Abstract

In this paper, we present a signal processing approach to improve the resolution of a spectrometer with a fixed number of low-cost, non-ideal filters. We aim to show that the resolution can be improved beyond the limit set by the number of filters by exploiting the sparse nature of a signal spectrum. We consider an underdetermined system of linear equations as a model for signal spectrum estimation. We design a non-negativeL1norm minimization algorithm for solving the system of equations. We demonstrate that the resolution can be improved multiple times by using the proposed algorithm.

© 2012 OSA

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References

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  1. D. J. Brady, Optical Imaging and Spectroscopy, (John Wiley and Sons, 2009).
  2. S. W. Wang, X. Chen, W. Lu, L. Wang, Y. Wu, and Z. Wang, “Integrated optical filter arrays fabricated by using the combinatorial etching technique,” Opt. Lett. 31(3), 332–334 (2006).
    [CrossRef] [PubMed]
  3. S. W. Wang, C. Xia, X. Chen, W. Lu, M. Li, H. Wang, W. Zheng, and T. Zhang, “Concept of a high-resolution miniature spectrometer using an integrated filter array,” Opt. Lett. 32(6), 632–634 (2007).
    [CrossRef] [PubMed]
  4. C. C. Chang and H. N. Lee, “On the estimation of target spectrum for filter-array based spectrometer,” Opt. Express 16(2), 1056–1061 (2008).
    [CrossRef]
  5. 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]
  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. H. N. Lee, Introduction to Compressed Sensing (Lecture notes; Spring Semester, GIST, South Korea, 2011).
  8. H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
    [CrossRef]
  9. H. Chen and H. Tang, “Application of miniature spectrometer in liquid signature analysis technology,” Appl. Opt. 50(26), 5093–5098 (2011).
    [CrossRef] [PubMed]
  10. N. J. Chanover, D. A. Glenar, D. G. Voelz, X. Xiao, R. Tawalbeh, P. J. Boston, W. B. Brinckerhoff, P. R. Mahaffy, S. Getty, I. Ten Kate, and A. McAdam, “An AOTF-LDTOF spectrometer suite for in situ organic detection and characterization,” in Proceedings of IEEE Aerospace Conference (2011), pp. 1–13.
  11. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
    [CrossRef]
  12. D. L. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse over complete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).
  13. D. L. Donoho, “For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest near solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
    [CrossRef]
  14. D. L. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52, 1289–1306 (2006).
  15. R. Baraniuk, “Compressive sensing,” IEEE Signal Process. Mag. 24(4), 118–121 (2007).
    [CrossRef]
  16. 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,” Proc. SPIE 6065, 43–52 (2006).
  17. 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 Signal Process. Mag. 25(2), 83–91 (2008).
    [CrossRef]
  18. D. L. Dohono and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory 54, 4789–4812 (2008).
  19. J. A. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).
  20. S. Boyd and L. Vandenberghe, Convex Optimization, (Cambridge University Press, 2009).
  21. S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).
  22. A. Forsgren, P. E. Gill, and M. H. Wright, “Interior methods for nonlinear optimization,” SIAM Rev. 44(4), 525–597 (2002).
    [CrossRef]

2011 (3)

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]

H. Chen and H. Tang, “Application of miniature spectrometer in liquid signature analysis technology,” Appl. Opt. 50(26), 5093–5098 (2011).
[CrossRef] [PubMed]

2009 (1)

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

2008 (3)

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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

D. L. Dohono and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory 54, 4789–4812 (2008).

C. C. Chang and H. N. Lee, “On the estimation of target spectrum for filter-array based spectrometer,” Opt. Express 16(2), 1056–1061 (2008).
[CrossRef]

2007 (3)

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

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

S. W. Wang, C. Xia, X. Chen, W. Lu, M. Li, H. Wang, W. Zheng, and T. Zhang, “Concept of a high-resolution miniature spectrometer using an integrated filter array,” Opt. Lett. 32(6), 632–634 (2007).
[CrossRef] [PubMed]

2006 (6)

S. W. Wang, X. Chen, W. Lu, L. Wang, Y. Wu, and Z. Wang, “Integrated optical filter arrays fabricated by using the combinatorial etching technique,” Opt. Lett. 31(3), 332–334 (2006).
[CrossRef] [PubMed]

D. L. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse over complete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).

D. L. Donoho, “For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest near solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
[CrossRef]

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

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,” Proc. SPIE 6065, 43–52 (2006).

J. A. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).

2002 (1)

A. Forsgren, P. E. Gill, and M. H. Wright, “Interior methods for nonlinear optimization,” SIAM Rev. 44(4), 525–597 (2002).
[CrossRef]

2001 (1)

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

Baraniuk, R.

R. Baraniuk, “Compressive sensing,” IEEE Signal Process. 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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

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,” Proc. SPIE 6065, 43–52 (2006).

Baron, D.

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,” Proc. SPIE 6065, 43–52 (2006).

Boyd, S.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

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. Express 16(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, H.

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[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]

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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

Dohono, D. L.

D. L. Dohono and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory 54, 4789–4812 (2008).

Donoho, D. L.

D. L. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse over complete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).

D. L. Donoho, “For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest near solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
[CrossRef]

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

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

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,” Proc. SPIE 6065, 43–52 (2006).

Elad, M.

D. L. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse over complete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).

Forsgren, A.

A. Forsgren, P. E. Gill, and M. H. Wright, “Interior methods for nonlinear optimization,” SIAM Rev. 44(4), 525–597 (2002).
[CrossRef]

Gill, P. E.

A. Forsgren, P. E. Gill, and M. H. Wright, “Interior methods for nonlinear optimization,” SIAM Rev. 44(4), 525–597 (2002).
[CrossRef]

Gorinevsky, D.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

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,” Proc. SPIE 6065, 43–52 (2006).

