Z. W. Pylyshyn and R. W. Storm (1988) have shown that human observers can accurately track four to five items at a time. However, when a threshold paradigm is used, observers are unable to track more than a single trajectory accurately (S. P. Tripathy & B. T. Barrett, 2004). This difference between the two studies is examined systematically using substantially suprathreshold stimuli. The stimuli consisted of one (Experiment 1) or more (Experiments 2 and 3) bilinear target trajectories embedded among several linear distractor trajectories. The target trajectories deviated clockwise (CW) or counterclockwise (CCW) (by 19°, 38°, or 76° in Experiments 1 and 2 and by 19°, 38°, or 57° in Experiment 3), and observers reported the direction of deviation. From the percentage of correct responses, the “effective” number of tracked trajectories was estimated for each experimental condition. The total number of trajectories in the stimulus and the number of deviating trajectories had only a small effect on the effective number of tracked trajectories; the effective number tracked was primarily influenced by the angle of deviation of the targets and ranged from four to five trajectories for a ±76° deviation to only one to two trajectories for a ±19° deviation, regardless of whether the different magnitudes of deviation were blocked (Experiment 2) or interleaved (Experiment 3). Simple hypotheses based on “averaging of orientations,” “preallocation of resources,” or pop-out, crowding, or masking of the target trajectories are unlikely to explain the relationship between the effective number tracked and the angle of deviation of the target trajectories. This study reconciles the difference between the studies cited above in terms of the number of trajectories that can be tracked at a time.

^{2}, background luminance 2.4 cd/m

^{2}, angular subtense 5′ × 5′) underwent apparent motion across the computer screen in a left-to-right direction. The dots moved with a speed of 4 deg/s along either a linear or a bilinear trajectory (i.e., a trajectory with a single deviation in direction), and the entire trajectory presentation lasted for a total of 51 frames (850 ms). Two vertical markers indicated the midline of the screen, and observers were aware that any deviation that may arise in a trajectory would take place when the dot was exactly halfway along its trajectory, that is, between the vertical markers (Figures 1 and 2).

*T*) presented on each trial. For each combination of deviation angle and

*T,*observers completed a practice block followed by two further blocks, and the percentage of correct responses was determined from these 200 trials. For each of the three deviations employed (± 19°, ± 38°, and ± 76°), data were gathered for values of

*T*= 1, 2, 3, 4, 6, 8, and 10 trajectories. The number of distractor trajectories present was always one less than

*T*.

*T*) was held constant and the number of deviating trajectories (

*D*) was varied. The number of distractor trajectories was therefore given by (

*T*−

*D*). As in Experiment 1, the magnitude of the deviation angle within a block of trials was fixed, as were

*D*and

*T*. Observers knew in advance how many trajectories (

*T*) would be presented in the block of trials, how many of these would deviate (

*D*), and the magnitude of the deviation. Data were collected for

*T*= 10 for deviations of ±19°, ±38°, and ±76° for values of

*D*= 1, 2, 3, 5, 7, and 10. As in the previous experiment, the percentage of correct responses for each combination of deviation angle and

*D*was determined from two blocks yielding 200 trials. The experiment was also conducted for the same deviation angles when the total number of trajectories was eight (i.e.,

*T*= 8, with

*D*= 1, 2, 3, 5, 6, and 8) and six (i.e.,

*T*= 6, with

*D*= 1, 2, 3, 4, and 6).

*T*) was held constant and the number of deviating trajectories (

*D*) was varied from trial to trial, as was the magnitude of the deviation angle. Observers knew in advance how many trajectories (

*T*) would be presented in the block of trials, but did not know how many of these would deviate (

*D*), or the magnitude of the deviation. Only a restricted range of values of

*D*was used and the range of deviation angles was also reduced; this was done to reduce any potential cues from the average orientation of the trajectories (see Results and discussion section, Experiment 2). One set of data was collected for

*T*= 10 with deviations of ±19°, ±38°, and ±57° for values of

*D*= 1 and 2 (the six combinations from three orientations and the two values of

*D*were interleaved). Another set of data was collected for

*T*= 10 with deviations of ±19°, ±38°, and ±57° when

*D*= 1, with deviations of ±19° and ±38° when

*D*= 2, and with deviations of ±19° when

*D*= 3. In this experiment, each block consisted of 120 trials, with 20 trials for each of the six combinations of deviation angle and the value of

*D*interleaved within a block. The percent correct responses for each combination of deviation angle and

*D*was derived from 15 blocks yielding 300 trials for each stimulus condition (which was larger than the 200 trials for each stimulus condition in the previous experiment).

