Abstract

In imaging situations where observations are made in spatial-frequency space, it is often desirable to lower the number of observations to fewer than that imposed by the Nyquist criterion. It is shown that patterns of regular spectral undersampling lead to aliasing that can be partially eliminated from some regions of a limited-extent image. An algorithm is presented for determining which regions are recoverable and which are not for a given pattern. Noniterative recovery, analogous to that proposed by Walsh Nielsen-Delaney [J. Opt. Soc. Am. A11, 572 (1994)], is shown to be feasible in cases of regular undersampling. The work has particular relevance to magnetic resonance imaging and aperture synthesis telescopy.

© 2001 Optical Society of America

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    [CrossRef]
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    [CrossRef]

1996 (1)

R. K. S. Kwan, A. C. Evans, G. B. Pike, “An extensible MRI simulator for post-processing evaluation,” Lect. Notes Comput. Sci. 1131, 135–140 (1996) (see also http://www.bic.mni.mcgill.ca/brainweb/ ).
[CrossRef]

1995 (3)

S. J. Reeves, L. P. Heck, “Selection of observations in signal reconstruction,” IEEE Trans. Signal Process. 43, 788–791 (1995).
[CrossRef]

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

T. J. Connolly, K. A. Landman, L. R. White, “On Gerchberg’s method for the Fourier inverse problem,” J. Aust. Math. Soc. B Appl. Math. 37, 26–44 (1995).
[CrossRef]

1994 (1)

1990 (1)

1986 (1)

1984 (1)

B. J. Sullivan, 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]

1978 (1)

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
[CrossRef]

1975 (2)

W. N. Brouw, “Aperture synthesis,” Methods Comput. Phys. 14, 131–175 (1975).

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[CrossRef]

1974 (1)

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

1973 (1)

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Benedetto, J. J.

J. J. Benedetto, H. Wu, “A multidimensional irregular sampling algorithm and applications,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 2039–2042.

Bracewell, R. N.

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Brouw, W. N.

W. N. Brouw, “Aperture synthesis,” Methods Comput. Phys. 14, 131–175 (1975).

Cheung, K. F.

Colvin, R. S.

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Connolly, T. J.

T. J. Connolly, K. A. Landman, L. R. White, “On Gerchberg’s method for the Fourier inverse problem,” J. Aust. Math. Soc. B Appl. Math. 37, 26–44 (1995).
[CrossRef]

D’Addario, L. R.

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Debatin, J. F.

J. F. Debatin, G. C. McKinnon, Ultrafast MRI: Techniques and Applications (Springer-Verlag, New York, 1998).

Evans, A. C.

R. K. S. Kwan, A. C. Evans, G. B. Pike, “An extensible MRI simulator for post-processing evaluation,” Lect. Notes Comput. Sci. 1131, 135–140 (1996) (see also http://www.bic.mni.mcgill.ca/brainweb/ ).
[CrossRef]

Gerchberg, R. W.

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

Grebenkemper, C. J.

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Heck, L. P.

S. J. Reeves, L. P. Heck, “Selection of observations in signal reconstruction,” IEEE Trans. Signal Process. 43, 788–791 (1995).
[CrossRef]

Hunt, B. R.

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

Kwan, R. K. S.

R. K. S. Kwan, A. C. Evans, G. B. Pike, “An extensible MRI simulator for post-processing evaluation,” Lect. Notes Comput. Sci. 1131, 135–140 (1996) (see also http://www.bic.mni.mcgill.ca/brainweb/ ).
[CrossRef]

Landman, K. A.

T. J. Connolly, K. A. Landman, L. R. White, “On Gerchberg’s method for the Fourier inverse problem,” J. Aust. Math. Soc. B Appl. Math. 37, 26–44 (1995).
[CrossRef]

Lauterbur, P. C.

Z. P. Liang, P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective (Institute of Electrical and Electronics Engineers, New York, 1999).

Liang, Z. P.

Z. P. Liang, P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective (Institute of Electrical and Electronics Engineers, New York, 1999).

Liu, B.

B. J. Sullivan, 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]

Luenberger, D. G.

