Abstract

Phase dominance refers to the observation that the phase of the Fourier transform of an image carries more information than does the amplitude. We show that apparent counterexamples to phase dominance are not in fact counterexamples since the phase functions used are not independent of the true phase function. Phase dominance appears to be a general phenomenon.

© 2003 Optical Society of America

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References

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  1. G. N. Ramachandran, R Srinivasan, Fourier Methods in Crystallography (Wiley, New York, 1970), pp. 60–71.
  2. T. S. Huang, J. W. Burnett, A. G. Deczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 529–542 (1975).
    [CrossRef]
  3. W. A. Pearlman, R. M. Gray, “Source coding of the discrete Fourier transform,” IEEE Trans. Inf. Theory IT-24, 683–692 (1978).
    [CrossRef]
  4. A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
    [CrossRef]
  5. L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
    [CrossRef] [PubMed]
  6. R. P. Millane, “Phase retrieval in crystallography and optics,” J. Opt. Soc. Am. A 7, 394–411 (1990).
    [CrossRef]
  7. A. W. Lohmann, D. Mendlovic, G. Shabtay, “Significance of phase and amplitude in the Fourier domain,” J. Opt. Soc. Am. A 14, 2901–2904 (1997).
    [CrossRef]
  8. M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
    [CrossRef]
  9. J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]

1997 (1)

1990 (1)

1982 (3)

J. R. Fienup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[CrossRef] [PubMed]

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

1981 (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

1978 (1)

W. A. Pearlman, R. M. Gray, “Source coding of the discrete Fourier transform,” IEEE Trans. Inf. Theory IT-24, 683–692 (1978).
[CrossRef]

1975 (1)

T. S. Huang, J. W. Burnett, A. G. Deczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 529–542 (1975).
[CrossRef]

Burnett, J. W.

T. S. Huang, J. W. Burnett, A. G. Deczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 529–542 (1975).
[CrossRef]

Campbell, F. W.

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

Deczky, A. G.

T. S. Huang, J. W. Burnett, A. G. Deczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 529–542 (1975).
[CrossRef]

Fienup, J. R.

Gray, R. M.

W. A. Pearlman, R. M. Gray, “Source coding of the discrete Fourier transform,” IEEE Trans. Inf. Theory IT-24, 683–692 (1978).
[CrossRef]

Hayes, M. H.

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

Huang, T. S.

T. S. Huang, J. W. Burnett, A. G. Deczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 529–542 (1975).
[CrossRef]

Lim, J. S.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Lohmann, A. W.

Mendlovic, D.

Millane, R. P.

Oppenheim, A. V.

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Pearlman, W. A.

W. A. Pearlman, R. M. Gray, “Source coding of the discrete Fourier transform,” IEEE Trans. Inf. Theory IT-24, 683–692 (1978).
[CrossRef]

Piotrowski, L. N.

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

Ramachandran, G. N.

G. N. Ramachandran, R Srinivasan, Fourier Methods in Crystallography (Wiley, New York, 1970), pp. 60–71.

Shabtay, G.

Srinivasan, R

G. N. Ramachandran, R Srinivasan, Fourier Methods in Crystallography (Wiley, New York, 1970), pp. 60–71.

Appl. Opt. (1)

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

M. H. Hayes, “The reconstruction of a multidimensional sequence from the phase or magnitude of its Fourier transform,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 140–154 (1982).
[CrossRef]

T. S. Huang, J. W. Burnett, A. G. Deczky, “The importance of phase in image processing filters,” IEEE Trans. Acoust. Speech Signal Process. ASSP-23, 529–542 (1975).
[CrossRef]

IEEE Trans. Inf. Theory (1)

W. A. Pearlman, R. M. Gray, “Source coding of the discrete Fourier transform,” IEEE Trans. Inf. Theory IT-24, 683–692 (1978).
[CrossRef]

J. Opt. Soc. Am. A (2)

Perception (1)

L. N. Piotrowski, F. W. Campbell, “A demonstration of the visual importance and flexibility of spatial-frequency amplitude and phase,” Perception 11, 337–346 (1982).
[CrossRef] [PubMed]

Proc. IEEE (1)

A. V. Oppenheim, J. S. Lim, “The importance of phase in signals,” Proc. IEEE 69, 529–541 (1981).
[CrossRef]

Other (1)

G. N. Ramachandran, R Srinivasan, Fourier Methods in Crystallography (Wiley, New York, 1970), pp. 60–71.

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Figures (6)

Fig. 1
Fig. 1

Test images (a) f 1 ( x ,   y ) and (b) f 2 ( x ,   y ) : Left, real part; right, imaginary part.

Fig. 2
Fig. 2

Reconstructed images (a) f 1 a ( x ,   y ) and (b) f 1 ϕ ( x ,   y ) , for ϕ ( u ,   v ) = 0 .

Fig. 3
Fig. 3

Reconstructed images (a) f 2 a ( x ,   y ) and (b) f 2 ϕ ( x ,   y ) , for ϕ ( u ,   v ) = 0 . Left, real part; right, imaginary part.

Fig. 4
Fig. 4

Reconstructed images f 1 ϕ ( x ,   y ) for (a) ϕ ( u ,   v ) = π / 2 and (b) ϕ ( u ,   v ) random. Left, real part; right, imaginary part.

Fig. 5
Fig. 5

Reconstructed images f 2 ϕ ( x ,   y ) for (a) ϕ ( u ,   v ) = π / 2 and (b) ϕ ( u ,   v ) random. Left, real part; right, imaginary part.

Fig. 6
Fig. 6

Test images (a) f 1 ( x ,   y ) and (b) g ( x ,   y ) and the reconstructed images (c) f 1 ϕ g a ( x ,   y ) and (d) f 1 a g ϕ ( x ,   y ) .

Tables (1)

Tables Icon

Table 1 Normalized Rms Errors for Images Generated by Retaining Either Fourier Amplitude or Fourier Phase (for the Examples Described in the Text)

Equations (5)

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

F ( u ,   v ) = f ( x ,   y ) exp ( i 2 π ( ux + vy ) ) d x d y = F { f ( x ,   y ) } ,
f a ( x ,   y ) = F   - 1 { | F ( u ,   v ) | exp ( i ϕ ( u ,   v ) ) } ,
f ϕ ( x ,   y ) = F   - 1 { a ( u ,   v ) exp ( i Φ F ( u ,   v ) ) } ,
f a   g ϕ ( x ,   y ) = F   - 1 { | F ( u ,   v ) | exp ( i Φ G ( u ,   v ) ) }
f ϕ   g a ( x ,   y ) = F   - 1 { | G ( u ,   v ) | exp ( i Φ F ( u ,   v ) ) } .

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