Abstract

Design and optimization of lensless phase-retrieval optical system with phase modulation of free-space propagation wavefront is proposed for subpixel imaging to achieve super-resolution reconstruction. Contrary to the traditional super-resolution phase-retrieval, the method in this paper requires a single observation only and uses the advanced Super-Resolution Sparse Phase Amplitude Retrieval (SR-SPAR) iterative technique which contains optimized sparsity based filters and multi-scale filters. The successful object imaging relies on modulation of the object wavefront with a random phase-mask, which generates coded diffracted intensity pattern, allowing us to extract subpixel information. The system’s noise-robustness was investigated and verified. The super-resolution phase-imaging is demonstrated by simulations and physical experiments. The simulations included high quality reconstructions with super-resolution factor of 5, and acceptable at factor up to 9. By physical experiments 3 $\def\upmu{\unicode[Times]{x03BC}}\upmu$m details were resolved, which are 2.3 times smaller than the resolution following from the Nyquist-Shannon sampling theorem.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2019 (1)

2018 (6)

H. Kwon, E. Arbabi, S. M. Kamali, M. Faraji-Dana, and A. Faraon, “Computational complex optical field imaging using a designed metasurface diffuser,” Optica 5(8), 924–931 (2018).
[Crossref]

M. Rostykus, M. Rossi, and C. Moser, “Compact lensless subpixel resolution large field of view microscope,” Opt. Lett. 43(8), 1654–1657 (2018).
[Crossref]

I. Shevkunov, V. Katkovnik, N. V. Petrov, and K. Egiazarian, “Super-resolution microscopy for biological specimens: lensless phase retrieval in noisy conditions,” Biomed. Opt. Express 9(11), 5511–5523 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Multiwavelength surface contouring from phase-coded noisy diffraction patterns: wavelength-division optical setup,” Opt. Eng. 57(08), 1 (2018).
[Crossref]

E. Achimova, V. Abaskin, D. Claus, G. Pedrini, I. Shevkunov, and V. Katkovnik, “Noise minimised high resolution digital holographic microscopy applied to surface topography,” Comput. Opt. 42(2), 267–272 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. Petrov, and K. Eguiazarian, “Multiwavelength absolute phase retrieval from noisy diffractive patterns: wavelength multiplexing algorithm,” Appl. Sci. 8(5), 719 (2018).
[Crossref]

2017 (6)

2016 (2)

2015 (2)

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Analysis 39(2), 277–299 (2015).
[Crossref]

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

2013 (1)

E. McLeod, W. Luo, O. Mudanyali, A. Greenbaum, and A. Ozcan, “Toward giga-pixel nanoscopy on a chip: a computational wide-field look at the nano-scale without the use of lenses,” Lab Chip 13(11), 2028–2035 (2013).
[Crossref]

2011 (1)

2010 (2)

2009 (1)

2007 (1)

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

2006 (1)

2000 (1)

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

1995 (1)

A. Kirkland, W. Saxton, K.-L. Chau, K. Tsuno, and M. Kawasaki, “Super-resolution by aperture synthesis: tilt series reconstruction in ctem,” Ultramicroscopy 57(4), 355–374 (1995).
[Crossref]

1994 (1)

1984 (1)

1982 (1)

1974 (1)

R. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21(9), 709–720 (1974).
[Crossref]

1972 (1)

R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Abaskin, V.

E. Achimova, V. Abaskin, D. Claus, G. Pedrini, I. Shevkunov, and V. Katkovnik, “Noise minimised high resolution digital holographic microscopy applied to surface topography,” Comput. Opt. 42(2), 267–272 (2018).
[Crossref]

Achimova, E.

E. Achimova, V. Abaskin, D. Claus, G. Pedrini, I. Shevkunov, and V. Katkovnik, “Noise minimised high resolution digital holographic microscopy applied to surface topography,” Comput. Opt. 42(2), 267–272 (2018).
[Crossref]

Agrawal, A.

A. Agrawal and R. Raskar, “Resolving objects at higher resolution from a single motion-blurred image,” in 2007 IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2007), pp. 1–8.

