01M2-6989/91 S3.00+ 0.00 PergamonPressplc
Vi&n Res. Vol. 31, No. 9, pp. 16194626,1991 Printedin GreatBritain
VERNIER ACUITY WITH STATIONARY MOVING GABORS RUSSELLL. DE VALOISand KAREN K. DE VALOIS Physiological Optics Group and Department of Psychology, University of California, Berkeley, CA 94720, U.S.A. (Received 14 Augur 1990) Abstract-We examined the ability of observers to determine the vertical alignment of three Gabor patches (cosine gratings tapered in X and Y by Gaussians) when the grating within the middle patch was moving right or left. The comparison patches were flickered in counterphase, as was the test patch in a control condition. In all conditions, the Gabor patch itself (the envelope) was stationary. Vernier acuity (i.e. sensitivity) was almost as good with the moving as with the flickering Gabors, but there was a very pronounced positional bias in the case of the patterns in which the internal gratings were moving. The (stationary) patches appeared to be displaced in the direction of the grating movement. Thus if the grating were drifting rightwards, the observer would see the patches as being aligned only when the test patch position in fact was shifted far over to the left. This movement-related bias increased rapidly with retinal eccentricity, reaching 15 min at 8 deg eccentricity. The bias was greatest at 4-g Hz temporal frequency, and at low spatial frequencies. Whether the patterns were on the horizontal or the vertical meridian was largely irrelevant, but larger biases were found with patterns moving towards or away from the fovea than with those moving in a tangential direction. Vernier acuity
Motion
Gabors
Quadrature movement
INTRODUCTION
One’s ability to align patterns in a vernier-acuity test is to a large extent independent of the nature of the patterns. Thus, for instance, one can not only precisely align fine lines, as in the classical vernier tests, but even two large, diffuse Gaussian blobs (Toet & Koenderink, 1988). We have previously examined vernier acuity with Gabor patches (cosine gratings tapered in both X and Y by Gaussians), and have found that it matters little if the patches to be aligned differ from each other in orientation, spatial frequency or color (Kooi, De Valois & Switkes, 1987). Similar results have been reported by others (Burbeck, 1988). These findings suggest that one has very precise information about what retinotopicallyrelated location on the striate cortex is activated by a pattern, regardless of what particular cell types in this location may be most stimulated. Thus, positional information seems to be largely independent of the character of the visual object. A striking exception to this occurs, however, when there is movement within the stimulus pattern. We have examined this phenomenon in the experiments reported below. A paper by Thorson, Lange and BiedermanThorson (1969) first indicated that there is
something strange about the localization of moving patterns, although it is not entirely clear how the “fine-grain motion illusion” they discovered is related to the phenomenon we are examining. They showed that if two nearby points in the periphery were successively stimulated, observers report seeing apparent movement and that the movement extends much further than the distance between the stimulated points. So, for instance, if points 1 and 2 along an imaginary scale were successively stimulated with a bright dot, the dot might appear to move from 1 to 4. The apparent movement, then, was perceptually considerably displaced, or at least extended, in the direction of movement. This illusory movement was found in the periphery, but not with central viewing. More recently, Anstis (1989) and Ramachandran and Anstis (1991) have been examining motion illusions which are doubtless the same phenomenon we have studied using moving Gabor patterns. They have shown that a stationary window containing moving patterns can, under some circumstances, be seen to be displaced in position. One such situation is a window filled with a moving texture seen against a flickering textured background; another is a window containing a moving grating when the
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RUSSELLL. DE VALOISand KAREN K. DE VALOIS
background luminance is modulated in phase with the grating. Those situations and our Gabor patterns may all have in common the absence of clear information about the location of the borders of the windows that contain the moving patterns. We have examined the effect of variations in eccentricity, spatial frequency and temporal frequency on this movement-related positional displacement. We have also studied how it varies as a function of the trajectory of movement through space. METHODS
Apparatus and stimuli
Patterns were presented on a Sony RGB monitor under control of a Sun 3-160 computer with a TAAC graphics unit having four independent graphics channels. The gratings in the Gabor patterns were made to move within the stationary envelope by quadrature movement. That is, to produce a moving Gabor, two Gabor patches containing gratings in spatial sine and cosine phase, respectively, were produced in separate graphics channels and superimposed. These two patterns were then modulated in contrast by a temporal sine and cosine, respectively (by changing their respective look-up tables each frame), to produce quadrature movement, taking advantage of the trigonometric identity: cos(wt -fx)
= cos(ot)
x cos(fx) + sin(ot)
x sin(fx).
Thus, a moving sine wave grating is identical to the sum of two counterphased gratings in spatial and temporal quadrature relationship. To reverse the direction of drift of the grating, the temporal frequency of one of the patterns was negated. Only one channel was used to produce the counterphase-flickered comparison patterns and the counterphased test pattern, in cosine spatial and temporal phase. The luminance output of the monitor was linearized under software control, and the spatiotemporal mean luminance and chromaticity of the patterns and of the background were maintained at a white (x = 0.253; y = 0.289) of 40 cd/m2. The monitor was viewed through a circular aperture in a large screen of approximately the same luminance and chrominance as the monitor. At the 172 cm viewing distance used, the aperture subtended 7 deg. The Gabor patterns used for most of the experiments were three vertically displaced
patches containing vertical gratings. The center of each comparison patch was displaced by 1 deg from the center of the test patch, one above and one below. A fixation cross was various distances to the right of the central Gabor patch. In other experiments, different configurations of this basic pattern were used. In one, the tests were run with three horizontal patches in a horizontal configuration, that is, with the Gabor group rotated 90deg but with the fixation point still to the right. Two other sets of tests had these same two configurations of Gabor patches, horizontal and vertical, but with the patterns above rather than to the left of the fixation point, thus along the vertical rather than the horizontal meridian. The nominal total extent of each Gabor patch was 1 by 1 deg, but since the Gaussians in most experiments had a sigma of 8 min in X and I’, the contrast fell to zero short of this. Most of the data were collected with a grating in the Gabor of 2 c/deg; for other spatial frequencies the sigma of the Gaussian envelope was changed in inverse relation to the frequency in order to maintain a constant bandwidth. That is, the patches were geometrically similar, with the same number of cycles, though of different size. Experimental procedures
A two-alternative forced-choice paradigm was used in conjunction with the method of constant stimuli. On each trial, patterns were presented in which the middle test Gabor was either aligned with the two comparison patterns or displaced by some amount to the left or to the right. The observer pressed one of two buttons to indicate whether the test patch was to the left or to the right. No feedback was given. The patterns were ramped on and off (while temporally modulating) with a temporal Gaussian ramp of 0.5 set duration and a maximumcontrast plateau of 1 sec. Thus, the total stimulus duration was 2 sec. A session comprised 75 presentations, 25 each of counterphased, leftward-moving, and rightward-moving patterns. Each group of 25 consisted of 5 trials at each of 5 different lateral displacements, chosen in each case to span the range from nearly all judged too far left to nearly all judged too far right. Stimuli were presented in random order, and the data were collected and analyzed under computer control. Ten sessions were run with each stimulus condition, so the estimates of threshold and bias are based on 250 trials each.
