Abstract

In this paper, we demonstrate two image reconstruction schemes for continuously self-imaging gratings (CSIGs). CSIGs are diffractive optical elements that generate a depth-invariant propagation pattern and sample objects with a sparse spatial frequency spectrum. To compensate for the sparse sampling, we apply two methods with different regularizations for CSIG imaging. The first method employs continuity of the spatial frequency spectrum, and the second one uses sparsity of the intensity pattern. The two methods are demonstrated with simulations and experiments.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]

2012 (1)

2011 (2)

2009 (2)

2008 (3)

G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. 33, 366–368 (2008).
[CrossRef]

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

2007 (7)

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

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

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 671208 (2007).
[CrossRef]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, P. Chavel, and J. Taboury, “Nonparaxial analysis of continuous self-imaging gratings in oblique illumination,” J. Opt. Soc. Am. A 24, 3379–3387 (2007).
[CrossRef]

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2007).
[CrossRef]

A. Burvall, K. Kolacz, A. V. Goncharov, Z. Jaroszewicz, and C. Dainty, “Lens axicons in oblique illumination,” Appl. Opt. 46, 312–318 (2007).
[CrossRef]

2006 (1)

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

2004 (1)

2003 (3)

2001 (1)

2000 (3)

1998 (1)

1997 (1)

1996 (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]

1987 (1)

1967 (1)

1958 (1)

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. Ser. A 248, 93–106 (1958).
[CrossRef]

1954 (1)

Bandres, M. A.

Baraniuk, R.

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

Bin, Z.

Bioucas-Dias, J. M.

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

Blanchard, P. M.

Boyd, S. P.

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2007).
[CrossRef]

Bunk, O.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Burvall, A.

Candes, E. J.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2007).
[CrossRef]

Carcole, E.

Chavel, P.

Chávez-Cerda, S.

Cottrell, D. M.

da Silva, P.

Dainty, C.

David, C.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Davis, J. A.

de Bougrenet, J.-L.

Dierolf, M.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Donoho, D.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Donoho, D. L.

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

Druart, G.

Durnin, J.

Dyson, J.

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. Ser. A 248, 93–106 (1958).
[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. T.

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

Fisher, D. J.

Friberg, A. T.

Goncharov, A. V.

Gray, J.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Greenaway, A. H.

Guérineau, N.

M. Piponnier, R. Horisaki, G. Druart, N. Guérineau, A. Kattnig, and J. Primot, “Relevance of continuously self-imaging gratings for noise robust imagery,” Opt. Lett. 37, 3492–3494 (2012).
[CrossRef]

M. Piponnier, G. Druart, N. Guérineau, J.-L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19, 23054–23066 (2011).
[CrossRef]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36, 1398–1400 (2011).
[CrossRef]

G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. 33, 366–368 (2008).
[CrossRef]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 671208 (2007).
[CrossRef]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, P. Chavel, and J. Taboury, “Nonparaxial analysis of continuous self-imaging gratings in oblique illumination,” J. Opt. Soc. Am. A 24, 3379–3387 (2007).
[CrossRef]

N. Guérineau, S. Rommeluere, E. D. Mambro, I. Ribet, and J. Primot, “New techniques of characterisation,” C. R. Phys. 4, 1175–1185 (2003).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, and K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–413 (2001).
[CrossRef]

J. Primot and N. Guérineau, “Extended hartmann test based on the pseudoguiding property of a hartmann mask completed by a phase chessboard,” Appl. Opt. 39, 5715–5720 (2000).
[CrossRef]

Guo, C.-S.

Gutiérrez-Vega, J. C.

Haïdar, R.

Harchaoui, B.

He, J. L.

Heggarty, K.

Horisaki, R.

Idir, M.

Jaroszewicz, Z.

Jefimovs, K.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Johnson, I.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Kattnig, A.

Kolacz, K.

Lanzl, T.

Lu, L.-L.

Lustig, M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Maier, M.

Mambro, E. D.

N. Guérineau, S. Rommeluere, E. D. Mambro, I. Ribet, and J. Primot, “New techniques of characterisation,” C. R. Phys. 4, 1175–1185 (2003).
[CrossRef]

McLeod, J. H.

Mercère, P.

Momose, A.

Montgomery, W. D.

Niggl, 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]

Pauly, J. M.

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Pfeiffer, F.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Piponnier, M.

Primot, J.

