Abstract

The theory of compressed sensing (CS) shows that signals can be acquired at sub-Nyquist rates if they are sufficiently sparse or compressible. Since many images bear this property, several acquisition models have been proposed for optical CS. An interesting approach is random convolution (RC). In contrast with single-pixel CS approaches, RC allows for the parallel capture of visual information on a sensor array as in conventional imaging approaches. Unfortunately, the RC strategy is difficult to implement as is in practical settings due to important contrast-to-noise-ratio (CNR) limitations. In this paper, we introduce a modified RC model circumventing such difficulties by considering measurement matrices involving sparse non-negative entries. We then implement this model based on a slightly modified microscopy setup using incoherent light. Our experiments demonstrate the suitability of this approach for dealing with distinct CS scenarii, including 1-bit CS.

© 2016 Optical Society of America

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References

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  1. D. L Donoho, “Compressed sensing,” IEEE Trans on Info Theory,  52, 1289–1306 (2006).
    [Crossref]
  2. R. G Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Process. Mag.,  25, 83–91 (2008).
    [Crossref]
  3. G. Huang, H. Jiang, K. Matthews, and P. Wilford, “Lensless imaging by compressive sensing,” In Proc of IEEE Int Conf on Image Process, 2101–2105 (2013).
  4. R. Fergus, A. Torralba, and W. T Freeman, “Random lens imaging,” preprint (2006).
  5. R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express,  18, 23041–23053 (2010).
    [Crossref] [PubMed]
  6. A. Ashok and M. A Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46, 2256–2268 (2007).
    [Crossref] [PubMed]
  7. A. Bourquard, F. Aguet, and M. Unser, “Optical imaging using binary sensors,” Opt. Express,  18, 4876–4888 (2010).
    [Crossref] [PubMed]
  8. L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).
  9. J. Romberg, “Compressive sensing by random convolution,” SIAM J Imaging Sci,  2, 1098–1128 (2009).
    [Crossref]
  10. R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, Computational Imaging VII, 72460G (2009).
    [Crossref]
  11. W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Proc. SPIE 7744, 77440K (2010).
    [Crossref]
  12. Z. T. Harmany, R. F. Marcia, and R.M. Willett, “Compressive coded aperture keyed exposure imaging with optical flow reconstruction,” arXiv:1306.6281 (2013).
  13. A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” Journal of Display Technology,  3, 315–320 (2007).
    [Crossref]
  14. J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1960).
  15. A. Bourquard and M. Unser, “Binary compressed imaging,” IEEE Trans on Image Process,  22, 1042–1055 (2013).
    [Crossref]
  16. A. Bourquard, “Compressed optical imaging,” PhD Thesis, Ecole Polytechnique Fédérale de Lausanne, (2013).
  17. R. Horisaki and J. Tanida, “Preconditioning for multiplexed imaging with spatially coded PSFs,” Opt. Express,  19, 12540–12550 (2011).
    [Crossref] [PubMed]
  18. V. Chandar, “A negative result concerning explicit matrices with the restricted isometry property,” preprint, (2008).
  19. R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).
  20. R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” preprint, (2008).
  21. E. Ben-Eliezer, N. Konforti, B. Milgrom, and E. Marom, “An optimal binary amplitude-phase mask for hybrid imaging systems that exhibit high resolution and extended depth of field,” Opt. Express,  16, 20540–20561 (2008).
    [Crossref] [PubMed]
  22. D. L. Donoho, “For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution,” Comm. on Pure and Applied Math.,  59, 797–829 (2006).
    [Crossref]
  23. Y. Zhang, J. Yang, and W. Yin, Yall1: Your algorithms for l1, online at yall1.blogs.rice.edu . (2011).
  24. B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
    [Crossref]
  25. N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
    [Crossref] [PubMed]
  26. S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,  4, 1–39 (2011).
    [Crossref]

2013 (1)

A. Bourquard and M. Unser, “Binary compressed imaging,” IEEE Trans on Image Process,  22, 1042–1055 (2013).
[Crossref]

2011 (2)

S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,  4, 1–39 (2011).
[Crossref]

