Abstract

The number of cycles in a low-frequency sinusoidal display is a crucial variable in determining the visibility of the display. In particular, the threshold contrast is essentially independent of spatial frequency for these displays. We have extended the above experiments, using more cycles and a variety of targets and observer tasks. The results confirm previous findings; they also show that the type of target or task has little influence. For low-frequency sinusoids that contain up to about 3 cycles, the threshold contrast is determined by the number of cycles. For high-number-of-cycles targets with spatial frequencies above 6–10 cycles per degree, visibility is predominantly dependent on the spatial frequency. The results suggest that the low-frequency decrease in reported MTF’s is due to the decrease of the number of cycles used in determining them.

© 1975 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
    [CrossRef] [PubMed]
  2. A sine wave of limited extent possesses spectral energy over a range of spatial frequencies that includes the nominal frequency. So, strictly, it is incorrect to speak of, for example, a 3-cycle-per-degree sine wave when we mean a truncated sine wave with a nominal spatial frequency of 3 cycles per degree. However, because we will be referring to truncated sine waves throughout this paper, it will frequently be convenient to use the looser terminology.
  3. J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
    [CrossRef]
  4. O. H. Schade, J. Opt. Soc. Am. 46, 721 (1956).
    [CrossRef] [PubMed]
  5. J. J. DePalma and E. M. Lowry, J. Opt. Soc. Am. 52, 328 (1962).
    [CrossRef]
  6. M. Davidson, J. Opt. Soc. Am. 58, 1300 (1968).
    [CrossRef] [PubMed]
  7. F. W. Campbell and J. G. Robson, J. Physiol. 197, 551 (1968).
  8. We found that, with our apparatus, a 1-cycle target had the same threshold whether the left-hand edge was equal to the average liminance (sine phase) or the maximum luminance (cosine phase) of the sinusoidal region. Kelly (Ref. 9) showed that the edge created by a cosine phase target can determine threshold in the no-plateau condition. In our experiments, however, the edge between the mirror and the tube face never completely disappeared, so subjects could not use the creation of an edge as their principal cue for threshold; they needed to perceive some form.
  9. D. H. Kelly, J. Opt. Soc. Am. 60, 98 (1970).
    [CrossRef]
  10. The definition of bandwidth used in this discussion is width at half-power. For targets with a fixed w, this bandwidth and the peak value of the transform undergo negligible change as n changes except when n is very close to zero. In that case, overlap with the transform component centered at −f makes a significant contribution. However, there is only a 10% change of the height of the peak of the transform of the 12-cycletarget, the lowest number of cycles used in our experiments.
  11. M. Davidson and J. A. Whiteside, J. Opt. Soc. Am. 61, 530 (1971).
    [CrossRef] [PubMed]
  12. C. Blakemore and F. W. Campbell, J. Physiol. 203, 237 (1969).
  13. F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol. 203, 223 (1969).
  14. R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

1974 (2)

J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
[CrossRef] [PubMed]

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
[CrossRef]

1971 (1)

1970 (1)

1969 (2)

C. Blakemore and F. W. Campbell, J. Physiol. 203, 237 (1969).

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol. 203, 223 (1969).

1968 (2)

M. Davidson, J. Opt. Soc. Am. 58, 1300 (1968).
[CrossRef] [PubMed]

F. W. Campbell and J. G. Robson, J. Physiol. 197, 551 (1968).

1962 (1)

1956 (1)

Bilsen, F. A.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
[CrossRef]

Blakemore, C.

C. Blakemore and F. W. Campbell, J. Physiol. 203, 237 (1969).

Bracewell, R.

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

Campbell, F. W.

C. Blakemore and F. W. Campbell, J. Physiol. 203, 237 (1969).

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol. 203, 223 (1969).

F. W. Campbell and J. G. Robson, J. Physiol. 197, 551 (1968).

Cooper, G. F.

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol. 203, 223 (1969).

Davidson, M.

DePalma, J. J.

Enroth-Cugell, C.

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol. 203, 223 (1969).

Hall, J. A.

J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
[CrossRef] [PubMed]

Hoekstra, J.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
[CrossRef]

Kelly, D. H.

Lowry, E. M.

McCann, J. J.

J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
[CrossRef] [PubMed]

Robson, J. G.

F. W. Campbell and J. G. Robson, J. Physiol. 197, 551 (1968).

Savoy, R. L.

J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
[CrossRef] [PubMed]

Scarpetti, J. J.

J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
[CrossRef] [PubMed]

Schade, O. H.

van den Brink, G.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
[CrossRef]

van der Goot, D. P. J.

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
[CrossRef]

Whiteside, J. A.

J. Opt. Soc. Am. (5)

J. Physiol. (3)

C. Blakemore and F. W. Campbell, J. Physiol. 203, 237 (1969).

F. W. Campbell, G. F. Cooper, and C. Enroth-Cugell, J. Physiol. 203, 223 (1969).

F. W. Campbell and J. G. Robson, J. Physiol. 197, 551 (1968).

Vision Res. (2)

J. J. McCann, R. L. Savoy, J. A. Hall, and J. J. Scarpetti, Vision Res. 14, 917 (1974).
[CrossRef] [PubMed]

J. Hoekstra, D. P. J. van der Goot, G. van den Brink, and F. A. Bilsen, Vision Res. 14, 364 (1974).
[CrossRef]

Other (4)

R. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1965).

