Visual processing of the impending collision of a looming object: Time to collision revisited

Abstract

As an object approaches an observer's eye, the optical variable tau, defined as the inverse relative expansion rate of the object's image on the retina (D. N. Lee, 1976), approximates the time to collision (TTC). Many studies have provided support that human observers use TTC, but evidence for the exclusive use of TTC generated by tau remains inconclusive. In the present study, observers were presented with a visual display of two sequentially approaching objects and asked to compare their TTCs at the moment these objects vanished. Upon dissociating several variables that may have potentially contributed to TTC perception, we found that observers were most sensitive to TTC information when completing the task and less sensitive to non-time variables, such as those that specified distance to collision, speed, and object size. Moreover, when we manipulated presented variables to provide conflicting TTC information, TTC specified by tau was weighted much more than TTC derived from distance and speed. In conclusion, our results suggested that even in the presence of other monocular sources of information, observers still had a greater tendency to specifically use optical tau when making relative TTC judgments.

Introduction

Humans and animals constantly interact with a dynamic environment. These interactions can include an observer moving toward a stationary object, an object moving toward a stationary observer, or a combination of both thereof. Over the past few decades, numerous studies have tried to elucidate which sources of information are used to mediate these types of interactions. Given that the time required for performing an action is biologically constrained and often fixed (i.e., resulting from a sequence of muscle movements), it is possible that humans and animals use predictive timing information specified by visual information to guide their actions. One potential source of information often focused on in the literature is time to collision [alternatively time to contact (TTC)]. TTC is defined as the time remaining before contact between the observer and object and can be derived in several ways. 

For instance, TTC can be computed using the incoming object's distance from the observer [d, or distance to collision (DTC)] divided by the approach speed (v): 

$\mathrm{T}\mathrm{T}\mathrm{C}=\frac{d}{v}.$

(1)

While it is possible that TTC is derived through processing of these variables, studies in the past few decades have focused on alternative perceptual mechanisms. One well-studied source is the optical variable tau (τ), defined as the inverse of the relative rate of expansion of the incoming object's image on the retina (Lee, 1976; Lee et al., 2009). TTC derived from tau is represented as 

$\mathrm{T}\mathrm{T}\mathrm{C}\approx \theta /(\Delta \theta /\Delta t)=\tau ,$

(2)

where θ represents the projected angular size of the approaching object, and Δθ/Δt represents the image's rate of expansion. 

Tau is often regarded as an invariant, and as such, it is veridical and independent of other variables, such as the distance and velocity of the incoming object. Tau theory has argued that a tau strategy is necessary and sufficient to perceive TTC. Compared to the time-consuming and less accurate computation of distance and speed (e.g., Rushton & Duke, 2009), tau provides a more direct and accurate estimate of TTC (Hecht & Savelsbergh, 2004). The capability to perceive tau, as such, would be especially useful in situations that require immediate and accurate responses (e.g., playing table tennis; Bootsma & van Wieringen, 1990). 

Following the initial proposal (Lee, 1976), numerous studies have attempted to validate the use of tau in a variety of visual motor tasks. For instance, early behavioral observations on humans and other animals (e.g., Lee, 1980; Lee & Reddish, 1981; Lee, Young, Reddish, Lough, & Clayton, 1983; Wagner, 1982) claimed that coordinating or executing certain behaviors were most consistent with the use of a tau strategy. In these studies, however, observed activities occurred in natural environments and thus could only be shown to correlate with TTC, which may have been drawn directly from tau or inferred from other variables (e.g., Wann, 1996). 

Some laboratory studies (e.g., Kaiser & Mowafy, 1993; McLeod & Ross, 1983; Regan & Hamstra, 1993; Schiff & Detwiler, 1979; Schiff & Oldak, 1990; Sun & Frost, 1998; Todd, 1981; Wang & Frost, 1992) have suggested that tau is used to perform a variety of TTC estimation tasks. Others (e.g., DeLucia, 1991; DeLucia & Warren, 1994; Heuer, 1993; Kerzel, Hecht, & Kim, 1999; Law et al., 1993; Oberfeld & Hecht, 2008; Rushton & Wann, 1999; Smith, Flach, Dittman, & Stanard, 2001; for comprehensive reviews, see Tresilian, 1999; Wann, 1996), however, have argued that other variables can also impact these estimations. For instance, DeLucia (1991) demonstrated that when viewing two approaching objects simultaneously, observers had a tendency to perceive the larger object as arriving sooner even when the smaller object specified an earlier TTC. This phenomenon, termed the size-arrival effect (SAE), provided evidence that objects' relative sizes can bias TTC perception. 

To evaluate the contribution of individual sources to TTC perception, it is necessary to first dissociate different covarying variables (e.g., TTC, distance, and speed from Equation 1) that may have influenced TTC judgments. Among the studies that investigated the use of tau, two groups of studies deserve special considerations for their methods used in controlling visual information. The first group of studies, conducted by Regan et al. (Gray & Regan, 1998, 2006; Regan & Hamstra, 1993; Regan & Vincent, 1995), devised a novel and systematic approach to isolate tau from other related optical variables. For instance, in an attempt to dissociate tau and rate of expansion at the moment of object presentation (see Equation 2), Regan and Hamstra (1993) created a two-dimensional matrix in which the two variables were systematically varied, one along each dimension, at the start of object trajectory. Cells of the matrix then formed trials using the values of these two variables as parameters. By examining responses to relative judgments of TTC, results showed that the observer was consistent with the use of a tau strategy and that his judgment was independent of rate of expansion. Additionally, the observer was also able to specifically judge rate of expansion, which was done independently of tau. Regan and Hamstra thus concluded that separate and independent systems exist for estimating TTC and rate of expansion. This orthogonal matrix design was replicated and later used to dissociate several other optical variables that may have also influenced TTC estimations (Gray & Regan, 1998, 2006; Regan & Vincent, 1995). 

Regan et al., however, only applied their psychophysical manipulations in situations where image expansion was the sole cue available. It remained uncertain, therefore, whether tau would still be used if other information specifying time, such as distance over speed, was also present. In these situations, both tau and the distance–speed ratio would have provided congruent TTC information. Given that both these sources are veridical, observers could theoretically rely on either or both to accurately estimate TTC. Although some studies have suggested that observers are poor at perceiving distance or speed of motion in depth (e.g., Rushton & Duke, 2009), it is unknown whether tau would still contribute to TTC judgments under conditions containing other sources of information. To resolve this issue, a second group of studies initiated by Savelsbergh et al. (Savelsbergh, Whiting, & Bootsma, 1991; Savelsbergh, Whiting, Pijpers, & Santvoord, 1993; van der Kamp, 1999) directly manipulated object size to create a conflict between TTC specified by tau and that of other sources. In these studies, participants had to grasp an incoming ball whose physical size was covertly manipulated through inflation or deflation during approach. Results showed that observers' responses shifted in the direction predicted by tau. Later quantitative analyses (van der Kamp, 1999; Wann, 1996), however, found that these response shifts were much smaller than predicted by a tau-only strategy. A similar paradigm in an animal target-directed locomotion task also revealed comparable findings (Sun, Carey, & Goodale, 1992). Altogether, these results suggested that tau is likely useful but may not be the only factor influencing TTC perception. 

While observers may not entirely rely on a tau-only strategy (Tresilian, 1999), tau may still contribute to TTC-related tasks (Rushton & Gray, 2006) along with other sources of information. Current research questions therefore were directed to reexamine the contributions of TTC and various non-time variables during TTC estimation. In addition, if TTC was specifically used in this type of task, we wanted to determine how different TTC sources, specifically tau versus distance–speed, contributed. To address these questions, the present study examined participants' performance in a relative TTC judgment task. Observers were presented with two sequential simulations of a spherical object (the target) approaching head-on at eye level. In each approach, the target would vanish en route, and the observer's task was to judge which of these two targets would arrive earlier from the moment of their disappearance. 

Sensitivity to time and non-time variables

As described above, the purpose of the current study was twofold. The first was to examine whether observers were sensitive to various sources of information (time or otherwise) when making relative judgments of TTC. Regan and Hamstra (1993) showed that when only image expansion was presented, TTC specified by tau was especially informative. However, in the real world, seldom is image expansion the only available source of information. The current study therefore investigated more inclusive situations where image cues were presented alongside information that specified distance to collision (DTC) and speed. 

In order to dissociate different variables that may have influenced TTC judgments, we adopted the orthogonal matrix design similar to the one used by Regan and Hamstra (1993). Using this method, we manipulated vanishing TTC and vanishing DTC (“vanishing” refers to at the moment of image disappearance) in an orthogonal fashion (i.e., TTC along rows and DTC along columns; see Table 1a). Even though participants were asked to perform a TTC judgment task, we could examine individual contributions of TTC and DTC by comparing psychometric functions generated by collapsing responses from columns and rows. However, due to the mathematical relationship between TTC, DTC, and speed (see Equation 1), if we varied TTC and DTC along the two dimensions of a single matrix, values for speed would unavoidably be varied along both dimensions. For instance, as shown in the rows of Table 1a, when DTC was held constant, speed covaried with TTC. Consequently, obtained sensitivity to TTC information could have been partially contributed by this speed variation. This problem would have also occurred when evaluating sensitivity to DTC. In both cases, the non-orthogonal speed variable was a potential confound for the analysis of TTC or DTC sensitivity. 

