Abstract

We derive a Fourier-domain Wiener filter for the reconstruction of undersampled imagery. The filter differs from previous implementations in that it permits adjustment of the trade-offs between sharpness of the reconstruction, noise amplification, and aliasing artifact suppression. Additionally, a net transfer function that characterizes the combined effects of the imaging system and the reconstruction process is derived. This net transfer function is valid for both unaliased and aliased spatial frequencies. The expression for the net transfer function is applicable to more general linear image sharpening algorithms.

© 2009 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Wiley, 1950).
  2. H. W. Bode and C. E. Shannon, “A simplified derivation of linear least square smoothing and prediction theory,” Proc. IRE 38, 417-425 (1950).
    [CrossRef]
  3. C. W. Helstrom, “Image reconstruction by the method of least squares,” J. Opt. Soc. Am. 57, 297-303 (1967).
    [CrossRef]
  4. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).
  5. H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
    [CrossRef]
  6. E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181-194 (1915).
  7. J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
    [CrossRef]
  8. R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
    [CrossRef]
  9. J. C. Leachtenauer, W. Malila, J. Irvine, L. Colburn, and N. Salvaggio, “General image-quality equation: GIQE,” Appl. Opt. 36, 8322-8328 (1997).
    [CrossRef]
  10. P. G. J. Barten, Contrast Sensitivity of the Human Eye and Its Effects on Image Quality (SPIE, 1999), Chap. 8.
    [CrossRef]
  11. S. T. Thurman and J. R. Fienup, “Wiener filtering of aliased imagery,” Proc. SPIE 7076, 70760J (2008).
    [CrossRef]
  12. See the Appendix of S. T. Thurman and J. R. Fienup, “Signal-to-noise ratio trade-offs associated with coarsely sampled Fourier transform spectroscopy,” J. Opt. Soc. Am. A 24, 2817-2821 (2007).
    [CrossRef]
  13. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 59.
  14. This image is in the public domain and is available through the URL http://seamless.usgs.gov/.
  15. N. G. Deriugin, “The power spectrum and the correlation function of the television signal,” Telecommunications 1, 1-12 (1956).
  16. G. J. Burton and I. R. Moorhead, “Color and spatial structure in natural scenes,” Appl. Opt. 26, 157-170 (1987).
    [CrossRef] [PubMed]
  17. D. J. Field, “Relations between the statistics of natural images and the response properties of cortical cells,” J. Opt. Soc. Am. A 4, 2379-2394 (1987).
    [CrossRef] [PubMed]
  18. D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229-232 (1992).
    [CrossRef] [PubMed]
  19. D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814-817 (1994).
    [CrossRef] [PubMed]
  20. A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
    [CrossRef] [PubMed]
  21. R. L. Lucke, “Fourier-space properties of photon-limited noise in focal plane array data, calculated with the discrete Fourier transform,” J. Opt. Soc. Am. A 18, 777-790 (2001).
    [CrossRef]
  22. D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).
  23. S. T. Thurman and J. R. Fienup, “Noise histogram regularization for iterative image reconstruction algorithms,” J. Opt. Soc. Am. A 24, 608-617 (2007).
    [CrossRef]

2008 (1)

S. T. Thurman and J. R. Fienup, “Wiener filtering of aliased imagery,” Proc. SPIE 7076, 70760J (2008).
[CrossRef]

2007 (2)

2005 (1)

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

2002 (1)

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

2001 (1)

1999 (1)

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

1997 (1)

1996 (1)

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

1994 (1)

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

1992 (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

1987 (2)

1967 (1)

1956 (1)

N. G. Deriugin, “The power spectrum and the correlation function of the television signal,” Telecommunications 1, 1-12 (1956).

1950 (1)

H. W. Bode and C. E. Shannon, “A simplified derivation of linear least square smoothing and prediction theory,” Proc. IRE 38, 417-425 (1950).
[CrossRef]

1928 (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

1915 (1)

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181-194 (1915).

