Information used in detecting upcoming collision

Perception, 2003, volume 32, pages 525 ^ 544

DOI:10.1068/p3433

Information used in detecting upcoming collision

Reinoud J Bootsma, Cathy M Craig

UMR Mouvement & Perception, Universite¨ de la Me¨diterraneë, 163 avenue de Luminy ^ CP 910, 13288 Marseille cedex 09, France; e-mail: [email protected] Received 25 July 2002, in revised form 16 December 2002; published online 23 April 2003

Abstract. In four experiments we examined the nature of the information used in judging whether events would or would not give rise to a collision in the near future. Observers were tested in situations depicting approaches between two objects (lateral approaches) and approaches between an object and the point of observation (head-on approaches), with objects moving according to constant deceleration or sinusoidal deceleration patterns. Judgments were found to be based, to a large extent, on the (in)sufficiency of current deceleration to avoid upcoming collision, as specified optically by tau-dot (_t). However, the information specified by tau (t), that is the current (first-order) time remaining until contact, was also found to play a significant role. We deduce that judgment of upcoming collision is based on the detection of t and its evolution over time, suggesting that observers are sensitive to Dt rather than to t_ itself.

1 Introduction Driving around the city of Marseille(s) (1) during the rush hour is a task of considerable complexity, likely to induce a state of traffic shock in the unsuspecting foreign visitor (Mayle 1999). All around you bikes, cars, vans, and trucks of varying size, shape, colour, and age continuously accelerate and decelerate, with the apparent goal of minimising the proportion of unoccupied tarmac. Yet, notwithstanding the commonly shared ö and often passionately expressedöopinion that all others are terrible drivers, accidents are relatively rare. In fact, every day the overwhelming majority of participants arrive at their destination without a scratch. In his seminal contribution published a quarter of a century ago, Lee (1976) elaborated a conceptual framework for understanding the regulation of action in this type of situation, based on the identification of action-relevant optical states. Relative approach between an observer and an object, like a car in front of us, is specified by an increase in the (solid) optical angle subtended by the object (Schiff 1965). However, as demonstrated by Lee (1976), the spatiotemporal properties of the change in angle produced by relative approach contain much more information that may be useful for the control of action. For instance, the rate of dilation of the optical angle is inversely proportional to the time remaining until the object will be reached if the current approach velocity were to be maintained. Put more formally, a particular property of the change in the optical angle f subtended by the object at the point of observation _ specifies the first-order time remaining [that is, TC (X) ÿX=X_ ] [that is, t(f) f=f] 1 until distance X becomes zero. The first-order nature of this temporal relation has often been taken to imply considerable limitations in its usefulness (eg Tresilian 1994; Wann 1996). However, as demonstrated by Lee (1976), the very fact that the optical variable t specifies a first-order temporal relation allows the rate of change over time of t to be informative about the possibility of an upcoming collision. Because the optical variable t specifies the first_ the rate of change of t (ie t_ or tau-dot) specifies XX= X_ 2 ÿ 1. order time TC1 (X ) ÿX=X, (1) The city of Massalia was founded more than 2600 years ago by the Phoceans, a people from minor Asia. As there is only one such city, we prefer the current French spelling without an s at the end.

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_ the deceleration X is sufficient to avoid collision For current distance X and velocity X, 2 _ when X 5 X =2X, that is when t_ 5 ÿ 0:5. Thus, t_ ÿ0:5 separates situations in which the current deceleration is sufficient to avoid upcoming collision (_t 5 ÿ 0:5) from situations in which it is not (_t 5 ÿ 0:5). Understood in this way, the value t_ ÿ0:5 constitutes a critical value [see Kaiser and Phatak (1993) for another interpretation]. Surprisingly, to date the number of studies that have addressed the two major questions that the identification of such an informational variable immediately evokes ö the question of observer sensitivity to the information carried in the optical variable t_ and the question of its use in the regulation of actionöare few and far between. Perhaps one of the reasons for this is to be found in the pernicious confusion of these two questions that has resulted from Lee's (1976) suggestion that the (active) control of deceleration could be based on maintaining the relevant optical variable (ie t_ ) at a constant value. Thus, in the majority of studies (eg Lee et al 1991, 1992, 1993; Wann et al 1993; Zaal and Bootsma 1995), the question addressed has been whether the control of deceleration is based on maintaining t_ at a constant value rather than the question whether (the information provided by) t_ is used in the control of deceleration. The exception confirming the rule is a study by Yilmaz and Warren (1995), the only existing study that we know of providing evidence that braking is regulated on the basis of a deviation from the critical t_ value of ÿ0:5. As argued above, t_ carries information about the sufficiency of current deceleration to avoid upcoming collision, with the value of ÿ0:5 separating events that will result in collision from events that will not result in collision (if current deceleration is maintained). Thus, sensitivity to t_ could allow observers to distinguish between such events (Bardy and Warren 1997). In a recent study, Andersen et al (1999) demonstrated that human observers, presented with simulated self-motion decelerating at a constant rate during approach to a stationary obstacle, were indeed capable of distinguishing situations that would result in collisions with the obstacle from situations that would not. Observers watched displays simulating forward motion of the observer through a 3-D scene consisting of a textured ground plane and roadway with three vertical signs located in the centre of the roadway. After a 6.5 s period of constant velocity, terminated by a warning signal, forward motion decelerated at a constant rate, sufficient or not sufficient to bring the observer to a stop before reaching the signs on the roadway. Following a variable time interval (2.0 to 3.5 s) after onset of deceleration, the displays went blank (at varying distances from the target signs) and observers were asked to indicate öby pressing the appropriate mouse buttonöwhether a collision would occur if motion continued as it had before. Although Andersen et al (1999) report that judgments were, to some degree, influenced by other variables (initial velocity, obstacle size, and distance from the obstacle at which the visual displays were blanked-out), their results support the use of information carried by t_ as a determinant of judgment behaviour. Andersen et al (1999) did not examine the predictive power of (combinations of ) influential variables in detail, and concluded that observers would use different sources of information. It is important to realise that constant-deceleration approaches, as the ones used by Andersen et al (1999), lead to continuously changing values of t_ (except for the particular deceleration value that, for the given initial conditions, gives rise to t_ ÿ0:5). Constant decelerations that will bring approach velocity to zero before contact has occurred (ie non-collision events) will lead to t_ values moving toward plus infinity, while constant decelerations that will bring approach velocity to zero after contact has occurred (ie collision events) will lead to t_ values moving towards ÿ1 (see figure 1). Thus, even though t_ continuously carries information about the sufficiency of current deceleration, t_ values vary during constant-deceleration approaches (except for the particular value of t_ ÿ0:5 ), indicating the continuously evolving nature of the (in)sufficiency of current deceleration. To understand the reasons underlying this,

