Abstract

In this paper, we introduce a method for improving the resolution of miniature spectrometers. Our method is based on using filters with random transmittance. Such filters sense fine details of an input signal spectrum, which, when combined with a signal processing algorithm, aid in improving resolution. We also propose an approach for designing filters with random transmittance using optical thin-film technology. We demonstrate that the improvement in resolution is 7-fold when using the filters with random transmittance over what was achieved in our previous work.

© 2013 OSA

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References

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    [CrossRef] [PubMed]
  6. C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
    [CrossRef]
  7. C. C. Chang and H. N. Lee, “On the estimation of target spectrum for filter-array based spectrometer,” Opt. Express16(2), 1056–1061 (2008).
    [CrossRef]
  8. U. Kurokawa, B. I. Choi, and C.-C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sens. J.11(7), 1556–1563 (2011).
    [CrossRef]
  9. C. Bendjaballah, “Information rates in optical channels,” Opt. Commun.17(1), 55–58 (1976).
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    [CrossRef]
  14. E. Candes and J. Romberg, “11-magic: Recovery of sparse signals via convex programming,” Technical report (2005). http://users.ece.gatech.edu/~justin/l1magic/
  15. S. Park and H. N. Lee, “Designing an algorithm to solve basis pursuit denoising with a nonnegative constraint,” IEEE Sig. Proc. Letters. (submitted to).
  16. R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc., B58, 267–288 (1996).
  17. A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Ima. Sciences2(1), 183–202 (2009).
    [CrossRef]
  18. A. Juditsky and A. Nemirovski, “First Order Methods for Nonsmooth Convex Large-Scale Optimization, I: General Purpose Methods,” in Optimization for Machine Learning, S. Sra, S. Nowozin, and S.J. Write, eds. (MIT Press, 2011), pp. 1–28.
  19. S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
    [CrossRef]
  20. M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
    [CrossRef]
  21. R. Dikpal, A. Veeraraghavan, and R. Chellappa, “P2C2: Programmable pixel compressive camera for high speed imaging,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2011), pp. 329–336.
  22. C. Li, T. Sun, K. Kelly, and Y. Zhang, “A compressive sensing and unmixing scheme for hyperspectral data processing,” Technical report. ( http://www.caam.rice.edu/~zhang/reports/tr1101.pdf ).
  23. A. Rajwade, D. Kittle, T.-H. Tsai, D. Brady, and L. Carin, “Coded hyperspectral imaging and blind compressive sensing,” submitted (2012).
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  27. Z. B. Haim, Y. C. Eldar, and M. Elad, “Coherence-based performance guarantees for estimating a sparse vector under random noise,” IEEE Trans Sig. Proc.58, 5030–5043 (2010).
  28. C. Z. Microscopy, “Fundamentals of mercury arc lamps,” http://zeiss-campus.magnet.fsu.edu/articles/lightsources/mercuryarc.html .

2012 (1)

2011 (2)

C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
[CrossRef]

U. Kurokawa, B. I. Choi, and C.-C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sens. J.11(7), 1556–1563 (2011).
[CrossRef]

2010 (2)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

Z. B. Haim, Y. C. Eldar, and M. Elad, “Coherence-based performance guarantees for estimating a sparse vector under random noise,” IEEE Trans Sig. Proc.58, 5030–5043 (2010).

2009 (2)

2008 (2)

C. C. Chang and H. N. Lee, “On the estimation of target spectrum for filter-array based spectrometer,” Opt. Express16(2), 1056–1061 (2008).
[CrossRef]

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

2007 (2)

2006 (1)

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

1997 (1)

1996 (1)

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc., B58, 267–288 (1996).

1995 (1)

1987 (1)

Y. Aizu, K. Ogino, and T. Asakura, “A laser velocimeter using a random pattern,” Opt. Commun.64(3), 205–210 (1987).
[CrossRef]

1976 (1)

C. Bendjaballah, “Information rates in optical channels,” Opt. Commun.17(1), 55–58 (1976).
[CrossRef]

Aizu, Y.

Y. Aizu, K. Ogino, and T. Asakura, “A laser velocimeter using a random pattern,” Opt. Commun.64(3), 205–210 (1987).
[CrossRef]

Asakura, T.

Y. Aizu, K. Ogino, and T. Asakura, “A laser velocimeter using a random pattern,” Opt. Commun.64(3), 205–210 (1987).
[CrossRef]

Baraniuk, R.

R. Baraniuk, “Compressive sensing,” IEEE Sig. Proc. Mag.24(4), 118–121 (2007).
[CrossRef]

Baraniuk, R. G.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Barry, J. R.

Beck, A.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Ima. Sciences2(1), 183–202 (2009).
[CrossRef]

Bendjaballah, C.

