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A spatial integration effect in visual acuity
Vision Res. Vol.
9. pp.
157-166.
Pcrpuaon
A SPATIAL
Press
1969.
Printed
in Great
Britain.
INTEGRATION VISUAL ACUITY
EFFECT
IN
J. M. FINDLAY~ J. J. Thomson Physical Laboratory, Whiteknights Park, Reading, Berkshire (Received 27 August 1968) INTRODUCTION MUCH recent work on human visual acuity has been concerned with the determination of the modulation transfer function of the eye’s optics, and its Fourier transform, the point spread function (CAMPBELL and GREEN, 1965; CAMPBELL and GUBISCH, 1966; VAN NES, KOENDERINK, NAS and BOU~IAN, 1967). These results allow prediction of the spatial distribution of intensity in the retinal image of any object. To relate this to psychophysical measurements it is necessary to have a method for evaluating the neural responses. It is only possible to use Fourier techniques if the neural response is linear. The majority of psychophysical measurements are made in terms of a threshold, and it is necessary to determine which physical parameter of the retinal image can be associated with this threshold value. For the simplest target, a grating pattern of constant spatial frequency, the most obvious choice for such a parameter is the contrast in the retinal image. The quantitative measurement of this is the visibility of the image, defined by
I max-Icnin v= L,X+~,iIl
where I,,, and Zminare the maximum and minimum intensities present. Since the optical system of the eye is linear, this quantity is proportional to the visibility in the object for a fixed spatial frequency of the grating. However, it seems very probable that, in some sense, a threshold decision is a decision concerning the detection of a signal in the presence of noise. The signal to noise ratio for a regular pattern, such as a grating, will increase as the size of the pattern is increased. Consequently, if the eye’s detection system operates in the most efficient manner possible, the threshold contrast for grating detection would decrease as the size of the grating was increased. To achieve this optimum performance it would be necessary for the eye-brain system to combine information from different spatial regions. This paper describes an investigation in which the threshold contrast is found for gratings that have the same spatial frequency, but contain a varying number of lines. Certain effects occur when the number of lines is very small, but the chief results of interest are when this number is six or more. In various conditions it is shown that an integration effect occurs whereby the threshold contrast decreases as the number of lines is increased. It has proved convenient to use the term ‘pattern integration’ for this effect and this will be the meaning of the term as used in the paper. 1 Present address: Department of Psychology, University of Durham. 157
J. M. FINDLAY
158
APPARATUS
AND
METHODS
The gratings were formed as transparencies by photographing a large, hand drawn, grating pattern, which was then viewed in transmitted illumination. The grating had a square-wave variation of transmission and its spatial frequency, aa &ally viewed, was 30 cycles/deg. At this spatial frequency it may confdentfy be expected that the di&rences between square-wave modulation and sim.zsoidaIrnod~a~io~ will be small.
I’ 82 FIG. 1. Optical system for presentation of targets with variable contrast. The optical system is shown in Fig. 1. A conventional projection system illtm&atea the rprtino G whichispMedatthepri&palfocusofthelensL~. Thiaiensisalso~to~tbe~tsqwrce (sffectively apenure A) on to the eye. Thus the image is seen at in&&y in MW view, ttwruph an aMicialpupilofdiam@er4mm. BIand&are I:1 beamsplitterstosqparateandrc@ontbinetM,b@ms which i&am&ate the aMe of the sfiltino target and a u&Corm &de respectirpeiy. The m&&m stide is aho placed at the px%cipal focus of Iens L2. Thetwo~arcplan*polarkcdinpe~~o~bymteasof~fixedpdaroidshects PI and Pz. P is a sheet of pokuoid that can be rotated by means of a d.c. motor and its ang&r pc&ititm is indicated on a counter. This allows continuous variation of the contrast of the gnH+ The v&&i&y Vis~tothsaapto8betwantbepolarisationdinaidnsof~~dshatandthspafaroidinthc grating beam by the relation co@ (T1- Tz) V -_
2TsMB + (?.I+ T2) &Xi@ ~~TtandT~arcI$lafiacticnafintdlrpirics~~byth*lightanddar)c~ofthcgntino~~y and 2’ that transmitted in the uniform beam. The denominator $ proportional to the resllitant mean intensity and it is seen that if Tl+T2=2T, this is indemdcnt of 8, providing a constant luminance target. With the grating target and uniform target used, Tt=@jol, T~==@141,2’=@426. “Ibis re%dted in a 5 per cent luminance variation over the @tire range used, whasrr e&et ws co&de& Ileglilil’ble. The light source used was aa A.E.I. Ai/ quartz iodine projector bulb which alIowe& production of red, green, blue and white light of resultant luminance at the eye of IO cd/r&, the value used for ail the
Spatial Integration
in Visual Acuity
159
experiments described. Chance ORI. OGrl and OBlO filters with the appropriate neutral density filters were inserted at F. Fixed masks attached to both grating and uniform slides limited the field to a rectangle of lf by Y (the short axis being that of the lines of rhe grating), and these rectangles were aligned to coincide. The grating slide was rhen further limited to a number N cycles by one of a series of masks, cut to correspond to values of N of 3, 4, 6, 8, 12, 16 and 28. Thus the observer was presented with a targeet whose luminance variation is shown in Fig . 2. Further considerations involving this type of target are mentioned in the Discussion.
r-7 i Fig. 2. Distribution
1 of intensity in target.
The observer’s task was to rotate the Polaroid wheel to the point where the lines were no longer visible. The procedure used in practice was to set the wheel at some point and if the grating became visible in a time of approximately 10 set, to decrease the contrast slightly and repeat the procedure. The fact that, near threshold, the target is visible only intermittently can in part be ascribed to accommodation fluctuations. The experimenter presented the targets for different values of N in a random sequence and the subject made three or four settings before the target was changed. The sequence was repeated three times in a complete experimental run so that the subject made ten settings with each target. Two male subjects were used, JMF aged 25 and HRP aged 56. HRP required a f0.75 D lens placed above the artificial pupil P. In blue light JMF used a -0.50 D lens and HRP a +0.25 D lens. Threshold criterion
For gratings consisting of six lines or more, the subjects experienced little difficulty in making a threshold setting. The procedure described above is an approximation to the more thorough psychophysical procedure of determining the percentage of times a target of a given contrast was seen for a given duration of presentation. The standard error in the settings, expressed in terms of the visibility, was usually 20-30 per cent over the ten settings for any value of N, and it was established that the majority of this was not due to taking finite increments in the setting procedure. The variability between one sequence and the next was almost always less than the figure for the standard error given above. An interesting fact to emerge was that the subject was either able to see the entire grating, or nothing at all was visible. Exceptions to this were observed at times for the largest grating, N=28. For a number of reasons, difficulties arise when attempts are made to determine the threshold for gratings consisting of 3 or 4 cycles. It was found that the threshold settings were extremely variable, and in consequence of this, these results were not used in the subsequent analysis. Various causes of the variability can be suggested. Firstly, it was not possible to position the mask with sufficient accuracy to ensure that its edge fell at a known point on the period of the grating. However, even if this difficulty were overcome, a more fundamental effect occurs because of the production of Mach bands at the grating edges. The effect of lateral inhibition on a three line grating is shown in Fig. 3. Even when the grating modulation is well below threshold the two bright Mach bands combine to produce an apparent ‘shadow’ in the centre of the target and this is difficult to distinguish from true grating modulation.