Information in detecting collision

527

1.0

t_

0.5

0.0 no collision

ÿ0:5 collision ÿ1:0 0

2

4 6 Distance from obstacle=m

8

10

Figure 1. t_ as a function of distance remaining during constant-deceleration approaches (X 2:5 m sÿ2 ). For the initial conditions of distance X 20 m and velocity X_ ÿ10 m sÿ1 , this constant deceleration is exactly the deceleration required to bring X_ to zero when X reaches zero X_ 2 =2X ), giving rise to a (constant) value of t_ ÿ0:5 (thick line). For initial conditions that (X bring X_ to zero before X 0, the deceleration is sufficient to avoid collision and t_ goes to infinity during approach (upper three lines, with X 20:83, 20.83, and 20.00 m; and X_ 9:9, 10.0, and 9.9 m sÿ1 , respectively). For initial conditions that do not bring X_ to zero before X 0, the deceleration is not sufficient to avoid collision and t_ goes to ÿ1:0 during approach (lower three lines, with X 19:23 m; and X_ 9:9, 10.0, and 10.1 m sÿ1 , respectively).

it is useful to realise that a deceleration that was initially only slightly too low to avoid upcoming collision will become increasingly more insufficient as the approach unfolds, up to the point that it is totally insufficient at the last instant before contact. Kim et al (1993) also conclude in favour of observer sensitivity to the information carried in t_ , but their study differs notably from the study of Andersen et al (1999) in the type of stimulus proposed as well as the type of judgment required. Stimuli consisted of 300 yellow dots forming, in the centre of the screen, a square regionö against a black backgroundöthat expanded over time. Kim et al (1993) presented observers with approaches in which t_ was maintained at constant values between ÿ1:0 and ÿ0:1 throughout the simulated event. Observers were asked to judge (by pressing the appropriate key on a computer keyboard) whether such events would give rise to a `hard' or `soft' collision. If we ignore the characteristics of the trajectories corresponding to the optical states presented,(2) the study of Kim et al (1993) can be taken to allow a (2) Kaiser and Phatak (1993) argued that any approach in which t _ is truly kept constant at a value between ÿ1 and 0 will lead to a soft collision, that is a velocity of zero at the moment of contact. Simulations by Lee et al (1992), Kaiser and Phatak (1993), Yilmaz and Warren (1995), and Bardy and Warren (1997) demonstrate that, while maintaining t_ constant at a value of ÿ0:5 leads to a constant deceleration pattern, maintaining t_ constant at a value in the range of ÿ0:5 5 t_ 5 0 gives rise to an initially higher and gradually decreasing deceleration pattern, specific to the particular t_ value chosen. These simulations also demonstrate that maintaining t_ constant at a value in the range ÿ1 5 t_ 5 ÿ 0:5 leads to an initially moderate but exponentially increasing deceleration, going to infinity at the end of the approach. Thus, the latter strategy is impossible to realise for a given physical system, and attempting to do so would, at the end, lead to a `hard' collision. Yet, it is obvious that in this case t_ is not truly maintained constant throughout the approach (Kaiser and Phatak 1993). As the optical displays in the study of Kim et al (1993) terminated before the final phase in which deceleration explosively increases to infinity, it is not clear what aspect of the approach kinematics observers were actually judging. Given our analysis of the information carried in t_ , we suggestöin line with Bardy and Warren (1997)öthat observers most likely judged whether a collision would or would not occur on the assumption that terminal deceleration will be held constant.

528


direct evaluation of observer sensitivity to particular (non-evolving) t_ values. As such, their finding that conditions with ÿ1 5 t_ 5 ÿ 0:5 give rise to `hard collision' judgments and that conditions with ÿ0:5 5 t_ 5 0 give rise to `soft collision' judgments can be taken as evidence in favour of observer sensitivity to the information carried in t_ with respect to the (in)sufficiency of current deceleration to avoid upcoming collision. In line with the original analysis of Lee (1976), both Andersen et al (1999) and Kim et al (1993) presented observers with head-on approaches between the observer and an environmental object. However, evaluating the risk of collision between, for example, a car and a child playing on the road is also likely to have implications for action. Several authors have provided theoretical and empirical evidence in favour of an extension of Lee's (1976) original analysis of (first-order) time to contact to situations other than head-on approach (Bootsma 1988; Bootsma and Craig 2002; Bootsma and Oudejans 1993; Kim et al 1998b; Tresilian 1990), and it seems reasonable to assume that t-like variables are indeed available in a variety of situations. After all, the basic idea underlying Lee's (1976) original analysis is that particular optical patterns (ie ts) specify particular patterns of gap closure (Lee 1998). Recently, Kim et al (1998a) demonstrated that observers are capable of judging (3) whether a decelerating object will collide with another object under a variety of motion trajectories. In summary, what do we know about observer sensitivity to the information carried in t_ ? (4) Kim et al (1993, 1998a) have provided evidence that observers confronted with motion simulations that maintained t_ at constant values during the presentation can distinguish safe states (ie combinations of current distance, velocity, and deceleration that will not result in collision) from unsafe states (ie combinations of current distance, velocity, and deceleration that will result in collision), whether collision is defined with respect to the observer or with respect to another object. Andersen et al (1999) have provided evidence for the use of t_ in detecting upcoming collision during constantdeceleration head-on approaches, but suggest that other variables might also be used. In the present study, we sought to determine the role played by the information carried in t_ using different types of approach (constant and sinusoidally changing deceleration) and different types of situation (head-on approach between observer and an object and transversal approach between one object and another). Different types of approach (constant and sinusoidally changing deceleration) allowed testing of the use of the information carried in t_ under a variety of deceleration conditions. Use of different types of situation (head-on approach between observer and an object and transversal approach between one object and another) allowed testing of the generality of the information identified under a variety of approach trajectories. The experiments reported in this study required participants to observe the computer-simulated motion of objects displayed on the screen in front of them. The objects in these displays disappeared before any potential collision would have taken place. Observers were then asked to judge whether a collision would have taken place or not by clicking the appropriate mouse button. The experiments differ according (3) While Kim et al (1998a) once again presented observers with approaches in which the relevant t_ was maintained at a constant value and asked them to judge the severity of upcoming collision, they note that t_ specifies whether ``the current level of deceleration is adequate to stop in front of the surface'' and add that the latter condition is `òften described in the literature as a soft collision'' (page 95). Leaving the latter affirmation for what it is, we again suggest that the judgment of soft versus hard collision most probably was taken by subjects to correspond to the judgment of no-collision versus collision, given current deceleration. (4) For reasons of convenience we refer to t _ as a flow-field variable that is specific to the rate of change of any relevant first-order time-to-contact (ie time until a gap D will be closed if current closure rate is maintained) without worrying, for the present purposes, about the particular (combinations of ) optical angles that are involved in the specification of TC1 (D) ÿD=D_ and, a fortiori, in the specification of its rate of change.


