Abstract

We consider reconstruction of signals by a direct method for the solution of the discrete Fourier system. We note that the reconstruction of a time-limited signal can be simply realized by using only either the real part or the imaginary part of the discrete Fourier transform (DFT) matrix. Therefore, based on the study of the special structure of the real and imaginary parts of the discrete Fourier matrix, we propose a fast direct method for the signal reconstruction problem, which utilizes the numerically truncated singular value decomposition. The method enables us to recover the original signal in a stable way from the frequency information, which may be corrupted by noise and∕or some missing data. The classical inverse Fourier transform cannot be applied directly in the latter situation. The pivotal point of the reconstruction is the explicit computation of the singular value decomposition of the real part of the DFT for any order. Numerical experiments for 1D and 2D signal reconstruction and image restoration are given.

© 2006 Optical Society of America

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  1. H. P. Baltes, Inverse Source Problems in Optics Vols. 9 and 10 of Topics in Current Physics (Springer, 1978 and 1980).
  2. M. Z. Nashed, "Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory," IEEE Trans. Antennas Propagat. AP-29, 220-231 (1981).
    [CrossRef]
  3. R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system," IEEE Trans. Antennas Propagat. AP-16, 712-717 (1968).
    [CrossRef]
  4. R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system-II," IEEE Trans. Antennas Propagat. AP-18, 64-68 (1970).
    [CrossRef]
  5. J. B. Abbis, C. DeMol, and H. Dhadwal, "Regularized iterative and noniterative procedure for object restoration from experimental data," Opt. Acta 30, 107-124 (1983).
    [CrossRef]
  6. R. N. Bracewell and S. J. Wernecke, "Image reconstruction over a finite field of view," J. Opt. Soc. Am. 65, 1342-1347 (1975).
    [CrossRef]
  7. M. Bertero, C. DeMol, and G. A. Viano, "On the problem of object restoration and image extrapolation in optics," J. Math. Phys. 20, 509-521 (1979).
    [CrossRef]
  8. J. M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 2-120 (1989).
  9. J. M. Bertero, C. DeMol, and G. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics, H.Baltes, ed., Vol. 20 of Topics in Current Physics (Springer, 1980), pp. 161-214.
    [CrossRef]
  10. R. J. Bell, Introduction to Fourier Transform Spectroscopy (Academic, 1972).
  11. S. Kawata, K. Minami, and S. Minami, "Superresolution of Fourier transform spectroscopy data by the maximum entropy method," Appl. Opt. 22, 3593-3598 (1983).
    [CrossRef] [PubMed]
  12. D. Slepian and H. O. Pollak, "Prolate spherical wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43-64 (1961).
  13. X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
    [CrossRef] [PubMed]
  14. S. J. Reeves and L. P. Heck, "Selection of obervations in signal reconstruction," IEEE Trans. Signal Process. 43, 788-791 (1995).
    [CrossRef]
  15. A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830-845 (1981).
    [CrossRef]
  16. B. J. Sullivan and B. Liu, "On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation," IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1201-1212 (1984).
    [CrossRef]
  17. S. R. Degraaf, "SAR imaging via modern 2-D spectral estimation methods," IEEE Trans. Image Process. 17, 729-761 (1998).
    [CrossRef]
  18. D. O. Walsh and P. A. Nielsen-Delaney, "Direct method for superresolution," J. Opt. Soc. Am. A 11, 572-579 (1994).
    [CrossRef]
  19. D. J. Wingham, "The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition," IEEE Trans. Signal Process. 40, 559-570 (1992).
    [CrossRef]
  20. M. Çetin and W. C. Karl, "Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization," IEEE Trans. Image Process. 10, 623-631 (2001).
    [CrossRef]
  21. W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, 1995).
  22. J. Li and P. Stoica, "An adaptive filtering approach to spectral estimation and SAR imaging," IEEE Trans. Signal Process. 44, 1469-1484 (1996).
    [CrossRef]
  23. M. Z. Nashed, Generalized Inverses and Applications (Academic, 1976).
  24. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).
  25. B. Noble, "Methods for computing the Moore-Penrose generalized inverse and related matters," in Generalized Inverses and Applications, M. Z. Nashed, ed. (Academic, 1976), pp. 245-301.
  26. G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, 1996).
  27. J. M. Varah, "On the numerical solution of ill-conditioned linear systems with application to ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 549-565 (1973).
  28. R. W. Gerchberg, "Super-resolution through energy reduction," Opt. Acta 21, 709-720 (1974).
    [CrossRef]
  29. N. N. Abdelmalek and T. Kasvand, "Image restoration by Gauss LU decomposition," Appl. Opt. 18, 1684-1686 (1979).
    [CrossRef] [PubMed]
  30. N. N. Abdelmalek, T. Kasvand, J. Olmstead, and M. -M. Tremblay, "Direct algorithm for digital image restoration," Appl. Opt. 20, 4227-4233 (1981).
    [CrossRef] [PubMed]
  31. T. S. Huang and P. M. Narendra, "Image restoration by singular value decomposition," Appl. Opt. 14, 2213-2216 (1975).
    [CrossRef] [PubMed]