Almoro, P.

Arbabi, E.

Bernet, S.

Bespalov, V. G.

N. V. Petrov, V. G. Bespalov, and A. A. Gorodetsky, “Phase retrieval method for multiple wavelength speckle patterns,” in Speckle 2010: Optical Metrology, vol. 7387 (International Society for Optics and Photonics, 2010), p. 73871T

Burger, H. C.

H. C. Burger and S. Harmeling, “Improving denoising algorithms via a multi-scale meta-procedure,” in Joint Pattern Recognition Symposium, (Springer, 2011), pp. 206–215.

Bürgi, T.

Camacho, L.

Candes, E. J.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Analysis 39(2), 277–299 (2015).
[Crossref]

Chapman, H. N.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Chau, K.-L.

A. Kirkland, W. Saxton, K.-L. Chau, K. Tsuno, and M. Kawasaki, “Super-resolution by aperture synthesis: tilt series reconstruction in ctem,” Ultramicroscopy 57(4), 355–374 (1995).
[Crossref]

Chen, H.-C.

Cheng, C.-J.

Claus, D.

E. Achimova, V. Abaskin, D. Claus, G. Pedrini, I. Shevkunov, and V. Katkovnik, “Noise minimised high resolution digital holographic microscopy applied to surface topography,” Comput. Opt. 42(2), 267–272 (2018).
[Crossref]

D. Claus, G. Pedrini, and W. Osten, “Iterative phase retrieval based on variable wavefront curvature,” Appl. Opt. 56(13), F134–F137 (2017).
[Crossref]

Claverley, J.

Cohen, O.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Dabov, K.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

Dong, B.-z.

Egami, R.

Egiazarian, K.

I. Shevkunov, V. Katkovnik, N. V. Petrov, and K. Egiazarian, “Super-resolution microscopy for biological specimens: lensless phase retrieval in noisy conditions,” Biomed. Opt. Express 9(11), 5511–5523 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Multiwavelength surface contouring from phase-coded noisy diffraction patterns: wavelength-division optical setup,” Opt. Eng. 57(08), 1 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Computational super-resolution phase retrieval from multiple phase-coded diffraction patterns: simulation study and experiments,” Optica 4(7), 786–794 (2017).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

P. Kocsis, I. Shevkunov, V. Katkovnik, and K. Egiazarian, “Single exposure lensless phase imaging,” in EUVIP Workshop (2018).

P. Kocsis, I. Shevkunov, V. Katkovnik, and K. Egiazarian, “Single exposure lensless subpixel phase imaging,” in Digital Optical Technologies 2019, vol. 11062 (International Society for Optics and Photonics, 2019), p. 1106212.

Eguiazarian, K.

V. Katkovnik, I. Shevkunov, N. Petrov, and K. Eguiazarian, “Multiwavelength absolute phase retrieval from noisy diffractive patterns: wavelength multiplexing algorithm,” Appl. Sci. 8(5), 719 (2018).
[Crossref]

Eldar, Y. C.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Ersoy, O. K.

Faraji-Dana, M.

Faraon, A.

Fienup, J. R.

Foi, A.

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

Gabor, D.

D. Gabor, A new microscopic principle (Nature Publishing Group, 1948).

García, J.

Gerchberg, R.

R. Gerchberg, “Super-resolution through error energy reduction,” Opt. Acta 21(9), 709–720 (1974).
[Crossref]

Gerchberg, R. W.

R. W. Gerchberg, “A practical algorithm for the determination of phase from image and diffraction plane pictures,” Optik 35, 237–246 (1972).

Godden, T.

Goodman, J. W.

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

Gorodetsky, A. A.

N. V. Petrov, V. G. Bespalov, and A. A. Gorodetsky, “Phase retrieval method for multiple wavelength speckle patterns,” in Speckle 2010: Optical Metrology, vol. 7387 (International Society for Optics and Photonics, 2010), p. 73871T

Greenbaum, A.