529

to the presentation of the stimulus objects. In experiments 1a and 2a, observers watched the unfolding of a scenario in the lateral plane that did not directly involve them (ie from a spectator's point of view). In experiments 1b and 2b, the observers watched the approach of/to an object in their frontal plane that would or would not have collided with them (ie from a participant's point of view). 2 Experiment 1: Constant deceleration In experiment 1, observers were shown simulated constant-deceleration approaches, either between two objects (experiment 1a) or between the observer and an object (experiment 1b). Thus, experiment 1b is similar to the experiments reported by Andersen et al (1999). Apart from the comparison between approaches involving the observer as a spectator and approaches involving the observer as a participant, the experimental setup allowed for the assessment of the role played by different variables (event duration, initial and final approach velocity, as well as particular combinations of distance, velocity, and deceleration) that have been reported to influence judgment behaviour. 2.1 Method 2.1.1 Participants. Sixteen volunteers, ranging in age from 22 to 42 years, participated in experiments 1a and 1b. All reported normal or corrected-to-normal vision. 2.1.2 Task and procedure. Visual displays were generated with a Silicon Graphics INDY 4600 XZ system and were presented on a 19 inch monitor operating with a refresh rate of 100 Hz. Seated participants viewed the displays in darkened room binocularly from a distance of approximately 40 cm without any physical constraints on head movement being imposed. The visual displays, with an untextured dark-blue background, simulated motion of a white square (30 cm leg length) object moving laterally towards another, stationary white square (experiment 1a) or a white circular (30 cm diameter) object moving in depth towards the observer (experiment 1b).(5) The moving object could start from three different initial positions (19.2, 20.0, and 20.8 m from the target) with three possible initial velocities (9.9, 10.0, and 10.1 m sÿ1), and decelerated at one of three possible constant rates (2.4, 2.5, or 2.6 m sÿ2 ). Thus, it would either stop before reaching the target (twelve cases, with stopping distances ranging from ÿ1:99 to ÿ0:38 m), upon reaching the target (three cases), or beyond the target (twelve cases, with stopping distances ranging from 0.38 to 2.02 m). When the moving object reached one of two possible distances from the target (2 or 4 m), the displays were truncated and the participant judged, by pressing the appropriate mouse button, whether it would have collided with (`bumped into') the target (stationary square in experiment 1a or the observer in experiment 1b), had it continued to move as it was doing before truncation. In the lateral-motion conditions of experiment 1a, the simulated distance of the observer from the scene was 20 m, thus leading to a constant optical size of the moving object during the entire duration of the scene. Final t_ (that is, at the moment of display termination) varied between ÿ0:75 and ÿ0:54 for the collision events (between ÿ0:75 and ÿ0:58 for the 2 m cutoff and between ÿ0:67 and ÿ0:54 for the 4 m cutoff ) and between ÿ0:45 and 65:33 for the no-collision events (between ÿ0:38 and 65:33 for the 2 m cutoff and between ÿ0:45 and ÿ0:01 for the 4 m cutoff ). In total, 5 out of 54 experimental conditions had final t_ values larger than 0 (cf figure 1).

(5) Pilot studies demonstrated that circular objects gave rise to smoother visual displays for the head-on approach conditions than did square objects, which is why we chose to use circular objects in experiment 1b.

530


Feedback on the correctness of the response was given immediately after each trial, with a green square appearing for a correct response and a red square for an incorrect response. The 27 (36363) different combinations of initial conditions and the two truncation distances gave rise to 54 different trials. In experiment 1a, half the participants were confronted with displays showing the object moving from left to right and the other half with displays showing the object moving from right to left. In both experiments, participants performed two blocks of 54 trials, with conditions being presented in a randomised order within each block. Because experiment 1b was run several months after experiment 1a, the first block of trials served as a practice session in both experiments. 2.2 Results and discussion 2.2.1 Experiment 1a: Lateral motion. Figure 2 (left panel) presents the descriptive statistics (in the form of a box-plot) of the average percentage of correct responses to the total set of stimuli for all subjects. Between-participant performance varied considerably, with the lowest score being 53.7% correct and the highest 85.2% correct, for an overall between-participant mean of 72.6% correct. This performance differed significantly from chance (t15 11:74, p 5 0:0001). There was also a significant difference (t15 2:90, p 5 0:05) in performance on the 2 m truncation trials (76.6% correct, significantly different from chance, t15 9:36, p 5 0:001) and the 4 m truncation trials (68.5% correct, significantly different from chance, t15 10:35, p 5 0:001). Constant deceleration

100

Correct responses=%

90 80 70 60 50 40

lateral

Approach

head-on

Figure 2. Box plots of the percentage of correct responses in the lateral (left panel) and head-on motion (right panel) conditions of experiment 1. The outer limits of the boxes represent the 75th (upper hinge) and the 25th (lower hinge) percentiles, with the line within the box representing the 50th percentile (the median). The outer fences at the end of the lines that extend from the box represent the acceptable outer limits of the range, with the circles representing values falling outside these limits (outliers).

The goal of the present series of experiments was to determine what information is used in judging whether a collision will or will not take place in the nearby future. We expected to find a typical psychophysical function, that is an S-shape form in the data, when the percentage judged as collisions is presented as a function of the informational quantity used for making the judgments. If judgments were based on the information carried in t_ , events with (final) t_ values smaller than ÿ0:5 are expected to be judged as giving rise to collisions, while events with (final) t_ values larger than ÿ0:5 are expected to be judged as not giving rise to collisions. Moreover, values further removed from the critical value of ÿ0:5 are expected to give rise to a larger percentage of correct distinction between collision and no-collision events. On the assumption of a symmetrical distribution, the critical point at which 50% of the trials is judged to be a collision (corresponding to the critical informational value, separating collision from no-collision events) can be obtained by fitting the data