Barten, P. G. J.

P. G. J. Barten, Contrast Sensitivity of the Human Eye and Its Effects on Image Quality (SPIE, 1999), Chap. 8.
[CrossRef]

Bialek, W.

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

Bode, H. W.

H. W. Bode and C. E. Shannon, “A simplified derivation of linear least square smoothing and prediction theory,” Proc. IRE 38, 417-425 (1950).
[CrossRef]

Burton, G. J.

Calef, B.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Chao, T.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Colburn, L.

Deriugin, N. G.

N. G. Deriugin, “The power spectrum and the correlation function of the television signal,” Telecommunications 1, 1-12 (1956).

Field, D. J.

Fienup, J. R.

S. T. Thurman and J. R. Fienup, “Wiener filtering of aliased imagery,” Proc. SPIE 7076, 70760J (2008).
[CrossRef]

See the Appendix of S. T. Thurman and J. R. Fienup, “Signal-to-noise ratio trade-offs associated with coarsely sampled Fourier transform spectroscopy,” J. Opt. Soc. Am. A 24, 2817-2821 (2007).
[CrossRef]

S. T. Thurman and J. R. Fienup, “Noise histogram regularization for iterative image reconstruction algorithms,” J. Opt. Soc. Am. A 24, 608-617 (2007).
[CrossRef]

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

Fiete, R. D.

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

Gerwe, D. R.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Griffith, D.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

Harrington, L.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

Helstrom, C. W.

Irvine, J.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

Jain, M.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Kowalczyk, A. M.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

Leachtenauer, J. C.

Lucke, R. L.

Luna, C.

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Malila, W.

Mandel, L.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 59.

Miller, J. J.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

Mooney, J. A.

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

Moorhead, I. R.

Nyquist, H.

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

Ruderman, D. L.

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

Salvaggio, N.

Shannon, C. E.

H. W. Bode and C. E. Shannon, “A simplified derivation of linear least square smoothing and prediction theory,” Proc. IRE 38, 417-425 (1950).
[CrossRef]

Tadmor, Y.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Thurman, S. T.

Tolhurst, D. J.

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

van der Schaaf, A.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

van Hateren, J. H.

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

Whittaker, E. T.

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181-194 (1915).

Wiener, N.

N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Wiley, 1950).

Wolf, E.

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 59.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

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

Ophthalmic Physiol. Opt. (1)

D. J. Tolhurst, Y. Tadmor, and T. Chao, “Amplitude spectra of natural images,” Ophthalmic Physiol. Opt. 12, 229-232 (1992).
[CrossRef] [PubMed]

Opt. Eng. (1)

R. D. Fiete, “Image quality and λFN/p for remote sensing systems,” Opt. Eng. 38, 1229-1240 (1999).
[CrossRef]

Phys. Rev. Lett. (1)

D. L. Ruderman and W. Bialek, “Statistics of natural images: scaling in the woods,” Phys. Rev. Lett. 73, 814-817 (1994).
[CrossRef] [PubMed]

Proc. IRE (1)

H. W. Bode and C. E. Shannon, “A simplified derivation of linear least square smoothing and prediction theory,” Proc. IRE 38, 417-425 (1950).
[CrossRef]

Proc. R. Soc. Edinburgh (1)

E. T. Whittaker, “On the functions which are represented by the expansions of the interpolation theory,” Proc. R. Soc. Edinburgh 35, 181-194 (1915).

Proc. SPIE (3)

J. R. Fienup, D. Griffith, L. Harrington, A. M. Kowalczyk, J. J. Miller, and J. A. Mooney, “Comparison of reconstruction algorithms for images from sparse-aperture systems,” Proc. SPIE 4792, 1-8 (2002).
[CrossRef]

S. T. Thurman and J. R. Fienup, “Wiener filtering of aliased imagery,” Proc. SPIE 7076, 70760J (2008).
[CrossRef]

D. R. Gerwe, M. Jain, B. Calef, and C. Luna, “Regularization for non-linear image restoration using a prior on the object power spectrum,” Proc. SPIE 5896, 21-36 (2005).