E. McLeod, W. Luo, O. Mudanyali, A. Greenbaum, and A. Ozcan, “Toward giga-pixel nanoscopy on a chip: a computational wide-field look at the nano-scale without the use of lenses,” Lab Chip 13(11), 2028–2035 (2013).
[Crossref]

Grillo, R.

Gu, B.-y.

Gustafsson, M. G.

M. G. Gustafsson, “Surpassing the lateral resolution limit by a factor of two using structured illumination microscopy,” J. Microsc. 198(2), 82–87 (2000).
[Crossref]

Harm, W.

Harmeling, S.

H. C. Burger and S. Harmeling, “Improving denoising algorithms via a multi-scale meta-procedure,” in Joint Pattern Recognition Symposium, (Springer, 2011), pp. 206–215.

Horisaki, R.

Humphry, M.

Javidi, B.

Jesacher, A.

Joyce, L. S.

Kamali, S. M.

Kandel, M. E.

T. H. Nguyen, M. E. Kandel, M. Rubessa, M. B. Wheeler, and G. Popescu, “Gradient light interference microscopy for 3d imaging of unlabeled specimens,” Nat. Commun. 8(1), 210 (2017).
[Crossref]

Katkovnik, V.

I. Shevkunov, V. Katkovnik, N. V. Petrov, and K. Egiazarian, “Super-resolution microscopy for biological specimens: lensless phase retrieval in noisy conditions,” Biomed. Opt. Express 9(11), 5511–5523 (2018).
[Crossref]

E. Achimova, V. Abaskin, D. Claus, G. Pedrini, I. Shevkunov, and V. Katkovnik, “Noise minimised high resolution digital holographic microscopy applied to surface topography,” Comput. Opt. 42(2), 267–272 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Multiwavelength surface contouring from phase-coded noisy diffraction patterns: wavelength-division optical setup,” Opt. Eng. 57(08), 1 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. Petrov, and K. Eguiazarian, “Multiwavelength absolute phase retrieval from noisy diffractive patterns: wavelength multiplexing algorithm,” Appl. Sci. 8(5), 719 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Computational super-resolution phase retrieval from multiple phase-coded diffraction patterns: simulation study and experiments,” Optica 4(7), 786–794 (2017).
[Crossref]

K. Dabov, A. Foi, V. Katkovnik, and K. Egiazarian, “Image denoising by sparse 3-d transform-domain collaborative filtering,” IEEE Trans. on Image Process. 16(8), 2080–2095 (2007).
[Crossref]

P. Kocsis, I. Shevkunov, V. Katkovnik, and K. Egiazarian, “Single exposure lensless phase imaging,” in EUVIP Workshop (2018).

P. Kocsis, I. Shevkunov, V. Katkovnik, and K. Egiazarian, “Single exposure lensless subpixel phase imaging,” in Digital Optical Technologies 2019, vol. 11062 (International Society for Optics and Photonics, 2019), p. 1106212.

Kawasaki, M.

A. Kirkland, W. Saxton, K.-L. Chau, K. Tsuno, and M. Kawasaki, “Super-resolution by aperture synthesis: tilt series reconstruction in ctem,” Ultramicroscopy 57(4), 355–374 (1995).
[Crossref]

Kirkland, A.

A. Kirkland, W. Saxton, K.-L. Chau, K. Tsuno, and M. Kawasaki, “Super-resolution by aperture synthesis: tilt series reconstruction in ctem,” Ultramicroscopy 57(4), 355–374 (1995).
[Crossref]

Kocsis, P.

P. Kocsis, I. Shevkunov, V. Katkovnik, and K. Egiazarian, “Single exposure lensless subpixel phase imaging,” in Digital Optical Technologies 2019, vol. 11062 (International Society for Optics and Photonics, 2019), p. 1106212.

P. Kocsis, I. Shevkunov, V. Katkovnik, and K. Egiazarian, “Single exposure lensless phase imaging,” in EUVIP Workshop (2018).

Kohler, C.

Kojima, T.

Kwon, H.

Li, X.

