Abstract

A new imaging technique that combines compressive sensing and super-resolution techniques is presented. Compressive sensing is accomplished by capturing optically a set of Radon projections. Super-resolution measurements are simply taken by introducing a slanted two-dimensional array in the optical system. The goal of the technique is to overcome resolution limitation that occurs in imaging scenarios where dense pixels sensors with large number of pixels are not available or cannot be used. With the presented imaging technique, owing to the compressive sensing approach, we were able to reconstruct images with significantly more number of pixels than measured, and owing to the super-resolution design we have been able to achieve resolution significantly beyond that limited by the sensor's pixels size.

© 2013 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  3. Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” in Computational Optical Sensing and Imaging (Optical Society of America, 2009).
  4. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett.32(21), 3077–3079 (2007).
    [CrossRef] [PubMed]
  5. D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in Electronic Imaging 2006 (International Society for Optics and Photonics, 2006).
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  8. Y. August, C. Vachman, Y. Rivenson, and A. Stern, “Compressive hyperspectral imaging by random separable projections in both the spatial and the spectral domains,” Appl. Opt.52(10), D46–D54 (2013).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  10. S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process20(3), 21–36 (2003).
    [CrossRef]
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    [CrossRef]
  12. Z. Zalevsky and D. Mendlovic, Optical Superresolution, Springer (2004).
  13. H. Greenspan, “Super-resolution in medical imaging,” Comput. J.52(1), 43–63 (2008).
    [CrossRef]
  14. Y. Kashter, O. Levi, and A. Stern, “Optical compressive change and motion detection,” Appl. Opt.51(13), 2491–2496 (2012).
    [CrossRef] [PubMed]
  15. S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
    [CrossRef] [PubMed]
  16. P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Adv. Signal Process.2006, 1–15 (2006).
    [CrossRef]
  17. A. Stern, Y. Porat, A. Ben-Dor, and N. S. Kopeika, “Enhanced-resolution image restoration from a sequence of low-frequency vibrated images by use of convex projections,” Appl. Opt.40(26), 4706–4715 (2001).
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  18. J. H. Jørgensen, “Knowledge-based tomography algorithms” (Doctoral dissertation, Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark,2009).
  19. V. Farber, E. Eduard, Y. Rivenson, and A. Stern, “A study of the coherence parameter of the progressive compressive imager based on Radon transform,” in SPIE Defense, Security, and Sensing (International Society for Optics and Photonics,2013).
  20. J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process.16(12), 2992–3004 (2007).
    [CrossRef] [PubMed]

2013 (2)

2012 (2)

2010 (1)

2008 (1)

H. Greenspan, “Super-resolution in medical imaging,” Comput. J.52(1), 43–63 (2008).
[CrossRef]

2007 (2)

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process.16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett.32(21), 3077–3079 (2007).
[CrossRef] [PubMed]

2006 (2)

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Adv. Signal Process.2006, 1–15 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

2004 (1)

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

2003 (2)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process20(3), 21–36 (2003).
[CrossRef]

D. Capel and A. Zisserman, “Computer vision applied to super resolution,” IEEE Signal Process20(3), 75–86 (2003).
[CrossRef]

2001 (1)

August, Y.

Ben-Dor, A.

Bioucas-Dias, J. M.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process.16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

Capel, D.

D. Capel and A. Zisserman, “Computer vision applied to super resolution,” IEEE Signal Process20(3), 75–86 (2003).
[CrossRef]

Donoho, D. L.

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

Elad, M.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Evladov, S.

Farsiu, S.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Figueiredo, M. A.

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process.16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

Greenspan, H.

H. Greenspan, “Super-resolution in medical imaging,” Comput. J.52(1), 43–63 (2008).
[CrossRef]

Javidi, B.

Kang, M. G.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process20(3), 21–36 (2003).
[CrossRef]

Kashter, Y.

Kopeika, N. S.

Levi, O.

Milanfar, P.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Park, M. K.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process20(3), 21–36 (2003).
[CrossRef]

Park, S. C.

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process20(3), 21–36 (2003).
[CrossRef]

Porat, Y.

Rivenson, Y.

Robinson, M. D.

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

Stern, A.

Süsstrunk, S.

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Adv. Signal Process.2006, 1–15 (2006).
[CrossRef]

Vachman, C.

Vandewalle, P.

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Adv. Signal Process.2006, 1–15 (2006).
[CrossRef]

Vetterli, M.

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Adv. Signal Process.2006, 1–15 (2006).
[CrossRef]

Zisserman, A.

