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.

*d*, or distance to collision (DTC)] divided by the approach speed (

*v*):

*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

*θ*represents the projected angular size of the approaching object, and Δ

*θ*/Δ

*t*represents the image's rate of expansion.

*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.

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 |

*S*′) was calculated using the following equation:

*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 Δ

*t*were 0 s, 0.5 s, or −0.5 s for control, expansion, and contraction conditions, respectively.

*t*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).

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 |

*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.

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% |

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 |

*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.

*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.

_{ 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).

*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.

*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.

*θ*and

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

*S*refers to the size of the target sphere and

*v*refers to the target's movement speed.

*θ*and

*θ*′ can be expressed as the function of TTC and as the function of the ratio of size over speed (

*S*/

*v*).

*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.

*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 }.

*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*′.

*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).

*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.

*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.

*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.

_{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).

*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.

*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.

*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

*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.

*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:

*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