C. Bendjaballah, “Information rates in optical channels,” Opt. Commun.17(1), 55–58 (1976).
[CrossRef]

Boyd, S.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

Chang, C. C.

C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
[CrossRef]

C. C. Chang and H. N. Lee, “On the estimation of target spectrum for filter-array based spectrometer,” Opt. Express16(2), 1056–1061 (2008).
[CrossRef]

Chang, C.-C.

U. Kurokawa, B. I. Choi, and C.-C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sens. J.11(7), 1556–1563 (2011).
[CrossRef]

Chen, X.

Choi, B. I.

U. Kurokawa, B. I. Choi, and C.-C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sens. J.11(7), 1556–1563 (2011).
[CrossRef]

Choi, B. I. I.

C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
[CrossRef]

Chu, E.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

Davenport, M. A.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Donoho, D. L.

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

Duarte, M. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Eckstein, J.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

Elad, M.

Z. B. Haim, Y. C. Eldar, and M. Elad, “Coherence-based performance guarantees for estimating a sparse vector under random noise,” IEEE Trans Sig. Proc.58, 5030–5043 (2010).

Eldar, Y. C.

Z. B. Haim, Y. C. Eldar, and M. Elad, “Coherence-based performance guarantees for estimating a sparse vector under random noise,” IEEE Trans Sig. Proc.58, 5030–5043 (2010).

Haim, Z. B.

Z. B. Haim, Y. C. Eldar, and M. Elad, “Coherence-based performance guarantees for estimating a sparse vector under random noise,” IEEE Trans Sig. Proc.58, 5030–5043 (2010).

Kahn, J. M.

Kelly, K. F.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Kurokawa, U.

U. Kurokawa, B. I. Choi, and C.-C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sens. J.11(7), 1556–1563 (2011).
[CrossRef]

C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
[CrossRef]

Laska, J. N.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Lee, H. N.

Lee, W. B.

Li, M.

Lin, N. T.

C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
[CrossRef]

Lu, W.

Madanipour, K.

Ogino, K.

Y. Aizu, K. Ogino, and T. Asakura, “A laser velocimeter using a random pattern,” Opt. Commun.64(3), 205–210 (1987).
[CrossRef]

Ojeda-Castañeda, J.

Oliver, J.

Parikh, N.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

Park, S.

S. Park and H. N. Lee, “Designing an algorithm to solve basis pursuit denoising with a nonnegative constraint,” IEEE Sig. Proc. Letters. (submitted to).

Park, S. J.

Peleato, B.

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

Sauceda, A.

Sun, T.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Takhar, D.

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

Tavassoly, M. T.

Teboulle, M.

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Ima. Sciences2(1), 183–202 (2009).
[CrossRef]

Tibshirani, R.

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc., B58, 267–288 (1996).

Wang, H.

Wang, S. W.

Xia, C.

Zhang, T.

Zheng, W.

Appl. Opt. (2)

Founda. and Tren. Mach. Learn. (1)

S. Boyd, N. Parikh, E. Chu, B. Peleato, and J. Eckstein, “Distributed optimization and statistical learning via the alternating direction method of multipliers,” Founda. and Tren. Mach. Learn.3(1), 1–122 (2010).
[CrossRef]

IEEE Sens. J. (1)

U. Kurokawa, B. I. Choi, and C.-C. Chang, “Filter-based miniature spectrometers: spectrum reconstruction using adaptive regularization,” IEEE Sens. J.11(7), 1556–1563 (2011).
[CrossRef]

IEEE Sig. Proc. Letters. (1)

S. Park and H. N. Lee, “Designing an algorithm to solve basis pursuit denoising with a nonnegative constraint,” IEEE Sig. Proc. Letters. (submitted to).

IEEE Sig. Proc. Mag. (2)

M. F. Duarte, M. A. Davenport, D. Takhar, J. N. Laska, T. Sun, K. F. Kelly, and R. G. Baraniuk, “Single-pixel imaging via compressive sampling,” IEEE Sig. Proc. Mag.25(2), 83–91 (2008).
[CrossRef]

R. Baraniuk, “Compressive sensing,” IEEE Sig. Proc. Mag.24(4), 118–121 (2007).
[CrossRef]

IEEE Trans Sig. Proc. (1)

Z. B. Haim, Y. C. Eldar, and M. Elad, “Coherence-based performance guarantees for estimating a sparse vector under random noise,” IEEE Trans Sig. Proc.58, 5030–5043 (2010).

