Abstract

We compare noise and classification metrics for three aperture codes in dispersive spectroscopy. In contrast with previous theory, we show that multiplex codes may be advantageous even in systems dominated by Poisson noise. Furthermore, ill-conditioned codes with a regularized estimation strategy are shown to perform competitively with well-conditioned codes.

© 2011 OSA

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References

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  1. M. T. E. Golay, “Multi-slit spectrometry,” J. Opt. Soc. Am. 39(6), 437–437 (1949). URL http://www.opticsinfobase.org/abstract.cfm?URI=josa-39-6-437 .
    [CrossRef] [PubMed]
  2. M. Harwit and N. J. A. Sloane, Hadamard transform optics (Academic Press, 1979).
  3. A. Barducci, D. Guzzi, C. Lastri, V. Nardino, P. Marcoionni, and I. Pippi, “Radiometric and signal-to-noise ratio properties of multiplex dispersive spectrometry,” Appl. Opt. 49(28), 5366–5373 (2010). URL http://ao.osa.org/abstract.cfm?URI=ao-49-28-5366 .
    [CrossRef] [PubMed]
  4. A. A. Wagadarikar, M. E. Gehm, and D. J. Brady, “Performance comparison of aperture codes for multimodal, multiplex spectroscopy,” Appl. Opt. 46(22), 4932–4942 (2007). URL http://ao.osa.org/abstract.cfm?URI=ao-46-22-4932 .
    [CrossRef] [PubMed]
  5. S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, “Hadamard spectroscopy with a two-dimensional detecting array,” Appl. Opt. 32(34), 7095–7105 (1993). URL http://ao.osa.org/abstract.cfm?URI=ao-32-34-7095 .
    [CrossRef] [PubMed]
  6. M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45(13), 2965–2974 (2006). URL http://ao.osa.org/abstract.cfm?URI=ao-45-13-2965 .
    [CrossRef] [PubMed]
  7. 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(21), 14,013–14,027 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013 .
    [CrossRef]
  8. A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008). URL http://ao.osa.org/abstract.cfm?URI=ao-47-10-B44 .
    [CrossRef] [PubMed]
  9. R. A. DeVerse, R. M. Hammaker, and W. G. Fateley, “Realization of the hadamard multiplex advantage using a programmable optical mask in a dispersive flat-field near-infrared spectrometer,” Appl. Spectrosc. 54(12), 1751–1758 (2000). URL http://as.osa.org/abstract.cfm?URI=as-54-12-1751 .
    [CrossRef]
  10. B. Noble, Applied linear algebra (Prentice Hall, 1977). ID: DUKE004063101; Formats: Book; 2d ed.; xvii, 477 p. ; 24 cm.; M2: OCLC Number: 02985355 Bibliography: p. 459–463.Includes index.
  11. S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33–61 (1998).
    [CrossRef]
  12. J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992 –3004 (2007).
    [CrossRef] [PubMed]
  13. Z. Harmany, R. Marcia, and R. Willett, “This is SPIRAL-TAP: sparse poisson intensity reconstruction algorithms - theory and practice,” ArXiv e-prints (2010). 1005.4274.
  14. D. J. Brady, Optical imaging and spectroscopy (Wiley-interscience, 2008).
  15. D. Donoho, “Compressed sensing,” IEEE Trans. Inf. Theory 52(4), 1289 –1306 (2006).
    [CrossRef]
  16. A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004). URL http://dx.doi.org/10.1023/B:JMIV.0000011325.36760.1e .
    [CrossRef]
  17. D. Kittle, K. Choi, A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded aperture snapshot spectral imagers,” Appl. Opt. 49(36), 6824–6833 (2010). URL http://ao.osa.org/abstract.cfm?URI=ao-49-36-6824 .
    [CrossRef] [PubMed]
  18. T. T. Cai and L. Wang, “Orthogonal matching pursuit for sparse signal recovery,” Technical Report (2010).
  19. S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
    [CrossRef]
  20. Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO,” IEEE Trans. Geosci. Remote Sens. 42(11), 2535–2543 (2004).
    [CrossRef]
  21. K. B. Petersen and M. S. Pedersen, “The matrix cookbook,” (2008). URL http://matrixcookbook.com/ .
  22. J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
    [CrossRef]

