Abstract

The quality of many imaging devices can be characterized, within certain constraints, by means of the modulation transfer function (MTF) and the phase transfer function (PTF). In many cases, it is possible to estimate, qualitatively, the effect of the MTF on the appearance of objects, and much progress has been made in making quantitative predictions of the detectability of objects and features within objects. This is not the case, however, for the PTF, and its influence is often neglected, even though nonideal PTF’s obviously may degrade image quality. Experiments are described that attempt to assess the significance of the PTF for human visual performance. The effects of various PTF’s were simulated by means of a technique that maintained the modulation of the spatial-frequency components closely constant. The visual detectability of such phase changes was measured by a discrimination method, and phase threshold levels were evaluated. Simple linear patterns provided the targets for all measurements, although some images are shown that illustrate qualitatively the phase-shift results on real two-dimensional targets. The work enables an assessment to be made of the effects of phase changes produced by various imaging systems.

© 1981 Optical Society of America

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References

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  1. J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
    [Crossref] [PubMed]
  2. J. J. Kulikowski, “Effective contrast constancy and linearity of contrast sensation,” Vision Res. 16, 1419–1431 (1976).
    [Crossref] [PubMed]
  3. C. R. Carlson and R. W. Cohen, “Visibility of displayed information,” (July1978).
  4. G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cybernetics 40, 27–38 (1981).
    [Crossref]
  5. R. C. Gonzalez and P. Wintz, Digital Image Processing (Addison-Wesley, London, 1977).
  6. K. J. Rosenbruck and R. Gershler, “Die Bedeutung der Phasenübertragungsfunktion und der Modulationsübertragungsfunktion bei der Benutzung der OTF als Bildgüterkriterium,” Optik 55, 173–182 (1980).
  7. G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
    [Crossref] [PubMed]
  8. D. Burr, “Sensitivity to spatial phase,” Vision Res. 20, 391–396 (1980).
    [Crossref] [PubMed]
  9. B. H. Crawford, Physical Photometry, National Physical Laboratory Notes on Applied Science no. 29 (H. M. Stationery Office, London, 1962).
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    [Crossref] [PubMed]
  11. D. H. Kelly and R. E. Savoie, “A study of sine-wave contrast sensitivity by two psychophysical methods,” Percep. Psychophys. 14, 313–318 (1973).
    [Crossref]
  12. J. J. Mezrich, C. R. Carlson, and R. W. Cohen, “Image descriptors for displays,” (February1977).
  13. F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. London 197, 551–566 (1965).
  14. P. W. Cobb and F. K. Moss, “The four variables of the visual threshold,” J. Franklin Inst. 205, 831–847 (1928).
    [Crossref]
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    [Crossref]
  16. H. R. Blackwell, “Contrast thresholds of the human eye,” J. Opt. Soc. Am. 36, 624–643 (1946).
    [Crossref] [PubMed]
  17. L. C. Martin, Technical Optics. Vol. II (Pitman, London, 1960).
  18. M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
    [Crossref] [PubMed]
  19. I. Overington, Vision and Acquisition (Pentech, London, 1976).
  20. G. Hauske, W. Wolf, and U. Lupp, “Matched filters in human vision,” Biol. Cybernetics 22, 181–188 (1976).
    [Crossref]
  21. G. J. Burton, “Visual detection model for static targets,” presented at Symposium on Visual Modelling for Operational Research, Royal Military College of Science, Shrivenham, Wilts, United Kingdom, January, 1978.

1981 (1)

G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cybernetics 40, 27–38 (1981).
[Crossref]

1980 (2)

K. J. Rosenbruck and R. Gershler, “Die Bedeutung der Phasenübertragungsfunktion und der Modulationsübertragungsfunktion bei der Benutzung der OTF als Bildgüterkriterium,” Optik 55, 173–182 (1980).

D. Burr, “Sensitivity to spatial phase,” Vision Res. 20, 391–396 (1980).
[Crossref] [PubMed]

1978 (1)

G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
[Crossref] [PubMed]

1976 (2)

J. J. Kulikowski, “Effective contrast constancy and linearity of contrast sensation,” Vision Res. 16, 1419–1431 (1976).
[Crossref] [PubMed]

G. Hauske, W. Wolf, and U. Lupp, “Matched filters in human vision,” Biol. Cybernetics 22, 181–188 (1976).
[Crossref]

1974 (1)

J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[Crossref] [PubMed]

1973 (1)

D. H. Kelly and R. E. Savoie, “A study of sine-wave contrast sensitivity by two psychophysical methods,” Percep. Psychophys. 14, 313–318 (1973).
[Crossref]

1971 (1)

1967 (1)

1965 (1)

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. London 197, 551–566 (1965).

