Abstract

When a multidimensional signal is uniformly sampled, its spectrum is replicated. If the signal is band limited and the replications (1) contain regions that are identically zero and (2) are not aliased, then the samples are dependent. Indeed, lost samples can be regained from those remaining. In dimensions greater than one, there are spectral regions of support for which this is the case even when sampling is performed at the Nyquist (minimum) density (e.g., a circular spectral region of support in two dimensions). When the known samples are perturbed by additive noise, lost-sample restoration noise levels in certain cases can be obtained by simple geometrical observations in the frequency domain. The results are specifically applied to coherent and incoherent optical images of objects of finite extent obtained from imaging systems with circular pupils.

© 1986 Optical Society of America

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References

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  1. R. J. Marks, “Restoring lost samples from an oversampled bandlimited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 752–755 (1983).
    [CrossRef]
  2. R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled band-limited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1984).
    [CrossRef]
  3. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  4. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 1.
  5. D. P. Peterson, D. Middleton, “Sampling and reconstruction of wave-number-limited function in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
    [CrossRef]
  6. A. K. Jain, S. Ranganath, “Extrapolation algorithms for discrete signals with application in spectral estimation,”IEEE Trans. Acoust. Speech Signal Process, ASSP-29, 830–845 (1981).
    [CrossRef]
  7. K. M. Rege, “Min-max linear estimation of band-limited sequences from noisy observations: a deterministic approach,”IEEE Trans. Inf. Theory IT-29, 902–909 (1983).
    [CrossRef]
  8. J. L. C. Sanz, T. S. Huang, “A unified approach to noniterative linear signal restoration,”IEEE Trans. Acoust. Speech Signal Process, ASSP-32, 403–409 (1984).
    [CrossRef]
  9. R. J. Marks, D. K. Smith, “Gerchberg-type linear deconvolution and extrapolation algorithms,” Proc. Soc. Photo-Opt. Instrument. Eng. 373, 161–178 (1984).
  10. A. A. Melkman, C. A. Micchelli, “Optical estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal. 16, 87–105 (1979).
    [CrossRef]
  11. A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984), pp. 235–237.
  12. J. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 355–357.

1984 (3)

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled band-limited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1984).
[CrossRef]

J. L. C. Sanz, T. S. Huang, “A unified approach to noniterative linear signal restoration,”IEEE Trans. Acoust. Speech Signal Process, ASSP-32, 403–409 (1984).
[CrossRef]

R. J. Marks, D. K. Smith, “Gerchberg-type linear deconvolution and extrapolation algorithms,” Proc. Soc. Photo-Opt. Instrument. Eng. 373, 161–178 (1984).

1983 (2)

K. M. Rege, “Min-max linear estimation of band-limited sequences from noisy observations: a deterministic approach,”IEEE Trans. Inf. Theory IT-29, 902–909 (1983).
[CrossRef]

R. J. Marks, “Restoring lost samples from an oversampled bandlimited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 752–755 (1983).
[CrossRef]

1981 (1)

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

1979 (1)

A. A. Melkman, C. A. Micchelli, “Optical estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal. 16, 87–105 (1979).
[CrossRef]

1962 (1)

D. P. Peterson, D. Middleton, “Sampling and reconstruction of wave-number-limited function in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 1.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Huang, T. S.

J. L. C. Sanz, T. S. Huang, “A unified approach to noniterative linear signal restoration,”IEEE Trans. Acoust. Speech Signal Process, ASSP-32, 403–409 (1984).
[CrossRef]

Jacobs, I. M.

J. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 355–357.

Jain, A. K.

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

Marks, R. J.

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled band-limited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1984).
[CrossRef]

R. J. Marks, D. K. Smith, “Gerchberg-type linear deconvolution and extrapolation algorithms,” Proc. Soc. Photo-Opt. Instrument. Eng. 373, 161–178 (1984).

R. J. Marks, “Restoring lost samples from an oversampled bandlimited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 752–755 (1983).
[CrossRef]

Melkman, A. A.

A. A. Melkman, C. A. Micchelli, “Optical estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal. 16, 87–105 (1979).
[CrossRef]

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 1.

Micchelli, C. A.

A. A. Melkman, C. A. Micchelli, “Optical estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal. 16, 87–105 (1979).
[CrossRef]

Middleton, D.