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Vernier acuity with stationary moving Gabors
The subject’s head was stabilized by chin and forehead rests. Binocular viewing with natural pupils was employed. The subject fixated a small cross present all the time on the monitor, in the case of near-peripheral tests, or a fixation spot on the screen surrounding the monitor, for more eccentric measures. In the case of the fovea1 tests, the observers directly fixated the central test Gabor patch. Three subjects were tested, the two experimenters and one naive observer. Subjects with significant refractive error used appropriate spectacle correction. Psychometric functions relating test patch displacement to percent judged “left” were plotted. Thresholds (which were taken as half the distance from the 25 to the 75% point) were estimated by probit analysis (Finney, 197 1). The movement-induced bias, our principal measure of interest, was defined as half the distance between the 50% points for the rightward and leftward moving patterns. RESULTS
Experiment
1
In the first experiment, we examined the movement-displacement effect, and measured its variation with retinal eccentricity. These tests were run with black-white Gabors of 2 c/deg, at 50% contrast (L,,,, - L,,/2L-,), with a temporal frequency of 4 Hz, presented in the standard configuration: three vertical Gabors arranged in a vertical column with the fixation cross various distances to the right of the center, test, Gabor. Although no attempt was made to optimize conditions in order to obtain the lowest possible vernier acuities-we opted out of that particular Olympic event-our subjects made settings, in the case of fovea1 fixation, in the hyperacuity range. In Fig. 1 are psychometric functions for one observer for tests at 0,l and 2 deg eccentricity. With fovea1 fixation and a counterphased control pattern (middle function at the bottom), the judgements were very precise, with an estimated acuity of 24 set and no significant bias. When the gratings within the Gabors were moving to the left or right, however, although thresholds were nearly as low (28 and 34 set, respectively) the bias became significant (in this case 120 set, or 2 min). The observer consistently judged the patterns to be veridically aligned when the test patch was displaced in a direction opposite to its direction of movement. Even for this fovea1 presentation, where the
-12-10-6-6-4-20
2
4
6
6
1012
Displacement (min arc)
Fig. 1. Psychometric functions for one observer for leftwarddrifting, rightwarddrifting, and counterphased patterns at three different eccentricities, zero, 1 and 2deg eccentric, from bottom to top, respectively. At each eccentricity, the middle curve is for the counterphased control pattern, and the ones to the left and right are for the rightward- and leftward-moving patterns, respectively. The positional bias is reflected in the fact that a stationary Gabor with a rightward-moving grating appears to lx shifted to the right and thus has to be positioned to the left to appear aligned with the comparison patterns. It can be seen that the observer made settings of considerable precision, but with a movement-related bias that increased with eccentricity.
effect is relatively small, it can be seen that there is almost no overlap among the curves: a pattern which was judged 96% of the time to be too far to the left when it contained a leftward-moving grating was judged 98% of the time to be too far to the right when the grating was counterphased. With increasingly peripheral fixation, the acuity becomes somewhat worse (the slopes of the psychometric curves decrease), but the more striking change is a large increase in the positional bias (Fig. 1, upper portions). The variation in bias as a function of retinal eccentricity for each of the three observers is shown in Fig. 2. It can be seen that with
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RUSSELLL. DE VALOISand KARENK. DE VALOIS Vernier
Bias
of Moving
Gabor
201
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Eccentricity
(deg)
Fig. 2. The variation in movement bias as a function of retinal eccentricity for the three observers. The effect increases rapidly with increasing retinal eccentricity.
increasing eccentricity the rightward-moving patterns appear to be increasingly shifted to the right and the leftward ones to the left, to produce an increasingly large total bias. By 8 deg eccentricity, the change is almost 15 min, which, in the case of the 2 c/deg pattern used, is a 180 deg phase shift in each direction. Experiment 2 In this experiment we examined how the movement-related positional bias varied as a function of spatial and temporal frequency. The pattern configurations were the same as in expt 1. In the tests of different temporal frequencies, a 2c/deg, 50% contrast white-black pattern at 2 deg eccentricity was used. The data are shown in Fig. 3. It can be seen that the positional bias Vernier
Bias
of
Moving
Gabor
is greatest in the 4-8 Hz range. The curve is very similar to the well-known temporal contrast sensitivity function (Kelly, 1979), but it does not resemble the motion-sensitivity function, which is essentially low-pass (Nakayama, 1985). In another set of trials we briefly examined the effect of different spatial frequencies. In this case, the patterns were whiteblack Gabor patches of 4 Hz temporal frequency presented at 1 deg eccentricity, to allow the presentation of a wider range of spatial frequencies. For the 1 and 2 c/deg tests, the contrast was 50%; for the 8 c/deg pattern it was 70%. For technical reasons, we could only examine a limited spatial frequency range: at frequencies lower than 1 c/deg our standard 1 deg patch would not even contain one cycle of the grating; at spatial frequencies much higher than 8 c/deg the (in)visibility of the patterns became a problem. Figure 4 shows that the positional bias is largest with low spatial frequencies, over the limited range we examined. The drop-off with increasing spatial frequency is not, however, as much as one would expect if the positional bias were a constant phase shift of the pattern. The large positional biases seen at low spatial and (moderately) high temporal frequencies indicates that the effect is greatest with high speeds of movement. However, the data with different spatial and temporal frequencies do not indicate that the amount of positional bias is directly proportional to the speed. Experiment 3 In all the experiments discussed above, the three Gabor patches contained vertical gratings Vernier
Bias
of
Moving
Gabor
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1
4
Frequency
(Hz)
Fig. 3. The variation in movement bias as a function of temporal frequency. The largest biases are found in the middle temporal frequency range.
Spatial
2 Frequency
4 (c/deg)
Fig. 4. Movement bias for three different spatial frequencies, showing that the effect in general is greatest for low spatial freouencies.
