Abstract

In this paper, we propose generalized sampling approaches for measuring a multi-dimensional object using a compact compound-eye imaging system called thin observation module by bound optics (TOMBO). This paper shows the proposed system model, physical examples, and simulations to verify TOMBO imaging using generalized sampling. In the system, an object is sheared and multiplied by a weight distribution with physical coding, and the coded optical signal is integrated on to a detector array. A numerical estimation algorithm employing a sparsity constraint is used for object reconstruction.

© 2010 Optical Society of America

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  1. R. Ng, "Fourier slice photography," in "SIGGRAPH ’05: ACM SIGGRAPH 2005 Papers," (ACM, New York, NY, USA, 2005), pp. 735-744.
  2. J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, "Thin compound-eye camera," Appl. Opt. 44, 2949-2956 (2005).
    [CrossRef] [PubMed]
  3. R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, "Reinventing the camera," Opt. Photon. News 19, 32-37 (2008).
    [CrossRef]
  4. J. Tanida, T. Kumagai, K. Yamada, S. Miyatake, K. Ishida, T. Morimoto, N. Kondou, D. Miyazaki, and Y. Ichioka, "Thin observation module by bound optics (TOMBO): concept and experimental verification," Appl. Opt. 40, 1806-1813 (2001).
    [CrossRef]
  5. M. Shankar, N. P. Pitsianis, and D. J. Brady, "Compressive video sensors using multichannel imagers," Appl. Opt. 49, B9-B17 (2010).
    [CrossRef] [PubMed]
  6. R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007).
    [CrossRef]
  7. W. Zhou, and J. Leger, "Grin-optics-based hyperspectral imaging micro-sensor," Proc. SPIE 6765, 676502 (2007).
    [CrossRef]
  8. R. J. Plemmons, S. Prasad, S. Matthews, M. Mirotznik, R. Barnard, B. Gray, V. P. Pauca, T. C. Torgersen, J. van der Gracht, and G. Behrmann, "Periodic: Integrated computational array imaging technology," in "Computational Optical Sensing and Imaging," (2007), p. CMA1.
  9. R. Horstmeyer, R. Athale, and G. Euliss, "Modified light field architecture for reconfigurable multimode imaging," Proc. SPIE 7468, 746804 (2009).
    [CrossRef]
  10. E. J. Candes, and M. B. Wakin, "An introduction to compressive sampling," Signal Processing Magazine, IEEE 25, 21-30 (2008).
    [CrossRef]
  11. M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, "An architecture for compressive imaging," in "ICIP06," (2006), pp. 1273-1276.
  12. A. Wagadarikar, R. John, R. Willett, and D. Brady, "Single disperser design for coded aperture snapshot spectral imaging," Appl. Opt. 47, B44-B51 (2008).
    [CrossRef] [PubMed]
  13. D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, "Compressive holography," Opt. Express 17, 13040-13049 (2009).
    [CrossRef] [PubMed]
  14. K. Nitta, R. Shogenji, S. Miyatake, and J. Tanida, "Image reconstruction for thin observation module by bound optics by using the iterative back projection method," Appl. Opt. 45, 2893-2900 (2006).
    [CrossRef] [PubMed]
  15. Y. Tsaig, and D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
    [CrossRef]
  16. E. J. Candes, and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory 51, 4203-4215 (2005).
    [CrossRef]
  17. K. Choi, and T. J. Schulz, "Signal-processing approaches for image-resolution restoration for TOMBO imagery," Appl. Opt. 47, B104-B116 (2008).
    [CrossRef] [PubMed]
  18. E. Hecht, Optics (Addison Wesley, 2001), 4th ed.
  19. 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]
  20. L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60, 259-268 (1992).
    [CrossRef]
  21. W. H. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. 62, 55-59 (1972).
    [CrossRef]
  22. L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
    [CrossRef]
  23. Q. Huynh-Thu, and M. Ghanbari, "Scope of validity of PSNR in image/video quality assessment," Electron. Lett. 44, 800-801 (2008).
    [CrossRef]
  24. R. Gribonval, and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans. Inf. Theory 49, 3320-3325 (2003).
    [CrossRef]

