Abstract

Superresolution by data inversion is the extrapolation of measured Fourier data to regions outside the measurement bandwidth using postprocessing techniques. Here we characterize superresolution by data inversion for objects with finite support using the twin concepts of primary and secondary superresolution, where primary superresolution is the essentially unbiased portion of the superresolved spectra and secondary superresolution is the remainder. We show that this partition of superresolution into primary and secondary components can be used to explain why some researchers believe that meaningful superresolution is achievable with realistic signal-to-noise ratios, and other researchers do not.

© 2006 Optical Society of America

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References

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  1. M. Bertero and C. De Mol, "Superresolution by data inversion," in Progress in Optics XXXVI, E. Wolf, ed. (Elsevier, Amsterdam, 1996), 129-178
  2. S. Bhattacharjee and M. K. Sundareshan, "Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution," J. Opt. Soc. Am. A 20, 1516-1527 (2003).
    [CrossRef]
  3. B. R. Hunt, "Super-resolution of images: algorithms, principles, and performance," Int. J. Imaging Syst. Technol. 6, 297-304 (1995).
    [CrossRef]
  4. H. Liu, Y. Yan, Q. Tan, and G. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," J. Opt. Soc. Am. A 19, 2185-2193 (2002).
    [CrossRef]
  5. V. F. Canales, D. M. de Juana, and M. P. Cagigal, "Superresolution in compensated telescopes," Opt. Lett. 29, 935-937 (2004).
    [CrossRef] [PubMed]
  6. C. K. Rushforth and R. W. Harris, "Restoration, resolution, and noise," J. Opt. Soc. Am. 58, 539-545 (1968).
    [CrossRef]
  7. J. J. Green and B. R. Hunt, "Improved restoration of space object imagery," J. Opt. Soc. Am. A 16, 2859-2865 (1999).
    [CrossRef]
  8. B. R. Frieden, "Evaluation, design, and extrapolation methods for optical signals based on the use of prolate functions," in Progress in Optics IX, E. Wolf, ed. (North-Holland, Amsterdam, 1971), 313-407
  9. D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - I," Bell Syst. Tech. J. 40, 43-63 (1961).
  10. H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - II," Bell Syst. Tech. J. 40, 65-84 (1961).
  11. 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]
  12. M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, 1996), 49.
  13. W. P. Latham and M. L. Tilton, "Calculation of prolate functions for optical analysis," Appl. Opt. 26, 2653-2658 (1987).
    [CrossRef] [PubMed]
  14. B. Porat, Digital Processing of Random Signals, Theory and Methods (Prentice-Hall, Englewood Cliffs, 1994), 65-67.
  15. R. C. Gonzalez and R. E. Woods, Digital image processing (Addison-Wesley, Reading, 1992), chap. 5.
  16. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN, 2nd ed.,(Cambridge Press, Cambridge, 1996), 134-135.
  17. C. L. Matson, "Variance reduction in Fourier spectra and their corresponding images with the use of support constraints," J. Opt. Soc. Am. A 11, 97-106 (1994).
    [CrossRef]
  18. P. J. Sementilli, B. R. Hunt, and M. S. Nadar, "Analysis of the limit to superresolution in coherent imaging," J. Opt. Soc. Am. A 10, 2265-2276 (1993).
    [CrossRef]
  19. C. L. Matson, "Fourier spectrum extrapolation and enhancement using support constraints," IEEE Trans. Signal Process. 42, 156-163 (1994).
    [CrossRef]
  20. Y. L. Kosarev, "On the superresolution limit in signal reconstruction," Sov. J. Commun. Technol. Electron. 35, 90-108 (1990).

Appl. Opt. (1)

W. P. Latham and M. L. Tilton, "Calculation of prolate functions for optical analysis," Appl. Opt. 26, 2653-2658 (1987).
[CrossRef] [PubMed]

Bell Syst. Tech. J. (2)

D. Slepian and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - I," Bell Syst. Tech. J. 40, 43-63 (1961).

H. J. Landau and H. O. Pollak, "Prolate spheroidal wave functions, Fourier analysis and uncertainty - II," Bell Syst. Tech. J. 40, 65-84 (1961).

