Abstract

A natural field of application for compressive sensing theory is imaging. Indeed, numerous compressive imaging (CI) systems and applications have been developed during the last few years. This work addresses the quantization effect in CI, which is fundamental for most CI architectures. In this paper, the implications of sensor quantization on universal CI are investigated theoretically and demonstrated with numerical experiments. It is shown that employing a CI framework may set severe requirements on the quantization depth of the optical sensor used. The quantization depth overhead requirement may be prohibitive in many optical imaging scenarios employing typical CI architectures. Practical solutions that significantly alleviate this requirement are suggested.

© 2013 Optical Society of America

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2012 (6)

2011 (3)

F. Magalhães, F. M. Araújo, M. V. Correia, M. Abolbashari, and F. Farahi, “Active illumination single-pixel camera based on compressive sensing,” Appl. Opt. 50, 405–414 (2011).
[CrossRef]

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

M. Wang, W. Xu, and A. Tang, “A unique “nonnegative” solution to an underdetermined system: from vectors to matrices,” IEEE Trans. Signal Process. 59, 1007–1016 (2011).
[CrossRef]

2010 (5)

2009 (2)

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, 72460G (2009).

S. Gazit, A. Szameit, Y. C. Eldar, and M. Segev, “Super-resolution and reconstruction of sparse sub-wavelength images: erratum,” Opt. Express 17, 23920–23946 (2009).
[CrossRef]

2008 (4)

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

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

A. M. Bruckstein, M. Elad, and M. Zibulevsky, “On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations,” IEEE Trans. Inf. Theory 54, 4813–4820 (2008).
[CrossRef]

2007 (4)

M. A. Neifeld and J. Ke, “Optical architectures for compressive imaging,” Appl. Opt. 46, 5293–5303 (2007).
[CrossRef]

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

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Display Technology 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, 2992–3004 (2007).
[CrossRef]

2006 (4)

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

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

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

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Abolbashari, M.

Aguet, F.

Araújo, F. M.

Balber, S.

Baraniuk, R. G.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Baron, D.

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Belhumeur, P.

J. Gu, S. Nayar, E. Grinspun, P. Belhumeur, and R. Ramamoorthi, “Compressive structured light for recovering inhomogeneous participating media,” in Computer Vision–ECCV,” Vol. 5305 of Lecture Notes in Computer Science (Springer, 2008), pp. 845–858.

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]

Bourquard, A. Ú.

Brady, D. J.

Bruckstein, A. M.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, “On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations,” IEEE Trans. Inf. Theory 54, 4813–4820 (2008).
[CrossRef]

Candès, E. J.

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

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

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

Chan, W. L.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Charan, K.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Cho, M.

Choi, K.

Correia, M. V.

Cull, C. F.

Donoho, D. L.

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

Duarte, M. F.

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Elad, M.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, “On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations,” IEEE Trans. Inf. Theory 54, 4813–4820 (2008).
[CrossRef]

Eldar, Y. C.

Farahi, F.

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]

Gazit, S.

Gehm, M.

Grinspun, E.

J. Gu, S. Nayar, E. Grinspun, P. Belhumeur, and R. Ramamoorthi, “Compressive structured light for recovering inhomogeneous participating media,” in Computer Vision–ECCV,” Vol. 5305 of Lecture Notes in Computer Science (Springer, 2008), pp. 845–858.

Gu, J.

J. Gu, S. Nayar, E. Grinspun, P. Belhumeur, and R. Ramamoorthi, “Compressive structured light for recovering inhomogeneous participating media,” in Computer Vision–ECCV,” Vol. 5305 of Lecture Notes in Computer Science (Springer, 2008), pp. 845–858.

Hahn, J.

Harmany, Z. T.

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, 72460G (2009).

Horisaki, R.

Javidi, B.

Kashter, Y.

Ke, J.

Kelly, K.

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Kelly, K. F.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Lam, E. Y.

Laska, J.

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Levi, O.

Magalhães, F.

Mahalanobis, A.

Mait, J. N.

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, 072601 (2011).
[CrossRef]

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, 72460G (2009).

Mariano, A.

Mattheiss, M.

Mittleman, D. M.

W. L. Chan, M. L. Moravec, R. G. Baraniuk, and D. M. Mittleman, “Terahertz imaging with compressed sensing and phase retrieval,” Opt. Lett. 33, 974–976 (2008).
[CrossRef]

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Moravec, M. L.

