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 OSA

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References

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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,” Optical Review 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 backprojection method,” Appl. Opt. 45, 2893–2900 (2006).
    [Crossref] [PubMed]
  15. Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52, 1289–1306 (2006).
    [Crossref]
  16. E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. 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 Proc. 16, 2992–3004 (2007).
    [Crossref]
  20. L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. 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. Info. Theory 49, 3320–3325 (2003).
    [Crossref]

2010 (1)

2009 (2)

D. J. Brady, K. Choi, D. L. Marks, R. Horisaki, and S. Lim, “Compressive holography,” Opt. Express 17, 13040–13049 (2009).
[Crossref] [PubMed]

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

2008 (5)

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

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]

R. Athale, D. M. Healy, D. J. Brady, and M. A. Neifeld, “Reinventing the camera,” Opt. Photon. News 19, 32–37 (2008).
[Crossref]

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

2007 (3)

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Optical Review 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 Proc. 16, 2992–3004 (2007).
[Crossref]

2006 (2)

2005 (2)

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

J. Duparré, P. Dannberg, P. Schreiber, A. Bräuer, and A. Tünnermann, “Thin compound-eye camera,” Appl. Opt. 44, 2949–2956 (2005).
[Crossref] [PubMed]

2003 (1)

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. 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,” Phys. 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]

Baraniuk, R.

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.

Barnard, R.

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.

Baron, D.

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.

Behrmann, G.

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.

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.

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. Info. Theory 51, 4203–4215 (2005).
[Crossref]

Choi, K.

Dannberg, P.

Donoho, D. L.

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52, 1289–1306 (2006).
[Crossref]

Duarte, M.

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.

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,” 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]

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]

Gray, B.

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.

Gribonval, R.

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. 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]

Hecht, E.

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

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,” Optical Review 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,” Optical Review 14, 347–350 (2007).
[Crossref]

Ishida, K.

John, R.

Kelly, K.

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.

Kondou, N.

Kumagai, T.

Laska, J.

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.

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.

Matthews, S.

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.

Mirotznik, M.

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.

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]

Ng, R.

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

Nielsen, M.

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. 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,” Optical Review 14, 347–350 (2007).
[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]

Pauca, V. P.

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.

Pitsianis, N. P.

Plemmons, R. J.

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.

Prasad, S.

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.

Richardson, W. H.

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]

Sarvotham, S.

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.

Schreiber, P.

Schulz, T. J.

Shankar, M.

Shogenji, R.

Takhar, D.

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.

Tanida, J.

Tao, T.

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

Torgersen, T. C.

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.

Tsaig, Y.

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52, 1289–1306 (2006).
[Crossref]

Tünnermann, A.

van der Gracht, J.

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.

Wagadarikar, A.

Wakin, M.

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.

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

IEEE Trans. Info. Theory (2)

E. J. Candes and T. Tao, “Decoding by linear programming,” IEEE Trans. Info. Theory 51, 4203–4215 (2005).
[Crossref]

R. Gribonval and M. Nielsen, “Sparse representations in unions of bases,” IEEE Trans. Info. Theory 49, 3320–3325 (2003).
[Crossref]

IEEE Transactions on Information Theory (1)

Y. Tsaig and D. L. Donoho, “Compressed sensing,” IEEE Transactions on Information Theory 52, 1289–1306 (2006).
[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]

Optical Review (1)

R. Horisaki, S. Irie, Y. Ogura, and J. Tanida, “Three-dimensional information acquisition using a compound imaging system,” Optical Review 14, 347–350 (2007).
[Crossref]

Phys. D (1)

L. I. Rudin, S. Osher, and E. Fatemi, “Nonlinear total variation based noise removal algorithms,” Phys. 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. Ng, “Fourier slice photography,” in “SIGGRAPH ’05: ACM SIGGRAPH 2005 Papers,” (ACM, New York, NY, USA, 2005), pp. 735–744.

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.

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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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