FIG. 3. Distribution of intensity in three bar grating target (a) retinal image, (b) perceptual image. Another effect may be significant with the small targets as suggested by Professor H. H. Hopkins (private communication). The spatial frequencies present in such a target can be found from the convolution of the discrete frequencies present in an infinite grating with the spatial frequency spectrum of the grating limits (a s&/x function). The result of this is to cause a range of frequencies to be present, and this is then altered by the frequency transmission function corresponding to the optics of the eye. In the extreme case, when the frequency of the infinite grating is outside the resolvable ‘pass band’ of the eye’s optics, the finite grating may have a spectrum with components within this band. This would give spurious resolution.
L
160
J. M. FINDLAY
However, it seems a detailed calculation would be necessary to assess this effect in any particular case (see Discussion). Analysis procedure In each experimenta set of ten threshold readings for each of the seven values of N used was obtained. The mean and standard deviation were found for each set. For the reasons stated above, only those for values of N greater than six were used subsequently. The following empirical analysis procedure was then adopted. It was observed in preliminary experiments that an approximately linear relationship existed between V and log N, so valuesof N were chosen at approximately equal intervals on a logarithmic scale. In order to test the hypothesis that the threshold changes with line number, a regression line for V against log N was fitted to the 50 points on each run (from N=6 to N=28) and a correlation coe&ient calculated.: For a zero correlation coefficient the sampling values are normahy distributed and so that the hypothesis of the variable being uncorrelated can be tested. FISHER (1946) has published 5 per cent significance values for observed correlation coefficients, and for 48 deg of freedom the value is 0.280. Values higher than this have less than 5 per cent chance of being obtained with uncorrelated variables. RESULTS In all the results presented here the grating period was 30 cycle/deg and the luminance 10 cd/m* (within +20 per cent).
Co/our varMtion
The experiment was carried out with the grating bars vertical as viewed by the subject, using the colour filters described earlier. The results are presented in Table 1 and Fig. 4. The mean value of V for the whole range N=6 to N=28 is given, together with the change in V from N=6 to N=28, d V, as calculated from the regression line. Positive values of this quantity indicate that the threshold contrast decreases as the line number is increased. The correlation coefficient is also given; as shown earlier, values of this greater than 0.280 show correlation significant at the 5 per cent level. All the experimental runs made under these conditions are shown with the exception of two for which the scatter in the observations was exceptionally high. Considering first the values of the mean visibility, it is seen that the two subjects are roughly similar except in the case of red, where the threshold is lower for JMF. The absolute values obtained can be compared with figures of O-02 in red and green lights presented by VAN NES et al. (1%7) at a similar luminance and spatial frequency. It is evident that the threshold in blue light is much higher than in the other colours, and it has been often noted that unusual results are obtained with blue in the fovea (WALD, 1967). It has been suggested that the ‘blue’ mechanism has a much lower acuity and the possibility exists that the results obtained are due to excitation of the ‘green’ mechanism (see BRINDLEY,1954). If this were so the results obtained would be identical to those obtained by stimulating the green mechanism with a green light of considerably lower luminance. The hypothesis can be tested in this way provided the equivalent stimulation of the green mechanism can be found, and this may be achieved by use of the values of the spectral sensitivities of the different mechanisms determined by STILES (1949). The calculation shows that a stimulus of 10 cd/m* in blue (OB 10) light is equivalent to a stimulus of co. 1 cd/m* in green light for the green receptors. A rough check on the acuity at this luminance showed it to be similar to that obtained in the blue experiments. It was * In order to simplify the computations it was assumed that the values of N were distributed with equal intervals on the scale of log IV. The error introduced by this was checked in a number of cases and found never to exceed 5 per cent.
Spatial Integration TABLE 1. MEAX AND
CORRELATION
1GI
in Visual Acuity
THRESHOU) VNBIUTY, DIFFERENCE IN VISIB~TY BETWEEN N=6 AND N=28, COEFFICIEXTS. EACH 5ET OF VALLCXS REPRESEXIJ ONE EWERI!vlENTAL RUN
Change in V
Mean visibility
Colour
Subject
Av
Correlation coefficient
OX)08 0.015 0,009 0.016 0.014 0,012 0.026 0.038 0.024 0.025 0.020 0.030 0.003 0.017 -0.005 o.cMx 0.004 0.023 0.015 0.010
0.371 0.502 0.219 0.414 0.480 0.333 0.563 0.629 0.344 0.501 0.445 0.316 0.145 0.404 -0.130 0,338 0.102 0.208 0.447 0.356
V
JMF
White
HRP
White
JMF
Green
HRP
Green
JMF
BlUC
HRP
Blue
JMF
Red
HRP
Red
Subject
0.036 0.043 0.032 0.039 0.026 0.043 0.031 0.040 0.055 0.062 0.041 0.070 0.020 0.041 0.021 0.021 0.044 0.039 0,026 0.035
. x
JMF
0.1
WhitC Red
m
Green
.
Blue
Subject
.
HRP
0.1
White
x
Red
,
Green
.
Blue
.
. . .
.
.
.
. m
:
x
I
t
? x
0’
6
Number
d
linct
e Number
. . L
I2 of
lines
FIG. 4. Variation of contrast threshold
with number of lines for a 30 cycle/deg grating in illumination of various colours.
also possible to obtain some acuity measurements using an interference filter (20 nm bandwidth) centred at 450 nm, and an identical calculation showed that the results obtained were again similar to those obtained from stimulation of the green receptors at the equivalent luminance level. Thus it must be concluded that for 30 cycles/deg gratings, the acuity is so low for the blue mechanism that the green mechanism is always used preferentially under normal conditions, though selective adaptation of the two mechanisms might produce a condition in which the blue mechanism was used.
J.M.
