Abstract

In this work we propose a unique sampling scheme of Radon Projections and a non-linear reconstruction algorithm based on compressive sensing (CS) theory to implement a progressive compressive sampling imaging system. The progressive sampling scheme offers online control of the tradeoff between the compression and the quality of reconstruction. It avoids the need of a priori knowledge of the object sparsity that is usually required for CS design. In addition, the progressive data acquisition enables straightforward application of ordered-subsets algorithms which overcome computational constraints associated with the reconstruction of very large images. We present, to the best of our knowledge for the first time, a compressive imaging implementation of megapixel size images with a compression ratio of 20:1.

© 2012 OSA

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References

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  1. A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007).
    [CrossRef]
  2. E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
    [CrossRef]
  3. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
    [CrossRef]
  4. Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” Computational Optical Sensing and Imaging, COSI OSA Technical Digest (CD), paper CTuA4 (2009).
  5. R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011).
    [CrossRef]
  6. A. Stern, “Compressed imaging system with linear sensors,” Opt. Lett. 32(21), 3077–3079 (2007).
    [CrossRef] [PubMed]
  7. A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010).
    [CrossRef]
  8. E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
    [CrossRef]
  9. M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
    [CrossRef]
  10. H. Niederreiter, Uniform Distribution of Sequences (Dover Publications, 2006).
  11. M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).
  12. M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Broadway Books, 2003).
  13. H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994).
    [CrossRef] [PubMed]
  14. H. Zaidi, Quantitative Analysis in Nuclear Medicine Imaging (Springer, 2006).
  15. 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(12), 2992–3004 (2007).
    [CrossRef] [PubMed]

2011 (1)

R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011).
[CrossRef]

2010 (2)

A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010).
[CrossRef]

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

2008 (2)

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

2007 (3)

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (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(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)

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

1994 (1)

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994).
[CrossRef] [PubMed]

Bachmat, E.

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

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(12), 2992–3004 (2007).
[CrossRef] [PubMed]

Candes, E. J.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Donoho, D. L.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[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(12), 2992–3004 (2007).
[CrossRef] [PubMed]

Hudson, H. M.

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994).
[CrossRef] [PubMed]

Javidi, B.

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007).
[CrossRef]

Kleider, M.

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

Larkin, R. S.

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994).
[CrossRef] [PubMed]

Levi, O.

A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010).
[CrossRef]

Lustig, M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

Marcia, R. F.

R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011).
[CrossRef]

Nichols, J. M.

R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011).
[CrossRef]

Pauly, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

Rafaely, B.

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

Rivenson, Y.

A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010).
[CrossRef]

Romberg, J.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Santos, J. M.

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

Stern, A.

A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010).
[CrossRef]

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

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007).
[CrossRef]

Tao, T.

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

Wakin, M. B.

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

Weiss, B.

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

Willett, R. M.

R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011).
[CrossRef]

IEEE Signal Process. Mag. (2)

E. J. Candes and M. B. Wakin, “An introduction to compressive sampling,” IEEE Signal Process. Mag. 25(2), 21–30 (2008).
[CrossRef]

M. Lustig, D. L. Donoho, J. M. Santos, and J. M. Pauly, “Compressed sensing MRI,” IEEE Signal Process. Mag. 25(2), 72–82 (2008).
[CrossRef]

IEEE Trans. Audio, Speech, Lang. Process. (1)

M. Kleider, B. Rafaely, B. Weiss, and E. Bachmat, “Golden-Ratio sampling for scanning circular microphone arrays,” IEEE Trans. Audio, Speech, Lang. Process. 18, 2091–2098 (2010).

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(12), 2992–3004 (2007).
[CrossRef] [PubMed]

IEEE Trans. Inf. Theory (2)

E. J. Candes, J. Romberg, and T. Tao, “Robust uncertainty principles: Exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory 52(2), 489–509 (2006).
[CrossRef]

D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289–1306 (2006).
[CrossRef]

IEEE Trans. Med. Imaging (1)

H. M. Hudson and R. S. Larkin, “Accelerated image reconstruction using ordered subsets of projection data,” IEEE Trans. Med. Imaging 13(4), 601–609 (1994).
[CrossRef] [PubMed]

J. Disp. Technol. (1)

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Disp. Technol. 3(3), 315–320 (2007).
[CrossRef]

J. Phys. Conf. Ser. (1)

A. Stern, O. Levi, and Y. Rivenson, “Optically compressed sensing by under sampling the polar Fourier plane,” J. Phys. Conf. Ser. 206, 012019 (2010).
[CrossRef]

Opt. Eng. (1)

R. M. Willett, R. F. Marcia, and J. M. Nichols, “Compressed sensing for practical optical imaging systems: a tutorial,” Opt. Eng. 50(7), 072601 (2011).
[CrossRef]

Opt. Lett. (1)

Other (4)

Y. Rivenson and A. Stern, “An efficient method for multi-dimensional compressive imaging,” Computational Optical Sensing and Imaging, COSI OSA Technical Digest (CD), paper CTuA4 (2009).

H. Niederreiter, Uniform Distribution of Sequences (Dover Publications, 2006).

H. Zaidi, Quantitative Analysis in Nuclear Medicine Imaging (Springer, 2006).

M. Livio, The Golden Ratio: The Story of Phi, the World's Most Astonishing Number (Broadway Books, 2003).

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Figures (7)

Fig. 1
Fig. 1

The principle of optical Radon Projections. The detector S collects the ray sum of the object by means of the cylindrical lens.

Fig. 2
Fig. 2

(a) 20 angular samples over 2π taken at uniform (+) and golden angle (), (b) a scenario in which the sampling process is unintentionally stopped after 8 samples.

Fig. 3
Fig. 3

Arcs, a, and, b, are in a golden ratio relation. Golden angle θ ga is subtended by arc b.

Fig. 4
Fig. 4

Reconstruction quality as function of number of Radon projections. (a) Reconstruction in the wavelet domain. (b) Reconstruction in the space domain, (c) Synthetic image used in simulation (1280 x 1280 pixels)

Fig. 5
Fig. 5

Schematic diagram of the automatic CI optical system. Radon projections are recorded with the linear detector aligned with the Y axis and located in the image plane of the system.

Fig. 6
Fig. 6

(a-e) Progressive reconstructions from. 1.56%, 3.13%, 4.3%, 5.5% and 20% of the nominal samples needed for perfect reconstruction.

Fig. 7
Fig. 7

(a) is the reference image, (b) is the reconstruction in the golden sampling scheme, image (c) is the reconstruction in the uniform sampling scheme, image (d) is OS reconstruction using the explicit Radon transition matrix. All images are cropped to 1024 x 1024 pixels.

Equations (6)

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f(s,θ)=g(s,θ)= x y f(x,y)δ( sxcos(θ)ysin(θ) )dydx ,
J(a)= 1 2 gA(Ψa) 2 +λΦ(a),
θ q =2παq, q=0,1...,.
φ= a b = a+b a .
PSNR=20 log 10 ( MA X I MSE ),
MSE= 1 mn i=0 m1 j=0 n1 [ R(i,j)I(i,j) ] 2 .

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