Table 1

 

Two matrices used to dissociate the effects of (a) TTC and DTC or (b) TTC and speed. In (a), TTC values varied along the horizontal dimension but were keep constant along the vertical dimension. DTC values varied along the vertical dimension but were kept constant along the horizontal dimension. Speed values were determined by TTC and DTC and varied along both dimensions. Thus, in this matrix, TTC was orthogonal to DTC, but speed remained unconfounded with both TTC and DTC. In (b), TTC was orthogonal to speed, and as a result, DTC became the confounding variable.

To address this issue, a second complimentary matrix was also used to dissociate sensitivity to vanishing TTC and speed (see Table 1b). All trials generated from these two matrices were then pooled and presented in a randomized sequence in the same experiment. During data analyses, resegregating results from each trial into their respective matrices would allow the comparison of psychometric functions generated by the two arrays to examine if the third non-orthogonal variable contributed to responses. 

Meanwhile, another variable that may have contributed to observer estimations is the physical size of the target. Relative size has been shown to be an important source of information for depth perception (Bruno & Cutting, 1988; Landy, Maloney, Johnston, & Young, 1995) and TTC estimations (e.g., SAE as demonstrated by DeLucia, 1991; DeLucia, Kaiser, Bush, Meyer, & Sweet, 2003; DeLucia & Warren, 1994). As the object's physical size potentially influences the perception of DTC, TTC, and even speed, it was necessary to also dissociate object size from these other variables. We therefore varied object size along a third orthogonal dimension to the existing two, in order to independently evaluate the effects of TTC, DTC, speed, and the object's physical size on TTC perception. This three-dimensional array design was used throughout the entire experiment. 

In this study, the availability of distance information through the presence of ground was manipulated in order to investigate the effects of DTC and speed. When distance information was presented, the target approached in a direction parallel to the ground surface and projected a shadow directly underneath. In this situation, both target and its shadow provided potential depth information. However, as we varied target size between trials, DTC and speed information were most saliently provided by the contrast of the moving object and shadow along the ground, which could then be used to estimate TTCd. Responses in these conditions were then compared to when ground and shadow information were unavailable. 

Tau versus the distance–speed ratio

While systematic variations of movement parameters provided opportunities to examine an observer's sensitivity to certain information (e.g., TTC), results would still have been inconclusive to the effects attributed to tau. It would remain uncertain how different sources of TTC information could be combined for TTC perception. As such, the second purpose of the present study was to dissociate the use of TTC information specified by the image, such as tau, from TTC information specified by the distance–speed ratio. We henceforth refer to the TTC specified by tau (Equation 2) as tau-based TTC (TTCt) and the TTC specified by distance–speed (Equation 1) as distance-based TTC (TTCd). Because both TTCt and TTCd normally coexist and provide congruent time information, we used a method for manipulating in real time the size of the looming object during its approach in order to provide a conflict between these two sources of TTC. 

To achieve this, we used a method similar in principle to previous tau manipulation studies (Savelsbergh et al., 1991, 1993; van der Kamp, 1999) but with several improvements. In previous studies, the physical size of the looming object in tau manipulation conditions changed in a linear fashion as the object approached, either by inflating or deflating. This manipulation resulted in a different image expansion profile (of a non-constant speed) compared to that of non-manipulated objects, thus making it a less than ideal form of tau manipulation. In the present study, however, the approaching target that moved at a constant speed was manipulated in a virtual environment so that the size of the stimulus provided a TTCt that specified a certain time sooner or later than TTCd specified by depth cues (similar to cue-conflict scenarios performed by Heuer, 1993; Rushton & Wann, 1999). This method ensured that the rate of expansion for both manipulated and non-manipulated (control) targets would follow a natural course of image expansion typically experienced by objects moving at constant speed. 

The present study further utilized a perceptual judgment task that asked observers to provide a single response (i.e., which stimulus arrived earlier) to the virtual stimuli. This was in contrast to motor tasks commonly used in previous studies. While utilizing motor tasks can be advantageous in many situations, they also lead to temporally variable responses resulting from the complexities of behavior. For instance, initiation of grasping can range from the moment the approaching object is a short distance away from the hand (followed by a slower motion) to the moment that the object contacts the hand (requiring faster movement). Consequently, motor responses may not be the simplest and most direct indicator of visual perception (Tresilian, 1994). On the other hand, a relative judgment task results in simpler and more consistent responses, especially when the confounding variables are systematically controlled (as in the present study; also see Tresilian, 1995, 1999). A relative judgment task, thus, would have been a more suitable design that reflected the observer's estimation. 

In summary, through systematic variations and well-controlled dissociation of movement parameters, along with the incorporation of cue conflict between the two sources of information that specify TTC, the current study aimed to quantify the relative contributions of tau and other variables such as DTC, speed, and physical size of the incoming object during a TTC judgment task. In Experiment 1, the availability of distance information through the presence of ground was manipulated in order to investigate the effects of DTC and speed. In Experiments 2a–2c and 3, we continued to provide ground depth information but further manipulated the physical size of the object during some of the approaches. This TTCt manipulation led to inconsistencies between TTC specified by tau and TTC specified by the distance–speed ratio. Changes in responses due to these manipulations then allowed us to quantify the extent observers used tau. 

General methods

Apparatus

Each experiment was conducted in a dark room. The virtual scene containing the stimulus was projected onto a film screen through a rear projection system (model: JVC projector DLA-SX21). This display measured 246 × 182 cm and had a resolution and frame rate of 1024 × 768 at 60 Hz. Participants remained stationary and viewed the screen from a distance of 133 cm resulting in a field of view spanning 85.5° × 68.8°. 

Stimulus

In the virtual scene, a simulated red sphere (the target) approached at eye level toward the stationary observer in a trajectory within the sagittal plane parallel to the horizon. This target approached at a constant speed either in front of a uniform gray colored background or in the presence of a black and white ground surface with random dot texture (see Figure 1). In these with-ground situations, the horizon was located directly along the midline of the screen. In addition, the target cast a shadow onto the ground that moved directly underneath the object as it approached. The target's height above the ground (measured from the center) was always 2 m. This scenario thus simulated a shadow that was created from a light source directly above and from an infinite distance, in contrast to typical light sources that are generally specified distances away (Kersten, Knill, Mamassian, & Bulthoff, 1996; Kersten, Mamassian, & Knill, 1997; Mamassian, Knill, & Kersten, 1998). Following a brief viewing time between 1 and 2 s, the target vanished en route at a specified time before contact (time to contact, vanishing TTC) with the observer. 

Figure 1

 

Snapshots of the stimulus in the two different ground conditions in Experiment 1. (a) With-ground condition. (b) Without-ground condition. In the with-ground condition, the target was presented simultaneously with a noise dot textured ground surface and an artificial shadow that was cast directly underneath. The ground and shadow provided observers with additional distance and speed information. In the without-ground condition, only the target was visible to the observer.

It should be noted that two different types of TTC are described in the literature. The first, termed “initial TTC” (Heuer, 1993), is defined as the time duration starting from the object's initial approach until contact with the observer (Heuer, 1993; Regan & Hamstra, 1993). The other, which was here referred to as “vanishing TTC” (also known as “final TTC” in Heuer, 1993 or “extrapolation interval” in Oberfeld & Hecht, 2008), is defined as the time range between object disappearance and the moment the object would have reached the observer (Oberfeld & Hecht, 2008; Schiff & Oldak, 1990). 

Procedure

Each experiment employed a two-alternative forced choice task. Individual trials consisted of two sequential presentations of a single target approach separated by a 500-ms blank-screen interval. The participant's task was to compare from the moment of disappearance the remaining TTC of the first and second approaches and judge which of the two would have arrived sooner under assumption that the target remained moving at the same speed. Upon viewing the second approach, participants responded by pressing a key, after which the next trial began. Following every 72 trials, a 1-min break was given, which could be extended upon request. Prior to beginning the formal experiment, participants did 30 or more practice trials until they were comfortable with the task. Feedback was given on each practice trial to indicate whether responses were correct. No feedback, however, was provided in the formal experiment. Additionally, during practice trials, participants were instructed that the physical size of the target may vary between trials and that the target and shadow, if present, were always vertically aligned. Finally, throughout each experiment, participants were instructed to wear an eye patch covering one eye of their choice. 

Experimental design

As described, we employed a modified orthogonal matrix design similar to the one used by Regan and Hamstra (1993). To dissociate the effects of TTC, DTC, and speed on estimations of target's vanishing TTC, we created two orthogonal matrices, each consisting of a combination of these movement parameters (see Table 1). In addition, we varied the target's physical size along a third orthogonal dimension. Of the two resulting arrays, the first individually varied TTC, DTC, and target size along the three available dimensions (TTC–DTC–size array), whereas the second varied TTC, speed, and target size (TTC–speed–size array). The combination of these two arrays ensured that the effects of TTC, DTC, speed, and target size on TTC estimations could each be examined independently. Meanwhile, viewing duration was randomized between 1 and 2 s. As TTC and DTC were specified for the moment of target disappearance, their values at the start of the trajectory were also varied accordingly. This prevented observers from using unrelated information, such as visible target duration, to perform the present task. 

In each trial, the two target approaches were each assigned as a test or a reference approach, with the presentation order randomly chosen. In the reference approach, movement and physical parameters were held constant at a TTC of 2 s (again, counting from moment of target disappearance), DTC of 20 m (virtual unit, from the position of target disappearance), target approach speed of 10 m/s, and target diameter of 2 m. In the test approach, these parameters were instead chosen randomly without replacement from the stimulus pool containing all combinations from the two orthogonal arrays. An exhaustive set of these combinations composed a block of trials. 