161
FINDLAY
Since the Stiles IS~mechanism (‘green’) has an absolute sensitivity greater than the z5 (red) mechanism, it may be anticipated that the results in white light will also be due to stimulation of the green mechanism. CAMPBELLand GUBISCH (1967) have shown the effect of chromatic aberration to be small at this spatial frequency. Turning now to the results of the change in visibility between N=6 and N=28 it is seen that in all cases where the ‘green’ mechanism is used, there is a decrease in threshold contrast as the line number is increased and in all but one case this is statistically significant at the 5 per cent level. The magnitude of the change is such that there is, on average, a 30-40 per cent decrease in threshold contrast from a six bar grating to a 28 bar grating. The results for red are somewhat different. With one exception the correlation coefficients are positive but they are evidently lower than those in the other colours. Of the eight results obtained, half fall at each side of the 5 per cent significance level. In the case of JMF the low correlation coefficients are explained by lower values of AV, although in the case of HRP they may in part be attributed to larger scatter. The only conclusive result that can be drawn is that the red and green results show differences, itself a feature of some interest, but it seems at least possible that the pattern integration effect is present with the green mechanism and not with the red. Although the mean threshold for red is lower than for green, when the pattern integration effect is taken in account the threshold for a large number of lines is about the same in the two cases. It seems unlikely that two mechanisms can achieve the same acuity in different ways, but it is not impossible in view of the complexity of the eye’s nervous system. One factor limiting acuity is undoubtedly diffraction, and this has a greater effect for green light than for red. The conclusion of these experiments is that the eye’s green sensitive mechanism possesses pattern integration ability. This is possibly absent in the red sensitive mechanism, Orientation dependence
It is well known that for most subjects acuity is dependent on the orientation of the test grating, being generally better for horizontal and vertical positions of the grating than for oblique positions. CAMPBELL, KIJLIICOWSKIand LEVIN~ON (1966) have shown that this is not a property of the eye’s optics, and so an explanation of this effect must be found in the visual nervous system. In view of the pattern integration TABLE 2. MEAN THRESHOLD VBIBILITY,DIFFERENCEIN vxsl~m CORRELATIONCOEPFICZENTFORDIFFERENTORIENTATIONS
Subject
Orientation
JMF
0
HRP
0”
JMF
45’
HRP
45’
JMF
90”
HRP
90”
JMF
135”
HRP
135”
V 0.026 0.043 0.031 0.041 0.080 0.100 0.129 0.065 0.036 0.051 0.037 0.025 0.081 0.054 0038 0.039
Av 0.013 0,012 O+O26 0.039 -0.006 0.015 0.022 0.042 0.011 0.020 0.025 0.010 0014 0.009 O-012 0.020
AND
Correlation coefficient 0.480 0.333 0.563 0.629 -0~070 0.225 0.147 0.793 0444 0.638 0.662 o* 393 0.167 0.125 0,352 0.458
Spatisl Integration
163
in Visual Acuity
effect found in section (a), it seemed possible that the difference in acuity might be produced bv such integration being operative in certain directions only. This was tested by carrying out a series of expeknents with the target grating lines vertical (0’). horizontal (90”) and in two oblique positions (545”). The experiments were carried out with green light, target conditions being unaltered. The results are presented in Table 2. For subject HRP, if one erratic result is discounted, no significant difference emerges between his acuity for horizontal, vertical, and obhque orientations. Subject JMF shows an orientation dependent variation in acuity. His values of d V are similar at ail orientations, but because of the difference in absolute threshold, the relative changes in visibility are less in the oblique directions, and examination of the correlation coefficient shows that in all cases these changes are not significant at the 5 per cent level. Thus there is some evidence that, for this subject at least, spatial pattern integration is superior in the horizontat and vertical directions. Figure 5 shows the mean values of threshoId visibility plotted against IV. The convergence of the lines for small N provides some support for the idea outlined above. However. the results for HRP are entirely different and its seems no conclusion can be drawn without recourse to experiments on further subjects.
.
SubjectJMF
.
0.1
o*
.
456
x
PO0
.
135O
Subject 0.
HRP
I
A
0” OS0
x
9o”
.
135”
f
.
Fro. 5. Variation of contrast threshold with number of lines for a 30 cycle/deg grating at various orientations.
Limit of spatid pattern i~tegr~t~o~ The evaluation made above gives no information about the extent of the range over which the pattern integration effect operates This has been analysed as follows Considering just the results for two values of N, e.g. N= 16 and N=28, it is possible to evaluate for each run the difference n between the mean values of V (e.g. vr6- P’2s)and a standard deviation for this quantity s. The results for a series of runs, labelled 1, 2, . . . . . . . n may be combined to give a weighted mean difference
‘===+ lJsI
......-i
lJs2:
'Jsn
and a weighted standard deviation n S=
‘ls,+ ys24 . . . . . . 4- ‘Is.
164
J.M. FINDLAY
A Student’s t-test may then be performed on the statistic d d(n- I)/; to determine whether the mean difference in the thresholds is significant. This test was carried out for each subject on all the runs in the 0“ and 90’ orientations, excepting those in red. The results are presented in Table 3. This gives the value of the statistic together with the probability of obtaining this result if no difference existed between the means. TABLE 3. VALUES-OFt STATISTIC, AND SCHFICANCE LEVELS FORTHE K&WDEFERENCES lNTHRESHOLDVISIBILlTyWlTHVARIOUSLINENUMBERS N=16 to N=28 JMF HRP
2.74 (2 %) 0.40( )
N= 12 to N=28 3.44 (0.6 “/,) 2.62 (3 %)
N-12 to N=16
N=6 to N=8
N=8 to N=12
1.34 (25%) 1.04 (30 %)
1.00 (30%) 0.62 ( )
2-18 (5%) 2.23 (SO,?
