Abstract

Autoregressive model fitting with singular value decomposition is applied to the interferogram data measured by a Fourier transform spectrometer to estimate superresolving spectra. The interferogram data matrix is decomposed into singular values with eigenvectors and its generalized inverse matrix is used for estimating the coefficients of the autoregressive model. This method suppresses the noise component in the data and avoids the risk of producing spurious peaks, which were the problem inherent in our earlier work using the maximum entropy method [ Appl. Opt. 22, 3593 ( 1983)]. The experimental results of superresolving spectral estimation are shown with the data of a visible emission spectrum and infrared absorption spectra.

© 1985 Optical Society of America

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References

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  1. L. Mertz, Transformations in Optics (Wiley, New York, 1965).
  2. J. L. Harris, “Diffraction and Resolving Power,” J. Opt. Soc. Am. 54, 931 (1964).
    [CrossRef]
  3. D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I,” Bell Syst. Tech. J. 40, 43 (1961).
  4. R. W. Gerchberg, “Superresolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
    [CrossRef]
  5. A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
    [CrossRef]
  6. J. Connes, “Resolution Enhancement by Numerical Methods,” presented at 1983 International Conference on Fourier Transform Spectroscopy, Durham U., U.K. (Sept. 1983).
  7. S. Kawata, K. Minami, S. Minami, “Superresolution of Fourier Transform Spectroscopy Data by the Maximum Entropy Method,” Appl. Opt. 22, 3593 (1983).
    [CrossRef] [PubMed]
  8. T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
    [CrossRef]
  9. S. M. Kay, S. L. Marple, “Spectral Analysis—a Modern Perspective,” Proc. IEEE 69, 1380 (1981).
    [CrossRef]
  10. D. W. Tuft, R. Kumaresan, “Singular Value Decomposition and Improved Frequency Estimation Using Linear Prediction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 671 (1982).
    [CrossRef]
  11. C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).
  12. T. Okamoto, S. Kawata, S. Minami, “Fourier Transform Spectrometer with a Self-Scanning Photodiode Array,” Appl. Opt. 23, 269 (1984).
    [CrossRef] [PubMed]

1984 (1)

1983 (1)

1982 (1)

D. W. Tuft, R. Kumaresan, “Singular Value Decomposition and Improved Frequency Estimation Using Linear Prediction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 671 (1982).
[CrossRef]

1981 (1)

S. M. Kay, S. L. Marple, “Spectral Analysis—a Modern Perspective,” Proc. IEEE 69, 1380 (1981).
[CrossRef]

1975 (1)

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

1974 (1)

R. W. Gerchberg, “Superresolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

1964 (1)

1961 (1)

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I,” Bell Syst. Tech. J. 40, 43 (1961).

Connes, J.

J. Connes, “Resolution Enhancement by Numerical Methods,” presented at 1983 International Conference on Fourier Transform Spectroscopy, Durham U., U.K. (Sept. 1983).

Gerchberg, R. W.

R. W. Gerchberg, “Superresolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

Harris, J. L.

Kawata, S.

Kay, S. M.

S. M. Kay, S. L. Marple, “Spectral Analysis—a Modern Perspective,” Proc. IEEE 69, 1380 (1981).
[CrossRef]

Kumaresan, R.

D. W. Tuft, R. Kumaresan, “Singular Value Decomposition and Improved Frequency Estimation Using Linear Prediction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 671 (1982).
[CrossRef]

Marple, S. L.

S. M. Kay, S. L. Marple, “Spectral Analysis—a Modern Perspective,” Proc. IEEE 69, 1380 (1981).
[CrossRef]

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Minami, K.

Minami, S.

Mitra, S. K.

C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

Okamoto, T.

Ooe, M.

T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

Papoulis, A.

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

Pollak, H. O.

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I,” Bell Syst. Tech. J. 40, 43 (1961).

Rao, C. R.

C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

Slepian, D.

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I,” Bell Syst. Tech. J. 40, 43 (1961).

Tuft, D. W.

D. W. Tuft, R. Kumaresan, “Singular Value Decomposition and Improved Frequency Estimation Using Linear Prediction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 671 (1982).
[CrossRef]

Ulrych, T. J.