D. Capel and A. Zisserman, “Computer vision applied to super resolution,” IEEE Signal Process20(3), 75–86 (2003).
[CrossRef]

Appl. Opt. (4)

Comput. J. (1)

H. Greenspan, “Super-resolution in medical imaging,” Comput. J.52(1), 43–63 (2008).
[CrossRef]

EURASIP J. Adv. Signal Process. (1)

P. Vandewalle, S. Süsstrunk, and M. Vetterli, “A frequency domain approach to registration of aliased images with application to super-resolution,” EURASIP J. Adv. Signal Process.2006, 1–15 (2006).
[CrossRef]

IEEE Signal Process (2)

S. C. Park, M. K. Park, and M. G. Kang, “Super-resolution image reconstruction: a technical overview,” IEEE Signal Process20(3), 21–36 (2003).
[CrossRef]

D. Capel and A. Zisserman, “Computer vision applied to super resolution,” IEEE Signal Process20(3), 75–86 (2003).
[CrossRef]

IEEE Trans. Image Process. (2)

S. Farsiu, M. D. Robinson, M. Elad, and P. Milanfar, “Fast and robust multiframe super resolution,” IEEE Trans. Image Process.13(10), 1327–1344 (2004).
[CrossRef] [PubMed]

J. M. Bioucas-Dias and M. A. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process.16(12), 2992–3004 (2007).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (1)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory52(4), 1289–1306 (2006).
[CrossRef]

Opt. Express (2)

Opt. Lett. (1)

Other (6)

D. Takhar, J. N. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. F. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” in Electronic Imaging 2006 (International Society for Optics and Photonics, 2006).

Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” in Computational Optical Sensing and Imaging (Optical Society of America, 2009).

E. J. Candès, “Compressive sampling,” Proc. Int. Congress of Mathematics 3, 1433–1452, Madrid, Spain (2006).
[CrossRef]

Z. Zalevsky and D. Mendlovic, Optical Superresolution, Springer (2004).

J. H. Jørgensen, “Knowledge-based tomography algorithms” (Doctoral dissertation, Technical University of Denmark, DTU, DK-2800 Kgs. Lyngby, Denmark,2009).

V. Farber, E. Eduard, Y. Rivenson, and A. Stern, “A study of the coherence parameter of the progressive compressive imager based on Radon transform,” in SPIE Defense, Security, and Sensing (International Society for Optics and Photonics,2013).

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Figures (8)

Fig. 1
Fig. 1

Capturing subpixel resolution Radon projections. (a) Compressive imaging method using a line array of sensors as in [6], (b) Subpixel information measurement by using two staggered line-array of sensors, (c) Subpixel information measurement using a rotated 2D array instead of the staggered line-array of sensors.

Fig. 2
Fig. 2

Optical Radon projection system.

Fig. 3
Fig. 3

Relation between the HR projection grid and LR CCD grid (a) LR (CCD) grid is aligned with HR (virtual) grid ( β=0° ) . Note that LR projections captured by the CCD columns contain the same information. (b) After slating the CCD by β=14° its columns { z j, θ i } j=1 j=b contain subpixel-shifted data.

Fig. 4
Fig. 4

Operator T θ relating the 1D HR grid to the set of columns { z j, θ i } j=1 j=b , illustrated for the case where the number of columns is b=4 and number of LR sensors in each column is n=6 . The relative angle β is in accordance with Eq. (5)

Fig. 5
Fig. 5

The SR transformation matrices in the case where b=4 , n=6 and β is in accordance with Eq. (5). (a) Overall SR system matrix T (b) One projection dimension reduction matrix T θ , (c) Operator T ¯ j relating the high resolution projection g θ i with j th column of CCD pixels z j, θ i , where j=1 .

Fig. 6
Fig. 6

Original synthetic image) a) Original synthetic image of size 480×480 .(b) Restored synthetic image of size 480×480 from 90 projections of size 480×1 .(c) Restored synthetic image of size 120×120 from 90 projections of size 120×1 (d) Synthetic image of size 480×480 restored from 90 projections of size 120×4 by SR.

Fig. 7
Fig. 7

Restoration of images from 90 projections. (a) Image of size 120×120 restored using CI as in [6] from projections of size 120×1 . (b) HR image of size 480×480 using SRCI projections of size 120×4 .

Fig. 8
Fig. 8

Restored image of size 480×480 using only 45 projections of size 120×4 .

Equations (11)

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g=Φf+ε=ΦΨα+ε=Ωα+ε,
α est = argmin α { 1 2 Ωα-g 2 2 +λ α 1 },
z j = T j g+ ε j , j=1,..,b ,
z=[ z 1 z b ]=[ T 1 T b ]g+ε=Tg+ε,
β= tan 1 ( 1 b ),
z=Tg=T( Ωα+ε )=Hα+ ε ,
T= I beams T θ ,
x ˜ = argmin x { 0.5 yKx 2 2 +λc( x ) },
T= I beams T θ =[ T θ 0 0 0 T θ 0 0 0 0 0 T θ ],
t ¯ i,k ={ 1 2 if k=b( i1 )+1ork=bi+1 1 if b( i1 )+1<k<bi+1 0 otherwise , i=1,2,,n k=1,2,,bn .
t ¯ j i,k ={ t ¯ j i,k = t ¯ i,kj+1 if j1<kj1+bn 0otherwise , i=1,2,,n k=1,2,,bn j=1,...,b .

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