The results show that the differences in contrast threshold observed between adjacent values of N in the experiments cannot usually be given significance. However, the results for the difference between the mean threshold for N= 12 and N=28 show for both subjects a significant value of t, indicating that pattern integration is carried out beyond N= 12. It seems likely that for subject JMF that pattern integration exists beyond N= 16, but the results do not confirm this for subject HRP. DISCUSSION
The experiments have shown that under certain conditions the contrast threshold for detection of a grating decreases as the number of lines in the grating is increased. The possibilities will be considered in turn that this is an experimental artefact, a physical effect in the optics of the eye, or a neural effect. Non-homogeneities in the grating illumination or uniform background illumination might lead to certain parts of the target possessing greater modulation. However, visual inspection failed to reveal any such lack of homogeneity and a change of lun&ance of ca. 30 per cent over the grating would have been necessary to explain the results in this way. A second possible artefact lies in the threshold procedure used, which is somewhat crude by modem psychophysical standards, but it seems likely that the relative values obtained using this procedure will still possess validity. The physical effect on the retinal image of limiting the size of the grating target must be considered. The intensity distribution in the retinal image may be found by forming the convolution of the distribution in the object with the spread function of the optics of the eye. CAMPBELL and GUBISCH(1966) have evaluated the optical &spread function for the eye. A numerical computation would be necessary to determine completely the effect of the convolution. However, any possible difference would be produced by the step in illumination at the edge of the grating. An upper limit to this difference may be estimated by noting that the line spread function has a value of O-05 or less for displacements of 4 min arc or more (for a 3-S mm dia. pupil). The target containing 6 lines had a width of 12 min arc, thus in its centre the visibility in the retinal image is certainly identical to within 5 per cent of that in a larger target. Thus the greater part of the effect found must take place in the visual nervous system. The simplest process would be for the threshold evaluation to take place in two stages, some intermediate mechanism evaluating the target on the basis of a probability decision,
Spatial Integration
in Visual Acuity
165
taking into account all the available information, and then presenting to the consciousness either a grating picture or a plain field. In support of this is the subjective finding that the grating is either seen in its entirety or not at all. The subject is not conscious of the ‘noise’ which must provide the limiting factor. BARLOW(1957) has suggested a similar two-stage mechanism, in which a probability decision is made on the results of the first stage, to explain increment thresholds in scotopic vision. The idea that the visual system can spatially integrate information in order to reach a threshold decision is well known in certain particular conditions (e.g. Vernier acuity and Ricco’s Law). COLTMAN and ANDERSON (1960) have demonstrated an effect in conditions rather similar to those of this experiment. In the course of a study to optimise television presentation, they determined the dependence of threshold contrast for a regular line pattern on the number of lines in the presence of external optical noise. Their results show an integration effect up to a line number value of 7; however, direct comparison is not possible since the spatial frequency they used is not given and it seems probable that extra-fovea1 vision was being used. It is certainly conceivable that the visual thresholds determined in this paper represent a similar signal/noise detection process, and a discussion of this is given in the Appendix. It is thus apparent that under certain conditions the physical parameter determining the threshold for a grating image is not simply the visibility in the image, but is dependent on a spatial integration over the whole region available. The targets used have been confined to the fovea, so further measurements would be necessary to determine whether the results could be extended to extrafoveal targets. However, it is evidently necessary to exercise caution in interpreting acuity results from grating patterns containing only a small number of lines (see also NACHMIAS, 1968). CONCLUSIONS
The response of the eye to 30 cycles/deg grating patterns of limited size has been studied. Thresholds for such a grating are determined by a process which involves integrating information over all the cycles of the grating present, up to at least twelve cycles. The effect is more pronounced when using the ‘green’ mechanism than when using the ‘red’ and may possibly be absent in the latter. Investigation of the orientation dependence of the effect did not allow any delinite conclusions to be drawn, but indicated the possibility that orientational variations in visual acuity could be due to variations in integration ability. Acknowledgements-Theauthor thanks Professor R. W. DITCHBURN,F.R.S., for advice and encouragement, and also Mr. H. R. PESCOD, both for technical assistance and for acting as a subject. The research was carried out under M.R.C. Grant No. G/964/218/B. REFERENCES B-w,
H. B. (1957).
Increment thresholdsand visual noise. J. Physiol. 136,46!2-488. summation areas of human colour-receptive mechanisms
BIUNDLEY, G. S. (1954). The
at increment J. Physiol. D&400-408. CAMF-BELL, F. W. and GREEN, D. G. (1965). Optical and retinal factors affecting visual resolution 1. Physiol. 181, 576593. C-BELL, F. W. and GUB~~CH.R W. (1966). Optical quality of the human eye. J. Physiol. 186, 558-578. CAMPBELL, F. W. and GUBISCH, R. W. (1967). The effect of chromatic aberration on visual acuity. J. Physiol. 192, 345-358. CAMPBELL, F. W., KULIKOWSKI, J. J. and LE~~NSON,J. (1966). The effect of orientation on the visual resolution of gratings. J. Physiol. 187, 427436. threshold.
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166
COLTMAN, J. W. and ANDERSON,A. E. (1960). Noise limitations to resolving power in electronic imaging. Proc. I.R.E. 48, 858-866. FISHER,R. A. (1946). Statistical Methoa!s for Research Workers. 10th edition, Oliver and Boyd, Edinburgh. NACHMIAS, J. (1968). Visual resolution of two bar patterns and square wave gratings. J. opt. Sot. Am. 58.9-13. VAN N&s, F. L., KOENDERINK, J. J., NAS, H. and BOLJSL~N, M.A. (1967). Spatiotemporal modulation transfer in the human eye. J. opf. Sot. Am. 57,401-K%. STILES,W. S. (1949). Increment thresholds and the mechanisms of colour vision. Documenta Uphthal. 3, 13&165. WALD, 0. (1967). Blue blindness in the normal fovea. J. opf. Sot. Am. 57, 3289-1301.
APPENDIX It is of interest to compare the results obtained with those that would be obtained with an ideal physical detector. The Shannon result for the optimum value of the information capacity of a channel of bandwidth W for a signal of power P in the presence of a white Gaussian noise of power Q is P-+-Q c = w log2 -
Q
For a grating pattern of visibility Ycontaining N lines, P is proportional to (VW and Q is proportional to N. Thus C = W log2 (kY2Nfl) where k is a constant. In an ideal system with fixed channel capacity and bandwidth, the factor VZN would be constant. This expm the familiar 1/N r&tio&ip for detection of a coherent signal. The table gives a comparison of values of 1/N relative to N= 12 with the observed values (green). N
6 8 12 16 28
l/v/N
1.41 1.22 1.00 0.87 0.65
JMF 1.36 1.19 I.00 0.87 @79
HRP 1.33 1.16 1.00 0.84 0.68
The results appear to show that the integrating capacity shown by the eye is only slig&tly poorer than that of an ideal detector. However, this must be interpreted with caution in view of the approximations made. Ab&~ct--The ei%ct of grating ria on the threshold for a 30 cycle per degree grating pattan is investigated for various colours of illumination and qt orientations. It is shown under certain conditions that the &hruhofd w as the target size is increased, and thb is attributed to spatial integration of information. R&m&-on &tudie pour diverses couleurs de la lumi&e et d&sea orientation de la mire I’dfasurie~&ladimeosioad’uncmireB~~de3ocyckspardtqrc. Dam certaines conditions on constate que la scuil diminue quand la dimension de la mire crott, ce qu’on attribue B une int&ration spatiale de l’information. FDer Einfiul3 der Gittcrgri& auf $ie Schwelle bei einem Gitter mit 30 Pcriodcn/Grad wird Nr vcmcbi&$ne Farben I& F&rm$a Objahs unt_nsttch’. M obje!ktEswizdgez&t,~u&ezb&mmten~ g&e abnimmt. Dies wird der &&hen Integration dcr Information zupnchrieben.