Telecommunications (1)

N. G. Deriugin, “The power spectrum and the correlation function of the television signal,” Telecommunications 1, 1-12 (1956).

Trans. Am. Inst. Electr. Eng. (1)

H. Nyquist, “Certain topics in telegraph transmission theory,” Trans. Am. Inst. Electr. Eng. 47, 617-644 (1928).
[CrossRef]

Vision Res. (1)

A. van der Schaaf and J. H. van Hateren, “Modelling the power spectra of natural images: statistics and information,” Vision Res. 36, 2759-2770 (1996).
[CrossRef] [PubMed]

Other (5)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge U. Press, 1995), p. 59.

This image is in the public domain and is available through the URL http://seamless.usgs.gov/.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, 1989).

P. G. J. Barten, Contrast Sensitivity of the Human Eye and Its Effects on Image Quality (SPIE, 1999), Chap. 8.
[CrossRef]

N. Wiener, Extrapolation, Interpolation and Smoothing of Stationary Time Series (Wiley, 1950).

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 (6)

Fig. 1
Fig. 1

Diagram illustrating Fourier-domain contributions to the net system transfer function: (a) the unaliased transfer function of the imaging system S ( u ) , (b) the periodically extended version of the Wiener filter Ω ( u ) , and (c) the unaliased net system transfer function S net ( u ) . The vertical dashed lines indicate the extent of the unaliased portion of the Fourier domain, u < 1 ( 2 Δ x ) .

Fig. 2
Fig. 2

Diagram illustrating the interpretation of the net transfer function S net ( u ) . Passing object data through S ( u ) , downsampling, and reconstructing with W p (left) yields the same result as passing object data through S net ( u ) and downsampling (right).

Fig. 3
Fig. 3

Digital object [14] used for simulation.

Fig. 4
Fig. 4

Noisy aliased image g m , n .

Fig. 5
Fig. 5

Wiener filter reconstructions f ̂ m , n of Fig. 4, with c n = 1 and (a) c a = 0 , (b) c a = 1 , and (c) c a = 5 . The arrows indicate a region containing noticeable aliasing artifacts associated with the painted lines of the parking lot; note that the stripes appear to go in the wrong direction.

Fig. 6
Fig. 6

Net system transfer functions S net ( u , v ) corresponding to each of the reconstructions shown in Fig. 5. For comparison, the system transfer function before processing S ( u , v ) is also shown. The vertical dotted lines represent the extent of the unaliased portion of the object Fourier transform, u < 1 ( 2 Δ x ) .

Equations (29)