2001 (1)

M. Çetin and W. C. Karl, "Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization," IEEE Trans. Image Process. 10, 623-631 (2001).
[CrossRef]

1998 (1)

S. R. Degraaf, "SAR imaging via modern 2-D spectral estimation methods," IEEE Trans. Image Process. 17, 729-761 (1998).
[CrossRef]

1996 (2)

J. Li and P. Stoica, "An adaptive filtering approach to spectral estimation and SAR imaging," IEEE Trans. Signal Process. 44, 1469-1484 (1996).
[CrossRef]

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, 1996).

1995 (1)

S. J. Reeves and L. P. Heck, "Selection of obervations in signal reconstruction," IEEE Trans. Signal Process. 43, 788-791 (1995).
[CrossRef]

1994 (1)

1992 (1)

D. J. Wingham, "The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition," IEEE Trans. Signal Process. 40, 559-570 (1992).
[CrossRef]

1989 (1)

J. M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 2-120 (1989).

1988 (1)

X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
[CrossRef] [PubMed]

1984 (1)

B. J. Sullivan and B. Liu, "On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation," IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1201-1212 (1984).
[CrossRef]

1983 (2)

S. Kawata, K. Minami, and S. Minami, "Superresolution of Fourier transform spectroscopy data by the maximum entropy method," Appl. Opt. 22, 3593-3598 (1983).
[CrossRef] [PubMed]

J. B. Abbis, C. DeMol, and H. Dhadwal, "Regularized iterative and noniterative procedure for object restoration from experimental data," Opt. Acta 30, 107-124 (1983).
[CrossRef]

1981 (3)

M. Z. Nashed, "Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory," IEEE Trans. Antennas Propagat. AP-29, 220-231 (1981).
[CrossRef]

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830-845 (1981).
[CrossRef]

N. N. Abdelmalek, T. Kasvand, J. Olmstead, and M. -M. Tremblay, "Direct algorithm for digital image restoration," Appl. Opt. 20, 4227-4233 (1981).
[CrossRef] [PubMed]

1979 (2)

N. N. Abdelmalek and T. Kasvand, "Image restoration by Gauss LU decomposition," Appl. Opt. 18, 1684-1686 (1979).
[CrossRef] [PubMed]

M. Bertero, C. DeMol, and G. A. Viano, "On the problem of object restoration and image extrapolation in optics," J. Math. Phys. 20, 509-521 (1979).
[CrossRef]

1978 (1)

H. P. Baltes, Inverse Source Problems in Optics Vols. 9 and 10 of Topics in Current Physics (Springer, 1978 and 1980).

1975 (2)

1974 (1)

R. W. Gerchberg, "Super-resolution through energy reduction," Opt. Acta 21, 709-720 (1974).
[CrossRef]

1973 (1)

J. M. Varah, "On the numerical solution of ill-conditioned linear systems with application to ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 549-565 (1973).