R. Horisaki and J. Tanida, “Preconditioning for multiplexed imaging with spatially coded PSFs,” Opt. Express,  19, 12540–12550 (2011).
[Crossref] [PubMed]

2010 (3)

2009 (2)

J. Romberg, “Compressive sensing by random convolution,” SIAM J Imaging Sci,  2, 1098–1128 (2009).
[Crossref]

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, Computational Imaging VII, 72460G (2009).
[Crossref]

2008 (2)

2007 (2)

A. Ashok and M. A Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46, 2256–2268 (2007).
[Crossref] [PubMed]

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” Journal of Display Technology,  3, 315–320 (2007).
[Crossref]

2006 (2)

D. L. Donoho, “For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution,” Comm. on Pure and Applied Math.,  59, 797–829 (2006).
[Crossref]

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

2005 (1)

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

1996 (1)

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

Aguet, F.

Ashok, A.

Baraniuk, R. G

R. G Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Process. Mag.,  25, 83–91 (2008).
[Crossref]

Becker, S.

S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,  4, 1–39 (2011).
[Crossref]

Ben-Eliezer, E.

Berinde, R.

R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).

R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” preprint, (2008).

Bibet, A.

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Bobin, J.

S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,  4, 1–39 (2011).
[Crossref]

Bourquard, A.

A. Bourquard and M. Unser, “Binary compressed imaging,” IEEE Trans on Image Process,  22, 1042–1055 (2013).
[Crossref]

A. Bourquard, F. Aguet, and M. Unser, “Optical imaging using binary sensors,” Opt. Express,  18, 4876–4888 (2010).
[Crossref] [PubMed]

A. Bourquard, “Compressed optical imaging,” PhD Thesis, Ecole Polytechnique Fédérale de Lausanne, (2013).

Candès, E. J.

S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,  4, 1–39 (2011).
[Crossref]

Chandar, V.

V. Chandar, “A negative result concerning explicit matrices with the restricted isometry property,” preprint, (2008).

Dierickx, B.

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

Donoho, D. L

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

Donoho, D. L.

D. L. Donoho, “For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution,” Comm. on Pure and Applied Math.,  59, 797–829 (2006).
[Crossref]

Fergus, R.

R. Fergus, A. Torralba, and W. T Freeman, “Random lens imaging,” preprint (2006).

Freeman, W. T

R. Fergus, A. Torralba, and W. T Freeman, “Random lens imaging,” preprint (2006).

Gilbert, A. C

R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).

Gonzo, L.

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1960).

Gottardi, M.

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

Harmany, Z. T.

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, Computational Imaging VII, 72460G (2009).
[Crossref]

Z. T. Harmany, R. F. Marcia, and R.M. Willett, “Compressive coded aperture keyed exposure imaging with optical flow reconstruction,” arXiv:1306.6281 (2013).

Horisaki, R.

Huang, G.

G. Huang, H. Jiang, K. Matthews, and P. Wilford, “Lensless imaging by compressive sensing,” In Proc of IEEE Int Conf on Image Process, 2101–2105 (2013).

Indyk, P.

R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” preprint, (2008).

R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).

Jacques, L.

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Javidi, B.

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” Journal of Display Technology,  3, 315–320 (2007).
[Crossref]

Jiang, H.

G. Huang, H. Jiang, K. Matthews, and P. Wilford, “Lensless imaging by compressive sensing,” In Proc of IEEE Int Conf on Image Process, 2101–2105 (2013).

Karloff, H.

R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).

Konforti, N.

Leblebici, Y.

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Majidzadeh, V.

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Marcia, R. F.

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, Computational Imaging VII, 72460G (2009).
[Crossref]

Z. T. Harmany, R. F. Marcia, and R.M. Willett, “Compressive coded aperture keyed exposure imaging with optical flow reconstruction,” arXiv:1306.6281 (2013).

Marom, E.

Massari, N.

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

Matthews, K.

G. Huang, H. Jiang, K. Matthews, and P. Wilford, “Lensless imaging by compressive sensing,” In Proc of IEEE Int Conf on Image Process, 2101–2105 (2013).

Meynants, G.