Data analysis

Responses from each participant were separated into two data sets according to the array that each trial was drawn from. In each set, responses were converted to the frequency that the test stimulus was judged to have arrived earlier. The six psychometric functions were generated by collapsing responses along each of the three orthogonal dimensions for each array, consisting of TTC, DTC, and target size for the first array and TTC, speed, and target size for the second array. These psychometric functions were then fitted to a logit model (Cohen, Cohen, West, & Aiken, 2003), which was then used to calculate the relative discrimination threshold (Weber's fraction) and point of subjective equality (PSE). The relative discrimination threshold was defined as (X ₇₅ − X ₂₅) / 2, in which X ₇₅ and X ₂₅ represented the value of the independent variable for which participants had a 75% and 25% chance, respectively, of selecting the test approach as arriving earlier. 

Experiment 1

Rationale

Using the orthogonal array design, Experiment 1 investigated whether participants were sensitive to TTC, DTC, speed, or physical size when judging TTC. We were further interested in whether sensitivity to the different variables would change in the presence of a ground surface and target's shadow, the inclusion of which provided salient motion-in-depth information (Kersten et al., 1996, 1997; Mamassian et al., 1998). 

Methods

Stimulus

Experiment 1 consisted of with-ground and without-ground conditions. As described in the General methods section, the with-ground condition included a black and white ground surface with random dot texture. Additionally, a shadow was projected directly underneath the approaching target. In the without-ground condition, only the approaching target was presented (see Figure 1). 

Experimental design

For the test approaches, each of the four independent variables was varied at six levels (0.4, 0.59, 0.8, 1.25, 1.7, and 2.5) relative to values in the reference approach. As a result, each of the two arrays incorporated a 6 × 6 × 6 design, resulting in 432 (6 × 6 × 6 × 2 arrays) different trials, which contained an exhaustive combination of these movement and physical parameters. Each block of these 432 trials was repeated thrice for both with-ground and without-ground conditions. Blocks containing with-ground trials were completed prior to those containing without-ground trials in order to prevent participants from ignoring ground information once they became accustomed to without-ground scenarios. Each block took an hour to complete and was completed once a day for a total of 6 days. 

Participants

Four university students from McMaster University participated in the study. Participants SS and SB were males; HM and TD were females. All participants had normal or corrected-to-normal visual acuity and were naive to the purpose of the study. Each student was paid for their participation. This research was approved by the Human Ethics Review Board of McMaster University. 

Results

Figure 2 depicts the percentage of response that participant SB chose the test approach as arriving earlier than the reference. Fitted curves of the three independent variables for each array were plotted into one graph. The left and right panels represent results from the TTC–DTC–size and TTC–speed–size arrays, respectively. The top and bottom panels represent results from with-ground and without-ground conditions. 

Figure 2

 

The fitted curves of relative TTC judgments for participant SB in Experiment 1. Each fitted curve was based on the different variables (TTC, DTC, speed, and size) used in the experimental task. Results from the with-ground and without-ground conditions are presented in the top and bottom panels, respectively. Results from the TTC–DTC–size array and TTC–speed–size array are presented in the left and right panels.

For participant SB, fitted curves based on the TTC variable were steep compared to those based on DTC, speed, and size. Weber fractions for TTC ranged from 0.22 to 0.26, whereas those of non-TTC variables were ten or more times greater (between 2.8 and 28.5) and represented by the shallower curves. Furthermore, no observable difference was found between with-ground and without-ground conditions. In particular, Weber fractions for TTC in the with-ground condition (0.26 and 0.23 for the TTC–DTC–size array and TTC–speed–size array, respectively) were similar to those in the without-ground condition (0.26 and 0.22 for the TTC–DTC–size array and TTC–speed–size array, respectively). 

Despite varied sensitivities to each variable, all four participants showed the smallest discrimination thresholds for TTC information. Among observers, participant HM showed some sensitivity to non-TTC variables. Nonetheless, even for HM, discrimination thresholds for non-time variables were approximately two to five times greater than those for TTC. Moreover, this greater sensitivity to non-TTC variables was largely contributed by HM's responses from the first block of the with-ground condition (i.e., DTC was only 1.5 times less sensitive than TTC). In the subsequent two blocks, HM became mostly sensitive to time (i.e., she was more than twice as sensitive to TTC as DTC), which was similar to the other observers. Unlike HM, the other participants showed little response differences between blocks. Weber fractions for all participants are listed in Table 2. 

Table 2

 

List of participants' Weber fractions for the TTC estimation task in Experiment 1 separated by independent variables.

Participant		TTC–DTC–size			TTC–speed–size
		TTC	DTC	Size	TTC	Speed	Size
With-ground	SS	0.32	25.0	11.0	0.34	13.3	5.24
	SB	0.26	3.40	8.05	0.23	5.38	28.5
	HM	0.58	1.43	1.03	0.37	1.82	1.99
	TD	0.32	4.34	14.1	0.42	2.90	43.9
Without-ground	SS	0.36	23.4	5.59	0.41	12.8	20.4
	SB	0.26	2.80	2.94	0.22	2.84	3.26
	HM	0.37	4.56	3.63	0.34	4.28	2.94
	TD	0.29	5.72	15.0	0.26	7.61	7.32

Discussion

In Experiment 1, we investigated the contribution of four variables—TTC, DTC, target speed, and physical size—during a relative TTC judgment task. Our results showed that participants could accurately discriminate trial-to-trial TTC differences based on TTC information. Discrimination thresholds revealed that TTC information was the most effective means for making TTC judgments compared to the other three non-time variables and that results were consistent among all observers. 

Recall that we implemented two stimulus arrays (one for TTC–DTC–size and one for TTC–speed–size) because we were concerned that the third non-orthogonal variable would confound responses for each array. As demonstrated, however, the resulting TTC psychometric functions for both arrays were almost identical. Meanwhile, DTC and speed psychometric functions were both shallow suggesting that observers showed minimal sensitivities to these variables during estimations. It was therefore unlikely that the potential confounding non-orthogonal variable played a major role in the present findings. 

Moreover, there was no noticeable difference between responses in with-ground and without-ground conditions. Only participant HM showed slightly more sensitivity to non-TTC variables in with-ground conditions than without-ground conditions. However, as described, this was due to her tendency to use non-TTC variables during the first block of the experiment. Altogether, our results suggested that, for the most part, depth information provided by the ground surface and target's shadow did not influence TTC estimations. Thus, we concluded that DTC and speed information derived from ground presence affects little, if at all, the perception of TTC. 

Meanwhile, observed Weber fractions for TTC ranged from 0.22 to 0.58, slightly higher than the 0.05 to 0.22 previously reported (Gray & Regan, 2006; Regan & Hamstra, 1993; Todd, 1981). Such discrepancy may be explained by the use of vanishing TTC in the present study and also because targets disappeared much earlier (between 0.8 and 5 s) prior to contact with the observer. This was in contrast to the smaller values of TTC (0.33–1.33 s) used in previous studies. As was demonstrated by Schiff and Oldak (1990), observers had a tendency to less accurately estimate approaches at longer TTCs, which would have explained the reduced sensitivity found in our results. 

Furthermore, while we demonstrated that tau information was primarily used in our TTC judgment task, we also found that relative size contributed little to these estimations. This was in contrast to reports of the SAE by DeLucia et al. (DeLucia, 1991; DeLucia & Warren, 1994) and may have been due to several reasons. First, in the present study, instructions were provided prior to formal testing about the potential change in size between approaches. This may have directed participants to use information other than targets' relative sizes to make their estimations. In addition, target size between trials varied from 0.4 to 2.5 times the reference target, which ensured that participants would have noticed these differences. It is likely that when participants were aware of such variations, they abandoned a prior assumption that larger image sizes specified a closer distance and thus a shorter TTC. This would be consistent with observed SAE reductions when participants were made aware that approaching objects differed in physical size (DeLucia, 2005). 

It is also important to note that the shadow of the target in our study was programmed to be positioned directly underneath the target (i.e., created from a light source infinitely far away). As a result, motion information (i.e., TTC, DTC, speed) provided by the target and its shadow were consistent. As demonstrated by DeLucia (1991), the presence of prominent ground information also weakens the SAE. This shadow may have reduced the chances that participants used the less reliable relative size cue by providing necessary ground intercept and positional information. 

Lastly, it remained possible that the order of the group conditions may have also played a role in participants' judgments. Changes in target size may have been more noticeable in the with-ground condition due to distance–size scaling. As the without-ground condition was always presented after, observers may have continued to ignore target size changes due to this previous exposure. In other words, strategies used in the with-ground condition may have transferred to the without-ground condition. 

Experiment 2a

Rationale

Experiment 1 found that participants were most sensitive to TTC information and less sensitive to non-time variables when judging relative TTC. However, it remained uncertain whether tau information was specifically and exclusively used for these estimations. Our results are still inconclusive as to whether participants ignored the distance–speed ratio. In principle, as the two possible means for deriving TTC (TTCt and TTCd) provided identical TTC values, the orthogonal array design cannot, by itself, unconfound them. In order to dissociate TTC derived from the image (TTCt) from TTC derived from distance (TTCd), we created a situation where TTCt and TTCd were incongruent. To do this, we manipulated in real time the target's tau by changing the physical size of the approaching object. Consequently, TTCt of the target specified a time sooner or later than the TTCd, which remained unaltered. 

Methods

Stimulus

The stimulus in the with-ground condition was used for Experiment 2a along with the addition of the three different types of TTCt manipulations. In the control condition, for a given approach, the target's physical size was unchanged (the same as in Experiment 1) and therefore provided consistent TTCt and TTCd information. In expansion and contraction conditions, however, the target's physical size was constantly manipulated (either expanded or contracted) so that tau (TTCt) specified a time 0.5 s less (expansion) or greater (contraction) than TTC specified by distance–speed (TTCd). Importantly, the distance of the target and shadow remained unmanipulated, similar to control conditions (see Figure 3). 