531

with the logistic equation Percentage judged as collisions

100 , 1 expÿkc ÿ x

(1)

where x is the value of the informational quantity under scrutiny. The c value corresponds to the 50% point (ie the critical value) and k relates to the slope of the function at that point. A large value of k indicates that a rapid switch occurred around the critical point from events judged as leading to collision to events judged as not leading to collision. A lower value of k indicates a more gradual shift, reflecting a larger range of values around the critical point for which judgments were relatively uncertain (Bootsma et al 1992). With t_ as the informational quantity used, equation (1) provided a satisfactory fit 2 to the data (r54 0:901), indicating a critical value of t_ of ÿ0:456 with a k slope of ÿ7:956 [corresponding to a slope of 256(ÿ7:956 ÿ199% per unit t_ ; see Bootsma et al (1992)]. The critical point was thus close to the predicted value of ÿ0:5. Moreover, the judgments obtained under the 2 m truncation condition and the judgments obtained 2 under the 4 m truncation conditions fell onto similar curves (with r27 0:910, 2 c ÿ0:438, k ÿ7:714 for the 2 m truncation conditions; and r27 0:898, c ÿ0:467, k ÿ8:044 for the 4 m truncation conditionsösee figure 3a). As can be seen from Truncation 2m 4m

Judged as collisions=%

100 75 50 25 0 ÿ1:0

ÿ0:8

(a)

ÿ0:6

t_

ÿ0:4

ÿ0:2

0.0


100

(b)

75 50 25 0 0.0

0.1

0.2 0.3 0.4 Final velocity=m sÿ1

0.5

0.6


100

(c)

75 50 25 0 1.0

1.5

2.0 2.5 3.0 Event duration=s

3.5

4.0

Figure 3. The percentage judged as collisions as a function of t_ (a), final object velocity (b), and event duration (c) for the lateral approach conditions of experiment 1.

532


figures 3b and 3c, this was not the case when the percentage judged as collisions was plotted as a function of final object velocity or event duration, with both the variables giving rise to two distinct curves. This observation indicates that judgments were not (primarily) based on either final object velocity or event duration. In order to explore the roles played by different variables, multiple regression procedures were used. To simplify the analysis, we replaced the logistic regression by a linear regression that gave satisfactory results for the conditions in which the final value of t_ was smaller than zero (linear regression of the percentage judged as colli2 sions onto final t_ : r49 0:842, with the 50% point located at t_ ÿ0:437). Thus, 5 of the 54 experimental conditions (with final t_ values greater than zero) were excluded from further analysis. Because the rate of change of first-order time to contact (ie t_ ) X_ 2 ÿ 1, t_ reflects a particular combination for the distance variable X is equal to XX= of current distance, velocity, and deceleration. Linear regression of the percentage _ and X thus allows identification of judged as collisions onto (combinations) of X, X, the best predictors of judgment behaviour. Since Andersen et al (1999) reported an effect of initial velocity, we first tested the _ and X. Of all combinaexplanatory power of (combinations of ) initial values of X, X, X_ 2 (ie initial t_ modulated by a constant) provided tions tested, the combination XX= 2 the best fit (r49 0:839), while initial velocity X_ accounted for less than 8% of the 2 variance in judgments (r49 0:076). Similar analyses conducted on the final values _ and X revealed that, once again, the combination XX= X_ 2 of (combinations of ) X, X, 2 (6) explained the largest amount of variance (r49 0:842), with final velocity explaining 2 less than 22% (r49 0:214). Table 1, upper panel, reports the coefficients of correlation (r) and determination (r 2 ) for the simple regression analyses between the percentage judged as collisions and the most important (combinations of ) variables. _ and X demonstrated Stepwise multiple linear regression with (combinations of ) X, X, _ that variables other than t did not explain a sufficient amount of the variance remain X_ 2 was ing to enter into the regression (F 5 4). After the first step in which XX= entered, the variable with the highest partial correlation was X=X_ for an (insufficient) F value of 2.25. 2.2.2 Experiment 1b: Head-on motion. As can be seen from figure 2 (right panel), during head-on motion between-participant performance also varied considerably, with the lowest and the highest scores being 57.4% and 83.3% correct, respectively. The overall performance mean of 67.6% correct differed significantly from chance (t15 10:11, p 5 0:0001). Performance on the 2 m truncation trials (68.3% correct, significantly different from chance, t15 9:57, p 5 0:001) and the 4 m truncation trials (66.9% correct, significantly different from chance, t15 6:65, p 5 0:001) were not significantly different from each other (t15 0:49, ns). Although a within-participant comparison of performance (percentage of correct judgments) for the lateral-motion conditions of experiment 1a and the head-on motion conditions of experiment 1b showed that observers were significantly better in detecting upcoming collision when motion occurred in the lateral plane (experiment 1a) than when motion occurred in the frontal plane (experiment 1b) (t15 2:34, p 5 0:05), one should be very careful in interpreting such comparisons. Even though the distal defini_ and X ) were the same for the lateral and head-on motion tions (in terms of X, X, conditions, the situations differed greatly in terms of the promixal optical patterns. that initial t_ values varied between ÿ0:447 and ÿ0:548, while final (ie at display termination at 2 or 4 m from the target) t_ values varied between ÿ0:007 and ÿ0:751, in accordance with the logic depicted in figure 1. As demonstrated by the regression analysis, with initial and final t_ both explaining a similar amount of variance, the larger t_ differences at the end of the presentations evolved systematically from smaller differences at the start of the presentations. (6) Note


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Table 1. Correlation coefficients (and r 2 between parentheses) of simple linear regressions of the _ and percentage judged as collisions onto selected (combinations of ) final distance X, velocity X, deceleration X for all four experiments. In order to facilitate interpretation, correlations with factors including a velocity term are reported relative to approach velocity (ie ÿX_ ). Thus X is distance remaining, ÿX_ is approach velocity, X is deceleration (of approach velocity), ÿX=X_ is time _ X remaining until distance X would become zero if current velocity is maintained [ie TC1 (X)], ÿX= is time remaining until approach velocity ÿX_ would become zero if current deceleration is _ and XX= X_ 2 is equal to t_ 1. maintained [ie TC1 (X)], X

_ X ÿX=X_ ÿX=

X_ 2 XX=

Approach Truncation

N

1 (constant deceleration)

lateral

49 ÿ0.231 0.463 ÿ0.560 (0.053) (0.214) (0.313) 22 n.a. 0.908 ÿ0.505 (0.825) (0.255) 27 n.a. 0.950 ÿ0.594 (0.902) (0.353)

ÿ0.708 (0.502) ÿ0.920 (0.847) ÿ0.930 (0.864)

0.524 (0.274) 0.905 (0.820) 0.942 (0.887)

ÿ0.917 (0.842) ÿ0.920 (0.847) ÿ0.911 (0.831)

49 ÿ0.646 ÿ0.038 ÿ0.283 (0.417) (0.002) (0.080) 22 n.a. 0.693 ÿ0.054 (0.481) (0.003) 27 n.a 0.888 ÿ0.438 (0.787) (0.192)