IEEE Trans. Signal Process. (1)

C. L. Matson, "Fourier spectrum extrapolation and enhancement using support constraints," IEEE Trans. Signal Process. 42, 156-163 (1994).
[CrossRef]

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

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

J. Opt. Soc. Am. (2)

H. Liu, Y. Yan, Q. Tan, and G. Jin, "Theories for the design of diffractive superresolution elements and limits of optical superresolution," J. Opt. Soc. Am. A 19, 2185-2193 (2002).
[CrossRef]

C. K. Rushforth and R. W. Harris, "Restoration, resolution, and noise," J. Opt. Soc. Am. 58, 539-545 (1968).
[CrossRef]

J. Opt. Soc. Am. A (4)

J. J. Green and B. R. Hunt, "Improved restoration of space object imagery," J. Opt. Soc. Am. A 16, 2859-2865 (1999).
[CrossRef]

C. L. Matson, "Variance reduction in Fourier spectra and their corresponding images with the use of support constraints," J. Opt. Soc. Am. A 11, 97-106 (1994).
[CrossRef]

P. J. Sementilli, B. R. Hunt, and M. S. Nadar, "Analysis of the limit to superresolution in coherent imaging," J. Opt. Soc. Am. A 10, 2265-2276 (1993).
[CrossRef]

S. Bhattacharjee and M. K. Sundareshan, "Mathematical extrapolation of image spectrum for constraint-set design and set-theoretic superresolution," J. Opt. Soc. Am. A 20, 1516-1527 (2003).
[CrossRef]

Opt. Acta (1)

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. Lett. (1)

V. F. Canales, D. M. de Juana, and M. P. Cagigal, "Superresolution in compensated telescopes," Opt. Lett. 29, 935-937 (2004).
[CrossRef] [PubMed]

Progress in Optics IX (1)

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

Progress in Optics XXXVI (1)

M. Bertero and C. De Mol, "Superresolution by data inversion," in Progress in Optics XXXVI, E. Wolf, ed. (Elsevier, Amsterdam, 1996), 129-178

Sov. J. Commun. Technol. Electron. (1)

Y. L. Kosarev, "On the superresolution limit in signal reconstruction," Sov. J. Commun. Technol. Electron. 35, 90-108 (1990).

Other (4)

M. C. Roggemann and B. Welsh, Imaging Through Turbulence (CRC Press, Boca Raton, 1996), 49.

B. Porat, Digital Processing of Random Signals, Theory and Methods (Prentice-Hall, Englewood Cliffs, 1994), 65-67.

R. C. Gonzalez and R. E. Woods, Digital image processing (Addison-Wesley, Reading, 1992), chap. 5.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in FORTRAN, 2nd ed.,(Cambridge Press, Cambridge, 1996), 134-135.

Supplementary Material (4)

» Media 1: MOV (191 KB)     
» Media 2: MOV (87 KB)     
» Media 3: MOV (89 KB)     
» Media 4: MOV (37 KB)     

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Figures (13)

Fig. 1.
Fig. 1.

A movie of PSWF Fourier amplitudes plotted as a function of frequency for a SBP of 40. The frequency axis is normalized to one at the edge of the measurement bandwidth. Each movie frame corresponds to a PSWF for the PSWF index indicated in the movie. [Media 1]

Fig. 2.
Fig. 2.

A plot of the PSWF eigenvalues associated with the PSWFs in Fig. 1.

Fig. 3.
Fig. 3.

A movie of the Fourier variances defined by Eq.(3) for a SBP of 40 plotted as a function of frequency normalized to one at the edge of the measurement bandwidth. Each movie frame corresponds to increasing the finite summation limit in Eq.(3) by one. The plots on the left are for all frequencies included in the calculations, while the plots on the right are a magnified version of the plots on the left for frequencies within the measurement bandwidth. [Media 2]

Fig. 4.
Fig. 4.

A plot of the FIM eigenvalues.

Fig. 5.
Fig. 5.

A movie of the CRBs for any unbiased estimate of any Fourier spectra for a SBP of 40 plotted as a function of frequency normalized to one at the edge of the measurement bandwidth. Each movie frame corresponds to adding one additional eigenvalue to the pseudoinverse of the FIM. The plots on the left are for all frequencies included in the calculations, while the plots on the right are a magnified version of the plots on the left for frequencies within the measurement bandwidth. These plots are independent of the actual Fourier spectra under consideration because the measurement noise is signal independent. [Media 3]

Fig. 6.
Fig. 6.