1972 (1)

R. J. Bell, Introduction to Fourier Transform Spectroscopy (Academic, 1972).

1970 (1)

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system-II," IEEE Trans. Antennas Propagat. AP-18, 64-68 (1970).
[CrossRef]

1968 (1)

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system," IEEE Trans. Antennas Propagat. AP-16, 712-717 (1968).
[CrossRef]

1961 (1)

D. Slepian and H. O. Pollak, "Prolate spherical wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43-64 (1961).

Abbis, J. B.

J. B. Abbis, C. DeMol, and H. Dhadwal, "Regularized iterative and noniterative procedure for object restoration from experimental data," Opt. Acta 30, 107-124 (1983).
[CrossRef]

Abdelmalek, N. N.

Baltes, H. P.

H. P. Baltes, Inverse Source Problems in Optics Vols. 9 and 10 of Topics in Current Physics (Springer, 1978 and 1980).

Bell, R. J.

R. J. Bell, Introduction to Fourier Transform Spectroscopy (Academic, 1972).

Bertero, J. M.

J. M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 2-120 (1989).

J. M. Bertero, C. DeMol, and G. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics, H.Baltes, ed., Vol. 20 of Topics in Current Physics (Springer, 1980), pp. 161-214.
[CrossRef]

Bertero, M.

M. Bertero, C. DeMol, and G. A. Viano, "On the problem of object restoration and image extrapolation in optics," J. Math. Phys. 20, 509-521 (1979).
[CrossRef]

Bracewell, R. N.

Carrara, W. G.

W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, 1995).

Çetin, M.

M. Çetin and W. C. Karl, "Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization," IEEE Trans. Image Process. 10, 623-631 (2001).
[CrossRef]

Degraaf, S. R.

S. R. Degraaf, "SAR imaging via modern 2-D spectral estimation methods," IEEE Trans. Image Process. 17, 729-761 (1998).
[CrossRef]

DeMol, C.

J. B. Abbis, C. DeMol, and H. Dhadwal, "Regularized iterative and noniterative procedure for object restoration from experimental data," Opt. Acta 30, 107-124 (1983).
[CrossRef]

M. Bertero, C. DeMol, and G. A. Viano, "On the problem of object restoration and image extrapolation in optics," J. Math. Phys. 20, 509-521 (1979).
[CrossRef]

J. M. Bertero, C. DeMol, and G. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics, H.Baltes, ed., Vol. 20 of Topics in Current Physics (Springer, 1980), pp. 161-214.
[CrossRef]

Dhadwal, H.

J. B. Abbis, C. DeMol, and H. Dhadwal, "Regularized iterative and noniterative procedure for object restoration from experimental data," Opt. Acta 30, 107-124 (1983).
[CrossRef]

Fante, R. L.

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system-II," IEEE Trans. Antennas Propagat. AP-18, 64-68 (1970).
[CrossRef]

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system," IEEE Trans. Antennas Propagat. AP-16, 712-717 (1968).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, "Super-resolution through energy reduction," Opt. Acta 21, 709-720 (1974).
[CrossRef]

Golub, G. H.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, 1996).

Goodman, R. S.

W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, 1995).

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Heck, L. P.

S. J. Reeves and L. P. Heck, "Selection of obervations in signal reconstruction," IEEE Trans. Signal Process. 43, 788-791 (1995).
[CrossRef]

Hu, X.

X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
[CrossRef] [PubMed]

Huang, T. S.

Jain, A. K.

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830-845 (1981).
[CrossRef]

Karl, W. C.

M. Çetin and W. C. Karl, "Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization," IEEE Trans. Image Process. 10, 623-631 (2001).
[CrossRef]

Kasvand, T.