9. pp.
157-166.
Pcrpuaon
A SPATIAL
Press
1969.
Printed
in Great
Britain.
INTEGRATION VISUAL ACUITY
EFFECT
IN
J. M. FINDLAY~ J. J. Thomson Physical Laboratory, Whiteknights Park, Reading, Berkshire (Received 27 August 1968) INTRODUCTION MUCH recent work on human visual acuity has been concerned with the determination of the modulation transfer function of the eye’s optics, and its Fourier transform, the point spread function (CAMPBELL and GREEN, 1965; CAMPBELL and GUBISCH, 1966; VAN NES, KOENDERINK, NAS and BOU~IAN, 1967). These results allow prediction of the spatial distribution of intensity in the retinal image of any object. To relate this to psychophysical measurements it is necessary to have a method for evaluating the neural responses. It is only possible to use Fourier techniques if the neural response is linear. The majority of psychophysical measurements are made in terms of a threshold, and it is necessary to determine which physical parameter of the retinal image can be associated with this threshold value. For the simplest target, a grating pattern of constant spatial frequency, the most obvious choice for such a parameter is the contrast in the retinal image. The quantitative measurement of this is the visibility of the image, defined by
I max-Icnin v= L,X+~,iIl
where I,,, and Zminare the maximum and minimum intensities present. Since the optical system of the eye is linear, this quantity is proportional to the visibility in the object for a fixed spatial frequency of the grating. However, it seems very probable that, in some sense, a threshold decision is a decision concerning the detection of a signal in the presence of noise. The signal to noise ratio for a regular pattern, such as a grating, will increase as the size of the pattern is increased. Consequently, if the eye’s detection system operates in the most efficient manner possible, the threshold contrast for grating detection would decrease as the size of the grating was increased. To achieve this optimum performance it would be necessary for the eye-brain system to combine information from different spatial regions. This paper describes an investigation in which the threshold contrast is found for gratings that have the same spatial frequency, but contain a varying number of lines. Certain effects occur when the number of lines is very small, but the chief results of interest are when this number is six or more. In various conditions it is shown that an integration effect occurs whereby the threshold contrast decreases as the number of lines is increased. It has proved convenient to use the term ‘pattern integration’ for this effect and this will be the meaning of the term as used in the paper. 1 Present address: Department of Psychology, University of Durham. 157
J. M. FINDLAY
158
APPARATUS
AND
METHODS
The gratings were formed as transparencies by photographing a large, hand drawn, grating pattern, which was then viewed in transmitted illumination. The grating had a square-wave variation of transmission and its spatial frequency, aa &ally viewed, was 30 cycles/deg. At this spatial frequency it may confdentfy be expected that the di&rences between square-wave modulation and sim.zsoidaIrnod~a~io~ will be small.
I’ 82 FIG. 1. Optical system for presentation of targets with variable contrast. The optical system is shown in Fig. 1. A conventional projection system illtm&atea the rprtino G whichispMedatthepri&palfocusofthelensL~. Thiaiensisalso~to~tbe~tsqwrce (sffectively apenure A) on to the eye. Thus the image is seen at in&&y in MW view, ttwruph an aMicialpupilofdiam@er4mm. BIand&are I:1 beamsplitterstosqparateandrc@ontbinetM,b@ms which i&am&ate the aMe of the sfiltino target and a u&Corm &de respectirpeiy. The m&&m stide is aho placed at the px%cipal focus of Iens L2. Thetwo~arcplan*polarkcdinpe~~o~bymteasof~fixedpdaroidshects PI and Pz. P is a sheet of pokuoid that can be rotated by means of a d.c. motor and its ang&r pc&ititm is indicated on a counter. This allows continuous variation of the contrast of the gnH+ The v&&i&y Vis~tothsaapto8betwantbepolarisationdinaidnsof~~dshatandthspafaroidinthc grating beam by the relation co@ (T1- Tz) V -_
2TsMB + (?.I+ T2) &Xi@ ~~TtandT~arcI$lafiacticnafintdlrpirics~~byth*lightanddar)c~ofthcgntino~~y and 2’ that transmitted in the uniform beam. The denominator $ proportional to the resllitant mean intensity and it is seen that if Tl+T2=2T, this is indemdcnt of 8, providing a constant luminance target. With the grating target and uniform target used, Tt=@jol, T~==@141,2’=@426. “Ibis re%dted in a 5 per cent luminance variation over the @tire range used, whasrr e&et ws co&de& Ileglilil’ble. The light source used was aa A.E.I. Ai/ quartz iodine projector bulb which alIowe& production of red, green, blue and white light of resultant luminance at the eye of IO cd/r&, the value used for ail the
Spatial Integration
in Visual Acuity
159
experiments described. Chance ORI. OGrl and OBlO filters with the appropriate neutral density filters were inserted at F. Fixed masks attached to both grating and uniform slides limited the field to a rectangle of lf by Y (the short axis being that of the lines of rhe grating), and these rectangles were aligned to coincide. The grating slide was rhen further limited to a number N cycles by one of a series of masks, cut to correspond to values of N of 3, 4, 6, 8, 12, 16 and 28. Thus the observer was presented with a targeet whose luminance variation is shown in Fig . 2. Further considerations involving this type of target are mentioned in the Discussion.
r-7 i Fig. 2. Distribution
1 of intensity in target.
The observer’s task was to rotate the Polaroid wheel to the point where the lines were no longer visible. The procedure used in practice was to set the wheel at some point and if the grating became visible in a time of approximately 10 set, to decrease the contrast slightly and repeat the procedure. The fact that, near threshold, the target is visible only intermittently can in part be ascribed to accommodation fluctuations. The experimenter presented the targets for different values of N in a random sequence and the subject made three or four settings before the target was changed. The sequence was repeated three times in a complete experimental run so that the subject made ten settings with each target. Two male subjects were used, JMF aged 25 and HRP aged 56. HRP required a f0.75 D lens placed above the artificial pupil P. In blue light JMF used a -0.50 D lens and HRP a +0.25 D lens. Threshold criterion
For gratings consisting of six lines or more, the subjects experienced little difficulty in making a threshold setting. The procedure described above is an approximation to the more thorough psychophysical procedure of determining the percentage of times a target of a given contrast was seen for a given duration of presentation. The standard error in the settings, expressed in terms of the visibility, was usually 20-30 per cent over the ten settings for any value of N, and it was established that the majority of this was not due to taking finite increments in the setting procedure. The variability between one sequence and the next was almost always less than the figure for the standard error given above. An interesting fact to emerge was that the subject was either able to see the entire grating, or nothing at all was visible. Exceptions to this were observed at times for the largest grating, N=28. For a number of reasons, difficulties arise when attempts are made to determine the threshold for gratings consisting of 3 or 4 cycles. It was found that the threshold settings were extremely variable, and in consequence of this, these results were not used in the subsequent analysis. Various causes of the variability can be suggested. Firstly, it was not possible to position the mask with sufficient accuracy to ensure that its edge fell at a known point on the period of the grating. However, even if this difficulty were overcome, a more fundamental effect occurs because of the production of Mach bands at the grating edges. The effect of lateral inhibition on a three line grating is shown in Fig. 3. Even when the grating modulation is well below threshold the two bright Mach bands combine to produce an apparent ‘shadow’ in the centre of the target and this is difficult to distinguish from true grating modulation.