T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

Appl. Opt. (2)

Bell Syst. Tech. J. (1)

D. Slepian, H. O. Pollak, “Prolate Spheroidal Wave Functions, Fourier Analysis and Uncertainty-I,” Bell Syst. Tech. J. 40, 43 (1961).

IEEE Trans. Acoust. Speech Signal Process. (1)

D. W. Tuft, R. Kumaresan, “Singular Value Decomposition and Improved Frequency Estimation Using Linear Prediction,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 671 (1982).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A New Algorithm in Spectral Analysis and Bandlimited Extrapolation,” IEEE Trans. Circuits Syst. CAS-22, 735 (1975).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Acta (1)

R. W. Gerchberg, “Superresolution Through Error Energy Reduction,” Opt. Acta 21, 709 (1974).
[CrossRef]

Proc. IEEE (1)

S. M. Kay, S. L. Marple, “Spectral Analysis—a Modern Perspective,” Proc. IEEE 69, 1380 (1981).
[CrossRef]

Other (4)

C. R. Rao, S. K. Mitra, Generalized Inverse of Matrices and Its Applications (Wiley, New York, 1971).

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

J. Connes, “Resolution Enhancement by Numerical Methods,” presented at 1983 International Conference on Fourier Transform Spectroscopy, Durham U., U.K. (Sept. 1983).

T. J. Ulrych, M. Ooe, “Autoregressive and Mixed Autoregressive-Moving Average Models and Spectra,” in Nonlinear Methods of Spectral Analysis, S. Haykin, Ed. (Springer, Berlin, 1979).
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Interferogram of a Cd lamp with 256 sampling points.

Fig. 2
Fig. 2

Spectra reconstructed from the interferogram in Fig. 1 by (a) FFT, (b) MEM with P = 80, (c) MEM with P = 90, and (d)–(f) the proposed method using SVD with L = 12 and P = 90, 120, and 150, respectively.

Fig. 3
Fig. 3

Procedure of superresolving spectral estimation for absorption spectra.

Fig. 4
Fig. 4

(a) and (b) Spectra of benzene reconstructed from the interferogram of 512 sampling points by FFT and the proposed method (P = 120 and L = 30), respectively. (c) Spectrum reconstructed by FFT from the 4096-point data.

Fig. 5
Fig. 5

Spectra of cyclohexane reconstructed from the interferogram of 512 sampling points by (a) FFT; (b) and (c) the proposed method with L = 30 and L = 50, respectively, (P = 200); (d) spectrum reconstructed by FFT from the 4096-point interferogram.

Equations (23)

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x k = i = 1 P c P i x k i + w k , k = P , , N 1 ,
X c + w = x ,
X c x .
ĉ = X 1 x ,
X = i = 1 P λ i u i v i t .
X = U Λ V t ,
X + = i = 1 L 1 λ i v i u i t = ( v 1 , , v L , 0 0 ) [ 1 λ 1 0 1 λ 1 0 0 ] ( u 1 , , u L , 0 0 ) t ,
ĉ = X + x .
Ŝ ( ν ) = σ w 2 | 1 + i = 1 P ĉ P i exp ( j ν i ) | 2 ,
2 M P N 2 M ,
plim N ĉ = plim N X + x = X + X c + plim N X + w = c + plim N X + w .
plim N ĉ c .
X c x .
E ( X t X ) c E ( X t x ) .
R c r
[ r 0 r P 1 r P 1 r 0 ] [ c P P c P 1 ] [ r P r 1 ] .
r i = E ( x k x k + i ) 1 P k = 1 P | i | ( x k + 1 μ ) ( x k μ ) ,
σ w 2 E ( | x k i P c P i x k 1 | 2 ) = r 0 i = 1 P c Piri ,
[ r 0 r P 1 r P r P 1 r 0 r P r 0 ] [ c P c P 1 1 ] = [ 0 0 σ w 2 ] .
ĉ = R 1 r .
σ w 2 = E ( | x k i c P i x k i | 2 ) + E ( | x k i c P i x k + i | 2 ) ,
R = X t X , r = X t x .
ĉ = R + r = X + x .

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