Kawata, S.

Lauterbur, P. C.

X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
[CrossRef] [PubMed]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

Levin, D. N.

X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
[CrossRef] [PubMed]

Li, J.

J. Li and P. Stoica, "An adaptive filtering approach to spectral estimation and SAR imaging," IEEE Trans. Signal Process. 44, 1469-1484 (1996).
[CrossRef]

Liu, B.

B. J. Sullivan and B. Liu, "On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation," IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1201-1212 (1984).
[CrossRef]

Majewski, R. M.

W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, 1995).

Mayhan, J. T.

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system-II," IEEE Trans. Antennas Propagat. AP-18, 64-68 (1970).
[CrossRef]

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system," IEEE Trans. Antennas Propagat. AP-16, 712-717 (1968).
[CrossRef]

Minami, K.

Minami, S.

Narendra, P. M.

Nashed, M. Z.

M. Z. Nashed, "Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory," IEEE Trans. Antennas Propagat. AP-29, 220-231 (1981).
[CrossRef]

M. Z. Nashed, Generalized Inverses and Applications (Academic, 1976).

Nielsen-Delaney, P. A.

Noble, B.

B. Noble, "Methods for computing the Moore-Penrose generalized inverse and related matters," in Generalized Inverses and Applications, M. Z. Nashed, ed. (Academic, 1976), pp. 245-301.

Olmstead, J.

Pollak, H. O.

D. Slepian and H. O. Pollak, "Prolate spherical wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43-64 (1961).

Ranganath, S.

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830-845 (1981).
[CrossRef]

Reeves, S. J.

S. J. Reeves and L. P. Heck, "Selection of obervations in signal reconstruction," IEEE Trans. Signal Process. 43, 788-791 (1995).
[CrossRef]

Slepian, D.

D. Slepian and H. O. Pollak, "Prolate spherical wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43-64 (1961).

Spraggins, T. A.

X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
[CrossRef] [PubMed]

Stoica, P.

J. Li and P. Stoica, "An adaptive filtering approach to spectral estimation and SAR imaging," IEEE Trans. Signal Process. 44, 1469-1484 (1996).
[CrossRef]

Sullivan, B. J.

B. J. Sullivan and B. Liu, "On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation," IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1201-1212 (1984).
[CrossRef]

Tremblay, M. -M.

Van Loan, C. F.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, 1996).

Varah, J. M.

J. M. Varah, "On the numerical solution of ill-conditioned linear systems with application to ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 549-565 (1973).

Viano, G.

J. M. Bertero, C. DeMol, and G. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics, H.Baltes, ed., Vol. 20 of Topics in Current Physics (Springer, 1980), pp. 161-214.
[CrossRef]

Viano, G. A.

M. Bertero, C. DeMol, and G. A. Viano, "On the problem of object restoration and image extrapolation in optics," J. Math. Phys. 20, 509-521 (1979).
[CrossRef]

Walsh, D. O.

Wernecke, S. J.

Wingham, D. J.

D. J. Wingham, "The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition," IEEE Trans. Signal Process. 40, 559-570 (1992).
[CrossRef]

Adv. Electron. Electron Phys. (1)

J. M. Bertero, "Linear inverse and ill-posed problems," Adv. Electron. Electron Phys. 75, 2-120 (1989).

Appl. Opt. (4)

Bell Syst. Tech. J. (1)

D. Slepian and H. O. Pollak, "Prolate spherical wave functions, Fourier analysis and uncertainty-I," Bell Syst. Tech. J. 40, 43-64 (1961).