2010

2008

2007

J. Bioucas-Dias and M. 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. A. Wagadarikar, M. E. Gehm, and D. J. Brady, “Performance comparison of aperture codes for multimodal, multiplex spectroscopy,” Appl. Opt. 46(22), 4932–4942 (2007). URL http://ao.osa.org/abstract.cfm?URI=ao-46-22-4932 .
[CrossRef] [PubMed]

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(21), 14,013–14,027 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013 .
[CrossRef]

2006

S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
[CrossRef]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45(13), 2965–2974 (2006). URL http://ao.osa.org/abstract.cfm?URI=ao-45-13-2965 .
[CrossRef] [PubMed]

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

2004

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004). URL http://dx.doi.org/10.1023/B:JMIV.0000011325.36760.1e .
[CrossRef]

Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO,” IEEE Trans. Geosci. Remote Sens. 42(11), 2535–2543 (2004).
[CrossRef]

J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[CrossRef]

2000

1998

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33–61 (1998).
[CrossRef]

1993

1949

Barducci, A.

Bioucas-Dias, J.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992 –3004 (2007).
[CrossRef] [PubMed]

Brady, D.

Brady, D. J.

Cai, T. T.

T. T. Cai and L. Wang, “Orthogonal matching pursuit for sparse signal recovery,” Technical Report (2010).

Carin, L.

Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO,” IEEE Trans. Geosci. Remote Sens. 42(11), 2535–2543 (2004).
[CrossRef]

Chambolle, A.

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004). URL http://dx.doi.org/10.1023/B:JMIV.0000011325.36760.1e .
[CrossRef]

Chen, S. S.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33–61 (1998).
[CrossRef]

Choi, K.

Claflin, E. S.

DeVerse, R. A.

Donoho, D.

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

Donoho, D. L.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33–61 (1998).
[CrossRef]

Fateley, W. G.

Figueiredo, M.

J. Bioucas-Dias and M. Figueiredo, “A new TwIST: two-step iterative shrinkage/thresholding algorithms for image restoration,” IEEE Trans. Image Process. 16(12), 2992 –3004 (2007).
[CrossRef] [PubMed]

Gehm, M. E.

Golay, M. T. E.

Guzzi, D.

Hammaker, R. M.

Harmany, Z.

Z. Harmany, R. Marcia, and R. Willett, “This is SPIRAL-TAP: sparse poisson intensity reconstruction algorithms - theory and practice,” ArXiv e-prints (2010). 1005.4274.

Harwit, M.

M. Harwit and N. J. A. Sloane, Hadamard transform optics (Academic Press, 1979).

Hoi, S. C. H.

S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
[CrossRef]

Jin, R.

S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
[CrossRef]

John, R.

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008). URL http://ao.osa.org/abstract.cfm?URI=ao-47-10-B44 .
[CrossRef] [PubMed]

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(21), 14,013–14,027 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013 .
[CrossRef]

Kittle, D.

Lastri, C.

Liao, X.

Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO,” IEEE Trans. Geosci. Remote Sens. 42(11), 2535–2543 (2004).
[CrossRef]

Lyu, M. R.

S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
[CrossRef]

Marcia, R.

Z. Harmany, R. Marcia, and R. Willett, “This is SPIRAL-TAP: sparse poisson intensity reconstruction algorithms - theory and practice,” ArXiv e-prints (2010). 1005.4274.

Marcoionni, P.

McCain, S. T.

Mende, S. B.

Nardino, V.

Noble, B.

B. Noble, Applied linear algebra (Prentice Hall, 1977). ID: DUKE004063101; Formats: Book; 2d ed.; xvii, 477 p. ; 24 cm.; M2: OCLC Number: 02985355 Bibliography: p. 459–463.Includes index.

Pippi, I.

Pitsianis, N. P.

Potuluri, P.

Rairden, R. L.

Saunders, M. A.

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33–61 (1998).
[CrossRef]

Schulz, T. J.

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(21), 14,013–14,027 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013 .
[CrossRef]

Sloane, N. J. A.

M. Harwit and N. J. A. Sloane, Hadamard transform optics (Academic Press, 1979).

Sullivan, M. E.

Swenson, G. R.

Tropp, J. A.

J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[CrossRef]

Wagadarikar, A.