*P*) of a limited capacity hypothetical observer

*A*trajectories. If the capacity of this observer was large, then

*A*would be large. If this hypothetical observer was presented with a stimulus containing fewer than

*A*trajectories, then its performance for identifying the direction of deviation of the target trajectories would be perfect. However, if it was presented with stimuli containing more than

*A*trajectories, then its performance would be less than perfect.

*A*randomly selected trajectories out of

*T*and ignores the remaining (

*T*−

*A*) trajectories? To determine the performance of this LCHO, we assumed that the trajectory deviations are much larger than its deviation threshold for a single trajectory; that is, the deviation on a trial will be detected correctly if the deviating trajectory is among the

*A*trajectories allocated adequate computational resources.

*T*−

*A*) trajectories that are not allocated computational resources, the LCHO has inadequate information regarding the direction of deviation and must select the direction of deviation at random. All available computational resources are directed to only the

*A*randomly selected trajectories. The percentage of correct responses

*P*can be determined from the following equations from basic probability:

*A*/

*T*) is the probability that the deviating target is among the

*A*trajectories that are allocated adequate computational resources (i.e., the probability of

*knowing*the direction of deviation), whereas the second term ((1 −

*A*/

*T*) / 2) is the probability of correctly

*guessing*the direction of deviation. Based upon these assumptions, we have generated predicted performance levels for Experiment 1 for different numbers of trajectories effectively tracked (

*A*) by the LCHO. These predicted performance levels are shown in Figure 3. For example, if six trajectories are presented (

*T*= 6) and the LCHO can process only three trajectories (

*A*= 3) effectively, the probability that a correct response will be made is 0.75 [i.e., 100 × (3/6 + (1 − 3/6) / 2), or 75% correct in Figure 3].

*D*), the maximum number of trajectories that the LCHO can track perfectly (

*A*), and the total number of trajectories (

*T*). In this case, the assumption made is that the direction of deviation will be correctly identified if any of the

*D*deviating trajectories are among the

*A*trajectories that are tracked by the LCHO. When all the

*D*deviating trajectories are among the (

*T*−

*A*) trajectories that are not allocated computational resources, the LCHO has no information regarding the direction of deviation and must determine the direction randomly. The percentage of correct responses

*P*can be determined similarly from simple probability using the following equations:

*D*> (

*T*−

*A*), then the

*A*trajectories that are allocated computational resources must contain at least one deviating trajectory. In Equation 4, the first term in the rectangular parenthesis represents the probability that there will be at least one deviating trajectory among the

*A*trajectories allocated computational resources (i.e., the probability of

*knowing*the direction of deviation). The second term is the probability of correctly

*guessing*the direction of deviation; the LCHO must resort to guessing when there is no deviating trajectory among the

*A*trajectories that are allocated resources.

*T*) fixed at 10. The predictions are shown for different numbers of maximum trajectories tracked (

*A*) by the LCHO. For example, if 10 trajectories are presented (

*T*= 10), of which 5 deviate (

*D*= 5), but the LCHO can track only 3 trajectories (

*A*), the probability of a correct response is 0.96 [i.e., 100 × ((1 − 10 / 120) − (10 / 120) / 2), or 96% correct in Figure 4]. The percent correct curves in Figure 4 are shown as continuous, whereas Equation 4 is defined only for integer values of

*D*; this has been done in order to facilitate the visualization of the functions. Similar predictions can be generated when

*T*= 8 and

*T*= 6.

*A*trajectories perfectly, like our LCHO does, will perform better than another hypothetical observer that distributes its resources (attentional, computational) so that it tracks say

*A*+ 2 trajectories slightly less than perfectly.

