Abstract

Ultimately, the band-limited nature of imaging systems restricts image quality in measured data. However, prior knowledge can be employed to improve image quality beyond that available from measured data. Kinds of prior knowledge include knowledge of the support of the object and knowledge that the object has only positive intensities. In previous work it has been shown that prior knowledge increases image quality by two means: superresolution and improvements in the signal-to-noise ratio in the Fourier domain. However, after prior knowledge is enforced, the resulting filter that multiplies the Fourier data may unduly limit resolution in the constrained image. Here maximum achievable resolutions are derived for one- and two-dimensional filters. In addition, it is shown that requiring a signal to be positive results in lowering its maximum achievable resolution by as much as a factor of 2. As a result, algorithms that use positivity to improve the quality of Fourier-domain data may benefit from a final postprocessing step to increase resolution.

© 1998 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
    [CrossRef]
  2. M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
    [CrossRef]
  3. H. Stark, ed., Image Recovery: Theory and Application (Academic, Boston, Mass., 1987).
  4. D. L. Fried, “Analysis of the CLEAN algorithm and implications for superresolution,” J. Opt. Soc. Am. A 12, 853–860 (1995).
    [CrossRef]
  5. N. Miura, N. Baba, “Superresolution for a nonnegative band-limited image,” Opt. Lett. 21, 1174–1176 (1996).
    [CrossRef] [PubMed]
  6. E. M. Haacke, Z. Liang, S. H. Izen, “Superresolution reconstruction through object modeling and parameter estimation,” IEEE Trans. Acoust. Speech Signal Process. 37, 592–595 (1989).
    [CrossRef]
  7. Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. Statist. Soc. B. 55, 569–612 (1993).
  8. G. E. B. Archer, D. M. Titterington, “The iterative image space reconstruction algorithm (ISRA) as an alternative to the EM algorithm for solving positive linear inverse problems,” Stat. Sin. 5, 77–96 (1995).
  9. A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, 2nd ed. (Wiley, New York, 1980), p. 348.
  10. C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Acoust. Speech, Signal Process. 42, 156–163 (1994).
    [CrossRef]
  11. 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]
  12. C. L. Matson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3233–3236 (1994).
    [CrossRef]
  13. C. L. Matson, “The role of positivity for error reduction in images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 766–777 (1994).
    [CrossRef]
  14. A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
    [CrossRef]
  15. A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraint—interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
    [CrossRef]
  16. P. J. Sementilli, B. R. Hunt, M. S. Nadar, “Analysis of the limit to superresolution in incoherent imaging,” J. Opt. Soc. Am. A 10, 2265–2276 (1993).
    [CrossRef]
  17. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  18. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991), pp. 247–250 and 305.
  19. T. Wilson, S. J. Hewlett, “Superresolution in confocal scanning microscopy,” Opt. Lett. 16, 1062–1064 (1991).
    [CrossRef] [PubMed]
  20. T. D. Milster, C. H. Curtis, “Analysis of superresolution in magneto-optic data storage devices,” Appl. Opt. 31, 6272–6279 (1992).
    [CrossRef] [PubMed]
  21. T. Suhara, H. Nishihara, “Theoretical analysis of super-resolution readout of disc data by semiconfocal pickup heads,” Jpn. J. Appl. Phys. 31, 534–541 (1992).
    [CrossRef]
  22. F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
    [CrossRef]

1996

1995

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

G. E. B. Archer, D. M. Titterington, “The iterative image space reconstruction algorithm (ISRA) as an alternative to the EM algorithm for solving positive linear inverse problems,” Stat. Sin. 5, 77–96 (1995).

1994

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Acoust. Speech, Signal Process. 42, 156–163 (1994).
[CrossRef]

C. L. Matson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3233–3236 (1994).
[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]

1993

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

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. Statist. Soc. B. 55, 569–612 (1993).

1992

T. Suhara, H. Nishihara, “Theoretical analysis of super-resolution readout of disc data by semiconfocal pickup heads,” Jpn. J. Appl. Phys. 31, 534–541 (1992).
[CrossRef]

T. D. Milster, C. H. Curtis, “Analysis of superresolution in magneto-optic data storage devices,” Appl. Opt. 31, 6272–6279 (1992).
[CrossRef] [PubMed]

1991

1990

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

1989

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

E. M. Haacke, Z. Liang, S. H. Izen, “Superresolution reconstruction through object modeling and parameter estimation,” IEEE Trans. Acoust. Speech Signal Process. 37, 592–595 (1989).
[CrossRef]

1987

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraint—interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

1978

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Archer, G. E. B.

G. E. B. Archer, D. M. Titterington, “The iterative image space reconstruction algorithm (ISRA) as an alternative to the EM algorithm for solving positive linear inverse problems,” Stat. Sin. 5, 77–96 (1995).

Baba, N.

Casanove, M. J.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraint—interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

Curtis, C. H.

Demoment, G.

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

Fried, D. L.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

Haacke, E. M.