Wagadarikar, A. A.

Wang, L.

T. T. Cai and L. Wang, “Orthogonal matching pursuit for sparse signal recovery,” Technical Report (2010).

Willett, R.

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008). URL http://ao.osa.org/abstract.cfm?URI=ao-47-10-B44 .
[CrossRef] [PubMed]

Z. Harmany, R. Marcia, and R. Willett, “This is SPIRAL-TAP: sparse poisson intensity reconstruction algorithms - theory and practice,” ArXiv e-prints (2010). 1005.4274.

Willett, R. M.

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(21), 14,013–14,027 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013 .
[CrossRef]

Zhang, Y.

Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO,” IEEE Trans. Geosci. Remote Sens. 42(11), 2535–2543 (2004).
[CrossRef]

Zhu, J.

S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
[CrossRef]

Appl. Opt.

A. Barducci, D. Guzzi, C. Lastri, V. Nardino, P. Marcoionni, and I. Pippi, “Radiometric and signal-to-noise ratio properties of multiplex dispersive spectrometry,” Appl. Opt. 49(28), 5366–5373 (2010). URL http://ao.osa.org/abstract.cfm?URI=ao-49-28-5366 .
[CrossRef] [PubMed]

A. A. Wagadarikar, M. E. Gehm, and D. J. Brady, “Performance comparison of aperture codes for multimodal, multiplex spectroscopy,” Appl. Opt. 46(22), 4932–4942 (2007). URL http://ao.osa.org/abstract.cfm?URI=ao-46-22-4932 .
[CrossRef] [PubMed]

S. B. Mende, E. S. Claflin, R. L. Rairden, and G. R. Swenson, “Hadamard spectroscopy with a two-dimensional detecting array,” Appl. Opt. 32(34), 7095–7105 (1993). URL http://ao.osa.org/abstract.cfm?URI=ao-32-34-7095 .
[CrossRef] [PubMed]

M. E. Gehm, S. T. McCain, N. P. Pitsianis, D. J. Brady, P. Potuluri, and M. E. Sullivan, “Static two-dimensional aperture coding for multimodal, multiplex spectroscopy,” Appl. Opt. 45(13), 2965–2974 (2006). URL http://ao.osa.org/abstract.cfm?URI=ao-45-13-2965 .
[CrossRef] [PubMed]

A. Wagadarikar, R. John, R. Willett, and D. Brady, “Single disperser design for coded aperture snapshot spectral imaging,” Appl. Opt. 47(10), B44–B51 (2008). URL http://ao.osa.org/abstract.cfm?URI=ao-47-10-B44 .
[CrossRef] [PubMed]

D. Kittle, K. Choi, A. Wagadarikar, and D. J. Brady, “Multiframe image estimation for coded aperture snapshot spectral imagers,” Appl. Opt. 49(36), 6824–6833 (2010). URL http://ao.osa.org/abstract.cfm?URI=ao-49-36-6824 .
[CrossRef] [PubMed]

Appl. Spectrosc.

IEEE Trans. Geosci. Remote Sens.

Y. Zhang, X. Liao, and L. Carin, “Detection of buried targets via active selection of labeled data: application to sensing subsurface UXO,” IEEE Trans. Geosci. Remote Sens. 42(11), 2535–2543 (2004).
[CrossRef]

IEEE Trans. Image Process.

J. Bioucas-Dias and M. 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

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

J. A. Tropp, “Greed is good: algorithmic results for sparse approximation,” IEEE Trans. Inf. Theory 50, 2231–2242 (2004).
[CrossRef]

J. Math. Imaging Vision

A. Chambolle, “An algorithm for total variation minimization and applications,” J. Math. Imaging Vision 20, 89–97 (2004). URL http://dx.doi.org/10.1023/B:JMIV.0000011325.36760.1e .
[CrossRef]

J. Opt. Soc. Am.

Opt. Express

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(21), 14,013–14,027 (2007). URL http://www.opticsexpress.org/abstract.cfm?URI=oe-15-21-14013 .
[CrossRef]

Proceedings of the 23rd International Conference on Machine Learning (ICML

S. C. H. Hoi, R. Jin, J. Zhu, and M. R. Lyu, “Batch mode active learning and its application to medical image classification,” in Proceedings of the 23rd International Conference on Machine Learning (ICML (2006), pp. 417–424. URL http://doi.acm.org/10.1145/1143844.1143897 .
[CrossRef]