Nayar, S.

J. Gu, S. Nayar, E. Grinspun, P. Belhumeur, and R. Ramamoorthi, “Compressive structured light for recovering inhomogeneous participating media,” in Computer Vision–ECCV,” Vol. 5305 of Lecture Notes in Computer Science (Springer, 2008), pp. 845–858.

Neifeld, M. A.

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, 072601 (2011).
[CrossRef]

Osman, T.

Poon, P.

Ramamoorthi, R.

J. Gu, S. Nayar, E. Grinspun, P. Belhumeur, and R. Ramamoorthi, “Compressive structured light for recovering inhomogeneous participating media,” in Computer Vision–ECCV,” Vol. 5305 of Lecture Notes in Computer Science (Springer, 2008), pp. 845–858.

Rivenson, Y.

Romberg, J.

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

Romberg, J. K.

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Rosen, J.

Rot, A.

Sarvotham, S.

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Segev, M.

Shechtman, Y.

Stenner, M.

Stern, A.

Szameit, A.

Takhar, D.

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Tang, A.

M. Wang, W. Xu, and A. Tang, “A unique “nonnegative” solution to an underdetermined system: from vectors to matrices,” IEEE Trans. Signal Process. 59, 1007–1016 (2011).
[CrossRef]

Tanida, J.

Tao, T.

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

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

Townsend, D.

Unser, M.

Vera, E.

Wakin, M. B.

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

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

Wang, M.

M. Wang, W. Xu, and A. Tang, “A unique “nonnegative” solution to an underdetermined system: from vectors to matrices,” IEEE Trans. Signal Process. 59, 1007–1016 (2011).
[CrossRef]

Wehrwein, S.

Wikner, D. A.

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, 072601 (2011).
[CrossRef]

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, 72460G (2009).

Xu, W.

M. Wang, W. Xu, and A. Tang, “A unique “nonnegative” solution to an underdetermined system: from vectors to matrices,” IEEE Trans. Signal Process. 59, 1007–1016 (2011).
[CrossRef]

Zibulevsky, M.

A. M. Bruckstein, M. Elad, and M. Zibulevsky, “On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations,” IEEE Trans. Inf. Theory 54, 4813–4820 (2008).
[CrossRef]

Appl. Opt. (4)

Appl. Phys. Lett. (1)

W. L. Chan, K. Charan, D. Takhar, K. F. Kelly, R. G. Baraniuk, and D. M. Mittleman, “A single-pixel terahertz imaging system based on compressed sensing,” Appl. Phys. Lett. 93, 121105 (2008).
[CrossRef]

Commun. Pure Appl. Math. (1)

E. J. Candès, J. K. Romberg, and T. Tao, “Stable signal recovery from incomplete and inaccurate measurements,” Commun. Pure Appl. Math. 59, 1207–1223 (2006).
[CrossRef]

IEEE Signal Process. Mag. (1)

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

A. M. Bruckstein, M. Elad, and M. Zibulevsky, “On the uniqueness of nonnegative sparse solutions to underdetermined systems of equations,” IEEE Trans. Inf. Theory 54, 4813–4820 (2008).
[CrossRef]

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

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

IEEE Trans. Signal Process. (1)

M. Wang, W. Xu, and A. Tang, “A unique “nonnegative” solution to an underdetermined system: from vectors to matrices,” IEEE Trans. Signal Process. 59, 1007–1016 (2011).
[CrossRef]

J. Display Technology (1)

A. Stern and B. Javidi, “Random projections imaging with extended space-bandwidth product,” J. Display Technology 3, 315–320 (2007).
[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, 072601 (2011).
[CrossRef]

Opt. Express (7)

Opt. Lett. (5)

Proc. SPIE (2)

D. Takhar, J. Laska, M. B. Wakin, M. F. Duarte, D. Baron, S. Sarvotham, K. Kelly, and R. G. Baraniuk, “A new compressive imaging camera architecture using optical-domain compression,” Proc. SPIE 6065, 606509 (2006).

R. F. Marcia, Z. T. Harmany, and R. M. Willett, “Compressive coded aperture imaging,” Proc. SPIE 7246, 72460G (2009).

Other (2)

J. Gu, S. Nayar, E. Grinspun, P. Belhumeur, and R. Ramamoorthi, “Compressive structured light for recovering inhomogeneous participating media,” in Computer Vision–ECCV,” Vol. 5305 of Lecture Notes in Computer Science (Springer, 2008), pp. 845–858.