Kim, S. J.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

Koh, K.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

Kurokawa, U.

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]

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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

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,” Proc. SPIE 6065, 43–52 (2006).

Lee, H. N.

Li, B. B.

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

Li, M.

Lin, H. J.

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

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.

Lustig, M.

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

Niu, X. Y.

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

Pan, X. X.

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

Sarvotham, S.

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,” Proc. SPIE 6065, 43–52 (2006).

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

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 Signal Process. 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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

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,” Proc. SPIE 6065, 43–52 (2006).

Tang, H.

Temlyakov, V.

D. L. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse over complete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).

Tropp, J. A.

J. A. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).

Tsaig, Y.

D. L. Dohono and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory 54, 4789–4812 (2008).

Wakin, M. B.

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,” Proc. SPIE 6065, 43–52 (2006).

Wang, H.

Wang, L.

Wang, S. W.

Wang, Z.

Wright, M. H.

A. Forsgren, P. E. Gill, and M. H. Wright, “Interior methods for nonlinear optimization,” SIAM Rev. 44(4), 525–597 (2002).
[CrossRef]

Wu, Y.

Xia, C.

Ying, Y. B.

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

Yu, H. Y.

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

Zhang, T.

Zheng, W.

Appl. Opt. (1)

Commun. Pure Appl. Math. (1)

D. L. Donoho, “For most large underdetermined systems of linear equations, the minimal l1-norm solution is also the sparsest near solution,” Commun. Pure Appl. Math. 59, 907–934 (2006).
[CrossRef]

Food Chem. (1)

H. Y. Yu, X. Y. Niu, H. J. Lin, Y. B. Ying, B. B. Li, and X. X. Pan, “A feasibility study on on-line determination of rice wine composition by Vis-NIR spectroscopy and least-squares support vector machines,” Food Chem. 113(1), 291–296 (2009).
[CrossRef]

IEEE J. Sel. Top. Signal Process. (1)

S. J. Kim, K. Koh, M. Lustig, S. Boyd, and D. Gorinevsky, “An interior-point method for large-scale l1-regularized least squares,” IEEE J. Sel. Top. Signal Process. 1, 606–617 (2007).

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 Signal Process. 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 Signal Process. Mag. 25(2), 83–91 (2008).
[CrossRef]

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

IEEE Trans. Inf. Theory (4)

D. L. Donoho, M. Elad, and V. Temlyakov, “Stable recovery of sparse over complete representations in the presence of noise,” IEEE Trans. Inf. Theory 52, 6–18 (2006).

D. L. Dohono and Y. Tsaig, “Fast solution of L1-norm minimization problems when the solution may be sparse,” IEEE Trans. Inf. Theory 54, 4789–4812 (2008).

J. A. Tropp, “Just relax: Convex programming methods for identifying sparse signals,” IEEE Trans. Inf. Theory 52, 1030–1051 (2006).

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

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 (1)

Opt. Lett. (2)

Proc. SPIE (1)

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,” Proc. SPIE 6065, 43–52 (2006).

SIAM Rev. (2)

A. Forsgren, P. E. Gill, and M. H. Wright, “Interior methods for nonlinear optimization,” SIAM Rev. 44(4), 525–597 (2002).
[CrossRef]

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM Rev. 43(1), 129–159 (2001).
[CrossRef]

Other (4)

S. Boyd and L. Vandenberghe, Convex Optimization, (Cambridge University Press, 2009).

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

H. N. Lee, Introduction to Compressed Sensing (Lecture notes; Spring Semester, GIST, South Korea, 2011).

N. J. Chanover, D. A. Glenar, D. G. Voelz, X. Xiao, R. Tawalbeh, P. J. Boston, W. B. Brinckerhoff, P. R. Mahaffy, S. Getty, I. Ten Kate, and A. McAdam, “An AOTF-LDTOF spectrometer suite for in situ organic detection and characterization,” in Proceedings of IEEE Aerospace Conference (2011), pp. 1–13.

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Figures (7)

Fig. 1
Fig. 1

Schematic diagram of a typical miniature spectrometer.

Fig. 2
Fig. 2

Typical transmission functions of filters in an array.

Fig. 3
Fig. 3

(a) Original signal spectrum. (b) Components of original signal spectrum.

Fig. 4
Fig. 4

(a) Output samples of the spectral detector. (b) Estimated signal spectrum.

Fig. 5
Fig. 5

Estimated signal spectrum in the sparse domain (a) using NNLM. (b) using L 2 technique.

Fig. 6
Fig. 6

Reconstruction of generally shaped signal spectrum by various algorithms.

Fig. 7
Fig. 7

Reconstruction of generally shaped signal spectrum in sparse domain.

Tables (1)

Tables Icon

Table 1 Pseudo code to solve Eq. (8) by Non-Negative Minimization Method

Equations (9)

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

y=Dx+w
[ D 1 ( λ 1 ) D 1 ( λ 2 ) D 1 ( λ N ) D 2 ( λ 1 ) D 2 ( λ 2 ) D 2 ( λ N ) D M ( λ 1 ) D M ( λ 2 ) D M ( λ N ) ]
μ N := min μ{1,2,,N1} μsubject toMSEδ
y=Dx+w=DΨs+w
s ^ = min s s 0 suchthat DΨs-y 2 ε
s ^ = min s s 1 suchthat DΨs-y 2 ε
min s s 1 + τ 2 yAs 2 2
min s 1 T ssubjectto As-y 2 2 ε,s0
min s Asy 2 2 +λ i=1 n s i subjectto s i 0,i=1,2,n

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