Figure 3

 

Illustration of the TTCt manipulation used in the control and expansion condition in Experiments 2a–2c and 3. This diagram illustrated the visual image of the target as it approached the observer's eye at speed v. For control condition, the image of the target was depicted as O _c (black solid circle) with the physical size of the target held constant at S ₀ during approach. For the expansion condition, the physical size of the target was enlarged (O _e, red dashed circle) during the approach and the magnitude of the increase was made to simulate the image of the target (O _i, red dotted circle) with constant size of S ₀ (same as the control), moving at a distance (v * Δt) in front of the actual object. During TTCt manipulations, the shadow of the target (P _o) was made to be the same size (diameter) and at the same position as the manipulated target (O _e).

The target's instantaneous size (S′) was calculated using the following equation: 

$S^{\prime }=S_0*\frac{d}{d-v*\Delta t},$

(3)

where d represents the target's DTC, v represents its speed, and S ₀ represents its physical size at a specific time. The absolute values of Δt were 0 s, 0.5 s, or −0.5 s for control, expansion, and contraction conditions, respectively. 

It should be noted that these three TTCt manipulating conditions (constant, expansion, and contraction) were only applied to the test approach and that each trial also consisted of an unmanipulated reference approach. As the reference approach was kept at a TTC (for both TTCt and TTCd) of 2 s, the relative value of Δt was expressed as 0 (0-s TTCt shift/2-s TTCt), 0.25 (0.5-s TTCt shift/2-s TTCt), and −0.25 (−0.5-s TTCt shift/2-s TTCt). 

Experimental design

The orthogonal array design (as in Experiment 1) was again used. However, in Experiment 2a, while the levels of TTC variation remained the same, the levels of DTC, speed, and target size were reduced to four. As a result, each new array contained 96 (6 × 4 × 4) combinations of parameters. These four levels corresponded to values of 0.5, 0.7, 1.4, and 2.0 relative to reference values. Each trial from the TTC–DTC–size and TTC–speed–size arrays was presented once for each TTCt manipulation condition (expansion, control, and contraction). Therefore, each block contained 576 trials (6 × 4 × 4 × 2 arrays × 3 manipulations) presented in random order, which was completed thrice for a total of 1728 trials per participant. 

Participants

Participants JY (the first author, male with normal visual acuity), SB, HM, and TD completed Experiment 2a. SB, HM, and TD each completed Experiment 1 prior. 

Data analysis

Responses were first grouped according to their TTCt manipulation condition. Within each group, similar analyses as those performed in Experiment 1 were used. PSEs in the different TTCt manipulation groups were then compared to investigate the effects of tau manipulation on TTC estimations. 

Results

Figure 4 depicts the percentage of responses that the same male participant SB chose the test approach as arriving sooner than the reference. To compare the effects of TTCt manipulation on each variable, fitted curves for the three different TTCt manipulation conditions were plotted in the same graph. The left and right panels represent results from the TTC–DTC–size and TTC–speed–size arrays, respectively. Responses based on TTC, DTC, speed, and size variables are presented in the top, middle left, middle right, and bottom panels. 

Figure 4

 

The fitted curves of relative TTC judgments for participant SB in Experiment 2a. Each fitted curve was based on the different variables (TTC, DTC, speed, and size) used in the experimental task. To compare the effects of TTCt manipulation, responses based on each variable were further separated by manipulation conditions and plotted into the same figure. Left and right panels depict results from the TTC–DTC–size array and TTC–speed–size arrays, respectively.

In general, curves based on the TTC variable for all three TTCt manipulation conditions were much steeper than those based on DTC, speed, and size. For instance, as shown in Table 3, discrimination thresholds based on the TTC variable in control conditions for participant SB were 0.21 for the TTC–DTC–size array and 0.16 for the TTC–speed–size array, similar to the corresponding 0.26 and 0.23 values obtained in Experiment 1. In addition, like Experiment 1, Weber fractions of the other non-time variables were nearly ten times greater than those for TTC (ranging from 3.63 to 22.6). Weber fractions for all three TTCt manipulation conditions showed similar patterns of sensitivity (see Table 3 for Weber fractions and PSEs for all participants). 

Table 3

 

List of participants' Weber fractions and PSEs for judging TTC in Experiment 2a, separated by each independent variable and TTCt manipulation condition.

	Participant	TTC–DTC–size				TTC–speed–size
			Expansion	Control	Contraction		Expansion	Control	Contraction
Weber fraction	JY	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	0.27	0.26	0.30	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	0.29	0.25	0.25
	SB		0.27	0.21	0.32		0.21	0.16	0.19
	HM		0.29	0.28	0.40		0.21	0.19	0.23
	TD		0.23	0.24	0.35		0.24	0.29	0.32
	JY	D ⁢ T C t ⁢ e ⁢ s ⁢ t D ⁢ T C r ⁢ e ⁢ f .	5.62	3.73	6.65	S ⁢ p ⁢ e ⁢ e d t ⁢ e ⁢ s ⁢ t S ⁢ p ⁢ e ⁢ e d r ⁢ e ⁢ f .	7.73	5.99	10.7
	SB		3.07	22.6	3.94		3.46	5.56	3.16
	HM		2.34	2.24	2.04		7.49	56.5	5.94
	TD		20.4	6.75	7.76		9.45	2.86	1.64
	JY	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	13.1	9.38	2.10	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	2.08	4.56	6.36
	SB		45.2	3.63	2.97		11.3	4.05	3.60
	HM		3.04	5.66	1.86		3.25	7.63	3.40
	TD		34.0	21.3	4.08		1876	13.4	5.00
PSE	JY	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	1.41	1.11	0.89	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	1.30	1.20	0.86
	SB		1.46	1.12	0.97		1.35	1.08	0.93
	HM		1.51	1.24	0.89		1.36	1.13	0.99
	TD		1.36	1.15	0.90		1.37	1.28	0.97
	JY	D ⁢ T C t ⁢ e ⁢ s ⁢ t D ⁢ T C r ⁢ e ⁢ f .	1.85	1.34	1.60	S ⁢ p ⁢ e ⁢ e d t ⁢ e ⁢ s ⁢ t S ⁢ p ⁢ e ⁢ e d r ⁢ e ⁢ f .	0.64	0.01	0.65
	SB		1.75	3.16	1.86		1.38	1.44	1.72
	HM		1.76	1.68	1.21		1.73	7.59	2.67
	TD		0.13	0.30	1.65		0.44	0.34	0.85
	JY	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	2.78	0.71	1.01	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	1.30	0.29	0.85
	SB		7.44	0.83	0.62		1.87	0.95	5.24
	HM		0.37	0.14	1.11		0.91	0.29	0.28
	TD		3.31	3.86	0.90		143	2.60	2.05

For SB, PSEs for expansion, control, and contraction conditions were 1.46, 1.12, and 0.97 in the TTC–DTC–size array. PSE differences from control to expansion and control to contraction conditions were 0.34 and −0.15, respectively. These values corresponded to 136% and 60% shifts, relative to the expected value of 0.25 and −0.25 had the participant relied completely on tau for his estimation. From contraction to expansion, participant SB's PSE shift was 0.49 or 98% of the expected value of 0.5. In the TTC–speed–size array, the participant's PSEs for expansion, control, and contraction conditions were 1.35, 1.08, and 0.93. PSE differences from control to expansion and control to contraction conditions were 0.27 and −0.15, corresponding to relative values of 108% and 60%. From contraction to expansion, the PSE shift was 0.42, a relative value of 84%. 

ANOVA analyses of mean PSE shifts for all participants indicated that there were significant PSE differences between manipulation conditions, with F(2, 6) = 114, p < 0.01 for the TTC–DTC–size array and F(2, 6) = 53, p < 0.01 for the TTC–speed–size array. Moreover, PSE shifts were significant between TTCt manipulation conditions and control for both arrays combined, t(7) = 7.25, p < 0.01 for expansion and t(7) = 8.01, p < 0.01 for contraction. We further compared observed shifts (i.e., expansion to control, control to contraction, and expansion to contraction) with their expected shifts (0.25, 0.25, and 0.5, respectively) for all four participants and found no significant differences except in the expansion to contraction condition for the TTC–speed–size array (t(3) = −5.95, p < 0.01). Meanwhile, there were no DTC, speed, or size differences between conditions (all p > 0.1). Ratios of participants' TTC shifts relative to their expected shifts are listed in Table 4. 

Table 4

 

List of participants' relative shifts in point of subject equality (PSE) between different TTCt manipulation conditions in Experiment 2a. Values are expressed as percentage to the expected shift had participants relied entirely on a tau strategy (0.25 for both Expansion–Control and Contraction–Control conditions and 0.50 for Expansion–Contraction condition).

Participant	TTC–DTC–size			TTC–speed–size
	Expansion to control	Contraction to control	Expansion to contraction	Expansion to control	Contraction to control	Expansion to contraction
JY	120%	88%	104%	40%	136%	88%
SB	136%	60%	98%	108%	60%	84%
HM	108%	140%	124%	92%	56%	74%
TD	84%	100%	92%	36%	124%	80%

Discussion

Similar to Experiment 1, Experiment 2a showed that participants were most sensitive to trial-to-trial differences in TTC during a TTC estimation task and not sensitive to variations of DTC, speed, and target size. Weber fractions of curves based on TTC information were less than those produced by other variables. Additionally, we dissociated the effects of TTCt from TTCd. If observers relied on tau (TTCt) to guide their judgments, responses would have differed between the three TTCt manipulation conditions. Responses, on the other hand, would have remained unchanged if observers relied instead on TTC specified by distance (TTCd). Our results showed that responses were mostly affected by tau and that manipulating TTCt reliably influenced TTC judgments. Specifically, enlarging the physical size of the target during approach caused observers to perceive a sooner arriving object. In contrast, decreasing the physical size of the target during approach caused participants to view the object as arriving later. The extent that individuals utilized tau, however, varied. While the present experiment only examined four observers, it was demonstrated that, at least in the TTC–speed–size array, participants did not fully shift their responses to the extent we would expect had observers only used tau. 

