from 2176 (12’b1000 1000 0000) to 2560 (12’b1010 0000 0000). (d) The cut-lines of PTF with different bit-depth (12-bit, 16-bit and 32-bit) of illumination source segmentation for single annular illumination pattern.
Fig. 5
Fig. 5 Four selected index of illumination pattern on the LED array. Sub-figures in the first row are the actual LED source patterns in light path, and last two rows of figure are the 2D images and cut lines of PTFs.
Fig. 6
Fig. 6 (a) Schematic diagram of implementation. (b) Photograph of whole imaging system. The crucial parts of setup are highlighted with the gray boxes. (c) Four selected binary LED illumination patterns on the LED array and the respective pattern distribution in the Fourier plane. Scale bar represents 300 µm and 5 mm, respectively.
Fig. 7
Fig. 7 Quantitative phase reconstruction results based on control samples using the optimal illumination pattern (Nindex = 128). (a) Results of the micro polystyrene bead with 8 µm diameter. (b) Results of quantitative phase resolution target. The achievable experimental imaging resolution is 0.274 µm, corresponding resolution element 6 in group 10 while the theoretical achievable resolution bar width is 0.188 µm, corresponding resolution element 3 in group 11. Scale bar denotes 20 µm, 10 µm, and 2 µm respectively.
Fig. 8
Fig. 8 Experimental results of unstained HeLa cell under five selected LED illumination patterns. First three rows of sub-figures are the five selected LED illumination patterns, corresponding 2D PTF images, and cut-line profiles. Sub-figures in the fourth and fifth rows are the difference images and 2D Fourier spectrum images of HeLa cell under different illumination patterns. Last two rows of sub-figures show the recovered quantitative phase images of HeLa cell and enlarged sub-region of single cell during the interphase. Scale bar denotes 50 µm and 20 µm, respectively.

Equations (7)

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t ( r ) a ( r ) exp [ i ϕ ( r ) ] a ( r ) [ 1 + i ϕ ( r ) ] a ( r ) = a 0 + Δ a ( r ) a 0 + Δ a ( r ) + i a 0 ϕ ( r )
I ( r ) = t ( r ) t * ( r ) a 0 2 + a 0 [ Δ a ( r ) + Δ a ( r ) ] + i a 0 [ ϕ ( r ) + ϕ ( r ) ]
I ˜ ( u ) = B δ ( u ) + A ˜ ( u ) H A ( u ) + P ˜ ( u ) H P ( u )
H p ( u ) o b l = 1 2 | P ( u ρ s ) | sin [ k z ( 1 λ 2 | u ρ s | 2 1 λ 2 | ρ s | 2 ) ] + 1 2 | P ( u + ρ s ) | sin [ k z ( 1 λ 2 | u + ρ s | 2 1 λ 2 | ρ s | 2 ) ]
S ( u ) = δ ( u ρ s ) + δ ( u + ρ s )
| P ( u ) | = { 1 , if | u | ρ P 0 , if | u | > ρ P
S ( u ) = i = 0 n δ ( u u i ) , | ρ 1 | | u i | | ρ 2 |

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