Abstract

The spectra recorded by a dispersion spectrophotometer are usually distorted by the response function of the instrument. To improve the resolving power, double or triple cascade spectrophotometers with narrow slits have been employed, but the total flux of the radiation available decreases accordingly, resulting in a low signal-to-noise ratio and a longer measuring time. The actual spectra can be restored approximately by mathematically removing the effects of the measuring instruments. Based on the Shalvi–Weinstein criterion, a (6, 2)-order normalized cumulant-based blind deconvolution algorithm for Raman spectral data is proposed. The actual spectral data and the unit-impulse response of the measuring instruments can be estimated simultaneously. By conducting experiments on real Raman spectra of some organic compounds, it is shown that this algorithm has a robust performance and fast convergence behavior and can improve the resolving power and correct the relative intensity distortion considerably.

© 2005 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef]
  4. P. A. Jansson, Deconvolution with Application in Spectroscopy (Academic, 1981).
  5. O. Shalvi, E. Weinstein, “Universal methods for blind deconvolution,” in Blind Deconvolution, S. Haykin, ed. (Prentice-Hall, 1994), pp. 122–168.
  6. C. Feng, C. Chi, “Performance of cumulant-based inverse filters for blind deconvolution,” IEEE Trans. Signal Process. 47, 1922–1935 (1999).
    [CrossRef]
  7. M. Gu, L. Tong, “Domains of attraction of Shalvi–Weinstein receivers,” IEEE Trans. Signal Process. 49, 1397–1408 (2001).
    [CrossRef]
  8. P. Schniter, L. Tong, “Existence and performance of Shalvi–Weinstein estimators,” IEEE Trans. Signal Process. 49, 2031–2041 (2001).
    [CrossRef]
  9. A. Mansour, A. K. Barros, N. Ohnishi, “Comparison among three estimators for high order statistics,” presented at the Fifth International Conference on Neural Information Processing, Kitakyushu, Japan, 21–23 October 1998.
  10. J. A. Cadzow, “Blind deconvolution via cumulant extrema,” IEEE Signal Process. Mag. 13, 24–42 (1996).
    [CrossRef]
  11. X. Zhang, “High-order statistical analysis,” in Modern Signal Processing, 2nd ed., X. Zhang, ed. (Tsinghua University Press, 2003), pp. 263–348.

2001 (2)

M. Gu, L. Tong, “Domains of attraction of Shalvi–Weinstein receivers,” IEEE Trans. Signal Process. 49, 1397–1408 (2001).
[CrossRef]

P. Schniter, L. Tong, “Existence and performance of Shalvi–Weinstein estimators,” IEEE Trans. Signal Process. 49, 2031–2041 (2001).
[CrossRef]

1999 (1)

C. Feng, C. Chi, “Performance of cumulant-based inverse filters for blind deconvolution,” IEEE Trans. Signal Process. 47, 1922–1935 (1999).
[CrossRef]

1996 (1)

J. A. Cadzow, “Blind deconvolution via cumulant extrema,” IEEE Signal Process. Mag. 13, 24–42 (1996).
[CrossRef]

1984 (1)

1967 (1)

Barros, A. K.

A. Mansour, A. K. Barros, N. Ohnishi, “Comparison among three estimators for high order statistics,” presented at the Fifth International Conference on Neural Information Processing, Kitakyushu, Japan, 21–23 October 1998.

Cadzow, J. A.

J. A. Cadzow, “Blind deconvolution via cumulant extrema,” IEEE Signal Process. Mag. 13, 24–42 (1996).
[CrossRef]

Chi, C.

C. Feng, C. Chi, “Performance of cumulant-based inverse filters for blind deconvolution,” IEEE Trans. Signal Process. 47, 1922–1935 (1999).
[CrossRef]

Feng, C.

C. Feng, C. Chi, “Performance of cumulant-based inverse filters for blind deconvolution,” IEEE Trans. Signal Process. 47, 1922–1935 (1999).
[CrossRef]

Gu, M.

M. Gu, L. Tong, “Domains of attraction of Shalvi–Weinstein receivers,” IEEE Trans. Signal Process. 49, 1397–1408 (2001).
[CrossRef]

Helstrom, C. W.

Jansson, P. A.

P. A. Jansson, Deconvolution with Application in Spectroscopy (Academic, 1981).

Kawata, S.

Mansour, A.