*E*)

*effective number of tracked (attended?) trajectories*under a set of experimental conditions to be

*E,*if the performance of the human observer under those conditions is comparable to that of the LCHO with

*A*=

*E*. In other words,

*E*is the specific value of

*A*for which the performance of the LCHO matches that of the human observer under a set of experimental conditions. To determine

*E,*we need to compare the human observer's performance to those of the LCHO for different values of

*A*and find the value of

*A*that yields the closest match. One could interpret the effective number tracked to be the number of trajectories that are assigned adequate computational resources to be able to detect a deviation of the magnitude used in the experimental condition. However, we do not wish to imply that the human observer perfectly tracks

*E*trajectories and ignores the remaining (

*T*−

*E*) trajectories. We only imply that the human observer's performance is comparable to that of a hypothetical observer that perfectly tracks

*E*trajectories and ignores the remaining (

*T*−

*E*) trajectories. For example, if the human observer's performance is comparable to a hypothetical observer that tracks

*E*trajectories perfectly, it could also be the case that the observer's performance is comparable to another hypothetical observer that tracks, say (

*E*+ 2) trajectories but less than perfectly. In general, the effective number of tracked trajectories should be seen as a

*description*of human performance and not as an

*explanation*of human performance. However, if under a variety of conditions the estimated effective number of tracked trajectories remains roughly constant, then a parsimonious explanation for this is that this estimate is a reflection of the actual number of trajectories tracked; in other words, the observers could really be tracking

*E*trajectories and ignoring (

*T*−

*E*) trajectories, and

*E*might represent some internal capacity limit for tracking for this observer in the experimental condition being tested.

*E*) from human performance

*D*= 1. Let us consider the case when there are 10 trajectories, with 2 trajectories deviating (i.e.,

*T*= 10;

*D*= 2). The predicted performance for the LCHO that tracks

*A*trajectories perfectly can be determined for

*A*= 1, 2,…, 10 by taking a vertical slice through Figure 4 at

*D*= 2. The expected percent correct when

*A*= 0 is 50% and when

*A*= 10 is 100%. These data have been replotted in Figure 5a with the number of trajectories tracked by the LCHO along the abscissa and then fitted with an exponential curve. This enables the human observer's percentage of correct responses to be graphically mapped on to its corresponding number of tracked trajectories as shown in Figure 5a. This will yield the effective number of tracked trajectories (

*E*) when there are 10 trajectories in the stimulus with 2 deviating (

*D*). If two observers had performance levels of 84% and 80.5% correct responses under these experimental conditions (these percentages correspond to actual data taken from Experiment 2 for a 76° deviation, see Figure 7a, red symbols at

*D*= 2), then their corresponding numbers of tracked trajectories will be 4.10 and 3.48, respectively, as can be seen in Figure 5a. In other words, for the experimental condition described here, the performance of the first observer was comparable to that of the LCHO when it tracked 4.10 trajectories perfectly, and that for the second observer yielded 3.48 for the effective number of tracked trajectories. Note that the fractional numbers obtained here for the effective numbers of trajectories tracked perfectly are not counterintuitive. One can readily imagine a hypothetical observer that tracks four trajectories on most trials and tracks five trajectories on a few trials to yield an average of 4.10 trajectories tracked perfectly.

*D*(in the case of

*D*= 1, a straight-line fit was used instead of an exponential fit). For any performance level of the human observer obtained with

*T*= 10, the number of tracked trajectories can be estimated by projecting the observed performance level to the curve corresponding to the number of deviating trajectories (

*D*) and reading off the appropriate abscissa value. Similar figures (not shown) were constructed for

*T*= 8 and for

*T*= 6, and these were used for estimating the effective number of tracked trajectories under these conditions. This graphical technique yielded more accurate estimates for the effective number of tracked trajectories compared to estimates obtained visually from Figure 4.

*T*trajectories deviating

*T*) when there was only one trajectory deviating. Also shown (dotted lines) are the predictions for the performance of the LCHO that perfectly tracks one, two, three, or four trajectories. These predictions are a subset of the curves shown in Figure 3.

*T*) tested, the effective number of tracked trajectories (

*E*) was between one and two when the angle of deviation was ±19°, between two and three when the angle of deviation was ±38°, and between three and four when the angle of deviation was ±76°. The next experiment investigated the effective number of tracked trajectories more systematically.