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Vernier acuity with stationary moving Gabors
The results are easier to describe than are the stimulus arrays. The movement-induced positional shift did not require movement towards or away from the fovea and was not confined to movement along the horizontal meridian. However, radial movement towards or away from the fovea (labeled RAD in Fig. 5) gave larger biases-very much larger for two observers-than did the tangential movement. 0-l
HM
Rad
HM ‘VM
VM
I
Rod
Pattern Type Fig. 5. Movement bias for four different stimulus configurations. The two leftmost were along the horizontal meridian (HM); the two on the right along the vertical meridian (VM). The ones labeled RAD involved radial movement towards or away from the fovea. It can be seen that the largest bias is found with patterns which move towards or away from the fovea, along a radius. Considerable, but smaller biases are found for patterns in which the grating is moving at right angles to a radius out from the fovea.
and were displaced vertically, with the fixation at various distances to the right of the central patch. The moving Gabors were thus moving towards or away from the fovea along the horizontal meridian. This standard condition is labeled pattern 1, HM, RAD (for radial along the horizontal meridian) in Fig. 5. We were curious as to whether there was anything distinctive about movement along a radius (with respect to the fovea), as opposed to some other retinal direction. We therefore tested one particular condition (2 deg eccentricity, 2 c/deg, 4 Hz) with three other patterns. In one pattern, Fig. 5, pattern 2, the test gratings moved perpendicular to the horizontal meridian. In this case, the whole Gabor array was rotated 90deg so that the three Gabors were now horizontal and arranged in a row along the horizontal meridian, but the same fixation point to the right was used. The middle test pattern was thus now positioned various amounts above and below the two comparison patterns, and the observer’s task was to judge whether it was too high or too low. The other two novel arrangements were these same two arrays positioned along the vertical meridian, above the fixation point. In this case the horizontal array would be moving radially towards or away from the fovea (pattern 3, labeled VM, RAD), and the vertical array would be moving perpendicular to the vertical meridian (pattern 4).
Sensitivity
Our primary interest was in measuring the vernier bias produced by movement of the gratings within the Gabor patches, but we also measured the vernier thresholds under these various conditions. The relationship between the movement bias and sensitivity is not a simple one. Our data support the well-documented fact that vernier acuity declines drastically with retinal eccentricity (e.g. Levi, Klein & Aitsebaomo, 1985). Since our experiment indicates that the movement bias also increases rapidly with eccentricity, one might suppose that the two are related. However, the shape of the change with eccentricity is not the same for these two measures, and with variations in other parameters vernier sensitivity and movement bias do not covary at all. Vernier Thresholds
for Moving Gobors
Fig. 6. Vernier thresholds for expts 1 and 2. At top it can be seen that thresholds increase with eccentricity, slowly at first and then more rapidly. At bottom are shown the sensitivity variations with various spatial and temporal frequencies. In these situations, in which movement bias shows considerable variation, the threshold changes almost not at all.
RUSSELLL. DE VALOISand KARENK. DE VALOIS
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Movement bias increases approximately linearly with increasing eccentricity, on average, as shown in Fig. 2. The vernier thresholds in our situation, however, changed relatively little up to an eccentricity of about 2-4deg and then rapidly increased, see the top graph in Fig. 6. The variations with eccentricity of the two measures, then, were not that similar. Furthermore, movement bias varies considerably with both spatial and temporal frequency, but we found almost no change in vernier thresholds over the range of these variables we studied, (Fig. 6, bottom two graphs). DISCUSSION
A considerable body of recent literature indicates that one is able to localize visual objects with great accuracy and largely without regard to the specific characteristics of the object (Toet, 1987; Burbeck, 1988). There are many factors which influence vernier acuity, such as the proximity of the patterns to be aligned, the presence or absence of other patterns in the neighborhood, etc. (Westheimer & McKee, 1977). But with these factors held constant and the patterns to be aligned at some distance from each other, it makes very little difference what the nature of the test and comparison patterns are. Not surprisingly, one can precisely align small spots, or fine lines, but one can also quite precisely align big Gaussian blobs (Toet, 1987; Toet & Koenderink, 1988) or extensive Gabor patches (Kooi et al., 1987). Furthermore, it makes little difference whether the test and comparison patterns are the same or not. Thus vernier acuity is almost as good when one has to align a vertical with a horizontal Gabor patch as when a vertical is to be aligned with an identical vertical pattern, or when the test and comparison Gabors differ in spatial frequency or even color (Kooi et al., 1987; Burbeck, 1988). Given this, it is quite surprising that movement within a pattern leads to very large misalignments, as we show in this experiment and as Anstis (1989) has demonstrated with related patterns. If the grating within a Gabor patch is drifting to the right, the whole patch appears to be displaced rightwards, although the average luminance distribution of the pattern does not change. This effect increases greatly with retinal eccentricity, but it is quite obvious even with fovea1 fixation. Most of our models, or even our vague thoughts, about most aspects of spatial vision
center on the properties of cells in the striate cortex. One factor that makes these findings rather puzzling is that it seems quite unlikely that they can be accounted for on the basis of striate cells’ response characteristics. Striate cells carry information both about the nature of local stimuli-their spatial frequency, orientation and perhaps color and direction of movement-and also about location, since they have spatially-localized receptive fields. The spatial localization of receptive fields appears to be more precise at this level than at any known subsequent stage. It is therefore not too surprising that we can accurately determine the positions of objects independent of their stimulus selectivities, since an array of cells with the same receptive field center location but different spatial frequency and orientation peaks would all carry the same location signal. But the fact that this does not hold if the pattern is moving creates problems for this notion. Although no one that we are aware of has actually examined the question physiologically, it seems very unlikely that striate cells with different receptive field centers would be activated depending on whether a pattern is stationary, moving left, or moving right. Thus one is perforce reduced to the familiar ploy of attributing the effect to some process at a higher level. There are psychophysical data suggesting that movement towards or away from the fovea may play a special role in vision, with specialized mechanisms being involved in detecting centrifugal and centripetal movement (Regan, 1986). This would certainly be understandable since, for instance, the optic flow pattern produced by a fixated approaching object, surely a most important visual stimulus, produces movement in all directions along radii out from the fovea, There is also physiological evidence which points in the same direction, in the presence in prestriate areas MT and MST of cells specifically responsive to movement towards or away from the fovea (Albright, 1989; Tanaka & Saito, 1989). It is thus of interest that we find the movement-induced positional bias to be greatest for centrifugal or centripetal movement. The difference between the bias with movement towards or away from the fovea as opposed to movement at right angles to such a radius was very large for two observers and also present to a lesser extent in the third observer. Our stationary moving Gabor patterns are rather unusual, unnatural stimuli, particularly since the movement in the pattern is purely