IEEE Trans. Acoust. Speech Signal Process. (2)

A. K. Jain and S. Ranganath, "Extrapolation algorithms for discrete signals with application in spectral estimation," IEEE Trans. Acoust. Speech Signal Process. ASSP-29, 830-845 (1981).
[CrossRef]

B. J. Sullivan and B. Liu, "On the use of singular value decomposition and decimation in discrete-time band-limited signal extrapolation," IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 1201-1212 (1984).
[CrossRef]

IEEE Trans. Antennas Propagat. (3)

M. Z. Nashed, "Operator-theoretic and computational approaches to ill-posed problems with applications to antenna theory," IEEE Trans. Antennas Propagat. AP-29, 220-231 (1981).
[CrossRef]

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system," IEEE Trans. Antennas Propagat. AP-16, 712-717 (1968).
[CrossRef]

R. L. Fante and J. T. Mayhan, "Bounds on the electric field outside a radiating system-II," IEEE Trans. Antennas Propagat. AP-18, 64-68 (1970).
[CrossRef]

IEEE Trans. Image Process. (2)

M. Çetin and W. C. Karl, "Feature-enhanced synthetic aperture radar image formation based on nonquadratic regularization," IEEE Trans. Image Process. 10, 623-631 (2001).
[CrossRef]

S. R. Degraaf, "SAR imaging via modern 2-D spectral estimation methods," IEEE Trans. Image Process. 17, 729-761 (1998).
[CrossRef]

IEEE Trans. Signal Process. (3)

S. J. Reeves and L. P. Heck, "Selection of obervations in signal reconstruction," IEEE Trans. Signal Process. 43, 788-791 (1995).
[CrossRef]

D. J. Wingham, "The reconstruction of a band-limited function and its Fourier transform from a finite number of samples at arbitrary locations by singular value decomposition," IEEE Trans. Signal Process. 40, 559-570 (1992).
[CrossRef]

J. Li and P. Stoica, "An adaptive filtering approach to spectral estimation and SAR imaging," IEEE Trans. Signal Process. 44, 1469-1484 (1996).
[CrossRef]

J. Math. Phys. (1)

M. Bertero, C. DeMol, and G. A. Viano, "On the problem of object restoration and image extrapolation in optics," J. Math. Phys. 20, 509-521 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

Magn. Reson. Med. (1)

X. Hu, D. N. Levin, P. C. Lauterbur, and T. A. Spraggins, "SLIM: Spectral localization by imaging," Magn. Reson. Med. 8, 314-322 (1988).
[CrossRef] [PubMed]

Opt. Acta (2)

R. W. Gerchberg, "Super-resolution through energy reduction," Opt. Acta 21, 709-720 (1974).
[CrossRef]

J. B. Abbis, C. DeMol, and H. Dhadwal, "Regularized iterative and noniterative procedure for object restoration from experimental data," Opt. Acta 30, 107-124 (1983).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

J. M. Varah, "On the numerical solution of ill-conditioned linear systems with application to ill-posed problems," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 10, 549-565 (1973).

Other (8)

W. G. Carrara, R. S. Goodman, and R. M. Majewski, Spotlight Synthetic Aperture Radar Signal Processing Algorithms (Artech House, 1995).

M. Z. Nashed, Generalized Inverses and Applications (Academic, 1976).

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (Prentice-Hall, 1974).

B. Noble, "Methods for computing the Moore-Penrose generalized inverse and related matters," in Generalized Inverses and Applications, M. Z. Nashed, ed. (Academic, 1976), pp. 245-301.

G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. (Johns Hopkins University Press, 1996).

H. P. Baltes, Inverse Source Problems in Optics Vols. 9 and 10 of Topics in Current Physics (Springer, 1978 and 1980).

J. M. Bertero, C. DeMol, and G. Viano, "The stability of inverse problems," in Inverse Scattering Problems in Optics, H.Baltes, ed., Vol. 20 of Topics in Current Physics (Springer, 1980), pp. 161-214.
[CrossRef]

R. J. Bell, Introduction to Fourier Transform Spectroscopy (Academic, 1972).

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Figures (21)

Fig. 1
Fig. 1

Input true 1D signal.

Fig. 2
Fig. 2

Exact frequency information.