A sine wave of limited extent possesses spectral energy over a range of spatial frequencies that includes the nominal frequency. So, strictly, it is incorrect to speak of, for example, a 3-cycle-per-degree sine wave when we mean a truncated sine wave with a nominal spatial frequency of 3 cycles per degree. However, because we will be referring to truncated sine waves throughout this paper, it will frequently be convenient to use the looser terminology.

The definition of bandwidth used in this discussion is width at half-power. For targets with a fixed w, this bandwidth and the peak value of the transform undergo negligible change as n changes except when n is very close to zero. In that case, overlap with the transform component centered at −f makes a significant contribution. However, there is only a 10% change of the height of the peak of the transform of the 12-cycletarget, the lowest number of cycles used in our experiments.

We found that, with our apparatus, a 1-cycle target had the same threshold whether the left-hand edge was equal to the average liminance (sine phase) or the maximum luminance (cosine phase) of the sinusoidal region. Kelly (Ref. 9) showed that the edge created by a cosine phase target can determine threshold in the no-plateau condition. In our experiments, however, the edge between the mirror and the tube face never completely disappeared, so subjects could not use the creation of an edge as their principal cue for threshold; they needed to perceive some form.

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 (8)

FIG. 1
FIG. 1

Some of the differences between a typical target we used in a previous paper and a typical target used by DePalma and Lowry. In both cases, the targets were uniform in one direction and varied sinusoidally in the perpendicular direction. Our target, shown on the left, consisted of a small number of cycles at a low spatial frequency on a plateau. DePalma and Lowry’s target, shown on the right, consisted of many cycles of sinusoid against a background of the same average luminance.

FIG. 2
FIG. 2

Procedures used in the experiments. In I and II, the subjects adjust the contrast of the target until it is just distinguishable from the uniform display. In III and IV, the subjects adjust the contrast of the target until it is as visible as a 0.1 contrast, 1 1 2 - cycle standard. The procedures I, II, III, and IV are referred to as threshold with no plateau, threshold with plateau, contrast matching with no plateau, and contrast matching with plateau, respectively.

FIG. 3
FIG. 3

Apparatus used to generate the stimuli. The no-plateau targets were generated by use of the upper arrangement, where A, B, C, and D designate the source of uniform illumination, mirror for average background, rectangular hole, and source of sinusoidal illumination, respectively. The with-plateau targets were generated by use of the lower arrangement, where A and D are as above and E and F designate the dark-grey background and octagonally shaped, partially silvered mirror, respectively.

FIG. 4
FIG. 4

Data for subject RLS in each of the four experiments. The contrast sensitivity is plotted against the number of cycles in the stimulus. The numbers I, II, III, and IV refer to threshold with no plateau, threshold with plateau, contrast matching with no plateau, and contrast matching with plateau, respectively. These are the same designations as are used in Fig. 2. Filled circles, squares, and triangles refer to targets subtending 7.6°, 2.7°, and 0.83°, respectively. For stimuli with a small, fixed number of cycles, but different sizes, and hence different spatial frequencies, the contrast sensitivities depend on the number of cycles.

FIG. 5
FIG. 5

Same data as Fig. 4, but here the horizontal axis is the spatial frequency of the stimuli. Otherwise, the notation is the same as in Fig. 4. The principal observation to be made is that the curves for stimuli of various sizes start to converge at frequencies greater than 5 cycles per degree.

FIG. 6
FIG. 6

Data of Campbell and Robson. Filled squares and triangles refer to targets subtending 10° and 2°, respectively. They plotted the contrast sensitivity versus spatial frequency, as shown on the right. We have replotted their data as a function of number of cycles on the left to show the near coincidence of the two curves in the low-number-of-cycles region.

FIG. 7
FIG. 7

Luminance profiles and magnitude of the Fourier transforms of three targets similar to those used in the experiments. The luminance profiles are directly above the transforms. The arrows point to the luminance profiles of targets with equal widths (n), equal nominal spatial frequencies (f), or an equal number of cycles (n).

FIG. 8
FIG. 8

Straight-line fits to our data in the low-number-of-cycles region (top graph) and high-spatial-frequency region (second graph), respectively. These two straight lines are combined by Eq. (1). The lower two graphs show how well Eq. (1) fits the data.

Equations (11)

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

= 1 C 1 · ( NUM ) C 2 + C 3 · ( CPD ) C 4 .
L ( x , y ) = 9.3 + A · cos [ 2 π f ( x + w / 2 ) ] , for | x | , | y | < w / 2 , L ( x , y ) = 9.3 , otherwise .
L ( x , y ) = X ( x ) · Y ( y ) ,
X ( x ) = Π ( x / w ) · cos [ 2 π f ( x + w / 2 ) ] ,
Y ( y ) = Π ( y / w ) .
L * ( u , υ ) = X * ( u ) · Y * ( υ ) ,
X * ( u ) = e i π w f · ( | w | / 2 ) · sinc [ w ( u + f ) ] + e i π w f · ( | w | / 2 ) · sinc [ w ( u f ) ]
Y * ( υ ) = | w | · sinc ( w υ ) .
| L * ( u , υ ) | = | X * ( u ) | · | Y * ( υ ) | ,
| X * ( u ) | = ( | w | / 2 ) · { sin c 2 [ w ( u + f ) ] · sin c 2 [ w ( u f ) ] + 2 · sinc [ w ( u + f ) ] · sinc [ w ( u f ) ] · cos ( 2 π f w ) } 1 / 2
| Y * ( υ ) | = | w · sinc ( w υ ) | .