Abstract

We analyze the problem of optical superresolution (OSR) of a one-dimensional (1D) incoherent spatial signal from undersampled data when the support of the signal is known in advance. The present paper corrects and extends our previous work on the calculation of Fisher information (FI) and the associated Cramer-Rao lower bound (CRB) on the minimum error for estimating the signal intensity distribution and its Fourier components at spatial frequencies lying beyond the optical band edge. The faint-signal and bright-signal limits emerge from a unified noise analysis in which we include both additive noise of detection and shot noise of photon counting via an approximate Gaussian statistical distribution. For a large space-bandwidth product, we derive analytical approximations to the exact expressions for FI and CRB in the faint-signal limit and use them to argue why achieving any significant amount of unbiased bandwidth extension in the presence of noise is a uniquely challenging proposition. Unlike previous theoretical work on the subject of support-assisted bandwidth extension, our approach is not restricted to specific forms of the system transfer functions, and provides a unified analysis of both digital and optical superresolution of undersampled data.

© 2009 Optical Society of America

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  1. S. Prasad, "Digital superresolution and the generalized sampling theorem," J. Opt. Soc. Am. A 24, 311-325 (2007)
    [CrossRef]
  2. S. Prasad, "Digital and optical superresolution of low-resolution image sequences," Proc. SPIE 6712, 67120E 1-11 (2007).
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    [CrossRef]
  4. A. Papoulis, "A new algorithm in spectral analysis and band-limited extrapolation," IEEE Trans. Circuits Syst. CAS-22, 735-742 (1975).
    [CrossRef]
  5. S. Plevritis and A. Macovski, "Spectral extrapolation of spatially bounded images," IEEE Trans. Medical Imaging 14, 487-497 (1995).
    [CrossRef]
  6. W. Richardson, "Bayesian-based iterative method of image restoration," J. Opt. Soc. Am. 62, 55-59 (1972).
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    [CrossRef]
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    [CrossRef]
  12. E. Boukouvala and A. Lettington, "Restoration of astronomical images by an iterative superresolving algorithm," Astron. Astrophys. 399, 807-811 (2003).
    [CrossRef]
  13. B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Filtering, T. Huang, ed., (Springer-Verlag, NY, 1975), 177-248.
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    [CrossRef]
  15. J. Hogboom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astron. Astrophys. Supp. 15, 417-426 (1974).
  16. D. Fried, "Analysis of the CLEAN algorithm and implications for superresolution," J. Opt. Soc. Am. A 12, 853-860 (1995).
    [CrossRef]
  17. P. Magain, F. Courbin, and S. Sohy, "Deconvolution with correct sampling," Astrophys. J. 494, 472-477 (1998).
    [CrossRef]
  18. F. Pijpers, "Unbiased image reconstruction as an inverse problem," Mon. Not. Roy. Astron. Soc. 307, 659-668 (1999).
    [CrossRef]
  19. R. Puetter and R. Hier, "Pixon sub-diffraction space imaging," Proc. SPIE 7094, 709405-709405-12 (2008).
  20. J. Starck, E. Pantin, and F. Murtagh, "Deconvolution in astronomy: A review," Publ. Astron. Soc. Pacific 114, 1051-1069 (2002).
    [CrossRef]
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    [CrossRef]
  25. M. Bertero and C. De Mol, "Superresolution by data inversion," Progress in Optics 36, 129-178 (1996).
    [CrossRef]
  26. C. Matson and D. Tyler, "Primary and secondary superresolution by data inversion," Opt. Express 14, 456-473 (2006).
    [CrossRef] [PubMed]
  27. D. Robinson and P. Milanfar, "Statistical performance analysis of superresolution," IEEE Trans. Image Process. 15, 1413-1428 (2006).
    [CrossRef] [PubMed]
  28. P. Sementilli, B. Hunt, and M. Nader, "Analysis of of the limit of superresolution in incoherent imaging," J. Opt. Soc. Am. A 10, 2265-2276 (1993).
    [CrossRef]
  29. H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

2007 (1)

2006 (2)

C. Matson and D. Tyler, "Primary and secondary superresolution by data inversion," Opt. Express 14, 456-473 (2006).
[CrossRef] [PubMed]

D. Robinson and P. Milanfar, "Statistical performance analysis of superresolution," IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

2003 (1)