M. Piponnier, R. Horisaki, G. Druart, N. Guérineau, A. Kattnig, and J. Primot, “Relevance of continuously self-imaging gratings for noise robust imagery,” Opt. Lett. 37, 3492–3494 (2012).
[CrossRef]

M. Piponnier, G. Druart, N. Guérineau, J.-L. de Bougrenet, and J. Primot, “Optimal conditions for using the binary approximation of continuously self-imaging gratings,” Opt. Express 19, 23054–23066 (2011).
[CrossRef]

J. Rizzi, T. Weitkamp, N. Guérineau, M. Idir, P. Mercère, G. Druart, G. Vincent, P. da Silva, and J. Primot, “Quadriwave lateral shearing interferometry in an achromatic and continuously self-imaging regime for future x-ray phase imaging,” Opt. Lett. 36, 1398–1400 (2011).
[CrossRef]

G. Druart, J. Taboury, N. Guérineau, R. Haïdar, H. Sauer, A. Kattnig, and J. Primot, “Demonstration of image-zooming capability for diffractive axicons,” Opt. Lett. 33, 366–368 (2008).
[CrossRef]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 671208 (2007).
[CrossRef]

G. Druart, N. Guérineau, R. Haïdar, J. Primot, P. Chavel, and J. Taboury, “Nonparaxial analysis of continuous self-imaging gratings in oblique illumination,” J. Opt. Soc. Am. A 24, 3379–3387 (2007).
[CrossRef]

N. Guérineau, S. Rommeluere, E. D. Mambro, I. Ribet, and J. Primot, “New techniques of characterisation,” C. R. Phys. 4, 1175–1185 (2003).
[CrossRef]

N. Guérineau, B. Harchaoui, J. Primot, and K. Heggarty, “Generation of achromatic and propagation-invariant spot arrays by use of continuously self-imaging gratings,” Opt. Lett. 26, 411–413 (2001).
[CrossRef]

J. Primot and N. Guérineau, “Extended hartmann test based on the pseudoguiding property of a hartmann mask completed by a phase chessboard,” Appl. Opt. 39, 5715–5720 (2000).
[CrossRef]

Renker, D.

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

Ribet, I.

N. Guérineau, S. Rommeluere, E. D. Mambro, I. Ribet, and J. Primot, “New techniques of characterisation,” C. R. Phys. 4, 1175–1185 (2003).
[CrossRef]

Rizzi, J.

Rommeluere, S.

N. Guérineau, S. Rommeluere, E. D. Mambro, I. Ribet, and J. Primot, “New techniques of characterisation,” C. R. Phys. 4, 1175–1185 (2003).
[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]

Sauer, H.

Taboury, J.

Tanaka, T.

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

Thaning, A.

Tong, D. M.

Vincent, G.

Wakin, M. B.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2007).
[CrossRef]

Wang, H.-T.

Wei, G.-X.

Weitkamp, T.

Woods, S. C.

Yamamoto, S.

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

Zhu, L.

Appl. Opt. (7)

C. R. Phys. (1)

N. Guérineau, S. Rommeluere, E. D. Mambro, I. Ribet, and J. Primot, “New techniques of characterisation,” C. R. Phys. 4, 1175–1185 (2003).
[CrossRef]

IEEE Signal Process. Mag. (2)

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25, 21–30 (2008).
[CrossRef]

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24, 118–121 (2007).
[CrossRef]

IEEE Trans. Image Process. (1)

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

IEEE Trans. Inf. Theory (1)

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

J. Fourier Anal. Appl. (1)

E. J. Candes, M. B. Wakin, and S. P. Boyd, “Enhancing sparsity by reweighted ℓ1 minimization,” J. Fourier Anal. Appl. 14, 877–905 (2007).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

Magn. Reson. Med. (1)

M. Lustig, D. Donoho, and J. M. Pauly, “Sparse MRI: the application of compressed sensing for rapid MR imaging,” Magn. Reson. Med. 58, 1182–1195 (2007).
[CrossRef]

Opt. Commun. (1)

T. Tanaka and S. Yamamoto, “Comparison of aberration between axicon and lens,” Opt. Commun. 184, 113–118 (2000).
[CrossRef]

Opt. Express (2)

Opt. Lett. (6)

Phys. Rev. Lett. (1)

I. Johnson, K. Jefimovs, O. Bunk, C. David, M. Dierolf, J. Gray, D. Renker, and F. Pfeiffer, “Coherent diffractive imaging using phase front modifications,” Phys. Rev. Lett. 100, 155503 (2008).
[CrossRef]

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]

Proc. R. Soc. Lond. Ser. A (1)

J. Dyson, “Circular and spiral diffraction gratings,” Proc. R. Soc. Lond. Ser. A 248, 93–106 (1958).
[CrossRef]

Proc. SPIE (1)

G. Druart, N. Guérineau, R. Haïdar, J. Primot, A. Kattnig, and J. Taboury, “Image formation by use of continuously self-imaging gratings and diffractive axicons,” Proc. SPIE 6712, 671208 (2007).
[CrossRef]

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Figures (12)

Fig. 1.
Fig. 1.