Experiment 2b

Rationale

It is uncertain whether the lack of sensitivity for non-TTC variables found in Experiment 2a was because participants ignored non-time variables when making relative TTC judgments or because participants were unable to detect trial-to-trial differences in these variables. Furthermore, it was necessary to determine whether manipulation of physical size during approach also affected observers' perception of DTC and speed. We addressed these issues in Experiments 2b and 2c by testing participants' ability to make relative DTC and speed judgments during the three TTCt manipulation conditions. If participants showed high sensitivity to DTC in Experiment 2b, and to speed in Experiment 2c, it would confirm that participants simply weighted non-time variables less during TTC estimations. 

Methods

Experimental design

Experiment 2b was similar to Experiment 2a with two exceptions. First, in Experiment 2b, the orthogonal dimensions of the two arrays were changed to DTC–TTC–size and DTC–speed–size in order to dissociate the effects of DTC from other variables during the DTC judgment task. Second, six levels of DTC were used (0.4, 0.59, 0.8, 1.25, 1.7, and 2.5 relative to reference values) along with four levels of all other orthogonal variables (relative values of 0.5, 0.7, 1.43, and 2). Due to high accuracy shown in pilot studies, participants were only required to complete one block (6 × 4 × 4 × 2 arrays × 3 TTCt manipulations = 576 trials) of trials presented in random order. 

Procedure and data analysis

These were similar to the ones used in Experiment 2a. However, unlike Experiment 2a where participants were asked to judge relative TTC, they were instead instructed to judge which of the two approaches appeared closer in terms of DTC at the moment of disappearance. 

Participants

Participants JY, SB, and TD completed Experiment 2b. Participant JY only completed Experiment 2a prior to Experiment 2b, whereas participants SB and TD completed both Experiments 1 and 2a prior. 

Results

Figure 5 depicts the percentage of response that participant SB chose the test stimulus as disappearing closer than the reference. The left and right panels represent results from the DTC–TTC–size and DTC–speed–size arrays, respectively. Responses based on DTC, TTC, speed, and size variables are shown in the top, middle left, middle right, and bottom panels. 

Figure 5

 

The fitted curves of relative DTC judgments for participant SB in Experiment 2b. Each fitted curve was based on the different variables (DTC, TTC, speed, and size) used in the experimental task. To compare the effects of TTCt manipulation, responses based on each variable were further separated by manipulation conditions and plotted into the same figure. Left and right panels depict results from the DTC–TTC–size array and DTC–speed–size arrays, respectively.

For participant SB, fitted curves based on DTC were steeper compared to those based on TTC, speed, and size. Weber fractions of DTC ranged from 0.1 to 0.19, the smallest among all variables. These discrimination thresholds demonstrated that SB was most sensitive to DTC when making judgments of distance. For the same observer, the Weber fractions of other variables ranged from 1.67 to 107, which are represented by the shallow curves in Figure 5. 

For the same participant, we also compared the effects of TTCt manipulation on DTC estimations. From control to expansion and control to contraction, the differences in PSEs between DTC curves were 0.04 and −0.04 in the DTC–TTC–size array and 0.01 and −0.05 in the DTC–speed–size array. These values were relatively small compared to the actual manipulated target size difference (0.25). Other participants showed similar patterns of Weber fractions and PSE shifts. 

Experiment 2c

Methods

Experimental design

Experiment 2c was similar to Experiment 2b with two exceptions. First, in Experiment 2c, the orthogonal dimensions of the two arrays were changed to speed–TTC–size and speed–DTC–size in order to dissociate the effects of speed from other variables during the speed judgment task. Second, six levels of speed were used (0.4, 0.59, 0.8, 1.25, 1.7, and 2.5 relative to reference values) along with four levels of all other orthogonal variables (relative values of 0.5, 0.7, 1.43, and 2). Each participant completed three blocks for a total of 1728 trials (6 × 4 × 4 × 2 arrays × 3 TTCt manipulations × 3 blocks). Trials from individual blocks were presented in random order. 

Procedure and data analysis

These were again similar to those in Experiments 2a and 2b. This time, however, participants were asked to judge which one of the two approaches traveled faster in terms of speed. 

Participants

The same participants from Experiment 2b completed Experiment 2c. 

Results

Figure 6 depicts the percentage of response that participant SB chose the test approach as traveling faster than the reference. The left and right panels represent results from the speed–TTC–size and speed–DTC–size arrays, respectively. Responses based on speed, TTC, DTC, and size variables are shown in the top, middle left, middle right, and bottom panels. 

Figure 6

 

The fitted curves of relative speed judgments for participant SB in Experiment 2c. Each fitted curve was based on the different variables (speed, TTC, DTC, and size) used in the experimental task. To compare the effects of TTCt manipulation, responses based on each variable were further separated by manipulation conditions and plotted into the same figure. Left and right panels depict results from the speed–TTC–size array and the speed–DTC–size array, respectively.

As demonstrated in Figure 6, curves based on speed were steep compared to those based on TTC, DTC, and size. Weber fractions of speed ranged from 0.19 to 0.48, the lowest among all other variables. The Weber fractions for other variables ranged from 0.92 to 21.5 and were represented by the shallower curves. When we compared the effects of TTCt manipulation on target speed estimations, differences in speed PSEs from control to expansion and control to contraction were −0.02 and 0.04 in the speed–TTC–size array and −0.05 and −0.01 in the speed–DTC–size array. Results from other participants showed similar patterns of Weber fractions and PSE shifts. 

Discussions

Overall, results in Experiments 2b and 2c indicated that participants could in fact discriminate trial-to-trial differences in DTC and speed. Therefore, the lack of sensitivity to DTC and speed observed in Experiment 2a was not due to the inability to perceive the two variables. We also demonstrated that TTCt manipulations did not noticeably influence DTC and speed perception in the present study, thus confirming the validity of our manipulations. 

Experiment 3

Rationale

In Experiments 2a–2c, the proportion of TTCt manipulation trials (expansion and contraction) was equal to that of control (1:1:1). Given such high frequency of TTCt manipulations, it is possible that participants may have been affected by them and performed the task differently from normal. To investigate this potential issue, we returned to the TTC judgment task. This time, however, the ratio of TTCt manipulation trials to control was lowered to 1:5. Additionally, a larger sample of naive participants was tested for a shorter duration to reduce practice effects from hours of repeated trials. Lastly, to accommodate for time and also to reduce exposure to TTCt manipulations, each participant only experienced either expansion or contraction conditions. These implementations of a larger sample size, less frequent exposure to TTCt manipulation, and shorter test duration allowed us to ultimately examine more natural behavior. 

Methods

Experiment design

Like Experiment 2a, 2 three-dimensional orthogonal arrays (TTC–DTC–size and TTC–speed–size) were used in Experiment 3. However, only four levels of TTC, DTC, speed, and target size were used. Furthermore, unlike Experiment 2a, control trials were only paired with either expansion or contraction trials for any given participant. Each participant was randomly assigned so that half of them completed the expansion and control conditions (Expansion–Control group) and the other half completed the contraction and control conditions (Contraction–Control group). For each group, trials from each array were repeated six times, once with manipulation (either expansion or contraction) and five additional times without (control condition). In total, each participant completed a single block of 768 trials (4 × 4 × 4 × 2 arrays × (1 manipulation + 5 controls)). 

Procedure

The same procedures from Experiment 2a were used in Experiment 3. 

Participants

Forty undergraduate students (13 males and 27 females) participated in this experiment for course credit. All participants had normal or corrected-to-normal visual acuity and were naive to the purpose of the study. 

Results

Results from individual participants were first examined to ensure that observers were able to discriminate at least the largest TTC difference. Results from eight participants (one in Expansion–Control group and seven in Contraction–Control group) were consequently excluded from analyses. Figure 7 depicts, based on TTC information, the percentage of responses that the population of participants chose the test approach as arriving earlier than the reference. For the two different groups of participants, fitted curves from the expansion, contraction, and their corresponding controls (total of four curves) were plotted into the same graph. The left panel and right panel represent results from the TTC–DTC–size and TTC–speed–size arrays, respectively. 

Figure 7

 

The fitted curves of relative TTC judgments for both Expansion–Control and Contraction–Control groups in Experiment 3. Only the fitted curves based on TTC information are presented. To compare the effects of TTCt manipulation, responses from the two participant groups were plotted into one figure. Left and right panels depict results from the TTC–DTC–size array and the TTC–speed–size array, respectively.

We first examined discrimination thresholds for different psychometric functions. Results presented in Table 5 revealed that the Weber fractions for TTC information ranged from 0.48 to 0.64 for the TTC–DTC–size array and 0.39 to 0.44 for the TTC–speed–size array. Statistical analyses revealed that TTC Weber fractions were not significantly different between the Expansion–Control and Contraction–Control groups. Additionally, the Weber fractions of other non-TTC variables were greater, ranging from 1.49 to 7.54 for the TTC–DTC–size array and 2.72 to 16.4 for the TTC–speed–size array. These values demonstrated that participants were at least three times more sensitive to tau than other non-tau variables. 