FIG. 3. Distribution of intensity in three bar grating target (a) retinal image, (b) perceptual image. Another effect may be significant with the small targets as suggested by Professor H. H. Hopkins (private communication). The spatial frequencies present in such a target can be found from the convolution of the discrete frequencies present in an infinite grating with the spatial frequency spectrum of the grating limits (a s&/x function). The result of this is to cause a range of frequencies to be present, and this is then altered by the frequency transmission function corresponding to the optics of the eye. In the extreme case, when the frequency of the infinite grating is outside the resolvable ‘pass band’ of the eye’s optics, the finite grating may have a spectrum with components within this band. This would give spurious resolution.
L
160
J. M. FINDLAY
However, it seems a detailed calculation would be necessary to assess this effect in any particular case (see Discussion). Analysis procedure In each experimenta set of ten threshold readings for each of the seven values of N used was obtained. The mean and standard deviation were found for each set. For the reasons stated above, only those for values of N greater than six were used subsequently. The following empirical analysis procedure was then adopted. It was observed in preliminary experiments that an approximately linear relationship existed between V and log N, so valuesof N were chosen at approximately equal intervals on a logarithmic scale. In order to test the hypothesis that the threshold changes with line number, a regression line for V against log N was fitted to the 50 points on each run (from N=6 to N=28) and a correlation coe&ient calculated.: For a zero correlation coefficient the sampling values are normahy distributed and so that the hypothesis of the variable being uncorrelated can be tested. FISHER (1946) has published 5 per cent significance values for observed correlation coefficients, and for 48 deg of freedom the value is 0.280. Values higher than this have less than 5 per cent chance of being obtained with uncorrelated variables. RESULTS In all the results presented here the grating period was 30 cycle/deg and the luminance 10 cd/m* (within +20 per cent).
Co/our varMtion
The experiment was carried out with the grating bars vertical as viewed by the subject, using the colour filters described earlier. The results are presented in Table 1 and Fig. 4. The mean value of V for the whole range N=6 to N=28 is given, together with the change in V from N=6 to N=28, d V, as calculated from the regression line. Positive values of this quantity indicate that the threshold contrast decreases as the line number is increased. The correlation coefficient is also given; as shown earlier, values of this greater than 0.280 show correlation significant at the 5 per cent level. All the experimental runs made under these conditions are shown with the exception of two for which the scatter in the observations was exceptionally high. Considering first the values of the mean visibility, it is seen that the two subjects are roughly similar except in the case of red, where the threshold is lower for JMF. The absolute values obtained can be compared with figures of O-02 in red and green lights presented by VAN NES et al. (1%7) at a similar luminance and spatial frequency. It is evident that the threshold in blue light is much higher than in the other colours, and it has been often noted that unusual results are obtained with blue in the fovea (WALD, 1967). It has been suggested that the ‘blue’ mechanism has a much lower acuity and the possibility exists that the results obtained are due to excitation of the ‘green’ mechanism (see BRINDLEY,1954). If this were so the results obtained would be identical to those obtained by stimulating the green mechanism with a green light of considerably lower luminance. The hypothesis can be tested in this way provided the equivalent stimulation of the green mechanism can be found, and this may be achieved by use of the values of the spectral sensitivities of the different mechanisms determined by STILES (1949). The calculation shows that a stimulus of 10 cd/m* in blue (OB 10) light is equivalent to a stimulus of co. 1 cd/m* in green light for the green receptors. A rough check on the acuity at this luminance showed it to be similar to that obtained in the blue experiments. It was * In order to simplify the computations it was assumed that the values of N were distributed with equal intervals on the scale of log IV. The error introduced by this was checked in a number of cases and found never to exceed 5 per cent.
Spatial Integration TABLE 1. MEAX AND
CORRELATION
1GI
in Visual Acuity
THRESHOU) VNBIUTY, DIFFERENCE IN VISIB~TY BETWEEN N=6 AND N=28, COEFFICIEXTS. EACH 5ET OF VALLCXS REPRESEXIJ ONE EWERI!vlENTAL RUN
Change in V
Mean visibility
Colour
Subject
Av
Correlation coefficient
OX)08 0.015 0,009 0.016 0.014 0,012 0.026 0.038 0.024 0.025 0.020 0.030 0.003 0.017 -0.005 o.cMx 0.004 0.023 0.015 0.010
0.371 0.502 0.219 0.414 0.480 0.333 0.563 0.629 0.344 0.501 0.445 0.316 0.145 0.404 -0.130 0,338 0.102 0.208 0.447 0.356
V
JMF
White
HRP
White
JMF
Green
HRP
Green
JMF
BlUC
HRP
Blue
JMF
Red
HRP
Red
Subject
0.036 0.043 0.032 0.039 0.026 0.043 0.031 0.040 0.055 0.062 0.041 0.070 0.020 0.041 0.021 0.021 0.044 0.039 0,026 0.035
. x
JMF
0.1
WhitC Red
m
Green
.
Blue
Subject
.
HRP
0.1
White
x
Red
,
Green
.
Blue
.
. . .
.
.
.
. m
:
x
I
t
? x
0’
6
Number
d
linct
e Number
. . L
I2 of
lines
FIG. 4. Variation of contrast threshold
with number of lines for a 30 cycle/deg grating in illumination of various colours.
also possible to obtain some acuity measurements using an interference filter (20 nm bandwidth) centred at 450 nm, and an identical calculation showed that the results obtained were again similar to those obtained from stimulation of the green receptors at the equivalent luminance level. Thus it must be concluded that for 30 cycles/deg gratings, the acuity is so low for the blue mechanism that the green mechanism is always used preferentially under normal conditions, though selective adaptation of the two mechanisms might produce a condition in which the blue mechanism was used.
J.M.