D. P. Peterson, D. Middleton, “Sampling and reconstruction of wave-number-limited function in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984), pp. 235–237.

Peterson, D. P.

D. P. Peterson, D. Middleton, “Sampling and reconstruction of wave-number-limited function in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Radbel, D.

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled band-limited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1984).
[CrossRef]

Ranganath, S.

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

Rege, K. M.

K. M. Rege, “Min-max linear estimation of band-limited sequences from noisy observations: a deterministic approach,”IEEE Trans. Inf. Theory IT-29, 902–909 (1983).
[CrossRef]

Sanz, J. L. C.

J. L. C. Sanz, T. S. Huang, “A unified approach to noniterative linear signal restoration,”IEEE Trans. Acoust. Speech Signal Process, ASSP-32, 403–409 (1984).
[CrossRef]

Smith, D. K.

R. J. Marks, D. K. Smith, “Gerchberg-type linear deconvolution and extrapolation algorithms,” Proc. Soc. Photo-Opt. Instrument. Eng. 373, 161–178 (1984).

Wozencraft, J.

J. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 355–357.

IEEE Trans. Acoust. Speech Signal Process (2)

J. L. C. Sanz, T. S. Huang, “A unified approach to noniterative linear signal restoration,”IEEE Trans. Acoust. Speech Signal Process, ASSP-32, 403–409 (1984).
[CrossRef]

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

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

R. J. Marks, “Restoring lost samples from an oversampled bandlimited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 752–755 (1983).
[CrossRef]

R. J. Marks, D. Radbel, “Error of linear estimation of lost samples in an oversampled band-limited signal,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 648–654 (1984).
[CrossRef]

IEEE Trans. Inf. Theory (1)

K. M. Rege, “Min-max linear estimation of band-limited sequences from noisy observations: a deterministic approach,”IEEE Trans. Inf. Theory IT-29, 902–909 (1983).
[CrossRef]

Inf. Control (1)

D. P. Peterson, D. Middleton, “Sampling and reconstruction of wave-number-limited function in N-dimensional Euclidean spaces,” Inf. Control 5, 279–323 (1962).
[CrossRef]

Proc. Soc. Photo-Opt. Instrument. Eng. (1)

R. J. Marks, D. K. Smith, “Gerchberg-type linear deconvolution and extrapolation algorithms,” Proc. Soc. Photo-Opt. Instrument. Eng. 373, 161–178 (1984).

SIAM J. Numer. Anal. (1)

A. A. Melkman, C. A. Micchelli, “Optical estimation of linear operators in Hilbert spaces from inaccurate data,” SIAM J. Numer. Anal. 16, 87–105 (1979).
[CrossRef]

Other (4)

A. Papoulis, Probability, Random Variables and Stochastic Processes, 2nd ed. (McGraw-Hill, New York, 1984), pp. 235–237.

J. Wozencraft, I. M. Jacobs, Principles of Communication Engineering (Wiley, New York, 1965), pp. 355–357.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing, (Prentice-Hall, Englewood Cliffs, N.J., 1984), Chap. 1.

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Figures (7)

Fig. 1
Fig. 1

Sampling geometry corresponding to the sampling matrix in Eq. (1).

Fig. 2
Fig. 2

Spectrum replication from the sampling geometry of Fig. 1.

Fig. 3
Fig. 3

One cell of Fig. 2. The region of integration, , must contain the spectral support region, A, and must not infringe onto adjacent spectra. C is a cell region. The areas of the regions A, , and C are A, B, and C, respectively.

Fig. 4
Fig. 4

Top, densely packed circles correspond to Nyquist sampling of images with spectra of circular support. Note the hexagonal structure. Bottom, a single hexagonal cell with inscribed circular spectrum support.

Fig. 5
Fig. 5

Hexagonal sampling geometry required to pack circles densely as shown in Fig. 4.

Fig. 6
Fig. 6

Top, minimum density rectangular sampling of images with spectra of circular support yields circles packed as shown. Bottom, a single cell with inscribed circular spectrum support.

Fig. 7
Fig. 7

Plots of η 2 ( O ) ¯ / ξ 2 ¯ (filled circles) and η 2 ( O ) ¯ / ψ 2 ¯ (open circles) in dB [10 log10(·)]. The solid lines are for minimum density rectangular sampling and the dashed for Nyquist (hexagonal) sampling.

