Abstract

We propose a preconditioning method to improve the convergence of iterative reconstruction algorithms in multiplexed imaging based on convolution-based compressive sensing with spatially coded point spread functions (PSFs). The system matrix is converted to improve the condition number with a preconditioner matrix. The preconditioner matrix is calculated by Tikhonov regularization in the frequency domain. The method was demonstrated with simulations and an experiment involving a range detection system with a grating based on the multiplexed imaging framework. The results of the demonstrations showed improved reconstruction fidelity by using the proposed preconditioning method.

© 2011 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).
  2. D. J. Brady, Optical imaging and spectroscopy (Wiley-OSA, 2009).
    [CrossRef]
  3. D. L. Donoho, “Compressed sensing,” IEEE Trans. Info. Theory 52, 1289–1306 (2006).
    [CrossRef]
  4. R. Baraniuk, “Compressive sensing,” IEEE Sig. Process. Mag. 24, 118–121 (2007).
    [CrossRef]
  5. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Sig. Process. Mag. 25, 21–30 (2008).
    [CrossRef]
  6. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
    [CrossRef] [PubMed]
  7. M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15, 14013–14027 (2007).
    [CrossRef] [PubMed]
  8. R. Horisaki, K. Choi, J. Hahn, J. Tanida, and D. J. Brady, “Generalized sampling using a compound-eye imaging system for multi-dimensional object acquisition,” Opt. Express 18, 19367–19378 (2010).
    [CrossRef] [PubMed]
  9. M. Shankar, N. P. Pitsianis, and D. J. Brady, “Compressive video sensors using multichannel imagers,” Appl. Opt. 49, B9–B17 (2010).
    [CrossRef] [PubMed]
  10. A. Ashok and M. A. Neifeld, “Compressive light field imaging,” Proc. SPIE 7690, 76900Q (2010).
    [CrossRef]
  11. J. Romberg, “Compressive sensing by random convolution,” SIAM J. Imaging Sci. 2, 1098–1128 (2009).
    [CrossRef]
  12. R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2008),” (2008), pp. 833–836.
  13. Y. Rivenson, A. Stern, and B. Javidi, “Single exposure super-resolution compressive imaging by double phase encoding,” Opt. Express 18, 15094–15103 (2010).
    [CrossRef] [PubMed]
  14. J. Hahn, S. Lim, K. Choi, R. Horisaki, and D. J. Brady, “Video-rate compressive holographic microscopic tomography,” Opt. Express 19, 7289–7298 (2011).
    [CrossRef] [PubMed]
  15. R. Horisaki and J. Tanida, “Multi-channel data acquisition using multiplexed imaging with spatial encoding,” Opt. Express 18, 23041–23053 (2010).
    [CrossRef] [PubMed]
  16. N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
    [CrossRef]
  17. K. Choi, R. Horisaki, J. Hahn, S. Lim, D. L. Marks, T. J. Schulz, and D. J. Brady, “Compressive holography of diffuse objects,” Appl. Opt. 49, H1–H10 (2010).
    [CrossRef] [PubMed]
  18. A. Ashok and M. A. Neifeld, “Pseudorandom phase masks for superresolution imaging from subpixel shifting,” Appl. Opt. 46, 2256–2268 (2007).
    [CrossRef] [PubMed]
  19. A. Mahalanobis, M. Neifeld, V. K. Bhagavatula, T. Haberfelde, and D. Brady, “Off-axis sparse aperture imaging using phase optimization techniques for application in wide-area imaging systems,” Appl. Opt. 48, 5212–5224 (2009).
    [CrossRef] [PubMed]
  20. J. M. Bioucas-Dias and M. A. T. Figueiredo, “A new TwIST: Two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Proc. 16, 2992–3004 (2007).
    [CrossRef]
  21. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
    [CrossRef]
  22. “Spectral image database,” http://spectral.joensuu.fi/multispectral/spectralimages.php .

2011

2010

2009

2008

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

2007

R. Baraniuk, “Compressive sensing,” IEEE Sig. Process. Mag. 24, 118–121 (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 Proc. 16, 2992–3004 (2007).
[CrossRef]

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

M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, “Single-shot compressive spectral imaging with a dual-disperser architecture,” Opt. Express 15, 14013–14027 (2007).
[CrossRef] [PubMed]

2006

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

2001

N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
[CrossRef]

1992

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Ashok, A.