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

Milgrom, B.

Morgan, S.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Proc. SPIE 7744, 77440K (2010).
[Crossref]

Neifeld, M. A

Ogiers, W.

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

Romberg, J.

J. Romberg, “Compressive sensing by random convolution,” SIAM J Imaging Sci,  2, 1098–1128 (2009).
[Crossref]

Scheffer, D.

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

Schmid, A.

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Simoni, A.

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

Stern, A.

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” Journal of Display Technology,  3, 315–320 (2007).
[Crossref]

Stoppa, D.

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

Strauss, M. J

R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).

Tanida, J.

Torralba, A.

R. Fergus, A. Torralba, and W. T Freeman, “Random lens imaging,” preprint (2006).

Unser, M.

A. Bourquard and M. Unser, “Binary compressed imaging,” IEEE Trans on Image Process,  22, 1042–1055 (2013).
[Crossref]

A. Bourquard, F. Aguet, and M. Unser, “Optical imaging using binary sensors,” Opt. Express,  18, 4876–4888 (2010).
[Crossref] [PubMed]

Vandergheynst, P.

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Vlummens, J.

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

Wilford, P.

G. Huang, H. Jiang, K. Matthews, and P. Wilford, “Lensless imaging by compressive sensing,” In Proc of IEEE Int Conf on Image Process, 2101–2105 (2013).

Willett, R. M.

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, Computational Imaging VII, 72460G (2009).
[Crossref]

Willett, R.M.

Z. T. Harmany, R. F. Marcia, and R.M. Willett, “Compressive coded aperture keyed exposure imaging with optical flow reconstruction,” arXiv:1306.6281 (2013).

Yang, J.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Proc. SPIE 7744, 77440K (2010).
[Crossref]

Yin, W.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Proc. SPIE 7744, 77440K (2010).
[Crossref]

Zhang, Y.

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Proc. SPIE 7744, 77440K (2010).
[Crossref]

Appl. Opt. (1)

Comm. on Pure and Applied Math. (1)

D. L. Donoho, “For most large underdetermined systems of linear equations the minimal l1-norm solution is also the sparsest solution,” Comm. on Pure and Applied Math.,  59, 797–829 (2006).
[Crossref]

IEEE Sig. Process. Mag. (1)

R. G Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Process. Mag.,  25, 83–91 (2008).
[Crossref]

IEEE Trans on Image Process (1)

A. Bourquard and M. Unser, “Binary compressed imaging,” IEEE Trans on Image Process,  22, 1042–1055 (2013).
[Crossref]

IEEE Trans on Info Theory (1)

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

IEEE Trans. on Neural Networks (1)

N. Massari, M. Gottardi, L. Gonzo, D. Stoppa, and A. Simoni, “A CMOS image sensor with programmable pixel-level analog processing,” IEEE Trans. on Neural Networks,  16, 1673–1684 (2005).
[Crossref] [PubMed]

Journal of Display Technology (1)

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” Journal of Display Technology,  3, 315–320 (2007).
[Crossref]

Opt. Express (4)

Proc. SPIE (3)

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, Computational Imaging VII, 72460G (2009).
[Crossref]

W. Yin, S. Morgan, J. Yang, and Y. Zhang, “Practical compressive sensing with toeplitz and circulant matrices,” Proc. SPIE 7744, 77440K (2010).
[Crossref]

B. Dierickx, D. Scheffer, G. Meynants, W. Ogiers, and J. Vlummens, “Random addressable active pixel image sensors,” Proc. SPIE 2905, 2 (1996).
[Crossref]

SIAM J Imaging Sci (1)

J. Romberg, “Compressive sensing by random convolution,” SIAM J Imaging Sci,  2, 1098–1128 (2009).
[Crossref]

SIAM Journal on Imaging Sciences (1)

S. Becker, J. Bobin, and E. J. Candès, “NESTA: a fast and accurate first-order method for sparse recovery,” SIAM Journal on Imaging Sciences,  4, 1–39 (2011).
[Crossref]

Other (10)

L. Jacques, P. Vandergheynst, A. Bibet, V. Majidzadeh, A. Schmid, and Y. Leblebici, “Cmos compressed imaging by random convolution,” In Proc IEEE Int Conf Acoust Speech Signal Process, 1113–1116 (2009).