Veher
acuity with
stati~mwymoving Gabors
local, the pattern as a whole being stationary. Having a stationary envelope conveniently allowed us to make precise measures of the effect, but there is no reason to think that the basic finding-that moving patterns are mislocated-would not apply to patterns moving across space as well. Since it would appear to be important for organisms to localize accurately moving as well as stationary patterns, the existence of this illusion would appear to be potentially detrimental. Movement-induced positional bias could, however, help compensate for the visual system’s nontrivial latency. It takes 50-100msec for striate cells to respond to a stimulus, and of course even longer for cells further downstream. What we see, then, is not the world as it is now but as it was in the near past. In the case of a stationary pattern, this latency is unimportant, but to respond accurately to a moving stimulus the orga~sm must somehow compensate for this neural delay. To catch a ball, for instance, one must intercept it at its true position at time t, even though at time t the visual system will be processing an image that occurred at, say, time f - 100 msec. The positional misalignments we have shown to occur with moving stimuli are in the correct direction to compensate for the visual latency. Thus, at any given instant a moving object appears to have moved further along its trajectory than the retinal image being processed at that instant would indicate. The fact that the effect increases with temporal frequency (up to 4-8 Hz) and is greatest at low spatial frequencies would also be in the correct direction since this means that the maximum shift in apparent position occurs for patterns moving at high velocities. It has traditionally been thought that the compensation for visual and response latency that allows one to anticipate the correct position of moving objects occurs at some late point in sensory-motor coordination, The results presented here hint at the possibility that some of the compensation may take place early within the visual system itself. However, the fact that the shift in apparent position varies greatly with retinal eccentricity makes this a less than optimal contribution to the solution. Thorson et al. (1969) reported an illusion which may be related to the movement-induced apparent positional change we have investigated here. They noted that successive stimulation of two nearby points in the peripheral retina produces an impression of apparent movement
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(under conditions that would not produce it with central fixation), and that this movement seemed to extend over considerably longer dis-
tances than the distance between the two points stimulated. They termed this the “fine-grain motion illusion”. Our study does not bear on the apparent motion aspect of this illusion, but may be related to the elongation of movement distance they reported. If a movement of the grating in our Gabors appears to be longer in extent than it is, the center of gravity of the patch might be expected to be shifted, thus producing the observed positional bias. However, it is by no means certain that the same process is in fact involved, p~icufarly since our data clearly show the process we studied to operate centrally as well as in the periphery. There has been a long-lasting if not earthshaking dispute between baseball players and certain physicists and photographers who have addressed the question of whether curve balls curve. More precisely, since the trajectory of any thrown ball will curve due to gravity, the question is whether the additional spin given by the pitcher to a curve ball makes it curve more than it would otherwise do. A ball as light as a ping-pong ball can clearly be made to curve by putting a spin on it, but it is less obvious that this is true for a heavy baseball. Although early photographic attempts to verify a curvature failed, it is now generally accepted that a curve ball does indeed curve to some extent (Adair, 1990). But one thing that is absolutely clear from both theory and measurement is that a durve ball must curve continuously, on an arc; it does not “break”‘-suddenly change direction in midflight-just before it reaches the plate (although the deflection will be continuously increasing as it approaches the plate). Baseball players, however, are unanimous in saying that that is just what a curve ball does: it suddenly dips down and out just before reaching the plate. The attentive reader will by now have anticipated that we are going to suggest that the sudden “breaking” curve baseball players see is partly the result of the perceptual distortion we have been examining in this study. The direction of the perceived change in position of the baseball is correct to be accounted for in this way: the stitches in the spinning curve ball are moving down diagonally on the side towards the batter as the ball approaches. We suggest that when the curve ball gets close enough to the batter for him to perceive the movement of
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L. DE VALIXSand
the stitches, the movement illusion we have been studying would be triggered to make the ball suddenly appear to shift down and out in position, that is, to “break”. Baseball batters also often report that a “fast ball” curves upwards, or hops, a most unlikely physical possibility. But since a fast ball spins in the reverse direction as a curve ball, the same perceptual effect which accounts for a curve ball’s dipping would lead one to expect that a fast ball should suddenly appear to rise as it approaches the plate, despite the fact that it is actually falling due to gravity. REFERENCES Adair, R. K. (1990). The physics of baseball. New York: Harper & Row. Albright, T. D. (1989). Centrifugal directional bias in the middle temporal visual area (MT) of the macaque. Visual Neuroscience, 2, 177-188. Anstis, S. M. (1989). Kinetic edges become displaced, segregated, and invisible. In Neural mechanisms of visual perception (pp. 247-260). Houston: Portfolio/Gulf. Burbeck, C. A. (1988) Large-scale relative localization across spatial frequency channels. Vision Research, 28, 857-859. Finney, D. J. (1971). Probit analysis. Cambridge: Cambridge University Press.
KAREN
K. DE VALOIS
Kelly, D. H. (1979). Motion and vision-II Stabilized spatio-temporal threshold surface. Journal of the Optical Society of America, 69, 1340-l 349. Kooi, F. L., De Valois, R. L. & Switkes, E. (1987). Vernier acuity with Gabor patches. Investigative Ophthalmology and Visual Science, 28, 360. Levi, D. M., Klein, S. A. & Aitsebaomo, A. P. (1985). Vernier acuity, crowding and cortical magnification. Vision Research, 25, 963-977. Nakayama, K. (1985). Biological image motion processing: a review. Vision Research, 25, 625-660. Ramachandran, V. S. & An&s, S. M. (1991). Perception, in press. Regan, D. (1986). Visual processing of four kinds of relative motion. Vision Research, 26, 127-145. Tanaka, K. & Saito, H. (1989). Analysis of motion of the visual field by direction, expansion/contraction, and rotation cells clustered in the dorsal part of the medial superior temporal area of the macaque monkey. Jouma/ of Neurophysiology, 62, 626-641. Thorson, J., Lange, G. D. & Biederman-Thorson, M. (1969). Objective measure of the dynamics of a visual movement illusion. Science, 164, 1087-1088. Toet, A. (1987) Visual perception of spatial order. Doctoral dissertation, University of Utrecht. Toet, A. & Koenderink, J. J. 1988). Differential spatial displacement thresholds for Gabor patches. Vision Research, 28, 133-143. Westheimer, G. & McKee, S. P. (1977). Integration regions for visual hyperacuity. Vision Research, 17. 89-93.