Table 5

 

List of participants' Weber fractions and PSEs for judging TTC in Experiment 3, separated by each independent variable and TTCt manipulation condition.

		TTC–DTC–size					TTC–speed–size
		Expansion	Control	Control	Contraction		Expansion	Control	Control	Contraction
Weber fraction	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	0.48	0.57	0.57	0.64	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	0.39	0.40	0.44	0.44
	D ⁢ T C t ⁢ e ⁢ s ⁢ t D ⁢ T C r ⁢ e ⁢ f .	1.97	1.80	1.96	1.49	S ⁢ p ⁢ e ⁢ e d t ⁢ e ⁢ s ⁢ t S ⁢ p ⁢ e ⁢ e d r ⁢ e ⁢ f .	3.16	2.72	4.20	7.78
	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	4.21	3.46	5.06	7.54	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	16.4	10.0	15.4	13.9
PSE	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	1.44	1.25	1.21	0.99	T ⁢ T C t ⁢ e ⁢ s ⁢ t T ⁢ T C r ⁢ e ⁢ f .	1.36	1.16	1.15	0.95
	D ⁢ T C t ⁢ e ⁢ s ⁢ t D ⁢ T C r ⁢ e ⁢ f .	1.65	1.40	1.35	1.09	S ⁢ p ⁢ e ⁢ e d t ⁢ e ⁢ s ⁢ t S ⁢ p ⁢ e ⁢ e d r ⁢ e ⁢ f .	1.60	1.30	1.25	0.44
	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	0.13	0.69	0.68	1.49	S ⁢ i ⁢ z e t ⁢ e ⁢ s ⁢ t S ⁢ i ⁢ z e r ⁢ e ⁢ f .	1.13	0.64	0.82	2.44

Next, we compared PSEs across different conditions. In the TTC–DTC–size array, observers in the Expansion–Control group had average PSEs of 1.44 for expansion condition and 1.25 for control. Observers in the Contraction–Control group had average PSEs of 0.99 for contraction condition and 1.21 for control. PSE shifts were 0.19 (76% of the expected) and −0.22 (88% of the expected) for the two groups, respectively. Results from the TTC–speed–size array showed that PSEs for tau were 1.36 and 1.16 for expansion and control conditions and 0.95 and 1.15 for contraction and control. For both groups, these values corresponded to shifts of 0.20 (80% of expected shift). While the PSE shift, compared to no shift, was significant, t(63) = 10.19, p < 0.01 for combined arrays and groups, the observed PSE shift was marginally different from the expected (0.25) shift had observers based their estimation entirely on tau, t(63) = −1.87, p < 0.07. 

Discussion

Although no participant from Experiments 2a–2c reported that they noticed a change in target size during approaches, it remained possible that they may have eventually sensed these changes due to the many hours of engagement and high frequency of manipulations. If so, participants may have adopted various other strategies to judge relative TTC as they may have regarded the target's physical size as unreliable. We demonstrated here, however, that this was not the case. Even for a larger sample of naive participants, reducing the number of manipulations did not greatly change the pattern of results. Results consistently revealed that tau was the most effective and utilized source of information for judging TTC. When tau conflicted with ground-based depth information, these new participants continued to base their judgment to a large extent on tau. Tau by itself, however, could not account for the total difference in response following manipulation (approximately 80%). Statistical analyses showed that this observed shift was different, although only marginally, to the expected 100% shift we would expect if observers had based their judgments entirely on tau. This suggested that observers relied on tau primarily for TTC estimations but also used other variables, albeit to a smaller extent. 

General discussion

In the present study, we used an orthogonal design and cue-conflict paradigm to investigate how participants made relative TTC judgments when non-time variables and different sources of TTC were available. Results from Experiment 1 showed that when judging TTC of an approaching target, participants were most sensitive to TTC and much less sensitive to variations of other non-time variables such as the approaching target's DTC, speed, and physical size. Similar performances in with-ground and without-ground conditions further confirmed that participants relied mostly on TTC information during these estimations. Given that TTC information is used, we were also interested in what source of time information drove this sensitivity. Our results in Experiments 2a–2c and 3 demonstrated that when different sources provided conflicting TTC information, observers were largely influenced by tau and biased their judgments more toward the extent tau was manipulated. Altogether, we conclude that observers primarily used tau when making relative TTC judgments. 

Optical and non-optical variables

Variables that have been previously identified to affect perception of impending collisions can be roughly considered to fall under two categories. The optical category focuses on cues directly generated from the projected retinal image of the approaching object. The non-optical category, however, focuses on the approaching object's physical characteristics and its spatial relationship with the environment. Cues such as tau, angular size, rate of expansion, and binocular disparity information fall under the optical category, whereas variables such as distance, speed, and physical size of the object fall under the non-optical category. Regan et al. (Gray & Regan, 1998; Regan & Hamstra, 1993) previously implemented the orthogonal matrix design in order to dissociate several of the variables within this optical classification. The present study, on the other hand, dissociated several variables from both optical and non-optical categories. Our results suggested that TTC judgments remained “tau-centric” even in the presence of depth information provided by shadow and ground. Furthermore, our findings were for the most part consistent with previous studies that demonstrated little influence of distance and speed in TTC judgment tasks (Gray & Regan, 1999; McLeod & Ross, 1983; Schiff & Detwiler, 1979). 

Tau versus other optical cues during TTC perception

The present study aimed primarily to dissociate between inferring TTC via the distance–speed ratio and directly perceiving TTC from the looming image. In the latter case, however, it remained possible that image size (theta, θ) and rate of expansion (theta prime, θ′) were used instead by observers. While our experimental design was not primarily intended as a means to dissociate between the different types of image-based variables (as performed by Regan & Hamstra, 1993), this was nonetheless an important issue as image size and rate of expansion would have simultaneously covaried when TTC changed. As demonstrated by several studies (Caljouw, van der Kamp, & Savelsbergh, 2004; Hosking & Crassini, 2011; López-Moliner, Field, & Wann, 2007), rate of expansion is an especially potent image variable that may strongly influence perception of TTC. 

To investigate sensitivities to these optical variables, we examined, much like how TTC, DTC, and speed were dissociated, Weber fractions for the TTC judgment task as the contribution of each variable was systematically excluded (see 1). This was performed by analyzing subset of trials in Experiments 1, 2a, and 3, in which values of individual image-based variables were within a small range. Analyses on Experiment 1 revealed that observers were more sensitive to the tau variable and less sensitive to variations of image size and rate of expansion when completing the task. Specifically, removing cues provided by image size (TTC _θ=C ) or rate of expansion (TTC _θ′=C ) did not significantly affect observer performance (Figure A2, 1). On the other hand, when variations of tau were excluded (i.e., θ _τ=C and θ′ _τ=C ), most observers performed much worse. Similar findings were also found through this type of analyses conducted on results from Experiment 2a (see Figures A2C and A2D, 1). 

In Experiment 3, at least two-thirds of observers clearly relied on tau for their estimations. Three of the 32 observers, however, appeared to use rate of expansion instead of tau, while the remaining 10 or so participants had comparable sensitivities to both tau and rate of expansion. As was demonstrated in Experiments 1 and 2, testing duration may have explained why individuals did not always preferentially use tau for their estimates. Experiments 1 and 2a each involved four observers who individually completed more than 6 h of testing. Experiment 3, on the other hand, tested a larger subject pool (n = 32) for a shorter duration (2 h each). Like HM in Experiment 1, prolonged exposure to the stimulus may eventually bias observers' responses more toward tau. Nonetheless, results suggested that, for many individuals, tau was the most useful optical variable for judging TTC and was used much more than image size and rate of expansion. 

The role of shadow in the stimulus

It is important to note that, in the present study, the projected target image was used primarily as a source of tau (thus, TTC) information. The assumptions of tau, along with target size manipulation within and between trials, required that the center of the target be at eye-height level to ensure targets always approached at the same trajectory. Meanwhile, characteristics of the shadow on the ground were used to provide salient distance and speed information for the target. During each trial, however, images of both target and shadow expanded simultaneously. Therefore, it may have been possible that observers perceived different types of tau information (e.g., “local tau” and “global tau”; Tresilian, 1991) from either or both the looming target and expanding shadow. In 2, we compared the accuracy of the various time information specified by both target and shadow. Results demonstrated that tau specified by the target or various parts of the shadow provided similar accuracy (at least for the purposes of our relative judgment task). Thus, it would be difficult to ascertain which source of tau was used by observers to generate TTC information. Nevertheless, the use of either object tau or shadow “taus” would have validated the tau theory, as the purpose of the current study was to dissociate the use of tau-based TTC from distance–speed-based TTC. 

Observers may not only use tau to guide their estimations

While responses to TTCt manipulations demonstrated the use of tau, it does not, however, imply that observers fully and only used tau when performing TTC-related tasks. As mentioned, one observer (HM) among the four in Experiment 1 showed a tendency to use rate of expansion in earlier trials (see 1). Moreover, in Experiment 3, approximately one-third of participants showed some sensitivity to other optical variables. Quantitative analyses on TTCt manipulation conditions in Experiment 3 further showed that observed mean PSE shifts were only about 20% less than the expected shift (though only marginally significant) had observers fully used tau to guide their estimations. Data analysis revealed that this partial shift even existed (significant difference between observed and expected shift) for the subset of observers that primarily used tau. As 1 illustrated, these observers were not sensitive to image size or rate of expansion, and thus, non-tau optical variables likely did not contribute to their estimates. Instead, these observers may have potentially relied on, although to a smaller extent, derived scene variables such as distance or speed when making their judgments. 