Z. T. Harmany, R. F. Marcia, and R.M. Willett, “Compressive coded aperture keyed exposure imaging with optical flow reconstruction,” arXiv:1306.6281 (2013).

J. W. Goodman, Introduction to Fourier Optics, (McGraw-Hill, 1960).

A. Bourquard, “Compressed optical imaging,” PhD Thesis, Ecole Polytechnique Fédérale de Lausanne, (2013).

Y. Zhang, J. Yang, and W. Yin, Yall1: Your algorithms for l1, online at yall1.blogs.rice.edu . (2011).

G. Huang, H. Jiang, K. Matthews, and P. Wilford, “Lensless imaging by compressive sensing,” In Proc of IEEE Int Conf on Image Process, 2101–2105 (2013).

R. Fergus, A. Torralba, and W. T Freeman, “Random lens imaging,” preprint (2006).

V. Chandar, “A negative result concerning explicit matrices with the restricted isometry property,” preprint, (2008).

R. Berinde, A. C Gilbert, P. Indyk, H. Karloff, and M. J Strauss, “Combining geometry and combinatorics: A unified approach to sparse signal recovery,” Proc of Allerton Conf on Communication, Control, and Computing, 798–805 (IEEE2008).

R. Berinde and P. Indyk, “Sparse recovery using sparse random matrices,” preprint, (2008).

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Figures (15)

Fig. 1
Fig. 1 Illustration of our optical model.

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Fig. 2
Fig. 2 Examples of filters h and h̃ complying with our specifications: (a) Original PSF (filter h) consisting in 20 spatial-domain impulses of equal intensity (b) Binarized angles φ obtained from the Fourier transform ĥ of the filter h and used in our simplified mask model. (c) Alternate PSF (filter h̃) corresponding to the simplified mask model.

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Fig. 3
Fig. 3 Effect of phase-shift inaccuracies. A simulated PSF h̃ is generated from one same phase mask m̃, considering three distinct wavelengths inducing phase shifts of {0, 0.9π} (blue), {0, π} (green), and {0, 1.1π} (red). The following effects are observed when phase shifts are different from {0, π}: (1) a central impulse associated with a 0th-order-component artifact, (2) a scaling effect due to the wavelength modification, according to the Fourier-transform scaling in Eq. (2).

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Fig. 4
Fig. 4 First row (a, e): ideal sparse symmetrical PSF we would like to encode. Second row (b, f): Its Fourier transform, which is real due to the symmetry. Third row (c, g): same, but rendered as its argument and phase. Fourth row (d, h): Inverse Fourier transform if we only keep the phase, setting all the argument values to 1.

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Fig. 5
Fig. 5 (a) Signal of interest. (b) Sampled signal. The original signal is fully captured by 80 evenly spaced samples but most of its features get lost when taking only 40 regular samples, i.e., either the diamonds or the squares. (c) Same result as (b) but using prior convolution with the sparse 8-spike PSF shown in Fig. 4. In this sequence, many values are non-zero because the PSF linearly combines non-adjacent points of the original signal. (d) Reconstruction of the original signal from 40 samples (here, the squares) of the convolved signal (c) using NESTA L1 minimization endowed with a spatial-domain sparsity prior.

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Fig. 6
Fig. 6 (a) Original object. (b) Convolution of the object with a random non-negative PSF, built by taking the absolute value of a PSF with Gaussian noise. (c) Result with the same PSF, but keeping the negative values. (d–f) Convolution of the object with a non-negative sparse PSF containing 2, 30 and 1000 impulses, respectively.

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Fig. 7
Fig. 7 (a) Original object (b) Intermediate image obtained by sparse random convolution through the specified phase mask with a PSF containing 30 non-zero impulses (c) Final acquisition following finite-differentiation and quantization of (b) according to [15] (d) Reconstruction based on the algorithm of [15] (e) Reconstruction quality via the structural similarity index measure (SSIM) as a function of the amount of impulses—including symmetrized ones—given distinct noise levels σ and using ideal filters (dashed lines) h versus their approximations (solid lines) h̃ (f) Reconstruction SSIM as a function of the PSF size.