ÿ0.895 (0.802) ÿ0.755 (0.569) ÿ0.884 (0.783)

0.010 (0.000) 0.637 (0.405) 0.849 (0.721)

ÿ0.719 (0.517) ÿ0.747 (0.559) ÿ0.863 (0.745)

90 ÿ0.048 0.527 ÿ0.117 (0.002) (0.278) (0.014) 45 n.a. 0.749 ÿ0.043 (0.561) (0.002) 45 n.a. 0.590 ÿ0.191 (0.349) (0.036)

ÿ0.560 (0.313) ÿ0.722 (0.521) ÿ0.588 (0.346)

0.257 (0.066) 0.582 (0.339) 0.170 (0.029)

ÿ0.678 (0.460) ÿ0.796 (0.634) ÿ0.633 (0.401)

90 ÿ0.004 0.448 ÿ0.101 (0.000) (0.200) (0.010) 45 n.a. 0.564 ÿ0.137 (0.318) (0.019) 45 n.a. 0.529 ÿ0.068 (0.280) (0.005)

ÿ0.462 (0.214) ÿ0.590 (0.348) ÿ0.549 (0.302)

0.288 (0.083) 0.367 (0.135) 0.370 (0.137)

ÿ0.538 (0.289) ÿ0.521 (0.272) ÿ0.779 (0.607)

All 2m 4m

head-on

All 2m 4m

2 (sinusoidal deceleration)

lateral

All 2m 4m

head-on

All 2m 4m

X

ÿX_

Experiment

As can be seen from figure 4, the percentage judged as collisions appeared to vary systematically as a function of t_ (figure 4a), while this was not the case for final object velocity (figure 4b) and event duration (figure 4c). To see how performance in the head-on approach events related to the information carried in t_ , we first analysed the results by fitting the percentage judged as collisions to the logistic function of 2 equation (1). Using t_ as the independent variable gave rise to r54 0:575, with the point at which 50% of the trials were judged as collisions located at ÿ0:404 and a slope of the function at that point of ÿ132% per unit t_ (k ÿ5:288). Note the lower percentage of variance explained by t_ in the head-on approach events (57.5%) compared to the percentage of variance explained by t_ in the lateral approach events (90.1%). 2 A separate analysis for the 2 m truncation conditions (r27 0:836, c ÿ0:190, 2 k ÿ4:665) and the 4 m truncation conditions (r27 0:768, c ÿ0:513, k ÿ6:356) revealed that observers judged head-on approach events that continued until the object was close to the observer (2 m truncation) as leading to collision for a critical value of t_ of approximately ÿ0:2, while they adopted the predicted value of ÿ0:5 for approach events that terminated further away (4 m truncation) from the observer. In order to further explore the role of different variables underlying judgment behaviour, we again proceeded with linear regression analyses, excluding the five conditions with positive final t_ values. Linear regression of the percentage judged as

534


Truncation 2m 4m


100 75 50 25 0 ÿ1:0

(a)

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100 75 50 25 0 1.0

(c)

Figure 4. The percentage judged as collisions as a function of t_ (a), final object velocity (b), and event duration (c) for the head-on approach conditions of experiment 1. 1.5

2.0 2.5 3.0 Event duration=s

3.5

4.0

_ and X (see table 1) revealed that the collisions onto (combinations of ) final X, X, 2 2 _ t_ combination XX=X (r49 0:517) explained less of the variance than the combina2 tion X=X_ (r49 0:802) that emerged as the best predictor. Separate analyses conducted on the 2 m truncation and the 4 m truncation conditions demonstrated that in 2 X_ 2 explained slightly less variance than X=X_ (r22 each condition XX= 0:559 and 2 2 2 2 _ r27 0:745 for the XX=X combinations, and r22 0:569 and r27 0:783 for the X=X_ combinations, for the 2 m and the 4 m truncation conditions, respectively). In line with the simple-linear-regression analyses, stepwise multiple linear regression _ and X onto the percentage judged as collisions for the 4 m of (combinations of ) X, X, truncation conditions showed that X_ was entered first. For the second step, the variable _ X for an (insufficient) F value of 1.32. For with the highest partial correlation was X= _ X. the 2 m truncation conditions, X=X_ was entered first, followed by X= Thus, from the logistic and linear regression analyses, it appears that, in the head-on approach events, performance could be described as depending on t_ (albeit with a strong shift in the critical value adopted for the 2 m truncation events) or t (as this is the informational quantity specifying the combination X=X_ ). It should be noted that for the separate analyses of the 2 m and 4 m truncation conditions, where final distance X is constant, final velocity X_ and (first-order) time remaining X=X_ are highly


535

correlated. Inspection of table 1 for the conditions of experiment 1 reveals that, for the separate analyses of 2 m and 4 m truncation conditions, final velocity X_ sometimes correlated slightly more strongly with the percentage judged as collisions than either X_ 2, leading it to enter early on during some of the multiple-regression X=X_ or XX= analyses. However, the correlation of final velocity X_ with the percentage judged as collisions broke down when the full data set (with distance X varying over trials) was X_ 2. analysed (see figures 3b and 4b), while this did not occur for either X=X_ or XX= We conclude that the correlation with final velocity is largely artifactual. In short, at least for the late truncation events, it seems that judgments were influenced by the firstorder time-to-contact remaining at the end of presentation. Note that TC1 (X ) ÿX=X_ was, on average, 0.613 s (range 0.455 to 0.828 s) for the 2 m truncation condition and 0.916 s (range 0.744 to 1.236 s) for the 4 m truncation condition. Taken together, the results of experiments 1a and 1b provide evidence for the position X_ 2 ) are used in the elaboration of a judgment of that both t (ie X=X_ ) and t_ (ie XX= _ upcoming collision. t was found to prevail in the lateral approach events of experiment 1a, where the observer viewed the scene from a spectator's point of view. In the head-on approach events of experiment 1b, where collision involved the observer, t played a slightly stronger role than t_ . One might wonder whether the initial conditions of experiment 1 varied enough to adequately dissociate the variables we compared. From the correlations between the (combinations of ) variables used (see table 2), it appears that some variables correlated to quite some degree with distance X. However, by using both the full set of data as well as the sets of data for the 2 m and 4 m truncation conditions separately, confounding can be avoided [eg between velocity X_ and (first-order) time to contact X=X_ ]. _ X, all other relations Apart from the relatively high correlation between X_ and X= explained less than 50% of the variance in a variable, leaving sufficient possibilities for discrimination with the stepwise regression procedure. _ and Table 2. Correlation coefficients between selected (combinations of ) distance X, velocity X, for the conditions of experiment 1 where approach followed a constant deceleration. deceleration X