Vi&n Res. Vol. 31, No. 9, pp. 16194626,1991 Printedin GreatBritain
VERNIER ACUITY WITH STATIONARY MOVING GABORS RUSSELLL. DE VALOISand KAREN K. DE VALOIS Physiological Optics Group and Department of Psychology, University of California, Berkeley, CA 94720, U.S.A. (Received 14 Augur 1990) Abstract-We examined the ability of observers to determine the vertical alignment of three Gabor patches (cosine gratings tapered in X and Y by Gaussians) when the grating within the middle patch was moving right or left. The comparison patches were flickered in counterphase, as was the test patch in a control condition. In all conditions, the Gabor patch itself (the envelope) was stationary. Vernier acuity (i.e. sensitivity) was almost as good with the moving as with the flickering Gabors, but there was a very pronounced positional bias in the case of the patterns in which the internal gratings were moving. The (stationary) patches appeared to be displaced in the direction of the grating movement. Thus if the grating were drifting rightwards, the observer would see the patches as being aligned only when the test patch position in fact was shifted far over to the left. This movement-related bias increased rapidly with retinal eccentricity, reaching 15 min at 8 deg eccentricity. The bias was greatest at 4-g Hz temporal frequency, and at low spatial frequencies. Whether the patterns were on the horizontal or the vertical meridian was largely irrelevant, but larger biases were found with patterns moving towards or away from the fovea than with those moving in a tangential direction. Vernier acuity
Motion
Gabors
Quadrature movement
INTRODUCTION
One’s ability to align patterns in a vernier-acuity test is to a large extent independent of the nature of the patterns. Thus, for instance, one can not only precisely align fine lines, as in the classical vernier tests, but even two large, diffuse Gaussian blobs (Toet & Koenderink, 1988). We have previously examined vernier acuity with Gabor patches (cosine gratings tapered in both X and Y by Gaussians), and have found that it matters little if the patches to be aligned differ from each other in orientation, spatial frequency or color (Kooi, De Valois & Switkes, 1987). Similar results have been reported by others (Burbeck, 1988). These findings suggest that one has very precise information about what retinotopicallyrelated location on the striate cortex is activated by a pattern, regardless of what particular cell types in this location may be most stimulated. Thus, positional information seems to be largely independent of the character of the visual object. A striking exception to this occurs, however, when there is movement within the stimulus pattern. We have examined this phenomenon in the experiments reported below. A paper by Thorson, Lange and BiedermanThorson (1969) first indicated that there is
something strange about the localization of moving patterns, although it is not entirely clear how the “fine-grain motion illusion” they discovered is related to the phenomenon we are examining. They showed that if two nearby points in the periphery were successively stimulated, observers report seeing apparent movement and that the movement extends much further than the distance between the stimulated points. So, for instance, if points 1 and 2 along an imaginary scale were successively stimulated with a bright dot, the dot might appear to move from 1 to 4. The apparent movement, then, was perceptually considerably displaced, or at least extended, in the direction of movement. This illusory movement was found in the periphery, but not with central viewing. More recently, Anstis (1989) and Ramachandran and Anstis (1991) have been examining motion illusions which are doubtless the same phenomenon we have studied using moving Gabor patterns. They have shown that a stationary window containing moving patterns can, under some circumstances, be seen to be displaced in position. One such situation is a window filled with a moving texture seen against a flickering textured background; another is a window containing a moving grating when the
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RUSSELLL. DE VALOISand KAREN K. DE VALOIS
background luminance is modulated in phase with the grating. Those situations and our Gabor patterns may all have in common the absence of clear information about the location of the borders of the windows that contain the moving patterns. We have examined the effect of variations in eccentricity, spatial frequency and temporal frequency on this movement-related positional displacement. We have also studied how it varies as a function of the trajectory of movement through space. METHODS
Apparatus and stimuli
Patterns were presented on a Sony RGB monitor under control of a Sun 3-160 computer with a TAAC graphics unit having four independent graphics channels. The gratings in the Gabor patterns were made to move within the stationary envelope by quadrature movement. That is, to produce a moving Gabor, two Gabor patches containing gratings in spatial sine and cosine phase, respectively, were produced in separate graphics channels and superimposed. These two patterns were then modulated in contrast by a temporal sine and cosine, respectively (by changing their respective look-up tables each frame), to produce quadrature movement, taking advantage of the trigonometric identity: cos(wt -fx)
= cos(ot)
x cos(fx) + sin(ot)
x sin(fx).
Thus, a moving sine wave grating is identical to the sum of two counterphased gratings in spatial and temporal quadrature relationship. To reverse the direction of drift of the grating, the temporal frequency of one of the patterns was negated. Only one channel was used to produce the counterphase-flickered comparison patterns and the counterphased test pattern, in cosine spatial and temporal phase. The luminance output of the monitor was linearized under software control, and the spatiotemporal mean luminance and chromaticity of the patterns and of the background were maintained at a white (x = 0.253; y = 0.289) of 40 cd/m2. The monitor was viewed through a circular aperture in a large screen of approximately the same luminance and chrominance as the monitor. At the 172 cm viewing distance used, the aperture subtended 7 deg. The Gabor patterns used for most of the experiments were three vertically displaced
patches containing vertical gratings. The center of each comparison patch was displaced by 1 deg from the center of the test patch, one above and one below. A fixation cross was various distances to the right of the central Gabor patch. In other experiments, different configurations of this basic pattern were used. In one, the tests were run with three horizontal patches in a horizontal configuration, that is, with the Gabor group rotated 90deg but with the fixation point still to the right. Two other sets of tests had these same two configurations of Gabor patches, horizontal and vertical, but with the patterns above rather than to the left of the fixation point, thus along the vertical rather than the horizontal meridian. The nominal total extent of each Gabor patch was 1 by 1 deg, but since the Gaussians in most experiments had a sigma of 8 min in X and I’, the contrast fell to zero short of this. Most of the data were collected with a grating in the Gabor of 2 c/deg; for other spatial frequencies the sigma of the Gaussian envelope was changed in inverse relation to the frequency in order to maintain a constant bandwidth. That is, the patches were geometrically similar, with the same number of cycles, though of different size. Experimental procedures
A two-alternative forced-choice paradigm was used in conjunction with the method of constant stimuli. On each trial, patterns were presented in which the middle test Gabor was either aligned with the two comparison patterns or displaced by some amount to the left or to the right. The observer pressed one of two buttons to indicate whether the test patch was to the left or to the right. No feedback was given. The patterns were ramped on and off (while temporally modulating) with a temporal Gaussian ramp of 0.5 set duration and a maximumcontrast plateau of 1 sec. Thus, the total stimulus duration was 2 sec. A session comprised 75 presentations, 25 each of counterphased, leftward-moving, and rightward-moving patterns. Each group of 25 consisted of 5 trials at each of 5 different lateral displacements, chosen in each case to span the range from nearly all judged too far left to nearly all judged too far right. Stimuli were presented in random order, and the data were collected and analyzed under computer control. Ten sessions were run with each stimulus condition, so the estimates of threshold and bias are based on 250 trials each.
