Abstract

This paper describes a generalized theoretical framework for a multiplexed spatially encoded imaging system to acquire multi-channel data. The framework is confirmed with simulations and experimental demonstrations. In the system, each channel associated with the object is spatially encoded, and the resultant signals are multiplexed onto a detector array. In the demultiplexing process, a numerical estimation algorithm with a sparsity constraint is used to solve the underdetermined reconstruction problem. The system can acquire object data in which the number of elements is larger than that of the captured data. This case includes multi-channel data acquisition by a single-shot with a detector array. In the experiments, wide field-of-view imaging and spectral imaging were demonstrated with sparse objects. A compressive sensing algorithm, called the two-step iterative shrinkage/thresholding algorithm with total variation, was adapted for object reconstruction.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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2010 (1)

2009 (1)

2008 (3)

2007 (3)

R. Baraniuk, "Compressive sensing," IEEE Signal Process. Mag. 24, 118-121 (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]

M. E. Gehm, R. John, D. J. Brady, R. M. Willett, and T. J. Schulz, "Single-shot compressive spectral imaging with a dual-disperser architecture," Opt. Express 15, 14013-14027 (2007).
[CrossRef] [PubMed]

2006 (2)

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

M. Levoy, "Light fields and computational imaging," IEEE Computer 39, 46-55 (2006).

2005 (1)

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

2003 (1)

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

1997 (1)

A. Rajagopalan, and S. Chaudhuri, "A variational approach to recovering depth from defocused images," IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158-1164 (1997).
[CrossRef]

1995 (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)

Baraniuk, R.

R. Baraniuk, "Compressive sensing," IEEE Signal Process. Mag. 24, 118-121 (2007).
[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.

Candes, E. J.

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

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

Cathey, W. T.

Chaudhuri, S.

A. Rajagopalan, and S. Chaudhuri, "A variational approach to recovering depth from defocused images," IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158-1164 (1997).
[CrossRef]

Choi, K.

Donoho, D. L.

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

Dowski, E. R.

Eldeniz, C.

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]

Gehm, M. E.

Gribonval, R.

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

Hahn, J.

Horisaki, R.

John, R.

Kim, C.

Kim, J.

Levoy, M.

M. Levoy, "Light fields and computational imaging," IEEE Computer 39, 46-55 (2006).

Lim, S.

Lucy, L. B.

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

Marcia, R. F.

Marks, D. L.

Nielsen, M.

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

Rajagopalan, A.

A. Rajagopalan, and S. Chaudhuri, "A variational approach to recovering depth from defocused images," IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158-1164 (1997).
[CrossRef]

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]

Schulz, T. J.

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]

Wagadarikar, A.

Wakin, M. B.

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

Willett, R.

Willett, R. M.

Appl. Opt. (2)

Astron. J. (1)

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

IEEE Computer (1)

M. Levoy, "Light fields and computational imaging," IEEE Computer 39, 46-55 (2006).

IEEE Signal Process. Mag. (2)

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

R. Baraniuk, "Compressive sensing," IEEE Signal Process. Mag. 24, 118-121 (2007).
[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)

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]

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

IEEE Trans. Pattern Anal. Mach. Intell. (1)

A. Rajagopalan, and S. Chaudhuri, "A variational approach to recovering depth from defocused images," IEEE Trans. Pattern Anal. Mach. Intell. 19, 1158-1164 (1997).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Express (4)

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]

Other (9)

V. Treeaporn, A. Ashok, and M. A. Neifeld, "Increased field of view through optical multiplexing," in "Imaging Systems," (Optical Society of America, 2010), p. IMC4.

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

S. Hiura, and T. Matsuyama, "Depth measurement by the multi-focus camera," in "CVPR ’98: Proceedings of the IEEE Computer Society Conference on Computer Vision and Pattern Recognition," (IEEE Computer Society, Washington, DC, USA, 1998), p. 953.

A. Levin, R. Fergus, F. Durand, and W. T. Freeman, "Image and depth from a conventional camera with a coded aperture," in "SIGGRAPH ’07: ACM SIGGRAPH 2007 papers," (ACM, New York, NY, USA, 2007), p. 70.
[CrossRef]

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.