Equations (46)

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X ( Ω ) = t x ( t ) exp ( - j Ω t ) d t ,
t = t 1 t 2 t N ·
x ( t ) = 1 ( 2 π ) N Ω X ( Ω ) exp ( j Ω t ) d Ω .
V = [ v 1 v 2 v N ] ,
V = [ - 1 2 3 - 2 ] .
D = 1 det V samples ( unit length ) N .
x ^ ( t ) = n x ( Vn ) δ D ( t - Vn ) ,
X ^ ( Ω ) = D k X ( Ω - Uk ) ,
U V = 2 π I .
U = [ u 1 u 2 u N ] .
U = [ π 3 π / 2 π π / 2 ] .
X ( Ω ) = X ^ ( Ω ) F ( Ω ) ,
F ( Ω ) = { det V ; Ω B 0 ; Ω B .
x ( t ) = x ^ ( t ) * f ( t ) ,
f ( t ) = det V ( 2 π ) N B exp ( j Ω t ) d Ω .
x ( t ) = n x ( Vn ) f ( t - Vn ) .
n M { δ ( k - n ) - f [ V ( k - n ) ] } x ( Vn ) = n M x ( Vn ) f [ V ( k - n ) ] ;             k M
x ( O ) = [ 1 - f ( O ) ] - 1 n O x ( Vn ) f ( - Vn ) .
x ( t ) = n O x ( Vn ) [ f ( t - Vn ) + { 1 - f ( O ) } - 1 f ( - Vn ) f ( t ) ] .
x ( t ) = ( n M + n M ) x ( Vn ) f ( t - Vn ) .
x ( Vk ) = ( n M + n M ) x ( Vn ) f { V ( k - n ) } ;             k M .
f ( Vn ) = det V ( 2 π ) N C exp ( j Ω Vn ) d Ω . = δ ( n ) .
n M [ δ ( k - n ) - f { V ( k - n ) } ] η ( Vn ) = n M ξ ( Vn ) f { V ( k - n ) } .
E [ ξ ( Vn ) ξ * ( Vm ) ] = ξ 2 ¯ δ ( n - m ) ,
η ( O ) = [ 1 - f ( O ) ] - 1 n O ξ ( Vn ) f ( - Vn ) .
η 2 ( O ) ¯ / ξ 2 ¯ = [ 1 - f ( O ) ] - 2 n O f ( - Vn ) 2 ,
η 2 ( O ) ¯ = E [ η ( O ) 2 ] .
η 2 ( O ) ¯ / ξ 2 ¯ = f ( O ) 1 - f ( O ) .
f ( O ) = det V ( 2 π ) N B d Ω .
B = B d Ω = area of integration , B
C = C d Ω = area of cell , C = det U = ( 2 π ) N / det V ,
η 2 ( O ) ¯ ξ 2 = ( C B - 1 ) - 1 .
ψ 2 ¯ / ξ 2 ¯ = B / C .
η 2 ( O ) ¯ ψ 2 ¯ = [ 1 - B C ] - 1 ,
V = [ T - T T / 3 T / 3 ] ,
f ( t 1 , t 2 ) = W 2 π D J 1 [ W ( t 1 2 + t 2 2 ) 1 / 2 ] ( t 1 2 + t 2 2 ) 1 / 2 .
V = [ T 0 0 T ] ,
η 2 ( O ) ¯ ξ 2 ¯ = ( 4 π - 1 ) - 1 . 3.66
η 2 ( O ) ¯ ψ 2 ¯ = [ 1 - π 4 ] - 1 , 4.66
A = { 2 N π ( N - 1 ) / 2 ( N - 1 2 ) ! W N N ! odd N π N / 2 ( N 2 ) ! W N even N .
C = 2 3 W 2 .
η 2 ( O ) ¯ ξ 2 ¯ = ( 2 3 π - 1 ) - 1 9.74 ,
η 2 ( O ) ¯ ψ 2 ¯ = ( 1 - π 2 3 ) - 1 . 10.74
ψ ( t ) = n ξ ( Vn ) f ( t - Vn ) .
ψ 2 ( t ) ¯ = ξ 2 ¯ n f ( t - Vn ) 2 .
f * ( τ - t ) = n f * ( τ - Vn ) f ( t - Vn ) .

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