1621
Vernier acuity with stationary moving Gabors
The subject’s head was stabilized by chin and forehead rests. Binocular viewing with natural pupils was employed. The subject fixated a small cross present all the time on the monitor, in the case of near-peripheral tests, or a fixation spot on the screen surrounding the monitor, for more eccentric measures. In the case of the fovea1 tests, the observers directly fixated the central test Gabor patch. Three subjects were tested, the two experimenters and one naive observer. Subjects with significant refractive error used appropriate spectacle correction. Psychometric functions relating test patch displacement to percent judged “left” were plotted. Thresholds (which were taken as half the distance from the 25 to the 75% point) were estimated by probit analysis (Finney, 197 1). The movement-induced bias, our principal measure of interest, was defined as half the distance between the 50% points for the rightward and leftward moving patterns. RESULTS
Experiment
1
In the first experiment, we examined the movement-displacement effect, and measured its variation with retinal eccentricity. These tests were run with black-white Gabors of 2 c/deg, at 50% contrast (L,,,, - L,,/2L-,), with a temporal frequency of 4 Hz, presented in the standard configuration: three vertical Gabors arranged in a vertical column with the fixation cross various distances to the right of the center, test, Gabor. Although no attempt was made to optimize conditions in order to obtain the lowest possible vernier acuities-we opted out of that particular Olympic event-our subjects made settings, in the case of fovea1 fixation, in the hyperacuity range. In Fig. 1 are psychometric functions for one observer for tests at 0,l and 2 deg eccentricity. With fovea1 fixation and a counterphased control pattern (middle function at the bottom), the judgements were very precise, with an estimated acuity of 24 set and no significant bias. When the gratings within the Gabors were moving to the left or right, however, although thresholds were nearly as low (28 and 34 set, respectively) the bias became significant (in this case 120 set, or 2 min). The observer consistently judged the patterns to be veridically aligned when the test patch was displaced in a direction opposite to its direction of movement. Even for this fovea1 presentation, where the
-12-10-6-6-4-20
2
4
6
6
1012
Displacement (min arc)
Fig. 1. Psychometric functions for one observer for leftwarddrifting, rightwarddrifting, and counterphased patterns at three different eccentricities, zero, 1 and 2deg eccentric, from bottom to top, respectively. At each eccentricity, the middle curve is for the counterphased control pattern, and the ones to the left and right are for the rightward- and leftward-moving patterns, respectively. The positional bias is reflected in the fact that a stationary Gabor with a rightward-moving grating appears to lx shifted to the right and thus has to be positioned to the left to appear aligned with the comparison patterns. It can be seen that the observer made settings of considerable precision, but with a movement-related bias that increased with eccentricity.
effect is relatively small, it can be seen that there is almost no overlap among the curves: a pattern which was judged 96% of the time to be too far to the left when it contained a leftward-moving grating was judged 98% of the time to be too far to the right when the grating was counterphased. With increasingly peripheral fixation, the acuity becomes somewhat worse (the slopes of the psychometric curves decrease), but the more striking change is a large increase in the positional bias (Fig. 1, upper portions). The variation in bias as a function of retinal eccentricity for each of the three observers is shown in Fig. 2. It can be seen that with
1622
RUSSELLL. DE VALOISand KARENK. DE VALOIS Vernier
Bias
of Moving
Gabor
201
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16
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L
2 c/&q 4 Hz
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n
12-
k s :: a .” a
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4-
01
012345678
Eccentricity
(deg)
Fig. 2. The variation in movement bias as a function of retinal eccentricity for the three observers. The effect increases rapidly with increasing retinal eccentricity.
increasing eccentricity the rightward-moving patterns appear to be increasingly shifted to the right and the leftward ones to the left, to produce an increasingly large total bias. By 8 deg eccentricity, the change is almost 15 min, which, in the case of the 2 c/deg pattern used, is a 180 deg phase shift in each direction. Experiment 2 In this experiment we examined how the movement-related positional bias varied as a function of spatial and temporal frequency. The pattern configurations were the same as in expt 1. In the tests of different temporal frequencies, a 2c/deg, 50% contrast white-black pattern at 2 deg eccentricity was used. The data are shown in Fig. 3. It can be seen that the positional bias Vernier
Bias
of
Moving
Gabor
is greatest in the 4-8 Hz range. The curve is very similar to the well-known temporal contrast sensitivity function (Kelly, 1979), but it does not resemble the motion-sensitivity function, which is essentially low-pass (Nakayama, 1985). In another set of trials we briefly examined the effect of different spatial frequencies. In this case, the patterns were whiteblack Gabor patches of 4 Hz temporal frequency presented at 1 deg eccentricity, to allow the presentation of a wider range of spatial frequencies. For the 1 and 2 c/deg tests, the contrast was 50%; for the 8 c/deg pattern it was 70%. For technical reasons, we could only examine a limited spatial frequency range: at frequencies lower than 1 c/deg our standard 1 deg patch would not even contain one cycle of the grating; at spatial frequencies much higher than 8 c/deg the (in)visibility of the patterns became a problem. Figure 4 shows that the positional bias is largest with low spatial frequencies, over the limited range we examined. The drop-off with increasing spatial frequency is not, however, as much as one would expect if the positional bias were a constant phase shift of the pattern. The large positional biases seen at low spatial and (moderately) high temporal frequencies indicates that the effect is greatest with high speeds of movement. However, the data with different spatial and temporal frequencies do not indicate that the amount of positional bias is directly proportional to the speed. Experiment 3 In all the experiments discussed above, the three Gabor patches contained vertical gratings Vernier
Bias
of
Moving
Gabor
8, ;pzj-jz,__,y
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0 / I
I P'
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/
&--A.
Temporal
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1 deg ccc
2 degecc 2 c/deg
4 Hz
01 2
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1
4
Frequency
(Hz)
Fig. 3. The variation in movement bias as a function of temporal frequency. The largest biases are found in the middle temporal frequency range.
Spatial
2 Frequency
4 (c/deg)
Fig. 4. Movement bias for three different spatial frequencies, showing that the effect in general is greatest for low spatial freouencies.