P. Ye, J. L. Paredes, G. R. Arce, Y. Wu, C. Chen, and D. W. Prather, "Compressive confocal microscopy," in "ICASSP ’09: Proceedings of the 2009 IEEE International Conference on Acoustics, Speech and Signal Processing," (IEEE Computer Society, Washington, DC, USA, 2009), pp. 429-432.

A. Kak, and M. Slaney, Principles of Computerized Tomographic Imaging (IEEE Press, New York, 1988).

M. D. Stenner, P. Shankar, and M. A. Neifeld, "Wide-field feature-specific imaging," in "Frontiers in Optics," (Optical Society of America, 2007), p. FMJ2.

R. F. Marcia, C. Kim, J. Kim, D. J. Brady, and R. M. Willett, "Fast disambiguation of superimposed images for increased field of view," in "Proceedings of the IEEE International Conference on Image Processing," (San Diego, CA, 2008), pp. 2620-2623.

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Figures (13)

Fig. 1
Fig. 1

System model of multi-channel data acquisition using multiplexed imaging with spatial encoding.

Fig. 2
Fig. 2

Simulations with an object with a smaller sparsity s. (a) The object. The captured data and the reconstructed results in (b), (c) system A, (d), (e) system B, and (f), (g) system C. (h) The result reconstructed from Fig. (b) by the Richardson-Lucy method.

Fig. 3
Fig. 3

Simulations with an object with larger sparsity s. (a) The object, (b) the captured data, and (c) the reconstructed result in system A.

Fig. 4
Fig. 4

Plots of reconstruction PSNR from noisy measurements in the proposed system with selected parameters.

Fig. 5
Fig. 5

Experimental setup of wide field-of-view imaging with phase grating.

Fig. 6
Fig. 6

Impulse response from sub-field 1. The numbers indicate the index l of the copies of a printed dot.

Fig. 7
Fig. 7

Experimental results of wide field-of-view imaging with phase grating. (a) Captured data and (b) the reconstructed result.

Fig. 8
Fig. 8

Experimental setup of wide field-of-view imaging with multiple overlapping sub-fields. Dashed lines show fields-of-view.

Fig. 9
Fig. 9

Impulse response of wide field-of-view imaging with multiple overlapping sub-fields. The numbers indicate the index l of the copies of a printed dot.

Fig. 10
Fig. 10

Experimental results of wide field-of-view imaging with multiple overlapping sub-fields. (a) Captured data in the center sub-field, (b) captured data with all sub-fields, and (c) the reconstructed result.

Fig. 11
Fig. 11

Schematic diagrams of spectral imaging demonstration. The solid lines and the dashed lines show diffracted rays of the 0-th and the 1-st orders in (a) red and (b) green lights, respectively.

Fig. 12
Fig. 12

Impulse responses of (a) the red and (b) the green inks. The numbers indicate the index l of the copies of a printed dot.

Fig. 13
Fig. 13

Experimental results of spectral imaging. (a) Original data, (b) the captured data, and (c) the reconstructed result.

Tables (4)

Tables Icon

Table 1 Parameters of (a) system A, (b) system B, and (c) system C in the simulations

Tables Icon

Table 2 The parameters of the impulse responses for wide field-of-view imaging

Tables Icon

Table 3 The parameters of the impulse responses for wide field-of-view imaging with multiple overlapping sub-fields

Tables Icon

Table 4 The parameters of the impulse responses for spectral imaging

Equations (10)

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

g = Φ f = Φ Ψ β = Θ β ,
( 1 ɛ ) β Λ 2 Θ Λ β Λ 2 ( 1 + ɛ ) β Λ 2 ,
β ^ = argmin β β 1 subject to g = Θ β ,
𝒢 ( x ) = c = 0 N c 1 l = 0 ( c ) 1 𝒲 ( l , c ) ( x 𝒮 ( l , c ) , c ) ,
A l , c ( p , q ) = { 𝒲 ( l , c ) ( p = q + 𝒮 ( l , c ) ) , 0 ( p q + 𝒮 ( l , c ) ) ,
E c = l = 0 ( c ) 1 A l , c .
T = [ E 0 O O O E 1 O O O E N c 1 ] ,
M = [ I I I ] ,
g = Φ f = MTf = [ E 0 E 1 E N c 1 ] f ,
Σ c Σ x Σ y ( ( x + 1 , y , c ) ( x , y , c ) ) 2 + ( ( x , y + 1 , c ) ( x , y , c ) ) 2 .

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