161
FINDLAY
Since the Stiles IS~mechanism (‘green’) has an absolute sensitivity greater than the z5 (red) mechanism, it may be anticipated that the results in white light will also be due to stimulation of the green mechanism. CAMPBELLand GUBISCH (1967) have shown the effect of chromatic aberration to be small at this spatial frequency. Turning now to the results of the change in visibility between N=6 and N=28 it is seen that in all cases where the ‘green’ mechanism is used, there is a decrease in threshold contrast as the line number is increased and in all but one case this is statistically significant at the 5 per cent level. The magnitude of the change is such that there is, on average, a 30-40 per cent decrease in threshold contrast from a six bar grating to a 28 bar grating. The results for red are somewhat different. With one exception the correlation coefficients are positive but they are evidently lower than those in the other colours. Of the eight results obtained, half fall at each side of the 5 per cent significance level. In the case of JMF the low correlation coefficients are explained by lower values of AV, although in the case of HRP they may in part be attributed to larger scatter. The only conclusive result that can be drawn is that the red and green results show differences, itself a feature of some interest, but it seems at least possible that the pattern integration effect is present with the green mechanism and not with the red. Although the mean threshold for red is lower than for green, when the pattern integration effect is taken in account the threshold for a large number of lines is about the same in the two cases. It seems unlikely that two mechanisms can achieve the same acuity in different ways, but it is not impossible in view of the complexity of the eye’s nervous system. One factor limiting acuity is undoubtedly diffraction, and this has a greater effect for green light than for red. The conclusion of these experiments is that the eye’s green sensitive mechanism possesses pattern integration ability. This is possibly absent in the red sensitive mechanism, Orientation dependence
It is well known that for most subjects acuity is dependent on the orientation of the test grating, being generally better for horizontal and vertical positions of the grating than for oblique positions. CAMPBELL, KIJLIICOWSKIand LEVIN~ON (1966) have shown that this is not a property of the eye’s optics, and so an explanation of this effect must be found in the visual nervous system. In view of the pattern integration TABLE 2. MEAN THRESHOLD VBIBILITY,DIFFERENCEIN vxsl~m CORRELATIONCOEPFICZENTFORDIFFERENTORIENTATIONS
Subject
Orientation
JMF
0
HRP
0”
JMF
45’
HRP
45’
JMF
90”
HRP
90”
JMF
135”
HRP
135”
V 0.026 0.043 0.031 0.041 0.080 0.100 0.129 0.065 0.036 0.051 0.037 0.025 0.081 0.054 0038 0.039
Av 0.013 0,012 O+O26 0.039 -0.006 0.015 0.022 0.042 0.011 0.020 0.025 0.010 0014 0.009 O-012 0.020
AND
Correlation coefficient 0.480 0.333 0.563 0.629 -0~070 0.225 0.147 0.793 0444 0.638 0.662 o* 393 0.167 0.125 0,352 0.458
Spatisl Integration
163
in Visual Acuity
effect found in section (a), it seemed possible that the difference in acuity might be produced bv such integration being operative in certain directions only. This was tested by carrying out a series of expeknents with the target grating lines vertical (0’). horizontal (90”) and in two oblique positions (545”). The experiments were carried out with green light, target conditions being unaltered. The results are presented in Table 2. For subject HRP, if one erratic result is discounted, no significant difference emerges between his acuity for horizontal, vertical, and obhque orientations. Subject JMF shows an orientation dependent variation in acuity. His values of d V are similar at ail orientations, but because of the difference in absolute threshold, the relative changes in visibility are less in the oblique directions, and examination of the correlation coefficient shows that in all cases these changes are not significant at the 5 per cent level. Thus there is some evidence that, for this subject at least, spatial pattern integration is superior in the horizontat and vertical directions. Figure 5 shows the mean values of threshoId visibility plotted against IV. The convergence of the lines for small N provides some support for the idea outlined above. However. the results for HRP are entirely different and its seems no conclusion can be drawn without recourse to experiments on further subjects.
.
SubjectJMF
.
0.1
o*
.
456
x
PO0
.
135O
Subject 0.
HRP
I
A
0” OS0
x
9o”
.
135”
f
.
Fro. 5. Variation of contrast threshold with number of lines for a 30 cycle/deg grating at various orientations.
Limit of spatid pattern i~tegr~t~o~ The evaluation made above gives no information about the extent of the range over which the pattern integration effect operates This has been analysed as follows Considering just the results for two values of N, e.g. N= 16 and N=28, it is possible to evaluate for each run the difference n between the mean values of V (e.g. vr6- P’2s)and a standard deviation for this quantity s. The results for a series of runs, labelled 1, 2, . . . . . . . n may be combined to give a weighted mean difference
‘===+ lJsI
......-i
lJs2:
'Jsn
and a weighted standard deviation n S=
‘ls,+ ys24 . . . . . . 4- ‘Is.
164
J.M. FINDLAY
A Student’s t-test may then be performed on the statistic d d(n- I)/; to determine whether the mean difference in the thresholds is significant. This test was carried out for each subject on all the runs in the 0“ and 90’ orientations, excepting those in red. The results are presented in Table 3. This gives the value of the statistic together with the probability of obtaining this result if no difference existed between the means. TABLE 3. VALUES-OFt STATISTIC, AND SCHFICANCE LEVELS FORTHE K&WDEFERENCES lNTHRESHOLDVISIBILlTyWlTHVARIOUSLINENUMBERS N=16 to N=28 JMF HRP
2.74 (2 %) 0.40( )
N= 12 to N=28 3.44 (0.6 “/,) 2.62 (3 %)
N-12 to N=16
N=6 to N=8
N=8 to N=12
1.34 (25%) 1.04 (30 %)
1.00 (30%) 0.62 ( )
2-18 (5%) 2.23 (SO,?