1623
Vernier acuity with stationary moving Gabors
The results are easier to describe than are the stimulus arrays. The movement-induced positional shift did not require movement towards or away from the fovea and was not confined to movement along the horizontal meridian. However, radial movement towards or away from the fovea (labeled RAD in Fig. 5) gave larger biases-very much larger for two observers-than did the tangential movement. 0-l
HM
Rad
HM ‘VM
VM
I
Rod
Pattern Type Fig. 5. Movement bias for four different stimulus configurations. The two leftmost were along the horizontal meridian (HM); the two on the right along the vertical meridian (VM). The ones labeled RAD involved radial movement towards or away from the fovea. It can be seen that the largest bias is found with patterns which move towards or away from the fovea, along a radius. Considerable, but smaller biases are found for patterns in which the grating is moving at right angles to a radius out from the fovea.
and were displaced vertically, with the fixation at various distances to the right of the central patch. The moving Gabors were thus moving towards or away from the fovea along the horizontal meridian. This standard condition is labeled pattern 1, HM, RAD (for radial along the horizontal meridian) in Fig. 5. We were curious as to whether there was anything distinctive about movement along a radius (with respect to the fovea), as opposed to some other retinal direction. We therefore tested one particular condition (2 deg eccentricity, 2 c/deg, 4 Hz) with three other patterns. In one pattern, Fig. 5, pattern 2, the test gratings moved perpendicular to the horizontal meridian. In this case, the whole Gabor array was rotated 90deg so that the three Gabors were now horizontal and arranged in a row along the horizontal meridian, but the same fixation point to the right was used. The middle test pattern was thus now positioned various amounts above and below the two comparison patterns, and the observer’s task was to judge whether it was too high or too low. The other two novel arrangements were these same two arrays positioned along the vertical meridian, above the fixation point. In this case the horizontal array would be moving radially towards or away from the fovea (pattern 3, labeled VM, RAD), and the vertical array would be moving perpendicular to the vertical meridian (pattern 4).
Sensitivity
Our primary interest was in measuring the vernier bias produced by movement of the gratings within the Gabor patches, but we also measured the vernier thresholds under these various conditions. The relationship between the movement bias and sensitivity is not a simple one. Our data support the well-documented fact that vernier acuity declines drastically with retinal eccentricity (e.g. Levi, Klein & Aitsebaomo, 1985). Since our experiment indicates that the movement bias also increases rapidly with eccentricity, one might suppose that the two are related. However, the shape of the change with eccentricity is not the same for these two measures, and with variations in other parameters vernier sensitivity and movement bias do not covary at all. Vernier Thresholds
for Moving Gobors
Fig. 6. Vernier thresholds for expts 1 and 2. At top it can be seen that thresholds increase with eccentricity, slowly at first and then more rapidly. At bottom are shown the sensitivity variations with various spatial and temporal frequencies. In these situations, in which movement bias shows considerable variation, the threshold changes almost not at all.
RUSSELLL. DE VALOISand KARENK. DE VALOIS
1624
Movement bias increases approximately linearly with increasing eccentricity, on average, as shown in Fig. 2. The vernier thresholds in our situation, however, changed relatively little up to an eccentricity of about 2-4deg and then rapidly increased, see the top graph in Fig. 6. The variations with eccentricity of the two measures, then, were not that similar. Furthermore, movement bias varies considerably with both spatial and temporal frequency, but we found almost no change in vernier thresholds over the range of these variables we studied, (Fig. 6, bottom two graphs). DISCUSSION
A considerable body of recent literature indicates that one is able to localize visual objects with great accuracy and largely without regard to the specific characteristics of the object (Toet, 1987; Burbeck, 1988). There are many factors which influence vernier acuity, such as the proximity of the patterns to be aligned, the presence or absence of other patterns in the neighborhood, etc. (Westheimer & McKee, 1977). But with these factors held constant and the patterns to be aligned at some distance from each other, it makes very little difference what the nature of the test and comparison patterns are. Not surprisingly, one can precisely align small spots, or fine lines, but one can also quite precisely align big Gaussian blobs (Toet, 1987; Toet & Koenderink, 1988) or extensive Gabor patches (Kooi et al., 1987). Furthermore, it makes little difference whether the test and comparison patterns are the same or not. Thus vernier acuity is almost as good when one has to align a vertical with a horizontal Gabor patch as when a vertical is to be aligned with an identical vertical pattern, or when the test and comparison Gabors differ in spatial frequency or even color (Kooi et al., 1987; Burbeck, 1988). Given this, it is quite surprising that movement within a pattern leads to very large misalignments, as we show in this experiment and as Anstis (1989) has demonstrated with related patterns. If the grating within a Gabor patch is drifting to the right, the whole patch appears to be displaced rightwards, although the average luminance distribution of the pattern does not change. This effect increases greatly with retinal eccentricity, but it is quite obvious even with fovea1 fixation. Most of our models, or even our vague thoughts, about most aspects of spatial vision
center on the properties of cells in the striate cortex. One factor that makes these findings rather puzzling is that it seems quite unlikely that they can be accounted for on the basis of striate cells’ response characteristics. Striate cells carry information both about the nature of local stimuli-their spatial frequency, orientation and perhaps color and direction of movement-and also about location, since they have spatially-localized receptive fields. The spatial localization of receptive fields appears to be more precise at this level than at any known subsequent stage. It is therefore not too surprising that we can accurately determine the positions of objects independent of their stimulus selectivities, since an array of cells with the same receptive field center location but different spatial frequency and orientation peaks would all carry the same location signal. But the fact that this does not hold if the pattern is moving creates problems for this notion. Although no one that we are aware of has actually examined the question physiologically, it seems very unlikely that striate cells with different receptive field centers would be activated depending on whether a pattern is stationary, moving left, or moving right. Thus one is perforce reduced to the familiar ploy of attributing the effect to some process at a higher level. There are psychophysical data suggesting that movement towards or away from the fovea may play a special role in vision, with specialized mechanisms being involved in detecting centrifugal and centripetal movement (Regan, 1986). This would certainly be understandable since, for instance, the optic flow pattern produced by a fixated approaching object, surely a most important visual stimulus, produces movement in all directions along radii out from the fovea, There is also physiological evidence which points in the same direction, in the presence in prestriate areas MT and MST of cells specifically responsive to movement towards or away from the fovea (Albright, 1989; Tanaka & Saito, 1989). It is thus of interest that we find the movement-induced positional bias to be greatest for centrifugal or centripetal movement. The difference between the bias with movement towards or away from the fovea as opposed to movement at right angles to such a radius was very large for two observers and also present to a lesser extent in the third observer. Our stationary moving Gabor patterns are rather unusual, unnatural stimuli, particularly since the movement in the pattern is purely