X X_ X X=X_ _ X X= X_ 2 XX=

X

X_

±

0.719 ±

X 0.112 ÿ0.289 ±

X=X_

_ X X=

0.813 0.187 0.393 ±

0.659 0.988 ÿ0.430 0.114 ±

X_ 2 XX= 0.172 ÿ0.539 0.580 0.706 ÿ0.597 ±

3 Experiment 2: Sinusoidal deceleration Experiment 1 revealed that both t_ and t were found to influence judgments concerning upcoming collision, with the latter quantity taking on a more prominent role in head-on approaches that remained visible until shortly before (possible) contact. In experiment 2 we examined whether these findings resulted from the particular choice of motion conditions. The use of a sinusoidal deceleration pattern, with motion starting from different initial conditions, allowed us to evaluate the importance of t_ and t information under different conditions. At the same time, the use of a (sinusoidally) changing deceleration provides an interesting test for the t_ hypothesis, as t_ specifies whether a collision will occur provided that current deceleration will not change.

536


3.1 Method 3.1.1 Participants. Twelve volunteers, ranging in age from 18 to 24 years, participated in experiments 2a and 2b. None of them had participated in experiment 1. All reported normal or corrected-to-normal vision. 3.1.2 Task and procedure. The material and apparatus were identical to those used in experiment 1. The only difference was that the approach of the moving object was determined by the sinusoidal function X (A ÿ O) ÿ A sin (2pft), where A is the amplitude of motion (18, 20, 22 m), O is the offset (ÿ1:5, ÿ1:0, 0.0, 0.5, 1.0 m) that determined the location (relative to the target) where the motion would stop, and f is the frequency (0.04, 0.05, 0.06 Hz). Displays showed the object moving for a quarter of a cycle, from the moment of maximal velocity onwards. As was the case in experiment 1, the display terminated when the moving object was 2 or 4 m away from the target. This truncation variable (2) along with the different amplitudes (3), frequencies (3), and offsets (5) yielded 90 different trials in total. In experiment 2a, events were viewed from a spectator's point of view, with half the participants being confronted with displays showing the object moving from left to right and the other half with displays showing the object moving from right to left. In experiment 2b, displays showed head-on approach between the observer and the object. Participants performed two blocks of trials in both experiments, with the first block serving as practice. 3.2 Results and discussion 3.2.1 Experiment 2a: Lateral motion. As in experiment 1, between-participant performance was found to vary considerably, from a minimum of 54.4% correct to a maximum of 82.2% correct (see figure 5, left panel). The overall average of 69.0% correct was significantly different from chance (t11 6:48, p 5 0:001). Performance on the 2 m truncation trials (72.8%, significantly different from chance, t11 5:83, p 5 0:001) was significantly better (t11 2:29, p 5 0:05) than performance on the 4 m truncation trials (65.2%, significantly different from chance, t11 5:60, p 5 0:001). 100

Correct responses=%

90 80 70 60 50 40

lateral

Approach

head-on

Figure 5. Box plots of the percentage of correct responses in the lateral (left panel) and head-on motion (right panel) conditions of experiment 2. The outer limits of the boxes represent the 75th (upper hinge) and the 25th (lower hinge) percentiles, with the line within the box representing the 50th percentile (the median). The outer fences at the end of the lines that extend from the box represent the acceptable outer limits of the range.

Fitting the logistic equation (1) with t_ as the independent variable to the total data 2 2 set resulted in r90 0:517, c ÿ0:564, k ÿ6:056. Separate fits yielded r45 0:663, 2 c ÿ0:487, k ÿ5:071 for the 2 m truncation trials and r45 0:402, c ÿ0:564, k ÿ6:888 for the 4 m truncation trials. As can be seen from figure 6a, a trend for the percentage judged as collisions to vary as a function of t_ was visible in the data,


537


100

(a)

Truncation 2m 4m

80 60 40 20 0 ÿ1:0

ÿ0:8

ÿ0:6

t_

ÿ0:4

ÿ0:2

0.0


100 80 60 40 20 0 0.0

0.1

0.2

(b)

0.3 0.4 Final velocity=m sÿ1

0.5

0.6


100 80 60 40 20 0 1.5

(c)

2.5

3.5 Event duration=s

4.5

5.5

Figure 6. The percentage judged as collisions as a function of t_ (a), final object velocity (b), and event duration (c) for the lateral approach conditions of experiment 2.

although variability was considerably higher than in the lateral-motion conditions of the previous experiments (as also indicated by the lower percentage of variance explained with the logistic fit). Final velocity (figure 6b) and event duration (figure 6c) were clearly not systematically related to the percentage judged as collisions. In line with the procedure followed for experiment 1, we proceeded with linear regression analyses (see table 1) for which we could now keep all data, as (final) t_ varied between ÿ0:745 and ÿ0:026. Simple linear regression on the complete data set resulted 2 in r90 0:460 with a critical t_ value of ÿ0:510. Stepwise multiple linear regression X_ 2 in a first step and X=X_ in a second step. The variable with the highest entered XX= _ X for an (insufficient) F value of 2.35. partial correlation remaining was X=