E. M. Haacke, Z. Liang, S. H. Izen, “Superresolution reconstruction through object modeling and parameter estimation,” IEEE Trans. Acoust. Speech Signal Process. 37, 592–595 (1989).
[CrossRef]

Harris, F. J.

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Hewlett, S. J.

Hunt, B. R.

Izen, S. H.

E. M. Haacke, Z. Liang, S. H. Izen, “Superresolution reconstruction through object modeling and parameter estimation,” IEEE Trans. Acoust. Speech Signal Process. 37, 592–595 (1989).
[CrossRef]

Lannes, A.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraint—interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

Lay, D. C.

A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, 2nd ed. (Wiley, New York, 1980), p. 348.

Lee, D.

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. Statist. Soc. B. 55, 569–612 (1993).

Liang, Z.

E. M. Haacke, Z. Liang, S. H. Izen, “Superresolution reconstruction through object modeling and parameter estimation,” IEEE Trans. Acoust. Speech Signal Process. 37, 592–595 (1989).
[CrossRef]

Matson, C. L.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Acoust. Speech, Signal Process. 42, 156–163 (1994).
[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]

C. L. Matson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3233–3236 (1994).
[CrossRef]

C. L. Matson, “The role of positivity for error reduction in images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 766–777 (1994).
[CrossRef]

Milster, T. D.

Miura, N.

Nadar, M. S.

Nishihara, H.

T. Suhara, H. Nishihara, “Theoretical analysis of super-resolution readout of disc data by semiconfocal pickup heads,” Jpn. J. Appl. Phys. 31, 534–541 (1992).
[CrossRef]

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991), pp. 247–250 and 305.

Roques, S.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraint—interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

Sementilli, P. J.

Sezan, M. I.

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Suhara, T.

T. Suhara, H. Nishihara, “Theoretical analysis of super-resolution readout of disc data by semiconfocal pickup heads,” Jpn. J. Appl. Phys. 31, 534–541 (1992).
[CrossRef]

Taylor, A. E.

A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, 2nd ed. (Wiley, New York, 1980), p. 348.

Tekalp, A. M.

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Titterington, D. M.

G. E. B. Archer, D. M. Titterington, “The iterative image space reconstruction algorithm (ISRA) as an alternative to the EM algorithm for solving positive linear inverse problems,” Stat. Sin. 5, 77–96 (1995).

Vardi, Y.

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. Statist. Soc. B. 55, 569–612 (1993).

Wilson, T.

Appl. Opt.

IEEE Trans. Acoust. Speech Signal Process.

E. M. Haacke, Z. Liang, S. H. Izen, “Superresolution reconstruction through object modeling and parameter estimation,” IEEE Trans. Acoust. Speech Signal Process. 37, 592–595 (1989).
[CrossRef]

G. Demoment, “Image reconstruction and restoration: overview of common estimation structures and problems,” IEEE Trans. Acoust. Speech Signal Process. 37, 2024–2036 (1989).
[CrossRef]

IEEE Trans. Acoust. Speech, Signal Process.

C. L. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Acoust. Speech, Signal Process. 42, 156–163 (1994).
[CrossRef]

J. Mod. Opt.

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. I. Partial deconvolution and spectral extrapolation with limited field,” J. Mod. Opt. 34, 161–226 (1987).
[CrossRef]

A. Lannes, S. Roques, M. J. Casanove, “Stabilized reconstruction in signal and image processing. II. Iterative reconstruction with and without constraint—interactive implementation,” J. Mod. Opt. 34, 321–370 (1987).
[CrossRef]

J. Opt. Soc. Am. A

J. Statist. Soc. B.

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. Statist. Soc. B. 55, 569–612 (1993).

Jpn. J. Appl. Phys.

T. Suhara, H. Nishihara, “Theoretical analysis of super-resolution readout of disc data by semiconfocal pickup heads,” Jpn. J. Appl. Phys. 31, 534–541 (1992).
[CrossRef]

Opt. Eng.

C. L. Matson, “Error reduction in images using high-quality prior knowledge,” Opt. Eng. 33, 3233–3236 (1994).
[CrossRef]

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. 29, 393–404 (1990).
[CrossRef]

Opt. Lett.

Proc. IEEE

F. J. Harris, “On the use of windows for harmonic analysis with the discrete Fourier transform,” Proc. IEEE 66, 51–83 (1978).
[CrossRef]

Stat. Sin.

G. E. B. Archer, D. M. Titterington, “The iterative image space reconstruction algorithm (ISRA) as an alternative to the EM algorithm for solving positive linear inverse problems,” Stat. Sin. 5, 77–96 (1995).

Other

A. E. Taylor, D. C. Lay, Introduction to Functional Analysis, 2nd ed. (Wiley, New York, 1980), p. 348.

H. Stark, ed., Image Recovery: Theory and Application (Academic, Boston, Mass., 1987).

C. L. Matson, “The role of positivity for error reduction in images,” in Image and Signal Processing for Remote Sensing, J. Desachy, ed., Proc. SPIE2315, 766–777 (1994).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991), pp. 247–250 and 305.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1

Plots of the one-dimensional high-pass zonal filter, defined in Eq. (3), with h(x)=0 as a function of n. Solid lines, n=1; dashed lines, n=2; dotted–dashed lines, n=5.