E. J. Candes, X. Li, and M. Soltanolkotabi, “Phase retrieval from coded diffraction patterns,” Appl. Comput. Harmon. Analysis 39(2), 277–299 (2015).
[Crossref]

Lin, Y.-c.

Liu, C.-Y.

Luo, W.

E. McLeod, W. Luo, O. Mudanyali, A. Greenbaum, and A. Ozcan, “Toward giga-pixel nanoscopy on a chip: a computational wide-field look at the nano-scale without the use of lenses,” Lab Chip 13(11), 2028–2035 (2013).
[Crossref]

Matsushima, K.

McLeod, E.

E. McLeod and A. Ozcan, “Microscopy without lenses,” Phys. Today 70(9), 50–56 (2017).
[Crossref]

E. McLeod, W. Luo, O. Mudanyali, A. Greenbaum, and A. Ozcan, “Toward giga-pixel nanoscopy on a chip: a computational wide-field look at the nano-scale without the use of lenses,” Lab Chip 13(11), 2028–2035 (2013).
[Crossref]

Miao, J.

Y. Shechtman, Y. C. Eldar, O. Cohen, H. N. Chapman, J. Miao, and M. Segev, “Phase retrieval with application to optical imaging: a contemporary overview,” IEEE Signal Process. Mag. 32(3), 87–109 (2015).
[Crossref]

Micó, V.

Moser, C.

Mudanyali, O.

E. McLeod, W. Luo, O. Mudanyali, A. Greenbaum, and A. Ozcan, “Toward giga-pixel nanoscopy on a chip: a computational wide-field look at the nano-scale without the use of lenses,” Lab Chip 13(11), 2028–2035 (2013).
[Crossref]

Muñiz-Piniella, A.

Nguyen, T. H.

T. H. Nguyen, M. E. Kandel, M. Rubessa, M. B. Wheeler, and G. Popescu, “Gradient light interference microscopy for 3d imaging of unlabeled specimens,” Nat. Commun. 8(1), 210 (2017).
[Crossref]

Osten, W.

Ozcan, A.

E. McLeod and A. Ozcan, “Microscopy without lenses,” Phys. Today 70(9), 50–56 (2017).
[Crossref]

E. McLeod, W. Luo, O. Mudanyali, A. Greenbaum, and A. Ozcan, “Toward giga-pixel nanoscopy on a chip: a computational wide-field look at the nano-scale without the use of lenses,” Lab Chip 13(11), 2028–2035 (2013).
[Crossref]

Pawley, J.

J. Pawley, Handbook of biological confocal microscopy (Springer Science & Business Media, 2010).

Pedrini, G.

Petrov, N.

V. Katkovnik, I. Shevkunov, N. Petrov, and K. Eguiazarian, “Multiwavelength absolute phase retrieval from noisy diffractive patterns: wavelength multiplexing algorithm,” Appl. Sci. 8(5), 719 (2018).
[Crossref]

Petrov, N. V.

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Multiwavelength surface contouring from phase-coded noisy diffraction patterns: wavelength-division optical setup,” Opt. Eng. 57(08), 1 (2018).
[Crossref]

I. Shevkunov, V. Katkovnik, N. V. Petrov, and K. Egiazarian, “Super-resolution microscopy for biological specimens: lensless phase retrieval in noisy conditions,” Biomed. Opt. Express 9(11), 5511–5523 (2018).
[Crossref]

V. Katkovnik, I. Shevkunov, N. V. Petrov, and K. Egiazarian, “Computational super-resolution phase retrieval from multiple phase-coded diffraction patterns: simulation study and experiments,” Optica 4(7), 786–794 (2017).
[Crossref]

N. V. Petrov, V. G. Bespalov, and A. A. Gorodetsky, “Phase retrieval method for multiple wavelength speckle patterns,” in Speckle 2010: Optical Metrology, vol. 7387 (International Society for Optics and Photonics, 2010), p. 73871T

Popescu, G.