E. Boukouvala and A. Lettington, "Restoration of astronomical images by an iterative superresolving algorithm," Astron. Astrophys. 399, 807-811 (2003).
[CrossRef]

2002 (1)

J. Starck, E. Pantin, and F. Murtagh, "Deconvolution in astronomy: A review," Publ. Astron. Soc. Pacific 114, 1051-1069 (2002).
[CrossRef]

1999 (1)

F. Pijpers, "Unbiased image reconstruction as an inverse problem," Mon. Not. Roy. Astron. Soc. 307, 659-668 (1999).
[CrossRef]

1998 (2)

1996 (1)

M. Bertero and C. De Mol, "Superresolution by data inversion," Progress in Optics 36, 129-178 (1996).
[CrossRef]

1995 (2)

D. Fried, "Analysis of the CLEAN algorithm and implications for superresolution," J. Opt. Soc. Am. A 12, 853-860 (1995).
[CrossRef]

S. Plevritis and A. Macovski, "Spectral extrapolation of spatially bounded images," IEEE Trans. Medical Imaging 14, 487-497 (1995).
[CrossRef]

1993 (1)

1989 (1)

1988 (1)

1986 (1)

R. Narayan and R. Nityananda,"Maximum entropy image restoration in astronomy," Ann. Rev. Astron. Astrophys. 24, 127-170 (1986).
[CrossRef]

1982 (2)

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction in positron emission tomography," IEEE Trans. Medical Imaging 1, 113-122 (1982).
[CrossRef]

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis: I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

1975 (1)

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

1974 (3)

L. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974)
[CrossRef]

R. Gerchberg, "Superresolution through error energy reduction," Opt. Acta 21, 709-721 (1974).
[CrossRef]

J. Hogboom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astron. Astrophys. Supp. 15, 417-426 (1974).

1972 (1)

1969 (1)

1968 (1)

Bertero, M.

M. Bertero and C. De Mol, "Superresolution by data inversion," Progress in Optics 36, 129-178 (1996).
[CrossRef]

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis: I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

Boukouvala, E.

E. Boukouvala and A. Lettington, "Restoration of astronomical images by an iterative superresolving algorithm," Astron. Astrophys. 399, 807-811 (2003).
[CrossRef]

Conchello, J.-A.

Courbin, F.

P. Magain, F. Courbin, and S. Sohy, "Deconvolution with correct sampling," Astrophys. J. 494, 472-477 (1998).
[CrossRef]

De Mol, C.

M. Bertero and C. De Mol, "Superresolution by data inversion," Progress in Optics 36, 129-178 (1996).
[CrossRef]

Fried, D.

Gerchberg, R.

R. Gerchberg, "Superresolution through error energy reduction," Opt. Acta 21, 709-721 (1974).
[CrossRef]

Harris, R.

Hogboom, J.

J. Hogboom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astron. Astrophys. Supp. 15, 417-426 (1974).

Holmes, T.

Hunt, B.

Lettington, A.

E. Boukouvala and A. Lettington, "Restoration of astronomical images by an iterative superresolving algorithm," Astron. Astrophys. 399, 807-811 (2003).
[CrossRef]

Liu, Y.-H.

Lucy, L.

L. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974)
[CrossRef]

Macovski, A.

S. Plevritis and A. Macovski, "Spectral extrapolation of spatially bounded images," IEEE Trans. Medical Imaging 14, 487-497 (1995).
[CrossRef]

Magain, P.

P. Magain, F. Courbin, and S. Sohy, "Deconvolution with correct sampling," Astrophys. J. 494, 472-477 (1998).
[CrossRef]

Matson, C.

Milanfar, P.

D. Robinson and P. Milanfar, "Statistical performance analysis of superresolution," IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

Murtagh, F.

J. Starck, E. Pantin, and F. Murtagh, "Deconvolution in astronomy: A review," Publ. Astron. Soc. Pacific 114, 1051-1069 (2002).
[CrossRef]

Nader, M.

Narayan, R.

R. Narayan and R. Nityananda,"Maximum entropy image restoration in astronomy," Ann. Rev. Astron. Astrophys. 24, 127-170 (1986).
[CrossRef]

Nityananda, R.

R. Narayan and R. Nityananda,"Maximum entropy image restoration in astronomy," Ann. Rev. Astron. Astrophys. 24, 127-170 (1986).
[CrossRef]

Pantin, E.