A movie of the total energy of the PSWFs included in the reconstruction process (normalized to one) as a function of the PSWF index for a SBP of 40. The dotted line corresponds to a value of 0.98. Each frame of the movie corresponds to a different superresolved frequency as indicated in the movie. [Media 4]

Fig. 7.
Fig. 7.

Plots of the increases in the sampling degrees of freedom in superresolved reconstructions brought about by including primary superresolution spectra. The dot-dash line is for a SBP of 1.27, the dashed line is for a SBP of 10 , the solid line is for a SBP of 20, and the dotted line is for a SBP of 40.

Fig. 8.
Fig. 8.

Plots of the Fourier amplitudes of a triple star for the parameters listed in the text. The solid line is a plot of the true Fourier amplitudes and the dotted line corresponds to a reconstruction of the true Fourier amplitudes using Eq.(2) with an upper limit of 46 and a SBP of 40. The frequency axis is normalized to one at the edge of the measurement bandwidth, and the dashed line indicates the upper boundary of the primary superresolution region.

Fig. 9.
Fig. 9.

Plots of the increases in the PSWF-defined degrees of freedom averaged over the entire support region brought about by including both primary and secondary superresolution in the reconstruction process. The dot-dash line is for a SBP of 1.27, the dashed line is for a SBP of 10, the solid line is for a SBP of 20, and the dotted line is for a SBP of 40. The asterisks denote values calculated in [1] for a SBP of 1.27.

Fig. 10.
Fig. 10.

Plot of the PSWF with index 65 for a SBP=40 demonstrating that the spacings of zeros for superresolving PSWFs in the support region decrease as the location in the support gets close to the support boundaries. Notice that the zeros are relatively evenly spaced in the center 75% of the object support.

Fig. 11.
Fig. 11.

Plots of the increases in the PSWF-defined degrees of freedom averaged over the center 75% of the support region brought about by including both primary and secondary superresolution in the reconstruction process. The dot-dash line is for a SBP of 1.27, the dashed line is for a SBP of 10, the solid line is for a SBP of 20, and the dotted line is for a SBP of 40. The vertical axis is the number of degrees of freedom that would be added to the entire support region if the resolution increase calculated in the center of the support region is the same throughout the support region.

Fig. 12.
Fig. 12.

Plots of the increases in the PSWF-defined degrees of freedom averaged over the 25% of the support region located at its edges brought about by including both primary and secondary superresolution in the reconstruction process. The dot-dash line is for a SBP of 1.27, the dashed line is for a SBP of 10 , the solid line is for a SBP of 20, and the dotted line is for a SBP of 40. The vertical axis is the number of degrees of freedom that would be added to the entire support region if the resolution increase calculated at the edges of the support region is the same throughout the support region.

Fig. 13.
Fig. 13.

Plots of reconstructions of the triple star whose Fourier amplitudes are shown in Fig. 8, where the solid line is a reconstruction using just the Fourier data inside the measurement bandwidth, the dashed line is a reconstruction using both the measured Fourier data and the primary superresolution part of the superresolved spectrum, and the dotted line is a reconstruction using the measured Fourier data and all of the superresolved spectrum.

Equations (9)

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i ( x ) = h ( x α ) o ( α ) + n ( x ) ,
O e ( f ) = m = 0 M λ m 1 φ m ( f ) f o f o O ( ξ ) φ m ( ξ ) + m = 0 M λ m 1 φ m ( f ) f o f o N ( ξ ) φ m ( ξ ) .
var { O e ( f ) } = Δ E [ O e ( f ) O e * ( f ) ] E [ O e ( f ) ] 2
= σ 2 m = 0 M λ m 1 φ m 2 ( f ) .
y = + η ,
f ( y ; θ ) = f η ( y ) ,
F p , q ( θ ) = E [ ln f ( y ; θ ) θ p ln f ( y ; θ ) θ q ] ,
F p , q ( o ) = 1 σ 2 k { ( ) k θ p ( ) k θ q }
= 1 σ 2 k h ( α k α p ) h ( α k α q )

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