Fig. 2
Fig. 2

Plots of hn(x) with h(x)=0, defined in Eq. (3), as a function of n. Solid curve, n=1; dashed curve, n=2; dotted–dashed curve, n=5; dotted curve, n=100.

Fig. 3
Fig. 3

Plots of h˜n(x) with h˜(x)=0, defined in Eq. (11), as a function of n. Solid curve, n=1; dashed curve, n=2; dotted–dashed curve, n=5; dotted curve, n=100.

Fig. 4
Fig. 4

Normalized location of the first zero of hn(r) with h(r) = 0, defined in Eq. (13), as a function of n.

Fig. 5
Fig. 5

Plots of hn(r) with h(r)=0, defined in Eq. (13), as a function of n. Solid curve, n=1; dashed curve, n=2; dotted–dashed curve, n=5; dotted curve, n=100.

Fig. 6
Fig. 6

Normalized location of the first zero of h˜n(r) with h˜(r) =0, defined in Eq. (15), as a function of n.

Fig. 7
Fig. 7

Plots of h˜n(r) with h˜(r)=0, defined in Eq. (15), as a function of n. Solid curve, n=1; dashed curve, n=2; dotted–dashed curve, n=5; dotted curve, n=100.

Fig. 8
Fig. 8

Plots of slices of three PSF’s defined by their Fourier transforms in Eqs. (16)–(18). Solid curve, circular-aperture PSF; dashed curve, modified Hamming PSF; dotted–dashed curve, Riesz PSF.

Fig. 9
Fig. 9

Plots of slices of the absolute values of three PSF’s defined by their Fourier transforms in Eqs. (16)–(18), emphasizing the sidelobe magnitudes. Solid curve, circular-aperture PSF; dashed curve, Hamming PSF; dotted–dashed curve, Riesz PSF.

Fig. 10
Fig. 10

(a) True computer-rendered satellite image. (b) Blurred satellite image filtered with PSF1, which corresponds to the unobscured circular-aperture filter. (c) Blurred satellite image filtered with PSF2, which corresponds to the modified Hamming filter. (d) Blurred satellite image filtered with PSF3, which corresponds to the Riesz filter.

Equations (21)

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

h(x)=1h(0)-u0u0H(u)exp(j2πux)du=1h(0)-u0u0H(u)cos(2πux)du+jh(0)-u0u0H(u)sin(2πux)du.
h(x)=2h(0)0u0H(u)cos(2πux)du,
hn(x)=h(x)+2n2nu0n+1u0 cos(2πux)duKn,
n=0, 1, 2, ,
hn14u0+=h14u0++2n2nu0/(n+1)u0cos2πu14u0+duKn.
hn14u0+=h14u0++n2π14u0+×cos(2πu0)1-cosπ(1-4u0)2(n+1)-sin(2πu0)sinπ(1-4u0)2(n+1)Kn.
hn14u0+h14u0++1π14u0+×cos(2πu0)nπ(1-4u0)2(n+1)2-sin(2πu0)n2π(1-4u0)2(n+1)Kn.
Hn(u)=H(u)+nn rectu+uc2(n+1)+n rectu-uc2(n+1)Kn,
uc=uo-12(n+1)u0.
limn Hn(u)=limnH(u)+nn rectu+uc2(n+1)+n rectu-uc2(n+1)Kn=1/2[δ(u+u0)+δ(u-u0)].
ddxh(x)=ddx2h(0)0u0H(u)cos(2πux)du=-4πh(0)0u0uH(u)sin(2πux)du.
h˜n(x)=h˜(x)+2n2u02nn+1u0/2 cos(2πux)du2Kn,
n=0, 1, 2, ,
h(r)=2π0R0ρH(ρ)J0(2πrρ)dρ,
hn(r)=h(r)+4πn2nR0n+1R0ρJ0(2πrρ)dρKn=h(r)Kn+2n2R0rKnJ1(2πrR0)-nn+1J12πrR0nn+1,
n=0, 1, 2, ,
ddrh(r)=ddr2π0R0ρH(ρ)J0(2πrρ)dρ=-4π20R0ρ2H(ρ)J1(2πrρ)dρ.
h˜n(r)=h˜(r)+4πR0n22n+1×(4n+2)J1[2πR0r/(4n+2)]πR0r2×J0[4πnR0r/(2n+1)]+5.062R0n22n+1×(4n+2)J1[2πR0r/(4n+2)]πR0r2Kn.
H1(ρ)=2πcosρR0-ρR01-(ρ/R0)20ρR00otherwise.
H2(ρ)=0.625+0.375 cosπρR00ρR00otherwise.
H3(ρ)=1-ρR020ρR00otherwise.

Metrics