D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. (Addition-Wesley, Reading, Mass., 1984).

Marks, R. J.

McKinnon, G. C.

J. F. Debatin, G. C. McKinnon, Ultrafast MRI: Techniques and Applications (Springer-Verlag, New York, 1998).

Nielsen-Delaney, P. A.

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[CrossRef]

Pike, G. B.

R. K. S. Kwan, A. C. Evans, G. B. Pike, “An extensible MRI simulator for post-processing evaluation,” Lect. Notes Comput. Sci. 1131, 135–140 (1996) (see also http://www.bic.mni.mcgill.ca/brainweb/ ).
[CrossRef]

Price, K. M.

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Reeves, S. J.

S. J. Reeves, L. P. Heck, “Selection of observations in signal reconstruction,” IEEE Trans. Signal Process. 43, 788–791 (1995).
[CrossRef]

Stark, H.

H. Stark, Image Recovery, Theory and Application (Academic, Orlando, Fla., 1987).

Strang, G.

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988).

Sullivan, B. J.

B. J. Sullivan, 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]

Thompson, A. R.

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Walsh, D. O.

White, L. R.

T. J. Connolly, K. A. Landman, L. R. White, “On Gerchberg’s method for the Fourier inverse problem,” J. Aust. Math. Soc. B Appl. Math. 37, 26–44 (1995).
[CrossRef]

Wu, H.

J. J. Benedetto, H. Wu, “A multidimensional irregular sampling algorithm and applications,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 2039–2042.

Youla, D. C.

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
[CrossRef]

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

B. J. Sullivan, 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. Circuits Syst. (2)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[CrossRef]

D. C. Youla, “Generalized image restoration by the method of alternating orthogonal projections,” IEEE Trans. Circuits Syst. 25, 694–702 (1978).
[CrossRef]

IEEE Trans. Signal Process. (1)

S. J. Reeves, L. P. Heck, “Selection of observations in signal reconstruction,” IEEE Trans. Signal Process. 43, 788–791 (1995).
[CrossRef]

Int. J. Imaging Syst. Technol. (1)

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

J. Aust. Math. Soc. B Appl. Math. (1)

T. J. Connolly, K. A. Landman, L. R. White, “On Gerchberg’s method for the Fourier inverse problem,” J. Aust. Math. Soc. B Appl. Math. 37, 26–44 (1995).
[CrossRef]

J. Opt. Soc. Am. A (3)

Lect. Notes Comput. Sci. (1)

R. K. S. Kwan, A. C. Evans, G. B. Pike, “An extensible MRI simulator for post-processing evaluation,” Lect. Notes Comput. Sci. 1131, 135–140 (1996) (see also http://www.bic.mni.mcgill.ca/brainweb/ ).
[CrossRef]

Methods Comput. Phys. (1)

W. N. Brouw, “Aperture synthesis,” Methods Comput. Phys. 14, 131–175 (1975).

Opt. Acta (1)

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

Proc. IEEE (1)

R. N. Bracewell, R. S. Colvin, L. R. D’Addario, C. J. Grebenkemper, K. M. Price, A. R. Thompson, “The Stanford five-element radio telescope,” Proc. IEEE 61, 1249–1257 (1973).
[CrossRef]

Other (6)

Z. P. Liang, P. C. Lauterbur, Principles of Magnetic Resonance Imaging: A Signal Processing Perspective (Institute of Electrical and Electronics Engineers, New York, 1999).

H. Stark, Image Recovery, Theory and Application (Academic, Orlando, Fla., 1987).

G. Strang, Linear Algebra and Its Applications, 3rd ed. (Harcourt Brace Jovanovitch, San Diego, Calif., 1988).

D. G. Luenberger, Linear and Nonlinear Programming, 2nd ed. (Addition-Wesley, Reading, Mass., 1984).

J. F. Debatin, G. C. McKinnon, Ultrafast MRI: Techniques and Applications (Springer-Verlag, New York, 1998).

J. J. Benedetto, H. Wu, “A multidimensional irregular sampling algorithm and applications,” in Proceedings of the IEEE International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1999), pp. 2039–2042.

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