CTF of a CSIG is defined as the intersection between a Cartesian grid of pitch b0=1/a0 and a Montgomery’s ring of radius η/a0.

Fig. 2.
Fig. 2.

Sparse OTF of a binary CSIG.

Fig. 3.
Fig. 3.

PSF of a binary CSIG. The square indicated in the dashed line represents the elementary pattern of the PSF that we acquire to suppress the circulant effect of the FT. The shaded area indicates the maximum FOV of this system.

Fig. 4.
Fig. 4.

OTF peaks are separated by a mean distance bmean in the FD. The Shannon criterion defines, from this sampling pitch, the maximum FOV in the SD inside which the observed object must be placed so that its spectrum is well sampled by the CSIG.

Fig. 5.
Fig. 5.

Experimental setup. The transmissive object is illuminated by the light which is produced by the LED and collimated by the lens with the diffuser. The transmitted light is captured by the image sensor with the modulation of the CSIG.

Fig. 6.
Fig. 6.

Experimental and simulation results of a circle whose diameter is 5 mm when S is 139. (a) Captured image and reconstructions by (b) interpolation in the FD and by (c) TwIST (experimental). Simulated reconstructions by TwIST with (d) MSNR=40.0dB where RPSNR=48.7dB, and with (e) MSNR=dB where RPSNR=48.7dB.

Fig. 7.
Fig. 7.

Experimental and simulation results of a circle whose diameter is 10 mm when S is 309. (a) Captured image and reconstructions by (b) interpolation in the FD and by (c) TwIST (experimental). Simulated reconstructions by TwIST with (d) MSNR=40.0dB where RPSNR=28.0dB, and with (e) MSNR=dB where RPSNR=30.8dB.

Fig. 8.
Fig. 8.

Experimental and simulation results of a circle whose diameter is 20 mm when S is 549. (a) Captured image and reconstructions by (b) interpolation in the FD and by (c) TwIST (experimental). Simulated reconstructions by TwIST with (d) MSNR=40.0dB where RPSNR=18.9dB, and with (e) MSNR=dB where RPSNR=23.7dB.

Fig. 9.
Fig. 9.

Experimental and simulation results of a circle whose diameter is 25 mm when S is 719. (a) Captured image and reconstructions by (b) interpolation in the FD and by (c) TwIST (experimental). Simulated reconstructions by TwIST with (d) MSNR=40.0dB where RPSNR=15.2dB, and with (e) MSNR=dB where RPSNR=22.1dB.

Fig. 10.
Fig. 10.

Experimental results of a flower, where S is 538. (a) Captured image. Reconstructions with (b) interpolation in the FD and (c) TwIST.

Fig. 11.
Fig. 11.

Experimental results of two small objects, a ring (left) and a flower (center), where S is 132. The magnified objects are shown below each image. (a) Captured image. Reconstructions with (b) interpolation in the FD and (c) TwIST.

Fig. 12.
Fig. 12.

Experimental results of a square with fine textures on the surface, where S is 1857. The magnified object is shown below each image. (a) Captured image. Reconstructions with (b) interpolation in the FD and (c) TwIST.

Tables (2)

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Table 1. Summary of the Experimental Results

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Table 2. Limitations of the Two Reconstruction Methods

Equations (13)

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g=Pf,
=F1OFf,
g=F1OFAf,
F1WFg=F1WOFAf,
gwiener=F1OwienerFAf,
(W)p=conj((O)p)|(O)p|2+e,
bmean=4πη2a02·1N2/2+1,
2η2πNb0,
FOVnyquist=22bmean=1bmean1b0N2η2π,
FOVnyquistFOVmaxb0bmean=N2η2π,
f^=argminfgwienerHf2+αR(f),
Sc·M,
FOVaberration=2a0η(z2λ)1/2,

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