A. Mansour, A. K. Barros, N. Ohnishi, “Comparison among three estimators for high order statistics,” presented at the Fifth International Conference on Neural Information Processing, Kitakyushu, Japan, 21–23 October 1998.

Minami, K.

Minami, S.

Ohnishi, N.

A. Mansour, A. K. Barros, N. Ohnishi, “Comparison among three estimators for high order statistics,” presented at the Fifth International Conference on Neural Information Processing, Kitakyushu, Japan, 21–23 October 1998.

Schniter, P.

P. Schniter, L. Tong, “Existence and performance of Shalvi–Weinstein estimators,” IEEE Trans. Signal Process. 49, 2031–2041 (2001).
[CrossRef]

Senga, Y.

Shalvi, O.

O. Shalvi, E. Weinstein, “Universal methods for blind deconvolution,” in Blind Deconvolution, S. Haykin, ed. (Prentice-Hall, 1994), pp. 122–168.

Tong, L.

M. Gu, L. Tong, “Domains of attraction of Shalvi–Weinstein receivers,” IEEE Trans. Signal Process. 49, 1397–1408 (2001).
[CrossRef]

P. Schniter, L. Tong, “Existence and performance of Shalvi–Weinstein estimators,” IEEE Trans. Signal Process. 49, 2031–2041 (2001).
[CrossRef]

Weinstein, E.

O. Shalvi, E. Weinstein, “Universal methods for blind deconvolution,” in Blind Deconvolution, S. Haykin, ed. (Prentice-Hall, 1994), pp. 122–168.

Zhang, X.

X. Zhang, “High-order statistical analysis,” in Modern Signal Processing, 2nd ed., X. Zhang, ed. (Tsinghua University Press, 2003), pp. 263–348.

Zheng, S. X.

S. X. Zheng, Laser Raman Spectroscopy (Shanghai Scientific & Technical Publishers, 1985).

Appl. Opt. (1)

IEEE Signal Process. Mag. (1)

J. A. Cadzow, “Blind deconvolution via cumulant extrema,” IEEE Signal Process. Mag. 13, 24–42 (1996).
[CrossRef]

IEEE Trans. Signal Process. (3)

C. Feng, C. Chi, “Performance of cumulant-based inverse filters for blind deconvolution,” IEEE Trans. Signal Process. 47, 1922–1935 (1999).
[CrossRef]

M. Gu, L. Tong, “Domains of attraction of Shalvi–Weinstein receivers,” IEEE Trans. Signal Process. 49, 1397–1408 (2001).
[CrossRef]

P. Schniter, L. Tong, “Existence and performance of Shalvi–Weinstein estimators,” IEEE Trans. Signal Process. 49, 2031–2041 (2001).
[CrossRef]

J. Opt. Soc. Am. (1)

Other (5)

A. Mansour, A. K. Barros, N. Ohnishi, “Comparison among three estimators for high order statistics,” presented at the Fifth International Conference on Neural Information Processing, Kitakyushu, Japan, 21–23 October 1998.

X. Zhang, “High-order statistical analysis,” in Modern Signal Processing, 2nd ed., X. Zhang, ed. (Tsinghua University Press, 2003), pp. 263–348.

S. X. Zheng, Laser Raman Spectroscopy (Shanghai Scientific & Technical Publishers, 1985).

P. A. Jansson, Deconvolution with Application in Spectroscopy (Academic, 1981).

O. Shalvi, E. Weinstein, “Universal methods for blind deconvolution,” in Blind Deconvolution, S. Haykin, ed. (Prentice-Hall, 1994), pp. 122–168.

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Figures (3)

Fig. 1
Fig. 1

(a) 846 length Raman spectral data of (D+)-Glucopyranose from 155 to 1000 cm−1. (b) Blind deconvoluted data of (a) for Q = 5, ρ = 0.9. (c) Estimated unit-impulse response for Q = 5. (d) Estimated unit-impulse response for Q = 6. (e) Convergence curve for Q = 5.

Fig. 2
Fig. 2

(a) 1159 length Raman spectral data of Maltotriose from 534 to 1692 cm−1. (b) Blind deconvoluted spectral data for Q = 35, ρ = 0.9. (c) Estimated unit-impulse response for Q = 35. (d) Estimated unit-impulse response for Q = 30.

Fig. 3
Fig. 3

(a) 1491 length Raman spectral data of D-Glucuronic from 209 to 1699 cm−1. (b) Blind deconvoluted spectral data for Q = 9, ρ = 0.9.