Multiple sources of information and the perception of TTC

In the presence of only image-based monocular information, tau may be the most reliable source available for perceiving TTC. Thus, it should be expected that observer responses are influenced entirely by this optical variable (Regan & Hamstra, 1993). In our study, we showed that this was indeed the case, as demonstrated by observers' sensitivity to TTC information in the without-ground condition of Experiment 1. In natural situations, however, observers are presented with many cues that could potentially influence TTC perception. 

Although considerable studies have addressed the question about what sorts of information are detected and used by observers, only a few studies have examined how multiple sources of information are actually utilized to perceive TTC (e.g., DeLucia et al., 2003; Gray & Regan, 1998; Rushton & Wann, 1999). Additionally, these results have been inconsistent in terms of how different variables contribute to TTC perception. Some studies (DeLucia et al., 2003; Gray & Regan, 1998) have suggested that perceiving TTC may be, to some extent, similar to perceiving depth, in that different available cues are averaged together for perception (Bruno & Cutting, 1988; Landy et al., 1995). Others, however, argued that the variety of information involved in TTC perception render a simple average rule unlikely under many situations. 

The present study along with several others (Landy et al., 1995; Tresilian, 1999) suggested that observers may have largely based their responses on the most effective variable, either because of accessibility, reliability, efficiency, or usefulness. For instance, previous demonstrations of the SAE may have been indicative that observers used the most readily accessible, efficient, and useful variable, namely, size differences, to estimate TTC. When the present study, and others (DeLucia, 2005), removed the usefulness of this cue (via changes in target size between trials), effects of relative size were reduced. 

Furthermore, several studies have also demonstrated that depending on the circumstances, observers could change their use of information during TTC-related tasks. For instance, Heuer (1993) investigated the effects of target vergence and size of an approaching object and found that their relative use was dependent on the size of the approaching stimulus. His results showed that, for larger objects, changes in image size resulting from object motion were more influential in affecting observer responses. For smaller objects, however, vergence was much more salient. This change in strategy was interpreted to be because larger objects provided higher quality information (Heuer, 1993; Tresilian, 1994). Therefore, the larger the object, the more image size became a reliable source of TTC information. In contrast, smaller objects provided lower quality image information, and thus, target vergence would have instead served as a better source of TTC. In these situations, it seemed observers directed their response to be more aligned with the higher quality and thus more reliable source of information. 

Similarly, Rushton and Wann (1999) reported a switch in cue weighting between an object's optical looming and binocular information. Their findings suggested that the relative effectiveness of optical looming and binocular disparity cues was determined not only by the object's physical size but also by whichever information specified the earliest arrival. This would certainly be a reasonable feature as any cue that would have indicated a shorter TTC must necessarily be attended to in order to avoid a potentially harmful situation. In this case, it seemed that the use of information was dependent on the urgency and usefulness of available information. 

Our results demonstrated that observers used multiple variables when making TTC estimations. These non-tau variables, however, were used to a much smaller extent than tau. This suggested that tau may have been the most efficient source of information in the present task, likely due to its reliability (as distance information was often limited), accessibility (as it was directly available), or efficiency (obviating the perception of distance and speed). Overall, collective evidence suggested that observers used multiple sources of information in a given situation by changing cue weighting during perception and biasing their response based on the most effective source available. 

Consequently, caution should be taken when generalizing our results. As mentioned, the use of tau is likely dependent on the presence and quality of other sources of information, many of which were made unavailable in the present study. For instance, one such powerful and missing cue was binocular disparity. Research has demonstrated that binocular cues are important sources of TTC information (Gray & Regan, 1998, 2004; Heuer, 1993; Rushton & Wann, 1999), the presence of which has been shown to increase accuracy during TTC judgment tasks (Cavallo & Laurent, 1988) and affect perception of depth. Additionally, familiar size, a potent cue in natural situations, was also made absent and thus prevented prior information from influencing TTC perception. Therefore, it is uncertain whether the demonstrated primary use of tau would remain had binocular disparity or any other powerful cue also been present. Nonetheless, the present study demonstrated that even when provided with distance, speed, and size information, observers to a large extent preferentially used tau to guide their estimates of TTC. 

In summary, our study performed systematic and quantitative examinations on whether observers were sensitive to optical and non-optical variables when judging relative TTC. It was shown that, under the current paradigm, human observers were highly sensitive to TTC and mainly used tau when performing a relative TTC judgment task. However, while our results showed a strong capacity to use tau, we acknowledge that the actual weighting of variables in more natural situations remains to be tested. Finally, the hypothesis that multiple sources of information redistribute their weighting in different tasks and situations warrants further investigation especially in regards to how they integrate and change when observers interact with the physical environment (Tresilian, 1999; Warren, 2006). 

Appendix A

Analysis of image cues from the spherical object

In the present study, as TTC levels varied (which was approximated by tau, τ), image size (theta, θ) and rate of expansion (theta prime, θ′) also simultaneously covaried. In order to investigate which of these three optical variables was used to guide observer estimates of TTC, we derived Weber fractions for the TTC judgment task as we systematically excluded the contribution of each of these three cues. 

The values of θ and θ′ of all trials at the instant of target disappearance can be calculated using the following equations (Sun & Frost, 1998), respectively: 

$\theta \left(t\right)=2\times \arctan \left(\frac{S}{2\times v\times \mathrm{T}\mathrm{T}\mathrm{C}}\right),$

(A1)

 

${\theta }^{\prime }\left(t\right)=\frac{1}{\frac{v}{S}\times \mathrm{T}\mathrm{T}{\mathrm{C}}^2+\frac{S}{4v}},$

(A2)

where S refers to the size of the target sphere and v refers to the target's movement speed. 

As shown in Equations A1 and A2, θ and θ′ can be expressed as the function of TTC and as the function of the ratio of size over speed (S/v). 

Figure A1 illustrates how TTC, speed, and target physical size were manipulated. As described in the methods of Experiment 1, these three variables were orthogonally varied. Consequently, the psychometric function generated from one dimension was unaffected by the other two. 

Figure A1

 

Orthogonal array for the variation of TTC, speed, and physical size of the target object. From this array, selected cells with constant θ (but different TTC) and selected cells with constant TTC (but different θ) could be identified. Consequently, for those cells, variations of TTC and θ were independent.

In Figure A1, the cells on the same vertically oriented plane had the same value of S/v. Consequently, the two vertical planes contained cells at two levels of S/v (0.5 and 1). Meanwhile, the two horizontal planes represented two levels of TTC (0.4 and 0.8). Among the four lines intercepting the two vertical planes and two horizontal planes, based on Equation A1, cells on the two lines labeled as L2 and L3 had the same θ value. We could then imagine a curved plane overlapping these two lines as containing this θ value. When this plane intercepted with many horizontal planes (each with a particular τ), variations of τ at intercepting cells would have been independent of θ. Similarly, when many of those curved planes, each with a particular θ, intercepted with one horizontal plane with constant τ, variations of θ would have been independent of TTC. Although on this TTC–speed–size matrix, variations of θ was not orthogonal to TTC in terms of cell location, the θ variation was nonetheless independent of τ (orthogonal mathematically). Variations of θ′ independent of TTC could also be derived in a similar manner. 

Methods

As observers could theoretically use any combination of τ, θ, or θ′ when all were available, we labeled Weber fractions for psychometric functions varying TTC as TTC_all. These Weber fractions that served as a baseline for comparison were the same as those reported in Experiment 1 and control conditions of Experiments 2a and 3. The TTC-based psychometric function generated from a subset of trials where θ values at the moment of target disappearance were at the same level represented observers' sensitivity excluding the contributions of θ. We labeled Weber fraction derived from this curve as TTC _θ=C . Using this same principle, we also calculated the Weber fraction for the psychometric function varying θ from a subset of trials in which contributions of τ was excluded. This Weber fraction was labeled as θ _τ=C . 

If participants exclusively used τ, but not θ, to perform the given task, then TTC_all and TTC _θ=C would be similarly low and comparable. Meanwhile, θ _τ=C would be expectedly greater because the cue most useful for the task (τ) would not have differed between trials and was thus made uninformative. If, in contrast, participants used θ rather than τ in estimating TTC, then θ _τ=C would be low, while TTC _θ=C would be greater. Similar analyses could also be performed comparing the usage of τ versus θ′. 

Experiment 1

In Experiment 1, the target's image size (θ) at the moment of disappearance ranged from 0.07 to 12.12 (relative to image size in the reference approach). From overall results, we chose a subset of trials in which θ values were similar (within a small range) and obtained a psychometric function and Weber fraction based on variations of TTC. We further examined five other subsets of trials with different intervals of θ values, found their thresholds to be similar, and averaged them. These six intervals were 0.5–0.7, 0.7–0.9, 0.9–1.1, 1.1–1.3, 1.3–1.5, and 1.5–1.7 in relative values (relative to the θ in the reference approach), which in total included 579 trials for each participant (45% of all trials) and, thus, constituted a reasonable estimate of an observer's true performance. Values for θ′ at the moment of target disappearance ranged from 0.03 to 18.82. Again, we obtained and averaged Weber fractions from the same six intervals. These data sets totaled 465 trials for each participant (36% of all trials). 

Lastly, recall that six levels of TTC were used in Experiment 1. For Weber fractions representing the exclusion of τ, we chose the subset of trials where TTC at moment of disappearance was 0.8 (close to the reference). This subset included 216 trials for each participant (17% of all trials). With τ constant, Weber fractions for both theta (θ _τ=C ) and theta prime (θ′ _τ=C ) were obtained and compared with TTC_all to determine participants' sensitivity to these optical variables. 