1962 (1)

T. N. Cornsweet, “The staircase method in psychophysics,” Am. J. Psychol. 75, 485–491 (1962).
[Crossref] [PubMed]

1946 (1)

1928 (1)

P. W. Cobb and F. K. Moss, “The four variables of the visual threshold,” J. Franklin Inst. 205, 831–847 (1928).
[Crossref]

Blackwell, H. R.

Burr, D.

D. Burr, “Sensitivity to spatial phase,” Vision Res. 20, 391–396 (1980).
[Crossref] [PubMed]

Burton, G. J.

G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cybernetics 40, 27–38 (1981).
[Crossref]

G. J. Burton, “Visual detection model for static targets,” presented at Symposium on Visual Modelling for Operational Research, Royal Military College of Science, Shrivenham, Wilts, United Kingdom, January, 1978.

Campbell, F. W.

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. London 197, 551–566 (1965).

Carlson, C. R.

C. R. Carlson and R. W. Cohen, “Visibility of displayed information,” (July1978).

J. J. Mezrich, C. R. Carlson, and R. W. Cohen, “Image descriptors for displays,” (February1977).

Cobb, P. W.

P. W. Cobb and F. K. Moss, “The four variables of the visual threshold,” J. Franklin Inst. 205, 831–847 (1928).
[Crossref]

Cohen, R. W.

J. J. Mezrich, C. R. Carlson, and R. W. Cohen, “Image descriptors for displays,” (February1977).

C. R. Carlson and R. W. Cohen, “Visibility of displayed information,” (July1978).

Cornsweet, T. N.

T. N. Cornsweet, “The staircase method in psychophysics,” Am. J. Psychol. 75, 485–491 (1962).
[Crossref] [PubMed]

Crawford, B. H.

B. H. Crawford, Physical Photometry, National Physical Laboratory Notes on Applied Science no. 29 (H. M. Stationery Office, London, 1962).

Gershler, R.

K. J. Rosenbruck and R. Gershler, “Die Bedeutung der Phasenübertragungsfunktion und der Modulationsübertragungsfunktion bei der Benutzung der OTF als Bildgüterkriterium,” Optik 55, 173–182 (1980).

Gonzalez, R. C.

R. C. Gonzalez and P. Wintz, Digital Image Processing (Addison-Wesley, London, 1977).

Hauske, G.

G. Hauske, W. Wolf, and U. Lupp, “Matched filters in human vision,” Biol. Cybernetics 22, 181–188 (1976).
[Crossref]

Kelly, D. H.

D. H. Kelly and R. E. Savoie, “A study of sine-wave contrast sensitivity by two psychophysical methods,” Percep. Psychophys. 14, 313–318 (1973).
[Crossref]

Kulikowski, J. J.

J. J. Kulikowski, “Effective contrast constancy and linearity of contrast sensation,” Vision Res. 16, 1419–1431 (1976).
[Crossref] [PubMed]

Lupp, U.

G. Hauske, W. Wolf, and U. Lupp, “Matched filters in human vision,” Biol. Cybernetics 22, 181–188 (1976).
[Crossref]

Martin, L. C.

L. C. Martin, Technical Optics. Vol. II (Pitman, London, 1960).

Mezrich, J. J.

J. J. Mezrich, C. R. Carlson, and R. W. Cohen, “Image descriptors for displays,” (February1977).

Moss, F. K.

P. W. Cobb and F. K. Moss, “The four variables of the visual threshold,” J. Franklin Inst. 205, 831–847 (1928).
[Crossref]

Nachmias, J.

Overington, I.

I. Overington, Vision and Acquisition (Pentech, London, 1976).

Robson, J. G.

M. B. Sachs, J. Nachmias, and J. G. Robson, “Spatial-frequency channels in human vision,” J. Opt. Soc. Am. 61, 1176–1186 (1971).
[Crossref] [PubMed]

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. London 197, 551–566 (1965).

Rosenbruck, K. J.

K. J. Rosenbruck and R. Gershler, “Die Bedeutung der Phasenübertragungsfunktion und der Modulationsübertragungsfunktion bei der Benutzung der OTF als Bildgüterkriterium,” Optik 55, 173–182 (1980).

Sachs, M. B.

Sansbury, R. V.

J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[Crossref] [PubMed]

Savoie, R. E.

D. H. Kelly and R. E. Savoie, “A study of sine-wave contrast sensitivity by two psychophysical methods,” Percep. Psychophys. 14, 313–318 (1973).
[Crossref]

Westheimer, G.

G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
[Crossref] [PubMed]

Wintz, P.