J. Starck, E. Pantin, and F. Murtagh, "Deconvolution in astronomy: A review," Publ. Astron. Soc. Pacific 114, 1051-1069 (2002).
[CrossRef]

Papoulis, A.

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

Pijpers, F.

F. Pijpers, "Unbiased image reconstruction as an inverse problem," Mon. Not. Roy. Astron. Soc. 307, 659-668 (1999).
[CrossRef]

Pike, E. R.

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis: I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

Plevritis, S.

S. Plevritis and A. Macovski, "Spectral extrapolation of spatially bounded images," IEEE Trans. Medical Imaging 14, 487-497 (1995).
[CrossRef]

Prasad, S.

Richardson, W.

Robinson, D.

D. Robinson and P. Milanfar, "Statistical performance analysis of superresolution," IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

Rushforth, C.

Sementilli, P.

Shepp, L.

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction in positron emission tomography," IEEE Trans. Medical Imaging 1, 113-122 (1982).
[CrossRef]

Sohy, S.

P. Magain, F. Courbin, and S. Sohy, "Deconvolution with correct sampling," Astrophys. J. 494, 472-477 (1998).
[CrossRef]

Starck, J.

J. Starck, E. Pantin, and F. Murtagh, "Deconvolution in astronomy: A review," Publ. Astron. Soc. Pacific 114, 1051-1069 (2002).
[CrossRef]

Toraldo Di Francia, G.

Tyler, D.

Vardi, Y.

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction in positron emission tomography," IEEE Trans. Medical Imaging 1, 113-122 (1982).
[CrossRef]

Ann. Rev. Astron. Astrophys. (1)

R. Narayan and R. Nityananda,"Maximum entropy image restoration in astronomy," Ann. Rev. Astron. Astrophys. 24, 127-170 (1986).
[CrossRef]

Appl. Opt. (1)

Astron. Astrophys. (2)

J. Hogboom, "Aperture synthesis with a non-regular distribution of interferometer baselines," Astron. Astrophys. Supp. 15, 417-426 (1974).

E. Boukouvala and A. Lettington, "Restoration of astronomical images by an iterative superresolving algorithm," Astron. Astrophys. 399, 807-811 (2003).
[CrossRef]

Astron. J. (1)

L. Lucy, "An iterative technique for the rectification of observed distributions," Astron. J. 79, 745-754 (1974)
[CrossRef]

Astrophys. J. (1)

P. Magain, F. Courbin, and S. Sohy, "Deconvolution with correct sampling," Astrophys. J. 494, 472-477 (1998).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

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

IEEE Trans. Image Process. (1)

D. Robinson and P. Milanfar, "Statistical performance analysis of superresolution," IEEE Trans. Image Process. 15, 1413-1428 (2006).
[CrossRef] [PubMed]

IEEE Trans. Medical Imaging (2)

S. Plevritis and A. Macovski, "Spectral extrapolation of spatially bounded images," IEEE Trans. Medical Imaging 14, 487-497 (1995).
[CrossRef]

L. Shepp and Y. Vardi, "Maximum likelihood reconstruction in positron emission tomography," IEEE Trans. Medical Imaging 1, 113-122 (1982).
[CrossRef]

J. Opt. Soc. Am. (3)

J. Opt. Soc. Am. A (5)

Mon. Not. Roy. Astron. Soc. (1)

F. Pijpers, "Unbiased image reconstruction as an inverse problem," Mon. Not. Roy. Astron. Soc. 307, 659-668 (1999).
[CrossRef]

Opt. Acta (2)

R. Gerchberg, "Superresolution through error energy reduction," Opt. Acta 21, 709-721 (1974).
[CrossRef]

M. Bertero and E. R. Pike, "Resolution in diffraction-limited imaging, a singular value analysis: I. The case of coherent illumination," Opt. Acta 29, 727-746 (1982).
[CrossRef]

Opt. Express (1)

Progress in Optics (1)

M. Bertero and C. De Mol, "Superresolution by data inversion," Progress in Optics 36, 129-178 (1996).
[CrossRef]

Publ. Astron. Soc. Pacific (1)

J. Starck, E. Pantin, and F. Murtagh, "Deconvolution in astronomy: A review," Publ. Astron. Soc. Pacific 114, 1051-1069 (2002).
[CrossRef]

Other (5)

R. Puetter and R. Hier, "Pixon sub-diffraction space imaging," Proc. SPIE 7094, 709405-709405-12 (2008).

B. R. Frieden, "Image enhancement and restoration," in Picture Processing and Digital Filtering, T. Huang, ed., (Springer-Verlag, NY, 1975), 177-248.