T. H. Nguyen, M. E. Kandel, M. Rubessa, M. B. Wheeler, and G. Popescu, “Gradient light interference microscopy for 3d imaging of unlabeled specimens,” Nat. Commun. 8(1), 210 (2017).
[Crossref]

Raskar, R.

A. Agrawal and R. Raskar, “Resolving objects at higher resolution from a single motion-blurred image,” in 2007 IEEE Conference on Computer Vision and Pattern Recognition, (IEEE, 2007), pp. 1–8.

Ritsch-Marte, M.

Rivenson, Y.

Rockstuhl, C.

Root, W. L.

Rossi, M.

Rostykus, M.

Rubessa, M.

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Figures (9)

Fig. 1.
Fig. 1. (a) Lensless optical setup with phase-mask and laser illumination, the object-mask distance is $d_{1}$ and mask-sensor distance is $d_{2}$. (b) Cross-section of a pixel fragment of an ideal binary phase-mask, with pixel size of $\Delta _{m}$ and height of $\Delta h$.
Fig. 2.
Fig. 2. Flowchart of SR-SPAR method. The images on the left are showing the phases in the object plane before and after the BM3D filtering for given iterations.
Fig. 3.
Fig. 3. Phase images used in simulation tests (from left to right): binary USAF target, continuous cameraman and cell.
Fig. 4.
Fig. 4. RRMSE of the reconstruction (a) in different distances with good (green), bad (reddish) and no values (white), (b) in different noise-level.
Fig. 5.
Fig. 5. RRMSE of the reconstructed cameraman object (a) with phase-masks of different $\Delta \varphi$ phase-delay and $\Delta _m/\Delta _s$ mask-sensor pixel size ratio with super-resolution factor of $r_s = 5$. (b) RRMSEs with different $\Delta _m/\Delta _s$ ratio with super-resolution factors from 2 to 5 by keeping the phase-delay $\Delta \varphi =\pi$.
Fig. 6.
Fig. 6. USAF target phase reconstructions with super-resolution factors of 3 and 5, and the cross-sections of the smallest resolved target groups. These groups can be seen in the zoomed regions with the line width of 1.15 $\upmu$ m ($r_s=3$) and 0.69 $\upmu$ m ($r_s=5$)
Fig. 7.
Fig. 7. Quality imaging of continuous phase-only cell object with super-resolution factors of 3 and 9. Please pay attention that the scales are different because of the different computational pixel size ($\Delta _c$).
Fig. 8.
Fig. 8. RRMSE of the reconstructions (a) with different mask-sensor distance error, (b) with different object-sensor distance error.
Fig. 9.
Fig. 9. Object reconstruction. (a) Amplitude and (b) depth-map of the original Phasefocus target [39] obtained by Atomic force microscopy. (c) Amplitude and (d) depth-map of the reconstructed object which are calculated with the super-resolution factor of $r_s=3.45$, and (g) the cross-sections of the depth-values of the 6th element from group 7 (left plot) and 1st element from the group 9 (right plot). For comparison (e) amplitude and (f) depth-map of the reconstructed object which are calculated in pixel-wise resolution.

Equations (11)

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z = | P d 2 { M P d 1 { u o } } | 2 ,
M ( x , y ) = exp ( j ϕ ( x , y ) ) ,
u ( x , y , d ) = F 1 { H ( f x , f y , d ) F { u ( x , y , 0 ) } } ,
H ( f x , f y , d ) = { exp [ i 2 π λ d 1 λ 2 ( f x 2 + f y 2 ) ] , f x 2 + f y 2 1 λ 2 , 0 , otherwise.
u o ( x , y ) = A o ( x , y ) exp ( i φ o ( x , y ) ) ,
φ ^ o = BM3D ( φ o , t h φ ) ,
A ^ o = BM3D ( A o , t h A ) .
RRMSE = φ o φ o ^ F φ o F ,
SNR = 10 l o g 10 ( S N ) ,
Δ A b b e = λ N A 2 d λ N Δ s = 2.56   μ m ,
Δ h = Δ φ λ 2 π ( n 1 ) ,

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