IEEE Trans. Inf. Theory (1)

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

J. R. Stat. Soc., B (1)

R. Tibshirani, “Regression shrinkage and selection via the lasso,” J. R. Stat. Soc., B58, 267–288 (1996).

Opt. Commun. (2)

C. Bendjaballah, “Information rates in optical channels,” Opt. Commun.17(1), 55–58 (1976).
[CrossRef]

Y. Aizu, K. Ogino, and T. Asakura, “A laser velocimeter using a random pattern,” Opt. Commun.64(3), 205–210 (1987).
[CrossRef]

Opt. Eng. (1)

C. C. Chang, N. T. Lin, U. Kurokawa, and B. I. I. Choi, “Spectrum reconstruction for filter-array spectrum sensor from sparse template selection,” Opt. Eng.50(11), 114402 (2011).
[CrossRef]

Opt. Express (2)

Opt. Lett. (2)

SIAM J. Ima. Sciences (1)

A. Beck and M. Teboulle, “A fast iterative shrinkage-thresholding algorithm for linear inverse problems,” SIAM J. Ima. Sciences2(1), 183–202 (2009).
[CrossRef]

Other (10)

A. Juditsky and A. Nemirovski, “First Order Methods for Nonsmooth Convex Large-Scale Optimization, I: General Purpose Methods,” in Optimization for Machine Learning, S. Sra, S. Nowozin, and S.J. Write, eds. (MIT Press, 2011), pp. 1–28.

D. J. Brady, Optical Imaging and Spectroscopy (John and Wiley Sons, 2009).

W. L. Wolfe, Introduction to Imaging Spectrometers (SPIE, 1997).

H. N. Lee, Introduction to Compressed Sensing (Lecture notes; Spring Semester, GIST, Korea, 2011). http://infonet.gist.ac.kr/?page_id=843

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics Publishing, 2002).

E. Candes and J. Romberg, “11-magic: Recovery of sparse signals via convex programming,” Technical report (2005). http://users.ece.gatech.edu/~justin/l1magic/

R. Dikpal, A. Veeraraghavan, and R. Chellappa, “P2C2: Programmable pixel compressive camera for high speed imaging,” in Proceedings of IEEE Conference on Computer Vision and Pattern Recognition (IEEE, 2011), pp. 329–336.

C. Li, T. Sun, K. Kelly, and Y. Zhang, “A compressive sensing and unmixing scheme for hyperspectral data processing,” Technical report. ( http://www.caam.rice.edu/~zhang/reports/tr1101.pdf ).

A. Rajwade, D. Kittle, T.-H. Tsai, D. Brady, and L. Carin, “Coded hyperspectral imaging and blind compressive sensing,” submitted (2012).

C. Z. Microscopy, “Fundamentals of mercury arc lamps,” http://zeiss-campus.magnet.fsu.edu/articles/lightsources/mercuryarc.html .

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Figures (8)

Fig. 1
Fig. 1

Schematic of the proposed filter-array-based spectrometer.

Fig. 2
Fig. 2

Transmittances of (a) ideal filters (b) non-ideal filters.

Fig. 3
Fig. 3

Summary of analog and digital design first approaches.

Fig. 4
Fig. 4

(a) Random transmittances produced by the thin-film method. (b) ACF of Filter-5. (c) CCF between Filter-5 and Filter-20.

Fig. 5
Fig. 5

g.MSE against resolution (N) for K = 5 and 50,000 support sets with ρ=0.95 .

Fig. 6
Fig. 6

Reconstruction of a mercury arc signal spectrum by a non-ideal TF matrix in [5].

Fig. 7
Fig. 7

Reconstruction of a mercury arc signal spectrum by the proposed random TF matrix.

Fig. 8
Fig. 8

(a) Original sparse spectrum of the mercury lamp. (b) Estimated sparse spectrum by using TFs in [5]. (c) Estimated sparse spectrum by thin-film-based random TFs.

Tables (3)

Tables Icon

Table 1 Measurement Matrices and Their Implementation in Various Imaging Applications

Tables Icon

Table 2 Recursive Equations for Calculating Power Parameters

Tables Icon

Table 3 Summary of Filter Design Approaches and Corresponding Resolution

Equations (10)

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

y i = x( λ ) T i ( λ ) dλ+ w i ,i=1,2,,M,
g i = x( λ ) h( λiΔ )dλ,i=1,2,,M,
y=Dx+wwherex0.
y= DG A s+w=As+wwheres0.
s ^ = min s s 1 subject to DGs-y 2 ε,s0
μ= max ij | a i , a j |i,j=1,,N
δ( Δλ ) [ T i ( λ ) m i ] [ T i ( λ+Δλ ) m i ]dλ,i=1,2,,M,
T( λ,θ )=1 1 2 ( | ρ TE | 2 + | ρ TM | 2 )
g.MSE(A, σ 2 ,K)E[ (x x ) 2 2 ]=E[ e x T e x ]=tr( C x )=tr( σ 2 G k ( A k T A k ) 1 G k T ).
N max :=max{ N{M,M+1,}:Pr{g.MSEδ}ρ } Δ λ min := W λ N max

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