S. Prasad, "Digital and optical superresolution of low-resolution image sequences," Proc. SPIE 6712, 67120E 1-11 (2007).

B. R. Frieden, "Evaluation, design, and extrapolation methods for optical signals based on the use of the prolate functions," in Progress in Optics, E. Wolf, ed., (North-Holland, Amsterdam, 1971), Vol. 9, 313-407.

H. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968).

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Figures (8)

Fig. 1.
Fig. 1.

Normalized CRB for a Fourier sample plotted vs the sample index, . The different points, indicated by different marker symbols and joined by solid line segments, in each plot correspond to different numbers of Fourier samples that are jointly estimated, as indicated by the maximum index value, max , at the rightmost point. The dashed line segments join points obtained analytically by means of formula (20) valid for large Q and DR frequencies.

Fig. 2.
Fig. 2.

Normalized CRB for a Fourier sample plotted vs the index, , of the sample in the bright-source limit (SNR=103), all other parameters being the same as in Fig. 1.

Fig. 3.
Fig. 3.

Normalized CRB for a Fourier sample plotted vs its index, , in the bright-source limit, for different widths of the Gaussian source and Q=20, χ=1, M=1, SNR=103. The marker symbols have the same meaning as in Fig. 1.

Fig. 4.
Fig. 4.

Normalized CRB for a Fourier sample plotted vs its index, , in the mixed-noise case, for different widths of the Gaussian source and Q=20, χ=1, M=1, SNR=10. The marker symbols have the same meaning as in Fig. 1.

Fig. 5.
Fig. 5.

Same as Figs. 1, except M=5

Fig. 6.
Fig. 6.

CRB for a Fourier sample plotted vs the index, , of the sample for Q=20 and SNR=103, for different values of χ and M. The marker symbols have the same meaning as in Fig. 1.

Fig. 7.
Fig. 7.

A sinusoidal object with three pure frequencies, 8, 9, and 11, and its image for Q=20

Fig. 8.
Fig. 8.

Plots of normalized CRB for estimating image-domain pixel intensity vs pixel position, for Q=20, SNR=103, and six different values of the maximum spatial-frequency index, max , from (Q-2) to (Q+3), as indicated in the legend. Figures (b)–(d) refer to ×2 undersampled detection (χ=2) for M equal to 2, 3, and 10, respectively, while (a) refers to a single critically sampled image.

Equations (50)