*D*-of-

*T*trajectories deviating: conditions blocked

*A*trajectories, where

*A*= 1, 2,…, 5 and ignores the remaining trajectories. These predicted performance levels are a subset of the curves shown in Figure 4. The results in Figure 7a seem comparable to those in Experiment 1, with observers effectively tracking one to two trajectories when the deviation is ±19°, two to three trajectories when the deviation is ±38°, and three to five trajectories when the deviation is ±76°.

*D*from 2 to 7 resulted in the effective number of tracked trajectories changing from 1.57 to about 1.19. A 10-trajectory stimulus with 2 trajectories deviating is very different from one with 7 trajectories deviating. However, for both of these very different stimuli, the effective number of tracked trajectories is about 1.38. Likewise, for the ±38° deviation data, a 10-trajectory stimulus with 1 trajectory deviating is very different from a 10-trajectory stimulus with 7 trajectories deviating; however, the effective number of tracked trajectories (averaged over the three observers) in the two cases is 2.74 and 1.96. In the case of the ±76° deviation data, only a limited number of these could be converted to effective numbers of tracked trajectories because performance was close to 100% correct when the number of deviating trajectories was five or more.

*T*= 10, 8, and 6), for all of the ±38° deviation conditions, and for all of the ±76° deviation conditions. The actual number of trajectories in the stimulus has only a small effect on the effective number of tracked trajectories over the range of trajectories tested. Likewise, the number of deviating trajectories has only a small effect on the number of tracked trajectories. Most interestingly, it is the magnitude of the angle of deviation that primarily determines how many trajectories are effectively tracked.

*T,*

*D,*and angle of deviation, if either observer had a performance level better than 60% correct in either of the two conditions (left-half occluded or right-half occluded) then, for that combination of parameters, the performance of the observers in the main experiment may have been influenced by additional cues. (The cut-off limit from the binomial theorem should have been 57%. We relaxed this limit because of the repeated comparisons involved and because we were overly conservative in other aspects—the cut-off limit was considered to have been exceeded if

*any*of the four percentages obtained (percent correct for either of the two observers under either of the two occlusion conditions) exceeded the cut-off limit.) Conditions for which the performance was better than the cut-off level, that is, for which extraneous cues may have influenced judgment, are identified in Figures 7b, 7d, and 7f by filled symbols. Most data obtained with targets having a deviation of ±76°, or with more than three trajectories that deviate by ±19° or more, could potentially have been influenced by extraneous cues available. Hence, for some of the experimental conditions, Figure 7 may overestimate the number of trajectories tracked and should be seen as the upper limit to tracking performance in our task.

*D*-of-

*T*trajectories deviating: conditions interleaved

*D*and the range of the angles of deviation to ensure that the extraneous cues in the two halves of the stimulus did not influence the observers judgment of trajectory deviation. At the start of any trial, the observer was unaware of the number of deviating trajectories, or the angle of deviation, and could not preallocate resources to each trajectory based on the difficulty of the task. The difficulty of the task varied from trial to trial because of the random interleaving of the stimulus conditions. Under these circumstances, one anticipates that the effective number of tracked trajectories for the six stimulus conditions interleaved within a block will be fixed at some mean level; that is, unlike in Experiments 1 and 2, the effective number of tracked trajectories will be independent of stimulus conditions, in particular independent of the angle of deviation.

*The percentage of correct responses is a poor indicator of tracking performance.*A higher percentage of correct responses for the second stimulus does not necessarily imply that tracking performance was better for that stimulus, it could simply reflect the fact that the probability of detecting a trajectory deviation is higher when there are four trajectories deviating than when there is only one trajectory deviating. The effective number of tracked trajectories takes these different probabilities into account, thus permitting a direct comparison of performance across the different stimulus conditions.

*T*= 10, 8, and 6 for the ±76° deviation data all lying close to one another. This was found to be more the case when the deviation was ±38° and was most evident when the deviation was ±19°. When the number of trajectories in the stimulus was changed from 6 to 10, the corresponding change in the effective number of tracked trajectories was typically much less than one. The fact that the effective number of tracked trajectories did not change proportionately when the number of trajectories in the stimulus was changed from 6 to 10 suggests that, even when only six trajectories were presented, observers were only tracking a subset of the available trajectories.