SIAM J. Sci. Comput. (USA)

S. S. Chen, D. L. Donoho, and M. A. Saunders, “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput. (USA) 20, 33–61 (1998).
[CrossRef]

Other

T. T. Cai and L. Wang, “Orthogonal matching pursuit for sparse signal recovery,” Technical Report (2010).

Z. Harmany, R. Marcia, and R. Willett, “This is SPIRAL-TAP: sparse poisson intensity reconstruction algorithms - theory and practice,” ArXiv e-prints (2010). 1005.4274.

D. J. Brady, Optical imaging and spectroscopy (Wiley-interscience, 2008).

M. Harwit and N. J. A. Sloane, Hadamard transform optics (Academic Press, 1979).

B. Noble, Applied linear algebra (Prentice Hall, 1977). ID: DUKE004063101; Formats: Book; 2d ed.; xvii, 477 p. ; 24 cm.; M2: OCLC Number: 02985355 Bibliography: p. 459–463.Includes index.

K. B. Petersen and M. S. Pedersen, “The matrix cookbook,” (2008). URL http://matrixcookbook.com/ .

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Figures (4)

Fig. 1
Fig. 1

SVD spectra of the three measurement systems to be compared. The slit code has a flat spectrum and is not multiplexed. The pseudo-random code has a decaying spectrum and is multiplexed. The cyclic-s code has a flat spectrum and is multiplexed.

Fig. 2
Fig. 2

A sample spectrum drawn from each of libraries demonstrating the canonical sparsity of the sythetic data set, and the smooth variations of the reflectance data set.

Fig. 3
Fig. 3

A sample reconstruction via OMP showing the inaccurate placement of the weakest peak by both the pseudo-random and cyclic-s matrix.

Fig. 4
Fig. 4

ROC for data reconstructed from signals of total photon level of 100 photons. The cyclic-s matrix generates the highest ROC curves, but requires more labels to do so with a 9 peak reconstruction (see Table 4). The 8 peak reconstruction improves the quality of all of the multiplex ROCs, and allows for classification with the same number of labels as the identity matrix. Random pt XX refers to what fraction of singular values are kept when truncated-SVD is performed. Random full refers to the performance of the pseudo-random matrix without truncated-SVD.

Tables (4)

Tables Icon

Table 1 Confirmation of the linear unbiased estimator results of [2] for the Poisson only noise model, and demonstration of the multiplexing advantage for the mixed noise model.

Tables Icon

Table 2 Convex optimization reconstruction data showing the multiplexing advantage when convex optimization is used. % singular values refers to the percentage of singular values kept after performing truncated-SVD on the data.

Tables Icon

Table 3 Means and variances of two distributions on which classification is to be performed.

Tables Icon

Table 4 Number of basis functions and total labels necessary to meet the stopping criteria for each system’s data.

Equations (19)

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

g = Hf + n
f ^ = arg min f ˇ g H f ˇ 2 2 + τ Ψ 1 f ˇ 1
L ( g | f ˇ ) = 1 T H f ˇ i = 1 N g i log ( e i T H f ˇ )
f ^ = arg min f ˇ g ˜ H ˜ f ˇ 2 2 + τ Ψ 1 f ˇ 1
g ˜ = W g
H ˜ = W H
H = U S V T
W = S n U n T .
M S E = ( 1 / N ) i = 1 N ( f i f ^ i ) 2
t i = ϕ T ( x i ) w + n
ϕ ( x ) = [ 1 K ( x , b 1 ) K ( x , b 2 ) K ( x , b n ) ] T
K ( x , b i ) = exp ( γ x b i 2 2 ) .
.
A i = 1 N ϕ ( x i ) ϕ ( x i ) T
= Φ Φ T .
b = arg max b ˇ { det [ q b ˇ q b ˇ T q b ˇ Φ T ( Φ Φ T ) 1 Φ q b ˇ T ] }
q b = [ K ( x 1 , b ) , K ( x N , b ) ]
A L N L Φ L Φ L T + λ I
x L N L + 1 = arg max x { 1 + ϕ ( x ) T ( A L N L ) 1 ϕ ( x ) } ,

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