2010 (1)

2009 (2)

R. Horstmeyer, R. Athale, and G. Euliss, "Modified light field architecture for reconfigurable multimode imaging," Proc. SPIE 7468, 746804 (2009).
[CrossRef]

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, "Compressive holography," Opt. Express 17, 13040-13049 (2009).
[CrossRef] [PubMed]

2008 (5)

A. Wagadarikar, R. John, R. Willett, and D. Brady, "Single disperser design for coded aperture snapshot spectral imaging," Appl. Opt. 47, B44-B51 (2008).
[CrossRef] [PubMed]

K. Choi, and T. J. Schulz, "Signal-processing approaches for image-resolution restoration for TOMBO imagery," Appl. Opt. 47, B104-B116 (2008).
[CrossRef] [PubMed]

Q. Huynh-Thu, and M. Ghanbari, "Scope of validity of PSNR in image/video quality assessment," Electron. Lett. 44, 800-801 (2008).
[CrossRef]

E. J. Candes, and M. B. Wakin, "An introduction to compressive sampling," Signal Processing Magazine, IEEE 25, 21-30 (2008).
[CrossRef]

R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, "Reinventing the camera," Opt. Photon. News 19, 32-37 (2008).
[CrossRef]

2007 (3)

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007).
[CrossRef]

W. Zhou, and J. Leger, "Grin-optics-based hyperspectral imaging micro-sensor," Proc. SPIE 6765, 676502 (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]

2006 (2)

2005 (2)

2003 (1)

R. Gribonval, and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans. Inf. Theory 49, 3320-3325 (2003).
[CrossRef]

2001 (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]

1974 (1)

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

1972 (1)

Athale, R.

R. Horstmeyer, R. Athale, and G. Euliss, "Modified light field architecture for reconfigurable multimode imaging," Proc. SPIE 7468, 746804 (2009).
[CrossRef]

R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, "Reinventing the camera," Opt. Photon. News 19, 32-37 (2008).
[CrossRef]

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]

Brady, D.

Brady, D. J.

Bräuer, A.

Candes, E. J.

E. J. Candes, and M. B. Wakin, "An introduction to compressive sampling," Signal Processing Magazine, IEEE 25, 21-30 (2008).
[CrossRef]

E. J. Candes, and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory 51, 4203-4215 (2005).
[CrossRef]

Choi, K.

Dannberg, P.

Donoho, D. L.

Y. Tsaig, and D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

Duparré, J.

Euliss, G.

R. Horstmeyer, R. Athale, and G. Euliss, "Modified light field architecture for reconfigurable multimode imaging," Proc. SPIE 7468, 746804 (2009).
[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]

Ghanbari, M.

Q. Huynh-Thu, and M. Ghanbari, "Scope of validity of PSNR in image/video quality assessment," Electron. Lett. 44, 800-801 (2008).
[CrossRef]

Gribonval, R.

R. Gribonval, and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans. Inf. Theory 49, 3320-3325 (2003).
[CrossRef]

Healy, D. M.

R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, "Reinventing the camera," Opt. Photon. News 19, 32-37 (2008).
[CrossRef]

Horisaki, R.

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, "Compressive holography," Opt. Express 17, 13040-13049 (2009).
[CrossRef] [PubMed]

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007).
[CrossRef]

Horstmeyer, R.

R. Horstmeyer, R. Athale, and G. Euliss, "Modified light field architecture for reconfigurable multimode imaging," Proc. SPIE 7468, 746804 (2009).
[CrossRef]

Huynh-Thu, Q.

Q. Huynh-Thu, and M. Ghanbari, "Scope of validity of PSNR in image/video quality assessment," Electron. Lett. 44, 800-801 (2008).
[CrossRef]

Ichioka, Y.

Irie, S.

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007).
[CrossRef]

Ishida, K.