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

i ( x ) = s ( x ) o ( x ) ,
h m = i ( m Δ x ) = i ( x ) δ ( x m Δ x ) d x ,
g m = h m + n m ,
G p = 1 M m = M 2 M 2 1 g m exp ( i 2 π m p M ) ,
Δ u = 1 M Δ x .
G p = N p + 1 M Δ x k = { [ S ( u ) O ( u ) ] 1 Δ u sinc ( u k M Δ u Δ u ) } δ ( u p Δ u ) d u ,
G p = N p + H a , p + H 0 , p ,
H a , p = 1 M Δ x k = k 0 { [ S ( u ) O ( u ) ] 1 Δ u sinc ( u k M Δ u Δ u ) } δ ( u p Δ u ) d u 1 M Δ x k = k 0 S [ ( p k M ) Δ u ] O [ ( p k M ) Δ u ] ,
H 0 , p = 1 M Δ x [ S ( u ) O ( u ) 1 Δ u sinc ( u Δ u ) ] δ ( u p Δ u ) d u 1 M Δ x S ( p Δ u ) O ( p Δ u ) .
F ̂ p = W p G p ,
e = 1 M p = M 2 M 2 1 W p G p F p 2 ,
F p = 1 M Δ x [ O ( u ) 1 Δ u sinc ( u Δ u ) ] δ ( u p Δ u ) d u 1 M Δ x O ( p Δ u )
e = 1 M p = N 2 N 2 1 [ W p 2 ( N p 2 + N p H a , p * + N p * H a , p + H a , p 2 + N p H 0 , p * + N p * H 0 , p + H a , p H 0 , p * + H a , p * H 0 , p + H 0 , p 2 ) W p ( N p F p * + H a , p F p * + H 0 , p F p * ) W p * ( N p * F p + H a , p * F p + H 0 , p * F p ) + F p 2 ] .
O ( u ) O * ( u ) = Φ o ( u ) δ ( u u ) ,
H a , p 2 1 M Δ x 2 k = k 0 k = k 0 S [ ( p k M ) Δ u ] S * [ ( p k M ) Δ u ] × O [ ( p k M ) Δ u ] O * [ ( p k M ) Δ u ] 1 M Δ x 2 k = k 0 S [ ( p k M ) Δ u ] 2 Φ o [ ( p k M ) Δ u ] 1 M Δ x 2 Φ a ( p Δ u ) ,
Φ a ( u ) = k = k 0 S ( u k N Δ u ) 2 Φ o ( u k N Δ u ) .
e 1 M 2 Δ x 2 p = M 2 M 2 1 { W p 2 [ Φ n ( p Δ u ) + Φ a ( p Δ u ) ] + [ W p 2 S ( p Δ u ) 2 W p S ( p Δ u ) W p * S * ( p Δ u ) + 1 ] Φ o ( p Δ u ) } ,
N p 2 = 1 M Δ x 2 Φ n ( p Δ u ) .
W p = S * ( p Δ u ) Φ o ( p Δ u ) S ( p Δ u ) 2 Φ o ( p Δ u ) + c n Φ n ( p Δ u ) + c a Φ a ( p Δ u ) ,
W p = Ω ( u ) δ ( u p Δ u ) d u = Ω ( p Δ u ) ,
Ω ( u ) = { S * ( u ) Φ o ( u ) S ( u ) 2 Φ o ( u ) + c n Φ n ( u ) + c a Φ a ( u ) for M Δ u 2 u < M Δ u 2 Ω ( u M Δ u ) otherwise } .
F ̂ p = W p ( N p + H a , p + H 0 , p ) W p N p + 1 M Δ x k = W p S [ ( p k M ) Δ u ] O [ ( p k M ) Δ u ] = W p N p + 1 M Δ x k = Ω ( u ) S ( u ) O ( u ) δ [ u ( p k M ) Δ u ] d u .
S net ( u ) = Ω ( u ) S ( u ) ,
S net ( u ) = S ( u ) S * ( u k M Δ u ) Φ o ( u k M Δ u ) S ( u k M Δ u ) 2 Φ o ( u k M Δ u ) + c n Φ n ( u k M Δ u ) + c a Φ a ( u k M Δ u ) for ( k 1 2 ) M Δ u u < ( k + 1 2 ) M Δ u and all k { 0 , ± 1 , ± 2 , } .
Φ o ( u , v ) = { A 0 2 for ρ = 0 A 2 ρ 2 α for ρ 0 } ,
Φ i ( u , v ) = k = l = S ( u k M Δ u , v l M Δ v ) 2 Φ o ( u k M Δ u , v l M Δ v ) .
Φ n ( u , v ) = Φ n ,
P ( G p , q 2 ) = 1 Φ i ( p Δ u , q Δ v ) + Φ n exp [ G p , q 2 Φ i ( p Δ u , q Δ v ) + Φ n ] ,
L = ( p , q ) ( 0 , 0 ) G p , q 2 Φ i ( p Δ u , q Δ v ) + Φ n ln [ Φ i ( p Δ u , q Δ v ) + Φ n ] .

Metrics