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Fig. 8
Fig. 8 Photo of the 5″ Cr mask with the 9 patterns of different size (left). Photo of a single LED light taken through a phase mask patterned with the finest pattern, the one that has the strongest spreading effect (right).

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Fig. 9
Fig. 9 (a) Acquisitions (3 out of 6 shown) obtained with our microscope without phase mask (b) Corresponding acquisitions with phase mask (c) Spatial-domain filter hsol calibrated from the acquired image pairs

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Fig. 10
Fig. 10 The 3 objects used for visual evaluation of the reconstruction approaches as captured by the microscope camera, without the phase mask (upper row) and with the phase mask (lower row). (a)–(b) Circuit image (c)–(d). USAF target image. (e)–(f) Salt crystal image.

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Fig. 11
Fig. 11 Superresolution experiment on object USAF target. (a) Original image (b) 5 × 5 downsampled acquisition without phase mask (c) 5 × 5 downsampled CS acquisition with phase mask (d) Reconstruction from (c). All images, except for (c), have been cropped for better visualization.

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Fig. 12
Fig. 12 Unmasking experiment on object USAF target. (a) Original image (b) Masked acquisition without phase mask (c) Masked CS acquisition with phase mask (d) Reconstruction from (c). All images, except for (c), have been cropped for better visualization.

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Fig. 13
Fig. 13 Desaturation experiment on object salt crystal. (a) Original image not subjected to saturation (b) Acquisition without phase mask subject to saturation (c) CS acquisition with phase mask subject to saturation (d) Reconstruction from CS acquisition. In this case neither (b) nor (c) have been cropped, for a better comparison of the saturated areas.

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Fig. 14
Fig. 14 1-bit-CS experiment on objects circuit (left column) and salt crystal (right column) with corresponding acquisitions and reconstructions. (a)–(b) Original images (c)–(d) CS acquisitions with phase mask subjected to 1-bit quantization. (e)–(f) Reconstructions from corresponding CS acquisitions.

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Fig. 15
Fig. 15 Comparison of our 1-bit CS reconstruction result (b) with: (c) a simulated noiseless situation, in which we used the calibrated PSF to construct the convolved image and (d) the 1-bit reconstruction without any phase mask. (a) is the original object.

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Equations (16)

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y ( u ) = ( x * h ) ( u ) ,
h ( ⋅ ) = 𝒦 h | ℱ { q ( ξ ) }     ( NA ⋅ λ 0 ) | 2 ,
ℱ { q } ( ω ) = ∫ ℝ 2 q ( ⋅ ) exp { − j 2 π ω T ⋅ } d ⋅ .
q ( ξ ) = circ ( ξ ) m ( ξ ) ,
m ˜ ( ξ ) = exp { j φ ( ξ ) } ,
φ ( ξ ) = { 0 , arg ( m ( ξ ) ) < 0 , π , otherwise .
h sol = arg     min h ∈ Ω + ∑ i ‖ h ★ x i − y i ‖ ℓ 2 + Λ ‖ h ‖ 1 ,
γ = 𝒟 { N { x ★ h } } ,
γ = D ( Hx + n ) ,
γ = D ( 𝒬 { FHx + n } ) ,
( 𝒬 { v } ) i = { + 1 , v i ≥ 0 − 1 , otherwise .
𝒟 { x ˜ ★ h sol } ≈ γ ,
x ˜ = arg   min x     ‖ DH sol x − γ ‖ 2 + Λ 1 ‖ x ‖ TV ,
x ˜ = arg     min x ∑ i χ i [ k ] Ψ ( γ i ( DFH sol x ) i ) + Λ 2 ℛ ( x ) ,
Ψ ( t ) = { M − 1 − t , t < 0 M − 1 ( M 2 t 2 + M t + 1 ) − 1 , otherwise ,
f [ k ] = − δ [ k ] + 2 δ [ k − [ 0 ; 1 ] ] − δ [ k − [ 1 ; 1 ] ] ,

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