538


Separate analyses for the 2 m and 4 m truncation conditions showed similar results. 2 X_ 2 was entered first (simple r45 For the 2 m truncation conditions, XX= 0:634) in _ a stepwise multiple linear regression analysis, followed by X. The variable with the _ X for an (insufficient) F value of 3.88. highest remaining partial correlation was X= Stepwise multiple linear regression for the 4 m truncation conditions also entered 2 X_ 2 first (simple r45 _ X. The variable with the highest XX= 0:401), followed by X= remaining partial correlation was X=X_ for an (insufficient) F value of 0.02). 3.2.2 Experiment 2b: Head-on motion. Between-participant performance varied from a minimum of 45.6% correct to a maximum of 77.8% correct (see figure 5, right panel). Overall, performance (65.1% correct) was significantly better than chance (t11 5:51, p 5 0:001). Contrary to what was found in experiment 1, the difference in performance for judging upcoming collision during lateral approach (experiment 2a: 69.0% correct) and during head-on approach (experiment 2b: 65.1% correct) did not prove to be significant (t11 1:25, ns). For the particular conditions of experiment 2b, performance was better (t11 3:61, p 5 0:01) on the 4 m truncation trials (72.6% correct, significantly different from chance, t11 5:82, p 5 0:001) than on the 2 m truncation trials (57.6% correct, significantly different from chance, t11 2:59, p 5 0:05). Fitting the logistic equation (1) with t_ as the independent variable to the total data 2 2 0:292, c ÿ0:423, k ÿ3:028. Separate fits yielded r45 0:265, set resulted in r90 2 c ÿ0:313, k ÿ1:960 for the 2 m truncation trials, and r45 0:627, c ÿ0:511, k ÿ8:720 for the 4 m truncation trials. As is evident from the percentage of explained variance and clearly appears in figure 7a, the percentage judged as collisions varied largely across conditions (especially for the 2 m truncation conditions), although a trend to increase with decreasing t_ was still visible. Final velocity (figure 7b) and event duration (figure 7c) showed no systematic relation to the judgments provided. Linear regression of the percentage judged as collisions onto t_ for the complete data 2 set yielded r90 0:289, for a critical t_ value of ÿ0:400 (see table 1). Stepwise multiple _ X. Separate analyses X_ 2 first, followed by X=X_ and X= linear regression entered XX= 2 for the 2 m truncation conditions revealed an r45 0:272 for XX=X_ 2, while stepwise multiple linear regression entered X=X_ first. The variable with the highest remaining _ X for an (insufficient) F value of 2.61. For the 4 m truncation partial correlation was X= 2 2 _ _ conditions XX=X (simple r45 0:607) was entered first, followed by X and X. Taken together, the results from experiments 2a and 2b corroborate the findings of experiments 1a and 1b with respect to the role played by the information carried in t_ in judging upcoming collision. Notwithstanding the lower amount of variance explained by this variable (and all others) in the 2 m truncation condition of experi X_ 2 ) emerged as the best predictor ment 2b, as compared to the other conditions, t_ (XX= of judgment behaviour in the majority of cases studied. As had been found in experi_ was also found to play a role, especially under the conditions ments 1a and 1b, t (X=X) of head-on approach where motion continued until shortly before (possible) contact. The similarity in the pattern of results obtained in experiments 1 and 2 is particularly interesting in the light of the differences between these experiments. Inspection of table 2 revealed that, for the constant-deceleration approaches of experiment 1, non_ and X. For negligible correlations existed between (combinations of ) variables X, X, the sinusoidally varying decelerations of experiment 2, these inter-variable correlations were, more often than not, considerably smaller (see table 3). As the same pattern of results was obtained in both experiments, we may conclude that this pattern is independent of the particular experimental conditions chosen.


539


100

Truncation 2m 4m

80 60 40 20 0 ÿ1:0

ÿ0:8

ÿ0:6

t_

(a)

ÿ0:4

ÿ0:2

0.0


100 80 60 40 20 0 0.0

0.1


(b)

0.5

0.6


100 80 60 40 20 0 1.5

2.5

3.5

4.5

5.5

Event duration=s (c) Figure 7. The percentage judged as collisions as a function of t_ (a), final object velocity (b), and event duration (c) for the head-on approach conditions of experiment 2.

_ and decelerTable 3. Correlation coefficients between selected (combinations of) distance X, velocity X, ation X for the conditions of experiment 2 where approach followed a sinusoidally varying deceleration.

X X_ X X=X_ _ X X= X_ 2 XX=

X

X_

X

±

0.565 ±

0.172 0.402 ±

X=X_

_ X X=

0.586 0.314 ÿ0.585 ±

0.618 ÿ0.323 ÿ0.672 ÿ0.376 ±

X_ 2 XX= 0.211 ÿ0.620 ÿ0.180 0.383 ÿ0.644 ±

540


4 General discussion Although the ability to judge whether a given situation is likely to lead to a collision in the near future would seem to be essential under many circumstances, there is very little experimental work in which this issue has been addressed directly. Perhaps such a paucity of work is due to the (intuitive) assumption that such judgments would most likely result from strategies combining numerous sources of information, rendering an experimental approach to the subject rather difficult. However, Lee's (1976) identification of an informational quantity (_t) specifying the (in)sufficiency of current deceleration to avoid upcoming collision suggests that action-relevant information might be directly available in the interaction between observer and environment. In the present series of experiments, we analysed the role played by the information carried in t_ in judging whether a collision would occur, varying the characteristics of the approach kinematics (constant and sinusoidal deceleration) and the type of approach viewed (lateral or head-on). In all four experiments reported, participants wereöon averageöable to judge whether the situations presented would or would not lead to a collision. For the present purposes, however, the most interesting data from the present series of experiments are not those related to the correctness of the judgments provided, but those related to the informational basis underlying these judgments. Our analyses indicated that, more often than not, judgments of upcoming collision were based on the perception of a higher-order property of the environment ^ actor system (EAS, Bootsma et al 1997), X_ 2 , characterising the (in)sufficiency of current deceleration to avoid upcoming XX= collision. Although perfect performance on the task would necessarily implicate this characteristic in the constant-deceleration conditions of experiment 1, it is noteworthy that it often came out as the best predictor of judgment behaviour, even though performance was far from perfect. Moreover, under the sinusoidal deceleration conditions X_ 2 was not the task-relevant characteristic, as the requirement of of experiment 2, XX= X_ 2 came constant deceleration was not fulfilled. Yet, even under those conditions, XX= out as the best predictor of judgment behaviour in the majority of cases. Given that t-like variables specify the first-order time remaining until contact ÿX=X_ (Bootsma 1988; Bootsma and Craig 2002; Bootsma and Oudejans 1993; Lee 1976; Kim et al 1998b; X_ 2 ÿ 1, Tresilian 1990), the rate of change of such t-like variables (ie t_ ) specifies XX= that is to say the higher-order property of interest (modified by a constant). Thus, without detailing the optical angles involved in the specification, the present series of experiments corroborates the role played by the information carried in t_ in judging the possibility of upcoming collision. The notable difference in the percentage of the total variance explained by t_ (and other variables) in the lateral versus head-on motion conditions (for the logistic fit, 90.1% versus 57.5% in experiment 1 and 51.7% versus 29.2% in experiment 2) needs to be interpreted with extreme caution. For the lateral approaches, the information with respect to upcoming collision is contained in the pattern of closing of the optical angle formed by the inner edge of the moving object, the point of observation, and the inner edge of the target object. For the head-on approaches, the information with respect to upcoming collision is contained in the pattern of expansion of the optical angle subtended by the contours of the object at the point of observation. Thus, the situations differ both with respect to the optical support (pertinent angles) and with respect to the pattern of change (contraction versus dilation). It might well be that these differences in stimulus characteristics underlie the differences in performance. On the other hand, it could also be the case that the degree of observer involvement (as a spectator in the lateral case and as a `target' in the head-on case) played a non-negligible role, with observers taking a larger safety margin (with respect to the critical t_ value used and/or by judging not with respect to the point of observation, but with respect to a point