Baraniuk, R.

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

Bhagavatula, V. K.

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 Proc. 16, 2992–3004 (2007).
[CrossRef]

Brady, D.

Brady, D. J.

Candes, E. J.

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

Choi, K.

Donoho, D. L.

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

Fatemi, E.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. 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 Proc. 16, 2992–3004 (2007).
[CrossRef]

Gehm, M. E.

Golub, G.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
[CrossRef]

Haberfelde, T.

Hahn, J.

Horisaki, R.

Javidi, B.

John, R.

Kak, A.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Lim, S.

Mahalanobis, A.

Marks, D. L.

Member, S.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
[CrossRef]

Milanfar, P.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
[CrossRef]

Neifeld, M.

Neifeld, M. A.

Nguyen, N.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Pitsianis, N. P.

Rivenson, Y.

Romberg, J.

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

Rudin, L. I.

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Schulz, T. J.

Shankar, M.

Slaney, M.

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

Stern, A.

Tanida, J.

Wakin, M. B.

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

Willett, R. M.

Appl. Opt.

IEEE Sig. Process. Mag.

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

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

IEEE Trans. Image Proc.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, “A computationally efficient superresolution image reconstruction algorithm,” IEEE Trans. Image Proc. 10, 573–583 (2001).
[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 Proc. 16, 2992–3004 (2007).
[CrossRef]

IEEE Trans. Info. Theory

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

Opt. Express

Phys. D

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. D 60, 259–268 (1992).
[CrossRef]

Proc. SPIE

A. Ashok and M. A. Neifeld, “Compressive light field imaging,” Proc. SPIE 7690, 76900Q (2010).
[CrossRef]

SIAM J. Imaging Sci.

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

Other

R. F. Marcia and R. M. Willett, “Compressive coded aperture superresolution image reconstruction,” in “IEEE International Conference on Acoustics, Speech and Signal Processing (ICASSP 2008),” (2008), pp. 833–836.

“Spectral image database,” http://spectral.joensuu.fi/multispectral/spectralimages.php .

A. Kak and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, 1988).

D. J. Brady, Optical imaging and spectroscopy (Wiley-OSA, 2009).
[CrossRef]

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Objects in the simulations. (a) An object which is sparse in the two-dimensional TV domain and (b) an object which is sparse in the three-dimensional DWT domain.

Fig. 2
Fig. 2

Sets of PSFs. The sets are composed of (a) multiple delta functions and (b) single circular functions.

Fig. 3
Fig. 3

The captured data of (a) Fig. 1(a) with the PSF in Fig. 2(a), (b) Fig. 1(a) with the PSF in Fig. 2(b), (c) Fig. 1(b) with the PSF in Fig. 2(a), and (d) Fig. 1(b) with the PSF in Fig. 2(b)

Fig. 4
Fig. 4

The reconstructed data of Fig. 1(a) with (a) the original TwIST from Fig. 3(a), (b) the preconditioned TwIST from Fig. 3(a), (c) the original TwIST from Fig. 3(b), and (d) the preconditioned TwIST from Fig. 3(b).

Fig. 5
Fig. 5

The reconstructed data of Fig. 1(b) with (a) the original TwIST from Fig. 3(c), (b) the preconditioned TwIST from Fig. 3(c), (c) the original TwIST from Fig. 3(d), and (d) the preconditioned TwIST from Fig. 3(d).

Fig. 6
Fig. 6

Plots of reconstruction PSNRs from noisy measurements of (a) Fig. 1(a) and (b) Fig. 1(b) in the original TwIST and the preconditioned TwIST.

Fig. 7
Fig. 7

The experimental setup of a range detection system.

Fig. 8
Fig. 8

The experimental results. (a) the captured data and the reconstructed data with (b) the original TwIST and (c) the preconditioned TwIST.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

𝒢 ( x ) = c x 𝒫 ( x x , c ) × ( x , c ) ,
g = Φ f
= [ C 0 C 1 C N c 1 ] f
= F 1 [ D 0 D 1 D N c 1 ] [ F 0 0 0 F 0 0 0 0 0 F ] f ,
f ^ = argmin f | | Pg P Φ f | | 2 + α ( f ) ,
P = [ F 1 0 0 0 F 1 0 0 0 0 0 F 1 ] [ D 0 D 1 D N c 1 ] F ,
d i T = ( d i H d i + λ I ) 1 d i H ,

Metrics