*T*and

*D,*the tracking performance of human observers was comparable to a hypothetical observer that perfectly tracked about one (±19° deviation), two (±38° deviation), or four (±76° deviation) trajectories and ignored the remaining trajectories. It is possible that when the values of

*D*were greater than three, the observers' performance might have been compromised by extraneous cues. However, even when these extraneous cues were unavailable, the effective number of tracked trajectories increased systematically with increasing angle of deviation. Could this almost proportional increase in the effective number of trajectories with increase in the angle of deviation have resulted from observers averaging the orientations of the trajectories, both pre- and postdeviation, and basing their judgment on the difference between the two? Such an explanation is unlikely; when observers were encouraged to average the orientations of the trajectories by having half the distractors deviate CW by a fixed angle and the other half deviate CCW by the same angle, performance dropped rapidly when the magnitude of this angle was increased, in spite of the distractor deviations resulting in no change in the average pre- or postdeviation orientation ( Figure 10).

*at least*one, two, or four trajectories must be utilized in order to match human performance. If one prefers an attentional explanation for the current findings, then

*at least*one, two, or four trajectories must have been attended in order to explain the performance obtained for ±19°, ±38°, or ±76° deviations. Of course this does not rule out that attention may have been directed to more trajectories than this minimum number. If one prefers an alternative explanation to the current findings, such as an averaging of orientations explanation, then the current findings are still informative—the orientations of

*at least*one, two, or four trajectories must be averaged (perfectly) in order to explain the results for ±19°, ±38°, or ±76° deviations, respectively. However, the poor averaging performance indicated by Figure 10 suggests that the actual number of trajectories averaged will have to be greater than these minimum numbers (if trajectory orientations are indeed averaged).

*E*trajectories the internal representation could contain adequate information for discriminating the direction of deviation and for the remaining (

*T − E*) trajectories the information available might be inadequate for deviation discrimination. But an alternative explanation based on the transience of the traces of the trajectories is also plausible. In Narasimhan, Tripathy, and Barrett (2005, 2006b), we measured deviation thresholds along the lines of Tripathy and Barrett (2004), but we cued the target trajectory during the second half of the display to examine if the earlier parts of the cued (target) trajectory could be remembered. Thresholds were only slightly improved by cueing, suggesting poor recall of the early parts of the trajectories (for a similar experiment with static letters instead of trajectories, see Sperling, 1960). In another experiment, we introduced a delay between the first and second halves of the display on each trial (see Sperling, 1960, who used a similar technique to demonstrate the temporal characteristics of visual sensory memory or iconic memory for tachistoscopic stimuli with static letters). When there were three trajectories in the stimulus, a delay of 300–400 ms halfway through the trial resulted in thresholds being elevated by a factor of four in some observers. This deterioration of performance, similar to that seen in Sperling's (1960) experiments, suggests the involvement of visual sensory memory in tracking; trajectory traces in memory may be utilized by the visual system in these deviation detection tasks. The decision process could involve the sequential accessing of these rapidly decaying traces to determine the direction of deviation of the target(s). The effective number of tracked trajectories may be a reflection of the number of trajectory traces that can be scanned before they decay to the point of not being useful to the decision process anymore. Although both attentional and memory-based explanations are plausible, we currently favor the explanation that is based on the scanning of traces in memory; such an explanation would be consistent with the current findings and the findings of our other recent studies that suggest the involvement of memory in deviation detection with similar stimuli (Narasimhan, 2006; Narasimhan et al., 2005, 2006b). Interestingly, amblyopes show little or no deficit in tracking deviations in trajectories, whether the deviations are close to threshold (Levi & Tripathy, 2006a, 2006b) or are substantially suprathreshold, as in the current study (Tripathy & Levi, 2006a, 2006b, 2006c). Amblyopes have been shown to undercount features in a stimulus, suggesting a deficit in attentional processing when viewing with their amblyopic eye (Sharma, Levi, & Klein, 2000). The absence of an amblyopic deficit in the current task suggests that the limits to performance in the current task are probably not attentional.