R. C. Gonzalez and P. Wintz, Digital Image Processing (Addison-Wesley, London, 1977).

Wolf, W.

G. Hauske, W. Wolf, and U. Lupp, “Matched filters in human vision,” Biol. Cybernetics 22, 181–188 (1976).
[Crossref]

Am. J. Psychol. (1)

T. N. Cornsweet, “The staircase method in psychophysics,” Am. J. Psychol. 75, 485–491 (1962).
[Crossref] [PubMed]

Biol. Cybernetics (2)

G. Hauske, W. Wolf, and U. Lupp, “Matched filters in human vision,” Biol. Cybernetics 22, 181–188 (1976).
[Crossref]

G. J. Burton, “Contrast discrimination by the human visual system,” Biol. Cybernetics 40, 27–38 (1981).
[Crossref]

J. Franklin Inst. (1)

P. W. Cobb and F. K. Moss, “The four variables of the visual threshold,” J. Franklin Inst. 205, 831–847 (1928).
[Crossref]

J. Opt. Soc. Am. (3)

J. Physiol. London (1)

F. W. Campbell and J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. London 197, 551–566 (1965).

Optik (1)

K. J. Rosenbruck and R. Gershler, “Die Bedeutung der Phasenübertragungsfunktion und der Modulationsübertragungsfunktion bei der Benutzung der OTF als Bildgüterkriterium,” Optik 55, 173–182 (1980).

Percep. Psychophys. (1)

D. H. Kelly and R. E. Savoie, “A study of sine-wave contrast sensitivity by two psychophysical methods,” Percep. Psychophys. 14, 313–318 (1973).
[Crossref]

Vision Res. (4)

G. Westheimer, “Spatial phase sensitivity for sinusoidal grating targets,” Vision Res. 18, 1073–1074 (1978).
[Crossref] [PubMed]

D. Burr, “Sensitivity to spatial phase,” Vision Res. 20, 391–396 (1980).
[Crossref] [PubMed]

J. Nachmias and R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[Crossref] [PubMed]

J. J. Kulikowski, “Effective contrast constancy and linearity of contrast sensation,” Vision Res. 16, 1419–1431 (1976).
[Crossref] [PubMed]

Other (7)

C. R. Carlson and R. W. Cohen, “Visibility of displayed information,” (July1978).

R. C. Gonzalez and P. Wintz, Digital Image Processing (Addison-Wesley, London, 1977).

B. H. Crawford, Physical Photometry, National Physical Laboratory Notes on Applied Science no. 29 (H. M. Stationery Office, London, 1962).

J. J. Mezrich, C. R. Carlson, and R. W. Cohen, “Image descriptors for displays,” (February1977).

L. C. Martin, Technical Optics. Vol. II (Pitman, London, 1960).

I. Overington, Vision and Acquisition (Pentech, London, 1976).

G. J. Burton, “Visual detection model for static targets,” presented at Symposium on Visual Modelling for Operational Research, Royal Military College of Science, Shrivenham, Wilts, United Kingdom, January, 1978.

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Figures (10)

Fig. 1
Fig. 1

Photographs of the central area of the monitor screen showing examples of targets used in the experiments. The calibration bars are of length 10 min. Upper targets: Left- and right-hand targets depict the phase-modified and unmodified patterns; the unmodified target has a rectangular profile of width 5.5 min, and the contrast (on the original display) is −0.5. The simulated PTF has a center frequency of 5.4 cycles/deg, bandwidth of 1.1 octaves, and peak phase shift of 72 deg (Y = 0.37 deg, A = 8.18 deg−1). Lower targets: The unmodified target is a Cobb-type pattern with bar width 1.1 min and contrast as for the upper target. The phase-modified target was obtained for a PTF with center frequency of 21.8 cycles/deg, bandwidth of 1.1 octaves, and peak phase shift of 72 deg (Y = 0.092 deg, A = 32.7 deg−1).

Fig. 2
Fig. 2

Examples of theoretical functions illustrating the sequence in the derivation of simulated PTF’s. Values of the parameters in Eq. (3) are f0 = 10.9 cycles/deg, Y = 0.183 deg, and A = 16.4 deg−1. (A) continuous and broken curves show the functions o(x) and the convolution term 0.324/(0.644)2 × o(x) ⊗ o(x), respectively. These odd and even functions are defined by Eqs. (3) and (4). (B) the LSF derived for the specified parameters f0, Y, and A. The central spike in the LSF simulates the δ function [Eq. (4)] in the spatially quantisized representation required for the digital convolution. (C) the components of the corresponding complex transfer function in the spatial-frequency domain are shown by the real part R(f), the imaginary part I(f), and the modulus M(f). (D) the calculated phase transfer function. The curve is smooth and exhibits only one dominant extremum.