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

g ( x ) = h 0 ( x x ) f ( x ) d x .
g k = p ( x x k ) g ( x ) dx + n k
= d x d x p ( x x k ) h 0 ( x x ) f ( x ) + n k ,
g k ( m ) = du exp [ i 2 π u ( x k t m ) ] P ( u ) H 0 ( u ) F ( u ) + n k ( m ) ,
g k ( m ) = χ 2 1 1 d u ¯ exp [ i π u ¯ ( x ¯ k t ¯ m ) ] sinc ( u ¯ χ 2 ) ( 1 u ¯ ) F ( B 0 u ¯ ) + n k ( m ) ,
f ( x ) = 1 2 L = F exp ( i π x L ) , x [ L , L ] ,
F = L L f ( x ) exp { i 2 π [ ( 2 L ) ] x } d x .
F ( u ) = = F sinc ( 2 u L )
g k ( m ) = = c k ( m ) F + n k ( m ) ,
c k ( m ) = χ 2 1 1 d u ( 1 u ) sinc ( χ u 2 ) exp { i π u [ χ k t ¯ m ] } sinc ( Q ( u Q ) ) ,
J = l n P F ln P F * = 2 ln P F F * ,
J = 1 σ D 2 m , k c k ( m ) c k ( m ) * .
J = 1 σ D 2 m , k c ¯ k ( m ) c ¯ k ( m ) ,
J m , k c ¯ k ( m ) c ¯ k ( m ) σ D 2 + g k ( m ) .
c k ( m ) χ 2 ( 1 Q ) sinc ( χ 2 Q ) exp [ i π ( Q ) χ ( k t ¯ m ) ] d v exp [ i π v ( χ k t ¯ m ) ] sinc ( Q v ) .
sinc ( Q v ) = 1 2 1 1 d w exp ( i Q v π w ) ,
1 Q = 1 1 d w δ ( w χ Q ( k t ˜ m ) ) ,
c k ( m ) χ 2 Q ( 1 Q ) sinc ( χ 2 Q ) exp [ i π ( Q ) χ ( k t ˜ m ) ] Θ ( 1 χ Q k t ˜ m ) .
J χ 2 4 Q 2 ( 1 Q ) ( 1 Q ) sinc ( χ 2 Q ) sinc ( χ 2 Q )
× m , k exp [ i π ( ) χ ( k t ˜ m ) Q ] σ D 2 + g k ( m ) ,
J χ M 2 Q σ D 2 ( 1 Q ) 2 sinc 2 ( χ 2 Q ) δ , , < Q .
CRB ( ) 1 J
2 Q σ D 2 χ M ( 1 Q ) 2 sinc 2 ( χ 2 Q ) , , < Q ,
J ( 1 ) χ M 4 π 2 Q 2 σ D 2 1 1 du ( 1 u ) 2 sinc 2 ( χ u 2 ) ( u / Q ) ( u / Q ) , , Q 1 .
f ( x ) = K 2 π w 2 L 2 exp [ x 2 / ( 2 w 2 L 2 ) ] , x [ L , L ] ,
j k k = = max max 1 = max max 1 exp [ i 2 π ( k k ) / ( 2 max ) ] J ,
= max max 1 exp = [ i 2 π ( k k ) / ( 2 max ) ] = 2 max δ k k , k , k = max , . . . , max 1 ,
C k ( ph ) = 1 ( 2 max ) 2 = max max 1 = max max 1 exp [ i 2 π ( ) k / ( 2 max ) ] J 1 .
f ( x ) = f 0 + f 1 sin ( 2 π v 1 x ) + f 2 sin ( 2 π v 2 x ) + f 3 sin ( 2 π v 3 x ) ,
Y = X + N ,
Q Y ( λ ) = exp [ i λ X λ 2 ( σ 2 + X ) / 2 ] exp [ X n 3 ( i λ ) n / n ! ] .
P Y ( y | X ) = 1 2 π Q Y ( λ ) e i λ y d λ
1 2 π e i λ ( y X ) λ 2 ( σ 2 + X ) / 2 d λ
= 1 2 π ( σ 2 + X ) e ( y X ) 2 / [ 2 ( σ 2 + X ) ] ,
J ( Y | X ) = 2 In P Y ( y | X ) X 2
= 1 σ 2 + X + 1 2 ( σ 2 + X ) 2 1 σ 2 + X ,
J ( Y λ ̲ ) = 2 In P Y ( y X ) λ λ
= X λ · J ( Y X ) · X λ
X λ · 1 σ 2 + X · X λ .
J = X 2 4 σ D 2 1 1 du 1 1 du ( 1 u ) ( 1 u ) sinc ( χ u / 2 ) sinc ( χ u / 2 )
× sinc ( Q u ) sinc ( Q u ) S ( u ) ,
S ( u u ) = m = 0 M 1 k = Q / χ Q / χ 1 exp [ ( u u ' ) χ ( k t ˜ m ) ]
S ( u u ) = { 1 exp [ i π ( u u ) χ ] 1 exp [ ( u u ) χ / M ] } · exp [ i π ( u u ) Q ] { 1 exp [ i 2 π Q ( u u ) ] 1 exp [ i π ( u u ) χ ] }
= exp [ ( u u ) χ ( 1 1 2 M ) ] sin [ π Q ( u u ) ] sin [ π χ 2 M ( u u ) ] ,
J = M χ 2 σ D 2 1 1 d u ( 1 u ) 2 sinc 2 ( χ u / 2 ) sinc ( Q u ) sinc ( Q u ) .
sinc 2 ( Q u ) du = 1 Q ,
sin ( Q u ) sinc ( Q u ) = sin [ π ( Q u ) ] sin [ π ( Qu ) ] π 2 ( Q u ) ( Q u )
= cos [ π ( ) ] cos [ 2 π Q u ( + ) π ] 2 π 2 ( Q u ) ( Q u )
J M χ 4 π 2 σ D 2 ( 1 ) 1 1 du ( 1 u ) 2 sinc 2 ( χ u / 2 ) ( Q u ) ( Q u ) ,
J M χ 2 Q σ D 2 ( 1 2 / Q 2 ) sinc 2 ( χ 2 Q ) sinc ( ) = 0 ,

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