Fig. 3
Fig. 3

Noisy, fragmentary frequency.

Fig. 4
Fig. 4

Reconstructions.

Fig. 5
Fig. 5

Reconstructions for the noisy frequency.

Fig. 6
Fig. 6

Input true 1D odd signal.

Fig. 7
Fig. 7

Symmetrized true 1D odd signal.

Fig. 8
Fig. 8

Exact frequency information for the symmetric case.

Fig. 9
Fig. 9

Noisy, fragmentary frequency.

Fig. 10
Fig. 10

Reconstructions.

Fig. 11
Fig. 11

Reconstructions for the noisy frequency.

Fig. 12
Fig. 12

Input true 2D signal.

Fig. 13
Fig. 13

Exact frequency information.

Fig. 14
Fig. 14

Noisy, fragmentary frequency.

Fig. 15
Fig. 15

Reconstructions.

Fig. 16
Fig. 16

Noisy, fragmentary frequency.

Fig. 17
Fig. 17

Reconstructions.

Fig. 18
Fig. 18

Noisy, fragmentary frequency.

Fig. 19
Fig. 19

Reconstructions.

Fig. 20
Fig. 20

Noisy, fragmentary frequency.

Fig. 21
Fig. 21

Reconstructions.

Equations (85)

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

[ 1 1 1 1 ω N ω N       N 1 1 ω N       N 1 ω N       ( N 1 ) 2 ] [ f 0 f 1 f N 1 ] = [ F 0 F 1 F N 1 ] ,
( T f ) ( u ) = 1 1 exp ( i c u x ) f ( x ) d x = F ( u ) ,
   l = 0 N 1 ω N       k l f l + 1 = F k + 1 , k = 0 , 1 , , M 1 ,
f = [ f 1 , f 2 , , f N ] T , F = [ F 1 , F 2 , , F M ] T .
f = F ,
f = m = 0 M 1 n = 0 N 1 ω M N       l m , k n f ( m + 1 ) ( n + 1 ) = F ( l + 1 ) ( k + 1 ) ,
ω M N       l m , k n = exp [ i 2 π ( l m / M + k n / N ) ] ,
l = 0 , 1 , , M 1 ;         k = 0 , 1 , , N 1.
= r + i i ,
r f = F r
f = r     F r ,
σ nonzero ( r ) = [ 3.162 277 660 168 384 3.162 277 660 168 383 3.162 277 660 168 382 3.162 277 660 168 382 3.162 277 660 168 381 3.162 277 660 168 380 ] ,
σ nearly  zero ( r ) = [ 1.945 788 483 288 222 × 10 15 1.253 079 556 056 248 × 10 15 5.263 367 760 387 904 × 10 16 2.198 412 987 245 457 × 10 18 ] .
= [ 1 1 1 1 1 ω N ω N       2 ω N       N 1 1 ω N       2 ω N       4 ω N 2 ( N - 1 ) 1 ω N       M 1 ω N       2 ( M 1 ) ω N       ( M 1 ) ( N 1 ) ] M × N ( M > N ) ,
= r + i i .
F = U Σ V T = i = 1 N σ i u i v i     T ,
U T U = V T V = I N ,
σ 1 σ 2 σ N .
r δ = min { rank ( B δ + A ) : B δ M × N , B δ 2 δ } .
σ r δ + 1 δ < σ r δ .
min x R N Ax b 2 .
A = U Σ V T = i = 1 N σ i u i v i     T ,