Fig. 3
Fig. 3

Examples of PTF’s employed in the experiments shown for different values: (A) of the center frequency at a peak phase shift of 72 deg and bandwidth of 1.1 octaves, (B) of the bandwidth at a center frequency of 10.9 cycles/deg, and (C) of the peak phase shift at a center frequency of 10.9 cycles/deg and bandwidth of 1.1 octaves. Note that the three dimensions of PTF variation, center frequency, bandwidth, and peak phase shift interact only slightly. Thus only small changes in peak phase shift occur for the different bandwidths in (B), and the bandwidth increases only slightly in (C) as the peak phase shift decreases.

Fig. 4
Fig. 4

Discrimination threshold values of peak phase shift determined by the two observers for a bar target of width 5.5 min (Fig. 1). The PTF bandwidth was 1.1 octaves, and phase threshold values are shown for a range of center frequencies of the PTF and for three different contrasts. Error bars denote the mean and standard deviation derived from four staircases. Symbols with arrows denote lower bound values (see Section 2).

Fig. 5
Fig. 5

As for Fig. 4, except that the Cobb-type target was used and consisted of two fine lines of width 1.1 min and center-to-center separation equal to 2.2 min. The resolution contrast threshold level of this target is −0.18.

Fig. 6
Fig. 6

The effect on the phase threshold level of the width of a bar target having a rectangular luminance profile. The PTF center frequency and bandwidth were fixed at 10.9 cycles/deg and 1.1 octaves, respectively.

Fig. 7
Fig. 7

Phase threshold levels shown for different bandwidths of the PTF at each of three center frequencies, 5.4, 10.9, and 21.7 cycles/deg. The bar target with rectangular luminance profile was 5.5 min and contrast −0.5. Note that there is a tendency toward the occurrence of a minimum in phase threshold levels for bandwidths of about 0.8 octaves.

Fig. 8
Fig. 8

As for Fig. 7, except that a Cobb-type target was used, of bar width 1.1 min and contrast −0.5. Values of phase threshold are shown for a single center frequency of 21.7 cycles/deg.

Fig. 9
Fig. 9

Probability of discriminating the phase-modified and unmodified targets for different settings of the peak phase shift. The data show, therefore, the psychometric functions for phase-shift detection. The targets employed were (A) a rectangular profile bar target of width 5.5 min and contrast −0.5 and (B) a Cobb-type target of bar width 1.1 min and contrast −0.5. Values shown in (A) were measured for PTF’s with center frequencies of 5.4 and 15.4 cycles/deg and in (B) with a center frequency of 21.8 cycles/deg. In all cases the PTF bandwidth was 1.1 octaves. The symbols with horizontal error bars show values of the mean and standard deviation for the phase threshold levels determined by the staircase procedure. These values are replotted from Figs. 4 and 5. Using linear interpolation of the psychometric functions, the corresponding probabilities for the staircase values are indicated by the ordinate positions. For the three experimental conditions and the two observers the mean probability is 52%.

Fig. 10
Fig. 10

Photographs taken of the monitor screen illustrating phase-shift effects on vehicle targets. A, the original, phase-unmodified targets. B, the results of applying a phase shift only in the horizontal dimension. Center frequency 5.4 cycles/deg, bandwidth 1.1 octaves, peak phase shift 72 deg. C, as for B except that a center frequency of 21.7 cycles/deg was used. D and E, the effect of applying the phase shifts in both dimensions with center frequencies corresponding to those used for B and C, respectively.

Equations (10)

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R ( f ) = 1 - ϕ 2 ( f ) / 2 ,             I ( f ) = ϕ ( f ) ,
R ( f ) = 1.040 - 0.324 ϕ 2 ( f ) ,             I ( f ) = 0.644 ϕ ( f ) .
o ( x ) = A × sin ( 2 π f 0 x ) × cos ( π x / Y ) ,             x Y / 2.
e ( x ) = 1.040 δ ( x ) - ( 0.324 / 0.644 2 ) × o ( x ) o ( x ) ,
- exp [ j θ ( f - g ) ] × sin ( π g L ) / ( π g ) d g = exp [ j θ ( f ) ] .
exp [ j θ ( f - g ) ] = G ( g ) × exp [ j θ ( f ) ] ,
- exp ( - j 2 π K g ) × sin ( π g L ) / ( π g ) d g = 1 ,
- exp ( - j 2 π K g ) × - L / 2 L / 2 exp ( j 2 π g x ) d x d g = 1.
- L / 2 L / 2 - exp [ j 2 π g ( x - K ) ] d g d x = 1.
- L / 2 L / 2 δ ( x - K ) d x = 1.