The results show that the differences in contrast threshold observed between adjacent values of N in the experiments cannot usually be given significance. However, the results for the difference between the mean threshold for N= 12 and N=28 show for both subjects a significant value of t, indicating that pattern integration is carried out beyond N= 12. It seems likely that for subject JMF that pattern integration exists beyond N= 16, but the results do not confirm this for subject HRP. DISCUSSION
The experiments have shown that under certain conditions the contrast threshold for detection of a grating decreases as the number of lines in the grating is increased. The possibilities will be considered in turn that this is an experimental artefact, a physical effect in the optics of the eye, or a neural effect. Non-homogeneities in the grating illumination or uniform background illumination might lead to certain parts of the target possessing greater modulation. However, visual inspection failed to reveal any such lack of homogeneity and a change of lun&ance of ca. 30 per cent over the grating would have been necessary to explain the results in this way. A second possible artefact lies in the threshold procedure used, which is somewhat crude by modem psychophysical standards, but it seems likely that the relative values obtained using this procedure will still possess validity. The physical effect on the retinal image of limiting the size of the grating target must be considered. The intensity distribution in the retinal image may be found by forming the convolution of the distribution in the object with the spread function of the optics of the eye. CAMPBELL and GUBISCH(1966) have evaluated the optical &spread function for the eye. A numerical computation would be necessary to determine completely the effect of the convolution. However, any possible difference would be produced by the step in illumination at the edge of the grating. An upper limit to this difference may be estimated by noting that the line spread function has a value of O-05 or less for displacements of 4 min arc or more (for a 3-S mm dia. pupil). The target containing 6 lines had a width of 12 min arc, thus in its centre the visibility in the retinal image is certainly identical to within 5 per cent of that in a larger target. Thus the greater part of the effect found must take place in the visual nervous system. The simplest process would be for the threshold evaluation to take place in two stages, some intermediate mechanism evaluating the target on the basis of a probability decision,
Spatial Integration
in Visual Acuity
165
taking into account all the available information, and then presenting to the consciousness either a grating picture or a plain field. In support of this is the subjective finding that the grating is either seen in its entirety or not at all. The subject is not conscious of the ‘noise’ which must provide the limiting factor. BARLOW(1957) has suggested a similar two-stage mechanism, in which a probability decision is made on the results of the first stage, to explain increment thresholds in scotopic vision. The idea that the visual system can spatially integrate information in order to reach a threshold decision is well known in certain particular conditions (e.g. Vernier acuity and Ricco’s Law). COLTMAN and ANDERSON (1960) have demonstrated an effect in conditions rather similar to those of this experiment. In the course of a study to optimise television presentation, they determined the dependence of threshold contrast for a regular line pattern on the number of lines in the presence of external optical noise. Their results show an integration effect up to a line number value of 7; however, direct comparison is not possible since the spatial frequency they used is not given and it seems probable that extra-fovea1 vision was being used. It is certainly conceivable that the visual thresholds determined in this paper represent a similar signal/noise detection process, and a discussion of this is given in the Appendix. It is thus apparent that under certain conditions the physical parameter determining the threshold for a grating image is not simply the visibility in the image, but is dependent on a spatial integration over the whole region available. The targets used have been confined to the fovea, so further measurements would be necessary to determine whether the results could be extended to extrafoveal targets. However, it is evidently necessary to exercise caution in interpreting acuity results from grating patterns containing only a small number of lines (see also NACHMIAS, 1968). CONCLUSIONS
The response of the eye to 30 cycles/deg grating patterns of limited size has been studied. Thresholds for such a grating are determined by a process which involves integrating information over all the cycles of the grating present, up to at least twelve cycles. The effect is more pronounced when using the ‘green’ mechanism than when using the ‘red’ and may possibly be absent in the latter. Investigation of the orientation dependence of the effect did not allow any delinite conclusions to be drawn, but indicated the possibility that orientational variations in visual acuity could be due to variations in integration ability. Acknowledgements-Theauthor thanks Professor R. W. DITCHBURN,F.R.S., for advice and encouragement, and also Mr. H. R. PESCOD, both for technical assistance and for acting as a subject. The research was carried out under M.R.C. Grant No. G/964/218/B. REFERENCES B-w,
H. B. (1957).
Increment thresholdsand visual noise. J. Physiol. 136,46!2-488. summation areas of human colour-receptive mechanisms
BIUNDLEY, G. S. (1954). The
at increment J. Physiol. D&400-408. CAMF-BELL, F. W. and GREEN, D. G. (1965). Optical and retinal factors affecting visual resolution 1. Physiol. 181, 576593. C-BELL, F. W. and GUB~~CH.R W. (1966). Optical quality of the human eye. J. Physiol. 186, 558-578. CAMPBELL, F. W. and GUBISCH, R. W. (1967). The effect of chromatic aberration on visual acuity. J. Physiol. 192, 345-358. CAMPBELL, F. W., KULIKOWSKI, J. J. and LE~~NSON,J. (1966). The effect of orientation on the visual resolution of gratings. J. Physiol. 187, 427436. threshold.
J. M. FINDUY
166
COLTMAN, J. W. and ANDERSON,A. E. (1960). Noise limitations to resolving power in electronic imaging. Proc. I.R.E. 48, 858-866. FISHER,R. A. (1946). Statistical Methoa!s for Research Workers. 10th edition, Oliver and Boyd, Edinburgh. NACHMIAS, J. (1968). Visual resolution of two bar patterns and square wave gratings. J. opt. Sot. Am. 58.9-13. VAN N&s, F. L., KOENDERINK, J. J., NAS, H. and BOLJSL~N, M.A. (1967). Spatiotemporal modulation transfer in the human eye. J. opf. Sot. Am. 57,401-K%. STILES,W. S. (1949). Increment thresholds and the mechanisms of colour vision. Documenta Uphthal. 3, 13&165. WALD, 0. (1967). Blue blindness in the normal fovea. J. opf. Sot. Am. 57, 3289-1301.
APPENDIX It is of interest to compare the results obtained with those that would be obtained with an ideal physical detector. The Shannon result for the optimum value of the information capacity of a channel of bandwidth W for a signal of power P in the presence of a white Gaussian noise of power Q is P-+-Q c = w log2 -
Q
For a grating pattern of visibility Ycontaining N lines, P is proportional to (VW and Q is proportional to N. Thus C = W log2 (kY2Nfl) where k is a constant. In an ideal system with fixed channel capacity and bandwidth, the factor VZN would be constant. This expm the familiar 1/N r&tio&ip for detection of a coherent signal. The table gives a comparison of values of 1/N relative to N= 12 with the observed values (green). N
6 8 12 16 28
l/v/N
1.41 1.22 1.00 0.87 0.65
JMF 1.36 1.19 I.00 0.87 @79
HRP 1.33 1.16 1.00 0.84 0.68
The results appear to show that the integrating capacity shown by the eye is only slig&tly poorer than that of an ideal detector. However, this must be interpreted with caution in view of the approximations made. Ab&~ct--The ei%ct of grating ria on the threshold for a 30 cycle per degree grating pattan is investigated for various colours of illumination and qt orientations. It is shown under certain conditions that the &hruhofd w as the target size is increased, and thb is attributed to spatial integration of information. R&m&-on &tudie pour diverses couleurs de la lumi&e et d&sea orientation de la mire I’dfasurie~&ladimeosioad’uncmireB~~de3ocyckspardtqrc. Dam certaines conditions on constate que la scuil diminue quand la dimension de la mire crott, ce qu’on attribue B une int&ration spatiale de l’information. FDer Einfiul3 der Gittcrgri& auf $ie Schwelle bei einem Gitter mit 30 Pcriodcn/Grad wird Nr vcmcbi&$ne Farben I& F&rm$a Objahs unt_nsttch’. M obje!ktEswizdgez&t,~u&ezb&mmten~ g&e abnimmt. Dies wird der &&hen Integration dcr Information zupnchrieben.