U T U = V T V = I N ,
| σ 1 | | σ 2 | | σ N | .
x L S = i = 1 r 1 σ i ( u i     T b ) v i ,
min x R N A x b 2     2 = i = r + 1 N | u i     T b | 2 .
A r ˜ = i = 1 r ˜ σ i u i v i     T ,
min x N A r ˜ x b 2 ,
x L S   appr = A r ˜     b = i = 1 r ˜ 1 σ i ( u i     T b ) v i ,
= [ 1 1 1 1 1 ω M ω M       2 ω M       M 1 1 ω M       2 ω M       4 ω M       2 ( M 1 ) 1 ω M       M 1 ω M       2 ( M 1 ) ω M ( M - 1 ) 2 ] M × M ,
T = , * = * = M I M ,
r = U Σ 1 V T ,     i = U Σ 2 V T ,
Σ 1 = diag ( M , , M , r 0 , , 0 ) ,
Σ 2 = diag ( 0,  ,  0,   M , , M M r ) .
U = 1 M [ 1 2 2 1 0 0 1 2 a 1 2 a r 2 a r 1 2 b r 2 2 b 1 1 2 a M 1 2 a ( M 1 ) ( r 2 ) a ( r 1 ) ( M 1 ) 2 b ( M 1 ) ( r 2 ) 2 b M 1 ] ,
V = [ 1 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 2 2 0 0 0 0 2 2 0 2 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 0 2 2 0 0 0 0 2 2 2 2 0 0 2 2 0 0 2 2 ] ( p + 1 ) th   row,
U = 1 M [ 1 2 2 0 0 1 2 a 1 2 a r 1 2 b r 1 2 b 1 1 2 a M 1 2 a ( M 1 ) ( r 1 ) 2 b ( M 1 ) ( r 1 ) 2 b M 1 ] ,
V = [ 1 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 2 2 2 2 0 0 0 0 2 2 2 2 0 0 2 2 0 0 2 2 ] ( p + 1 ) th   row,
= U Σ V T .
P 1 r P = D ,
f = P D P T F ,
D = [ λ 1     1 0 0 0 λ r     1 0 0 0 0 ] ,
E 2 min .
J ( f ) def ¯ ¯ r f F 2 min .
vec ( U ) = [ U 11 ,   , U M x 1 , U 12 ,   , U M x 2 ,   , U 1 M y ,   , U M x M y ] T .
x y = ( A x + B x i ) ( A y + B y i )
= ( A x A y B x B y ) + ( A x B y B x A y ) i .
min K f F 2 ,
A x = U x [ Σ 1 x 0 0 0 ] V x     T ,     A y = U y [ Σ 1 y 0 0 0 ] V y     T ,
B x = U x [ 0 0 0 Σ 2 x ] V x     T ,     B y = U y [ 0 0 0 Σ 2 y ] V y     T ,
K = A x A y B x B y
= ( U x U y ) Σ ( V x V y ) T ,
Σ = [ Σ 1 x 0 0 0 ] [ Σ 1 y 0 0 0 ] [ 0 0 0 Σ 2 x ] [ 0 0 0 Σ 2 y ] .
f LS = K F = ( V x V y ) Σ ( U x U y ) T F ,
Σ = [ Σ 1 x         1 0 0 0 ] [ Σ 1 y         1 0 0 0 ] [ 0 0 0 Σ 2 x         1 ] [ 0 0 0 Σ 2 y         1 ] .
( U x U y ) T F = vec ( U y     T F U x ) = vec ( F )
Σ vec ( F ) = vec ( [ Σ 1 y     1 0 0 0 ] F [ Σ 1 x       1 0 0 0 ] [ 0 0 0 Σ 2 y       1 ] F [ 0 0 0 Σ 2 x       1 ] )
= vec ( [ F 1 0 0 - F 2 ] ) = vec ( F ) ,
F 1 = U 1 y T F U 1 x / M x M y , F 2 = U 2 y T F U 2 x / M x M y ,