D. J. Brady, Optical Imaging and Spectroscopy (Wiley-Interscience, 2009).

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Figures (7)

Fig. 1.
Fig. 1.

Block diagram of CS and reconstruction [7].

Fig. 2.
Fig. 2.

Schematic optical setup for sequential CI architecture: (a) by encoding the object’s image and (b) using structured illumination.

Fig. 3.
Fig. 3.

Example of a schematic optical setup for parallel CI architecture.

Fig. 4.
Fig. 4.

Uniform scalar quantizer input–output characteristics. g i is the quantized value of the i th component of A f , g i c .

Fig. 5.
Fig. 5.

Gray level distributions of g = A f for the “student image” [Fig. 6(h)].

Fig. 6.
Fig. 6.

Images used in the numerical experiment.

Fig. 7.
Fig. 7.

Number of sensor bits required as a function of the reconstruction RMSE with CI (upper curves) and conventional imaging (lower curves): (a) full dynamic range and (b) adapted dynamic range.

Equations (36)

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

g = A f + n .
f ^ f 2 C s ε ,
g = H f + n ,
g = γ I I f + n ,
g = γ CS A f + n .
qMSE I = E f ^ f 2 2 = E g γ I f 2 2 = E γ I f + n γ I f 2 2 = E n γ I 2 2 ,
qRMSE I = E n γ I 2 2 = E ( i = 1 N | n i γ I | 2 ) = ( i = 1 N E | n i γ I | 2 ) = N γ I · Δ I 12 ,
qRMSE I I C s ε = C s E n 2 2 = C s E i = 1 M ( n i γ CS ) 2 = C s γ CS · M · Δ I I 12 ,
Δ I I = 1 C s γ CS γ I N M Δ I .
B I = log 2 L I Δ I = log 2 γ I L o Δ I bits .
max g i = γ cs N L o M ,
L cs = 2 γ cs N L 0 M .
B cs = log 2 L cs Δ I I = log 2 2 γ cs N L o M Δ I I .
B cs = log 2 2 N L o C s γ I Δ I = log 2 L o γ I Δ I + log 2 2 N C s = B I + log 2 2 N C s .
σ g 2 = γ I 2 σ f 2 + Δ I 2 12 γ I 2 σ f 2 ,
B I = log 2 L I Δ I = log 2 r γ I σ f Δ I bits .
σ g 2 γ cs 2 σ f 2 N M .
L cs = r σ g = r γ cs σ f N M .
B cs = log 2 ( r C s γ I σ f Δ I ) = log 2 ( r γ I σ f Δ I ) + log 2 ( C s ) = B I + log 2 ( C s ) bits .
B I = log 2 D 2 T L 0 Δ I ,
B I = log 2 T D 2 r σ f Δ I
H = D 2 T e A ,
H = D 2 T e ( M 2 A + 1 2 ) .
γ cs = D 2 T e M 2 = D 2 T 2 M ,
qRMSE I I C s 2 M D 2 T · Δ I I 12 ,
Δ I I = N C s 2 M Δ I .
L cs = D 2 T L 0 N M ,
B cs = log 2 D 2 T L 0 N M Δ I I = log 2 2 C s D 2 T L o N Δ I bits .
B cs = log 2 r σ f D 2 T N 2 M Δ I I bits .
H = D 2 T e 1 M A ,
H = D 2 T e M ( M 2 A + 1 2 )
γ cs = D 2 T e 2 M .
σ g 2 = E [ ( γ CS j = 1 N a i , j f j + n ) ( γ CS l = 1 N a i , l f l + n ) ] = E ( γ CS 2 j l a i , j a i , l f j f l ) + E ( n ) E ( γ CS j a i , j f j ) + E ( n ) E ( γ CS l a i , l f l ) + E ( n 2 ) .
σ g 2 = γ CS 2 j l E [ a i , j a i , l ] E [ f j f l ] + Δ I I 2 12 .
σ g 2 = γ CS 2 j l σ a 2 δ [ j l ] σ f 2 + Δ I I 2 12 = γ CS N σ a 2 σ f 2 + Δ I I 2 12 .
σ g 2 = γ CS 2 σ f 2 N M + Δ I I 2 12 .

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