John, R.

Kondou, N.

Kumagai, T.

Leger, J.

W. Zhou, and J. Leger, "Grin-optics-based hyperspectral imaging micro-sensor," Proc. SPIE 6765, 676502 (2007).
[CrossRef]

Lim, S.

Lucy, L. B.

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

Marks, D. L.

Miyatake, S.

Miyazaki, D.

Morimoto, T.

Neifeld, M. A.

R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, "Reinventing the camera," Opt. Photon. News 19, 32-37 (2008).
[CrossRef]

Nielsen, M.

R. Gribonval, and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans. Inf. Theory 49, 3320-3325 (2003).
[CrossRef]

Nitta, K.

Ogura, Y.

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007).
[CrossRef]

Osher, S.

L. I. Rudin, S. Osher, and E. Fatemi, "Nonlinear total variation based noise removal algorithms," Physica D 60, 259-268 (1992).
[CrossRef]

Pitsianis, N. P.

Richardson, W. H.

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]

Schreiber, P.

Schulz, T. J.

Shankar, M.

Shogenji, R.

Tanida, J.

Tao, T.

E. J. Candes, and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory 51, 4203-4215 (2005).
[CrossRef]

Tsaig, Y.

Y. Tsaig, and D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

Tünnermann, A.

Wagadarikar, A.

Wakin, M. B.

E. J. Candes, and M. B. Wakin, "An introduction to compressive sampling," Signal Processing Magazine, IEEE 25, 21-30 (2008).
[CrossRef]

Willett, R.

Yamada, K.

Zhou, W.

W. Zhou, and J. Leger, "Grin-optics-based hyperspectral imaging micro-sensor," Proc. SPIE 6765, 676502 (2007).
[CrossRef]

Appl. Opt. (6)

Astron. J. (1)

L. B. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974).
[CrossRef]

Electron. Lett. (1)

Q. Huynh-Thu, and M. Ghanbari, "Scope of validity of PSNR in image/video quality assessment," Electron. Lett. 44, 800-801 (2008).
[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 (3)

Y. Tsaig, and D. L. Donoho, "Compressed sensing," IEEE Trans. Inf. Theory 52, 1289-1306 (2006).
[CrossRef]

E. J. Candes, and T. Tao, "Decoding by linear programming," IEEE Trans. Inf. Theory 51, 4203-4215 (2005).
[CrossRef]

R. Gribonval, and M. Nielsen, "Sparse representations in unions of bases," IEEE Trans. Inf. Theory 49, 3320-3325 (2003).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (1)

Opt. Photon. News (1)

R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, "Reinventing the camera," Opt. Photon. News 19, 32-37 (2008).
[CrossRef]

Opt. Rev. (1)

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, "Three-dimensional information acquisition using a compound imaging system," Opt. Rev. 14, 347-350 (2007).
[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. SPIE (2)

W. Zhou, and J. Leger, "Grin-optics-based hyperspectral imaging micro-sensor," Proc. SPIE 6765, 676502 (2007).
[CrossRef]

R. Horstmeyer, R. Athale, and G. Euliss, "Modified light field architecture for reconfigurable multimode imaging," Proc. SPIE 7468, 746804 (2009).
[CrossRef]

Signal Processing Magazine, IEEE (1)

E. J. Candes, and M. B. Wakin, "An introduction to compressive sampling," Signal Processing Magazine, IEEE 25, 21-30 (2008).
[CrossRef]

Other (4)

M. Wakin, J. Laska, M. Duarte, D. Baron, S. Sarvotham, D. Takhar, K. Kelly, and R. Baraniuk, "An architecture for compressive imaging," in "ICIP06," (2006), pp. 1273-1276.

R. J. Plemmons, S. Prasad, S. Matthews, M. Mirotznik, R. Barnard, B. Gray, V. P. Pauca, T. C. Torgersen, J. van der Gracht, and G. Behrmann, "Periodic: Integrated computational array imaging technology," in "Computational Optical Sensing and Imaging," (2007), p. CMA1.