Veher
acuity with
stati~mwymoving Gabors
local, the pattern as a whole being stationary. Having a stationary envelope conveniently allowed us to make precise measures of the effect, but there is no reason to think that the basic finding-that moving patterns are mislocated-would not apply to patterns moving across space as well. Since it would appear to be important for organisms to localize accurately moving as well as stationary patterns, the existence of this illusion would appear to be potentially detrimental. Movement-induced positional bias could, however, help compensate for the visual system’s nontrivial latency. It takes 50-100msec for striate cells to respond to a stimulus, and of course even longer for cells further downstream. What we see, then, is not the world as it is now but as it was in the near past. In the case of a stationary pattern, this latency is unimportant, but to respond accurately to a moving stimulus the orga~sm must somehow compensate for this neural delay. To catch a ball, for instance, one must intercept it at its true position at time t, even though at time t the visual system will be processing an image that occurred at, say, time f - 100 msec. The positional misalignments we have shown to occur with moving stimuli are in the correct direction to compensate for the visual latency. Thus, at any given instant a moving object appears to have moved further along its trajectory than the retinal image being processed at that instant would indicate. The fact that the effect increases with temporal frequency (up to 4-8 Hz) and is greatest at low spatial frequencies would also be in the correct direction since this means that the maximum shift in apparent position occurs for patterns moving at high velocities. It has traditionally been thought that the compensation for visual and response latency that allows one to anticipate the correct position of moving objects occurs at some late point in sensory-motor coordination, The results presented here hint at the possibility that some of the compensation may take place early within the visual system itself. However, the fact that the shift in apparent position varies greatly with retinal eccentricity makes this a less than optimal contribution to the solution. Thorson et al. (1969) reported an illusion which may be related to the movement-induced apparent positional change we have investigated here. They noted that successive stimulation of two nearby points in the peripheral retina produces an impression of apparent movement
1625
(under conditions that would not produce it with central fixation), and that this movement seemed to extend over considerably longer dis-
tances than the distance between the two points stimulated. They termed this the “fine-grain motion illusion”. Our study does not bear on the apparent motion aspect of this illusion, but may be related to the elongation of movement distance they reported. If a movement of the grating in our Gabors appears to be longer in extent than it is, the center of gravity of the patch might be expected to be shifted, thus producing the observed positional bias. However, it is by no means certain that the same process is in fact involved, p~icufarly since our data clearly show the process we studied to operate centrally as well as in the periphery. There has been a long-lasting if not earthshaking dispute between baseball players and certain physicists and photographers who have addressed the question of whether curve balls curve. More precisely, since the trajectory of any thrown ball will curve due to gravity, the question is whether the additional spin given by the pitcher to a curve ball makes it curve more than it would otherwise do. A ball as light as a ping-pong ball can clearly be made to curve by putting a spin on it, but it is less obvious that this is true for a heavy baseball. Although early photographic attempts to verify a curvature failed, it is now generally accepted that a curve ball does indeed curve to some extent (Adair, 1990). But one thing that is absolutely clear from both theory and measurement is that a durve ball must curve continuously, on an arc; it does not “break”‘-suddenly change direction in midflight-just before it reaches the plate (although the deflection will be continuously increasing as it approaches the plate). Baseball players, however, are unanimous in saying that that is just what a curve ball does: it suddenly dips down and out just before reaching the plate. The attentive reader will by now have anticipated that we are going to suggest that the sudden “breaking” curve baseball players see is partly the result of the perceptual distortion we have been examining in this study. The direction of the perceived change in position of the baseball is correct to be accounted for in this way: the stitches in the spinning curve ball are moving down diagonally on the side towards the batter as the ball approaches. We suggest that when the curve ball gets close enough to the batter for him to perceive the movement of
1626
RUSELL
L. DE VALIXSand
the stitches, the movement illusion we have been studying would be triggered to make the ball suddenly appear to shift down and out in position, that is, to “break”. Baseball batters also often report that a “fast ball” curves upwards, or hops, a most unlikely physical possibility. But since a fast ball spins in the reverse direction as a curve ball, the same perceptual effect which accounts for a curve ball’s dipping would lead one to expect that a fast ball should suddenly appear to rise as it approaches the plate, despite the fact that it is actually falling due to gravity. REFERENCES Adair, R. K. (1990). The physics of baseball. New York: Harper & Row. Albright, T. D. (1989). Centrifugal directional bias in the middle temporal visual area (MT) of the macaque. Visual Neuroscience, 2, 177-188. Anstis, S. M. (1989). Kinetic edges become displaced, segregated, and invisible. In Neural mechanisms of visual perception (pp. 247-260). Houston: Portfolio/Gulf. Burbeck, C. A. (1988) Large-scale relative localization across spatial frequency channels. Vision Research, 28, 857-859. Finney, D. J. (1971). Probit analysis. Cambridge: Cambridge University Press.
KAREN
K. DE VALOIS
Kelly, D. H. (1979). Motion and vision-II Stabilized spatio-temporal threshold surface. Journal of the Optical Society of America, 69, 1340-l 349. Kooi, F. L., De Valois, R. L. & Switkes, E. (1987). Vernier acuity with Gabor patches. Investigative Ophthalmology and Visual Science, 28, 360. Levi, D. M., Klein, S. A. & Aitsebaomo, A. P. (1985). Vernier acuity, crowding and cortical magnification. Vision Research, 25, 963-977. Nakayama, K. (1985). Biological image motion processing: a review. Vision Research, 25, 625-660. Ramachandran, V. S. & An&s, S. M. (1991). Perception, in press. Regan, D. (1986). Visual processing of four kinds of relative motion. Vision Research, 26, 127-145. Tanaka, K. & Saito, H. (1989). Analysis of motion of the visual field by direction, expansion/contraction, and rotation cells clustered in the dorsal part of the medial superior temporal area of the macaque monkey. Jouma/ of Neurophysiology, 62, 626-641. Thorson, J., Lange, G. D. & Biederman-Thorson, M. (1969). Objective measure of the dynamics of a visual movement illusion. Science, 164, 1087-1088. Toet, A. (1987) Visual perception of spatial order. Doctoral dissertation, University of Utrecht. Toet, A. & Koenderink, J. J. 1988). Differential spatial displacement thresholds for Gabor patches. Vision Research, 28, 133-143. Westheimer, G. & McKee, S. P. (1977). Integration regions for visual hyperacuity. Vision Research, 17. 89-93.