Experiment 2

Similar analyses were conducted on the control condition of Experiment 2a. However, as the levels of DTC and speed were reduced in Experiments 2a, intervals we chose for analysis were changed accordingly. For subsets excluding θ (TTC _θ=C ), the intervals were 0.2–0.4, 0.4–0.6, 0.6–0.8, 0.8–1, and 1.2–1.4, which in total consisted of 363 trials (62% of overall data). For subsets excluding θ′ (TTC _θ′=C ), the intervals were 0.3–0.5, 0.5–0.7, 0.7–0.9, and 1.1–1.3, which totaled 225 trials (39% of overall trials). Finally, for τ exclusion subsets (θ _τ=C and θ′ _τ=C ), we again chose τ = 0.8, which contained 96 trials (17% of overall), for comparison. 

Experiment 3

Image cue analyses for each observer were again conducted on results from control conditions. Weber fractions were derived, combined, and averaged. For subsets excluding θ (TTC _θ=C ), the intervals were 0.3–0.5, 0.7–0.9, 0.9–1.1, 1.3–1.5, and 1.9–2.1, which in total contained a sample size of 7680 trials (75% of all trials). For subsets excluding θ′ (TTC _θ′=C ), the intervals were 0.3–0.5, 0.5–0.7, 0.9–1.1, 1.3–1.5, and 1.9–2.1, which totaled 7360 trials (72% of all trials). Finally, for τ exclusion subsets (θ _τ=C and θ′ _τ=C ), we chose τ = 1.43, which contained 2560 trials (25% overall), for comparison. 

Results and discussion

For Experiment 1, Weber fractions for TTC containing all image cues (TTC_all) and for those that excluded the effects of different optical variables are depicted in Figure A2. Figure A2A represents the comparison between τ and θ, whereas Figure A2B represents the comparison between τ and θ′. As shown, Weber fractions were similarly low for both TTC_all (all image cues available) and TTC _θ=C (θ excluded) but much greater for θ _τ=C (τ excluded). A similar pattern between τ and θ′ was also observed with exception for participant HM, suggesting that observers had a tendency to use τ over other image cues when completing the TTC judgment task. These patterns of results were similar in Experiment 2a (Figures A2C and A2D). 

Figure A2

 

Individual participant's Weber fractions for image cues in Experiments 1, 2a, and 3 (refer to text for descriptions of terms on the x-axis). The figures on the left (A, C, E) represent comparisons between the effects of tau (τ) and image size (θ). The right figures (B, D, F) represent similar comparisons between the effects of tau and rate of expansion (θ′). In the bottom panels, participants whose Weber fractions for the variation of non-tau variables (i.e., tau was held constant) were at least twice that of TTC_all were plotted in green. Participants whose Weber fractions for non-theta variable (TTC _{θ = C} ) or non-theta prime variable (TTC _{θ′ = C} ) were twice that of TTC_all were plotted in red. Remaining participants that could not be categorized under these two patterns of responses were plotted in gray.

In Experiment 1, participant HM had results atypical of the other three observers, in that she was sensitive to θ′ during the earlier blocks. In blocks 1 to 3, her ratio of θ′ _τ=C over TTC_all were 0.63, 1, and 16.1, respectively. This demonstrated that, as HM progressed to block 3, the pattern of her Weber fractions became more similar to the other participants and that, specifically, her sensitivity to τ became much greater than her sensitivity to θ′. This suggested that HM switched from initially using θ′ to appropriately using τ for the remainder of her participation. Results from the subsequent experiment (Experiment 2a) supported this conclusion as HM's Weber fractions were congruent to her later performance in Experiment 1 and did not change between blocks. 

Figures A2E and A2F depict image cue Weber fractions from the 32 participants in Experiment 3. As shown, when tau was excluded, Weber fractions for most observers increased (21 for θ _τ=C and 17 for θ′ _τ=C , represented by green lines) to at least twice that of corresponding TTC_all, TTC _θ=C , or TTC _θ′=C . When image cues for θ or θ′ were excluded, three participants showed greater Weber fractions (at least twice as much) for TTC _θ=C compared to TTC_all (represented by red lines). Meanwhile, the same three participants showed low Weber fractions for θ′ _τ=C , suggesting that they were sensitive to θ′. Sensitivities for the remaining participants (11 for θ and 12 for θ′), however, seemed to follow diverse patterns (represented by gray lines) and may have been because these individuals combined and varied the use of several variables (including non-image ones such as distance and speed) during their estimations. Overall, our results suggested that for approximately two-thirds of participants, τ rather than θ or θ′ was used to guide TTC judgments. 

Appendix B

Shadow cue analysis

In the present study, observers, in theory, could have potentially based their TTC judgments on different parts of the image. These include TTC that is directly perceived from the looming spherical target and also the various forms of time-to-passage (TTP) information specified by the target's shadow. An illustration of the relationship between these variables is presented in Figure B1. 

Figure B1

 

An illustration of the different stimulus features that might have contributed to the perception of TTC. At the moment of target disappearance, the object's TTC (t), speed (v), and DTC (D) are related by the equation D = v * t. The diameter of the target is S, and the angle subtended on the observer's eye is θ. The vertical displacement from the center of the target to the shadow is h, and the image angle of this displacement as subtended on the eye is φ. The cast shadow on the ground had width S _{(a − b)}, and depth S _{(c − d)}, both equal to S in physical size. Note that points a, b, c, and d are actually on the same plane (as they correspond to the same shadow) but were separated for illustration purpose.

Features of the target's shadow

Along with the target, it is possible that participants relied on the projected shadow image to directly perceive TTP. Due to the shadow's irregular shape and non-collision trajectory, several distinct features may have been useful, such as the width of the shadow, depth of the shadow, and relative displacement between the shadow and target. 

Shadow width

During target approach, image expansion of the shadow's width (through points a and b in Figure B1) on the transverse plane is similar to the image expansion of the target's width. Tau derived from the local image expansion of the shadow's width provides a relatively accurate estimate of TTP and is a form of “local tau” as described by Tresilian (1991). According to the relationships depicted in Figure B1, we can generate equations needed to calculate the angle of the shadow width (θ(t)_width) using 

${\theta \left(t\right)}_{\mathrm{w}\mathrm{i}\mathrm{d}\mathrm{t}\mathrm{h}}=2*\frac{\arctan \left(\frac{S}{2}\right)}{\sqrt{(v*t{)}^2+h^2}},$

(B1)

where variables S, v, t, and h represent the target size (diameter), speed, TTC, and target–shadow displacement (distance between center of the target and shadow), respectively. 

Shadow depth

During target motion, image expansion also occurs for the shadow's depth (through points c and d in Figure B1), which potentially serves as an alternative “local tau” variable specifying TTP. Using Equation B2, we can calculate the angle subtended by shadow depth (θ(t)_depth), despite the projections of points c and d being asymmetrical around the center of the shadow image: 

${\theta \left(t\right)}_{\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{t}\mathrm{h}}=\arctan \left(\frac{h}{v*t-\frac{S}{2}}\right)-\arctan \left(\frac{h}{v*t+\frac{S}{2}}\right).$

(B2)

 

Target–shadow displacement (TSD)

Given that the shadow was cast on the ground, the angle φ (Figure B1) is subtended by the shadow's relative displacement to the center of the target, which also expands during target approach. This expansion, thus, generates a form of “global tau” as described by Tresilian (1991). We can calculate the angle of the TSD (φ(t)_height) using 

${\phi \left(t\right)}_{\mathrm{h}\mathrm{e}\mathrm{i}\mathrm{g}\mathrm{h}\mathrm{t}}=\arctan \left(\frac{h}{v*t}\right).$

(B3)

 

Tau accuracy comparison

Theoretically, tau generated from the above-described features could all have been used to estimate TTC. Using Equations A1 and B1–B3, we calculated the various taus specified by the target and its shadow features and computed relative tau values (tau for the comparison stimulus divided by the reference stimulus in the stimulus pair). Figure B2 illustrates the relative tau values of the three shadow features compared to the tau specified by the target for all trials in Experiment 1. Our results showed that with the exception of shadow-depth tau in a few trials, tau values for all shadow features were similar to the tau values specified by the target. In other words, the small and consistent errors between tau generated by the object and taus generated by the different shadow features made it impossible to identify which tau was actually used by observers during the TTC estimation task. 

Figure B2

 

Scatter plot representing the relationship between the tau values specified by different parts of the shadow and tau specified by the object. Object tau values (relative to the object tau in the reference stimulus) were plotted along the x-axis, while the three types of shadow taus (relative to the corresponding shadow tau in the reference stimulus) were plotted along the y-axis.

This problem, however, could be addressed if a large discrepancy was artificially created between information specified by these different features. Indeed, when comparing object tau and TSD tau, the TTCt manipulation conditions in Experiments 2a and 3 demonstrated that participants' responses shifted to a large extent based on how object tau was manipulated. TSD tau, however, remained unaltered, which suggested that object tau was more likely responsible for TTC estimates, at least more so than tau specified by TSD. 

Acknowledgments

This study was supported by the Natural Science and Engineering Research Council of Canada and National Key Discipline of Basic Psychology at Southwest University, China. 

We would like to thank three anonymous reviewers for their helpful comments on previous versions of this manuscript. 

Commercial relationships: none. 

Correspondence authors: Hong Li and Hong-Jin Sun. 

Emails: [e-mail]; [e-mail]. 

Addresses: School of Psychology, Southwest University, Chongqing 400715, China; Department of Psychology, Neuroscience and Behaviour, McMaster University, Hamilton, Ontario L8S 4K1, Canada.