E. Hecht, Optics (Addison Wesley, 2001), 4th ed.

R. Ng, "Fourier slice photography," in "SIGGRAPH ’05: ACM SIGGRAPH 2005 Papers," (ACM, New York, NY, USA, 2005), pp. 735-744.

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Figures (7)

Fig. 1.
Fig. 1.

Cross section view of TOMBO. ν, Ou , and Lu are the spatial dimension, the center position, and the position of a lenslet in the u-th unit, respectively.

Fig. 2.
Fig. 2.

Coding schemes in TOMBO. (a) Sheared integration and (b) weighted integration in a unit.

Fig. 3.
Fig. 3.

Cross section views of TOMBO for spectral imaging with (a) SI and (b) WI.

Fig. 4.
Fig. 4.

Top views of TOMBO for polarization imaging with (a) SI and (b) WI. Arrows, dots, circles, and shaded areas indicate directions of polarization, centers of shifted images, lenslets, and polarization plates, respectively.

Fig. 5.
Fig. 5.

Simulation results with total variation. (a) A four-dimensional object (∈ ℝ128×128×4×2), where indices of axial planes are shown under each axial plane, (b) a measurement data, (c) a reconstruction with TwIST, (d) a reconstruction with RL, and (e) a reconstruction with TwIST using a small ∣A 0∣.

Fig. 6.
Fig. 6.

Simulation results with discrete wavelet transform. (a) A five-dimensional object (∈ ℝ128×128×2×2×2), (b) a measurement data, (c) a reconstruction with TwIST, (d) a reconstruction with RL, and (e) a reconstruction with TwIST using a large h.

Fig. 7.
Fig. 7.

Plots of reconstruction PSNR from noisy measurements in the proposed system and a baseline sensing system which is a Gaussian random sensing matrix. (a) Plots with the object, the parameters, and the basis used in Fig. 5(c) and (b) plots with the object, the parameters, and the basis used in Fig. 6(c).

Equations (18)

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

g = Φ f = Φ Ψ β = Θ β ,
( 1 c s ) β Λ 2 2 Θ Λ β Λ 2 2 ( 1 + c s ) β Λ 2 2 ,
μ ( Φ , Ψ ) = N f max 1 i N g , 1 j N f Φ ( i , : ) , Ψ ( : , j )
N g c μ ( Φ , Ψ ) 2 s log N f ,
β ̂ = argmin β β 1 subject to g = Θ β ,
G u ( v ) = F ( v L u n S n , u ( z n ) , z 0 , , z N n 1 ) n W n , u ( z n ) dz n ,
( u = 0 , , N u 1 ) ,
G u ( v ) = F ( v L u S u ( z ) , z ) W u ( z ) dz .
G ˜ u ( j ) = i rect ( i x j v v ) m F ˜ ( i S ¯ u ( m ) , m ) W ¯ u ( m ) ,
C m , u ( p , q ) = { W ¯ u ( m ) ( p = q + S ¯ u ( m ) ) , 0 ( p q + S ¯ u ( m ) ) ,
C u = [ C 0 , u O O O C 1 , u O O O C N z 1 , u ] ,
Q = [ I I I ] ,
T = [ 1 T 0 T 0 T 0 T 1 T 0 T 0 T 0 T 1 T ] ,
Φ = [ T Q C 0 T Q C 1 T Q C N u 1 ] = [ T C 0,0 T C 1,0 TC N z 1,0 T C 0,1 T C 1,1 TC N z 1,1 TC 0 , N u 1 TC 1 , N u 1 TC N z 1, N u 1 ] .
ϕ m = [ T C m , 0 T C m , 1 T C m , N u 1 ] .
Φ = [ ϕ 0 ϕ 1 ϕ N z 1 ] ,
Ψ = [ ψ O O O ψ O O O ψ ] ,
Θ = Φ Ψ = [ ϕ 0 ψ ϕ 1 ψ ϕ N z 1 ψ ] ,

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