541

located somewhat in front of them) in the case of head-on approach. Noting that the issue of the informational power of the optical foundations of judgments cannot be addressed without further experimentation and analysis, we insist on the fact that, overall, the present results allow confirming the role played by t_ in detecting upcoming collision. A similar conclusion was reached by Andersen et al (1999), who reported that, under conditions of constantly decelerating self-motion toward an obstacle, t_ explained some 65% of the variance in the judgments obtained. However, Andersen et al found that other factors, such as approach speed and size of the obstacle also influenced judgment behaviour, leading them to propose a mechanism that involves a comparison between two estimated distances: the perceived distance until a stop, derived from perceived deceleration and velocity; and the perceived current distance, derived from (knowledge of ) object size and visual angle. In line with the results of Andersen et al (1999), our X_ 2 (ie t_ ) was not the only variable underlying judgment analyses indicated that XX= behaviour. X_ 2 always came out as one Inspection of table 1 indicates that, even though XX= of the better indicators of judgment behaviour, other variables should also be considered. The first thing to notice is that, while the task is one of judging sufficiency of itself did not seem to play a significant role. Variables that regularly deceleration, X X_ 2 were correlated with the percentage judged as collisions to the same extent as XX= _ As already noted in the discussion of experiment 1b, X_ can be taken as a X_ and X=X. reasonable predictor of the percentage judged as collisions only when the 2 m and 4 m truncation conditions are regarded separately. With distance X being constant under these conditions, the final velocity X_ and the (first-order) time remaining X=X_ are highly correlated, making it difficult to decide which of the two is the relevant variable used. As the correlation between the percentage judged as collisions and X_ broke down when the full data set was analysed in all four experiments, the conclusion that the correlation with final velocity X_ is artifactual seems warranted. Figures 3b, 4b, 6b, and 7b attest to this. We suggest that the conceptual leap of Andersen et al (1999), who concluded from the fact the velocity influenced judgment behaviour that information with respect to velocity was used in arriving at such judgments, suffers from the same artifact. Unfortunately, Andersen et al (1999) did not consider X=X_ as a potential variable of interest and we cannot, therefore, evaluate the pertinence of our position with respect to that data set. X_ 2 ) is equal to the rate of change Given that t_ (the information specifying XX= _ of t (the information specifying X=X ), the question whether t_ is detected as such, or registered through the detection of the change of t over time is not an innocent one. It is not because a formal analysis of proximal flow fields (optical or other) demonstrates that an informational property (such as t_ ) is specific to an EAS property X_ 2 ) that we may assume that the system is sensitive to the informational (such as XX= quantity in question (Bootsma 1998). Although the use of t-like information in the regulation of movement is subject to debate (see, for instance, Bootsma et al 1997; Tresilian 1997; Wann 1996), the sensitivity of biological systems to the information carried in t (ie first-order time remaining until contact) is undisputed (DeLucia 1991; Schiff and Detwiler 1979; Sun and Frost 1998; Todd 1981; Wang and Frost 1992). Three different experimental observations lead us to believe that t_ might not be detected as such (ie through a special, dedicated mechanism sensitive to this higherorder informational quantityöRuneson 1977). In the first place, Kim et al (1993) report an experiment in which observers were confronted with head-on (constant t_ ) approaches that lasted only two or three frames. Detection of t_ at least requires sensitivity to the rate of change of the rate of change (ie acceleration) of relevant optical angles. Thus, two-frame presentations do not allow detection of t_ , and it should therefore be expected that discriminative judgments

542


of upcoming collision based on the detection of t_ should not be possible under those conditions. Three-frame presentations, on the other hand, contain the minimal information necessary and should therefore be expected to lead to reasonably discriminative judgments, although a large variability might be expected. Inspection of the results obtained (Kim et al 1993, page 186, figure 6) indicates that, contrary to the results obtained in each of the five other experiments reported, reliable judgments did not emerge in either the two-frame or the three-frame presentation conditions. Apparently, longer sequences are required before the visual system can reliably extract the relevant information. In the second place, Yilmaz and Warren (1995) in their study on the active control of deceleration reported that deceleration was not adjusted continuously but rather intermittently. ``Deceleration is adjusted and then kept constant for a plateau period of one-half second or more, as though the observer is adjusting the brake, detecting the optical consequences and then readjusting the brake accordingly'' (page 1010). It is unclear why such deceleration plateaux should occur if observers were able to (continuously) detect the current value of t_ . On the other hand, such plateaux should be expected if t_ is detected over (extended periods of ) time. Finally, the results of the present series of experiments indicate that t plays a nonnegligible role in judging upcoming collision. The EAS property it specifies, first-order _ was among the better predictors of the percentage judged as time remaining ÿX=X, collisions under all experimental conditions. Under our interpretation of the confounding of velocity and (first-order) time remaining, the results of Andersen et al (1999) with respect to the influence of velocity can also be taken to point in this direction. Thus, we propose that observers may not be sensitive to t_ as a continuous secondorder variable. Rather, we suggest that observers continuously monitor the variable t and its evolution over time, leading to a quantal (Yilmaz and Warren 1995) Dt assessment of the EAS property of interest. Under this hypothesis, displays with short (first-order) times remaining at display termination are expected to lead to judgments biased towards upcoming collision, owing to the imminence of collision indicated by t. Obviously, the pertinence of imminent collision will be more pronounced in the case of head-on approach (implying the observer) than in the case of lateral approach. The results of the present series of experiments support such an hypothesis, as do the results of Andersen et al (1999). Obviously, more experimental work is needed before this issue can be definitively settled. By monitoring the evolution over time of the variable t (specific to X=X_ ), observers have access to the information specifying the (in)sufficiency of current deceleration to X_ 2 ). Noting that XX= X_ 2 is mathematically equivalent avoid upcoming collision (XX= _ _ to the ratio of X=X over X=X, a comparison with the type of model proposed by Andersen et al (1999) can be made. These authors suggested that observers might compare two distance estimates, one based on the distance until velocity will become zero and one based on the current distance from the target. Use of t_ information can be interpreted as giving rise to a comparison between two time estimates, one based _ X ) and one based on the on the time remaining until velocity will become zero (X= time remaining until the target will be reached (X=X_ ). Perhaps the most important difference between the two models is that the t_ (or Dt) model does not require explicit detection of the two constituent components, rendering it perhaps more robust than the model proposed by Andersen et al (1999). Acknowledgment. The authors gratefully acknowledge the helpful comments and suggestions of Bill Warren on an earlier draft of this paper.


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