( V x V y ) vec ( F ) = vec ( V y [ F 1 0 0 - F 2 ] V x       T )
=  vec ( [ V 1 y V 2 y ] [ F 1 0 0 - F 2 ] [ V 1 x V 2 x ] T )
=  vec ( V 1 y F 1 V 1 x T V 2 y F 2 V 2 x T ) .
f LS = vec ( V 1 y F 1 V 1 x T V 2 y F 2 V 2 x T ) .
F 1 = U 1 y T F U 1 x / M x M y ;
F 2 = U 2 y T F U 2 x / M x M y ;
f LS = vec ( V 1 y F 1 V 1 x T V 2 y F 2 V 2 x T ) .
f ( x ) = 1000 [ rect ( 16 ( x 3.5 32 ) ) + rect ( 16 ( x + 3.5 32 ) ) ] ,
F : = F noise = F true  +  rand ( size ( F true ) ) ,
f ( x ) = 1000 [ rect ( 16 ( x 3.5 32 ) ) rect ( 16 ( x + 3.5 32 ) ) ] ,
f ( x ) = { 1000 rect ( 16 ( x 3.5 32 ) ) 1000 rect ( 16 ( x + 3.5 32 ) )   for   x [ 4 , 4 ] , 1000 rect ( 16 ( x + 3.5 32 ) ) 1000 rect ( 16 ( x 3.5 32 ) )   for   x [ 4 , 12 ] .
K r = [ M 0 0 0 0 0 M 2 0 0 M 2 0 0 M 2 M 2 0 0 0 0 M 2 0 M 2 0 0 0 0 0 0 M 0 0 0 0 0 0 M 2 0 M 2 0 0 0 0 M 2 M 2 0 0 M 2 0 0 M 2 ] × ( p + 1 ) th   row ,
K i = [ 0 0 0 0 0 0 M 2 0 0 M 2 0 0 M 2 M 2 0 0 0 0 M 2 0 M 2 0 0 0 0 0 0 0 0 0 0 0 0 0 M 2 0 M 2 0 0 0 0 M 2 M 2 0 0 M 2 0 0 M 2 ] ( p + 1 ) th   row
K r = [ M 0 0 0 0 0 M 2 0 0 M 2 0 0 M 2 M 2 0 0 0 0 M 2 M 2 0 0 0 0 0 M 2 M 2 0 0 0 0 M 2 M 2 0 0 M 2 0 0 M 2 ] ( p + 1 ) th   row,
K i = [ 0 0 0 0 0 0 M 2 0 0 M 2 0 0 M 2 M 2 0 0 0 0 M 2 M 2 0 0 0 0 0 M 2 M 2 0 0 0 0 M 2 M 2 0 0 M 2 0 0 M 2 ] ( p + 1 ) th   row.
l = 0 M 1 cos 2 k l π M cos 2 k l π M = l = 0 M 1 sin 2 k l π M sin 2 k l π M = 0
l = 0 M 1 cos 2 2 k l π M = l = 0 M 1 sin 2 2 k l π M = M 2
t = 0 M 1 cos 2 k l π M sin 2 k l π M = 0
K r = V T Λ 1 V , K i = V T Λ 2 V ,
V = [ 1 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 2 2 0 0 0 0 2 2 0 2 2 0 0 0 0 0 0 1 0 0 0 0 0 0 2 2 0 2 2 0 0 0 0 2 2 2 2 0 0 2 2 0 0 2 2 ] ( p + 1 ) th   row
V = [ 1 0 0 0 0 0 2 2 0 0 2 2 0 0 2 2 2 2 0 0 0 0 2 2 2 2 0 0 0 0 0 2 2 2 2 0 0 0 0 2 2 2 2 0 0 2 2 0 0 2 2 ] ( p + 1 ) th   row
det ( K r λ I ) = ( λ ) M / 2 1 ( M λ ) M / 2 + 1 .
Σ 1 = diag ( M , , M r , 0 , , 0 ) ,
1 1 ψ i ( t ) ψ j ( t ) d t = { 0 , i j , 2 π c λ i 2 , i = j .
2 N exp ( i u k ) j = 0 N 1 exp ( i k t j ) f ( t j π 1 ) = F ( u k ) ,
k = 0 , , N 1 ,

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