Tables (2)

Tables Icon

Table 1 Ratios in Energy, Average Energy, and Intensity of Peak 397 to that of 406 cm−1 and of 424 to that of 406 cm −1 for Measured Data and Restored Data, Respectively

Tables Icon

Table 2 Calculating Time of Ten Trials

Equations (21)

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

x ( n ) = k f k w ( n k ) , n = 1 , 2 , , N ,
y ( n ) = k K g k x ( n k ) ,
y ( n ) = a w ( n m ) ,
h n = k f k g n k = a δ ( n m ) ,
δ ( n ) = { 1 , n = 0 , 0 , otherwise .
max { | c y ( p ) | } , subject to c y ( 2 ) = 1 ,
max { | c y ( p ) | [ c y ( 2 ) ] p / 2 } ,
x = [ x ( 1 ) , x ( 2 ) , , x ( N ) ] T ,
g = ( g 0 , g 1 , , g Q ) T ,
y ( n ) = t = 0 Q g t x ( n t ) , 1 n N .
μ y ( k ) = 1 N Q n = Q + 1 N [ y ( n ) m y ] k ,
m y = 1 N Q n = Q + 1 N y ( n ) .
g ( m + 1 ) = g ( m ) + α δ ,
| ĉ y ( m + 1 ) ( 6 , 2 ) | > | ĉ y ( m ) ( 6 , 2 ) |
δ ( m ) = g m | c y ( m ) ( 6 , 2 ) | = sgn [ c y ( m ) ( 6 , 2 ) ] g m [ c y ( m ) ( 6 , 2 ) ] .
ĉ y ( m ) ( 6 , 2 ) = μ ̂ y ( m ) ( 6 ) 15 μ ̂ y ( m ) ( 4 ) μ ̂ y ( m ) ( 2 ) 10 μ ̂ y ( m ) ( 3 ) 2 + 30 μ ̂ y ( m ) ( 2 ) 3 μ ̂ y ( m ) ( 2 ) 3 ,
ĉ y ( m ) ( 6 , 2 ) g t = 1 μ ̂ y ( m ) ( 2 ) 3 [ μ ̂ y ( m ) ( 6 ) ] g t 15 μ ̂ y ( m ) ( 2 ) 2 [ μ ̂ y ( m ) ( 4 ) ] g t 20 μ ̂ y ( m ) ( 3 ) μ ̂ y ( m ) ( 2 ) 3 [ μ ̂ y ( m ) ( 3 ) ] g t 3 μ ̂ y ( m ) ( 6 ) + 30 μ ̂ y ( m ) ( 4 ) μ ̂ y ( m ) ( 2 ) + 30 μ ̂ y ( m ) ( 3 ) 2 μ ̂ y ( m ) ( 2 ) 4 × [ μ ̂ y ( m ) ( 2 ) ] g t , t = 0 , 1 , , Q .
ĉ y ( m ) ( 6 , 2 ) g t = 6 ( N Q ) μ ̂ y ( m ) ( 2 ) 3 n = Q + 1 N { [ y ( n ) m ̂ Y ] 5 ξ } 60 ( N Q ) μ ̂ y ( m ) ( 2 ) 2 n = Q + 1 N { [ y ( n ) m ̂ Y ] 3 ξ } 60 μ ̂ y ( m ) ( 3 ) ( N Q ) μ ̂ y ( m ) ( 2 ) 3 n = Q + 1 N { [ y ( n ) m ̂ Y ] 2 ξ } 6 μ ̂ y ( m ) ( 6 ) + 60 μ ̂ y ( m ) ( 4 ) μ ̂ y ( m ) ( 2 ) + 60 μ ̂ y ( m ) ( 3 ) 2 ( N Q ) μ ̂ y ( m ) ( 2 ) 4 × n = Q + 1 N { [ y ( n ) m ̂ Y ] ξ } , t = 0 , 1 , , Q ,
ξ = x ( n t ) 1 ( N Q ) k = Q + 1 N x ( k t ) .
g m + 1 = g m + ( ρ ) l g m [ ĉ y ( m ) ( 6 , 2 ) ] sgn [ ĉ y ( m ) ( 6 , 2 ) ] .
e = | ĉ y ( m + 1 ) ( 6 , 2 ) | | ĉ y ( m ) ( 6 , 2 ) | ,

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