Abstract

In this Letter, we propose a preconditioning method to improve the convergence speed of iterative reconstruction algorithms in a compact, multidimensional, compound-eye imaging system called the thin observation module by bound optics. The condition number of the system matrix is improved by using a preconditioner matrix. To calculate the preconditioner matrix, the system model is expressed in the frequency domain. The proposed method is simulated by using a compressive sensing algorithm called the two-step iterative shrinkage/thresholding algorithm. The results showed improved reconstruction fidelity with a certain number of iterations for high signal-to-noise ratio measurements.

© 2011 Optical Society of America

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References

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  1. R. Horisaki, K. Choi, J. Hahn, J. Tanida, and D. J. Brady, Opt. Express 18, 19367 (2010).
    [CrossRef] [PubMed]
  2. A. D. Portnoy, N. P. Pitsianis, X. Sun, and D. J. Brady, Appl. Opt. 47, B76 (2008).
    [CrossRef] [PubMed]
  3. N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
    [CrossRef]
  4. J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
    [CrossRef]
  5. D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
    [CrossRef]
  6. “The USC-SIPI Image Database,” http://sipi.usc.edu/database/.
  7. J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 259–260.

2010 (1)

2008 (1)

2007 (1)

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

2006 (1)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

2001 (1)

N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Brady, D. J.

Choi, K.

Donoho, D. L.

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Figueiredo, M. A. T.

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

Golub, G.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 259–260.

Hahn, J.

Horisaki, R.

Member, S.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Milanfar, P.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Nguyen, N.

N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

Pitsianis, N. P.

Portnoy, A. D.

Sun, X.

Tanida, J.

Appl. Opt. (1)

IEEE Trans. Image Process. (2)

N. Nguyen, P. Milanfar, S. Member, and G. Golub, IEEE Trans. Image Process. 10, 573 (2001).
[CrossRef]

J. M. Bioucas-Dias and M. A. T. Figueiredo, IEEE Trans. Image Process. 16, 2992 (2007).
[CrossRef]

IEEE Trans. Inf. Theory (1)

D. L. Donoho, IEEE Trans. Inf. Theory 52, 1289 (2006).
[CrossRef]

Opt. Express (1)

Other (2)

“The USC-SIPI Image Database,” http://sipi.usc.edu/database/.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996), pp. 259–260.

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Figures (2)

Fig. 1
Fig. 1

Simulation results. (a) The original data f 3 D , (b) the captured image g , (c) the data reconstructed with the original TwIST, and (d) the data reconstructed with the preconditioned TwIST.

Fig. 2
Fig. 2

Plots of reconstruction PSNRs from noisy measurements in the original TwIST and the preconditioned TwIST.

Equations (9)

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g = Φ 2 D f 2 D = T 0 T 1 T N u 1 f 2 D ,
Φ 2 D = C 0 , 0 C 0 , 1 C 0 , N u 1 C 1 , 0 C 1 , 1 C 1 , N u 1 C N u 1 , 0 C N u 1 , 1 C N u 1 , N u 1 R ,
g = Φ 3 D f 3 D = Φ 2 D 0 Φ 2 D 1 Φ 2 D N z 1 f 3 D ,
C u , p z = circshift ( C u , p , S ( u , z ) / N u [ p < mod ( S ( u , z ) , N u ) ] ) × W ( u , z ) ,
Φ 3 D = C 0 , 0 0 C 0 , 1 0 C 0 , N u 1 N z 1 C 1 , 0 0 C 1 , 1 0 C 1 , N u 1 N z 1 C N u 1 , 0 0 C N u 1 , 1 0 C N u 1 , N u 1 N z 1 × R 0 0 0 R 0 0 0 R ,
C ˜ = F 1 0 0 0 F 1 0 0 0 F 1 × D 0 , 0 0 D 0 , 1 0 D 0 , N u 1 N z 1 D 1 , 0 0 D 1 , 1 0 D 1 , N u 1 N z 1 D N u 1 , 0 0 D N u 1 , 1 0 D N u 1 , N u 1 N z 1 × F 0 0 0 F 0 0 0 F ,
d ¯ i = ( d i H d i + λ I ) 1 d i H ,
d i = ( D 0 , 0 0 ) i ( D 0 , 1 0 ) i ( D 0 , N u 1 N z 1 ) i ( D 1 , 0 0 ) i ( D 1 , 1 0 ) i ( D 1 , N u 1 N z 1 ) i ( D N u 1 , 0 0 ) i ( D N u 1 , 1 0 ) i ( D N u 1 , N u 1 N z 1 ) i ,
f ^ 3 D = argmin f 3 D Pg P Φ 3 D f 3 D 2 + α R ( f 3 D ) ,

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