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Article  |   May 2011
The influence of optic expansion rates when judging the relative time to contact of familiar objects
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Journal of Vision May 2011, Vol.11, 20. doi:10.1167/11.6.20
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      Simon G. Hosking, Boris Crassini; The influence of optic expansion rates when judging the relative time to contact of familiar objects. Journal of Vision 2011;11(6):20. doi: 10.1167/11.6.20.

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Abstract

Previous research has shown that size-dependent errors in time-to-contact (TTC) judgments can be attenuated when approaching objects are familiar to the observer and have a known size. We describe two experiments that show that the effect of size on relative TTC judgments can be modeled on observers' reliance on the instantaneous optic expansion rates of the approaching objects. This reliance on optic expansion rates occurred independently of object familiarity and when the actual TTC of the approaching objects was relatively brief or relatively long. However, observers' sensitivity to differences in TTC was improved for familiar objects when TTC was large. These results are consistent with other research showing that optic expansion rate is a critical variable for judging TTC.

Introduction
Successful completion of interceptive actions requires coordination of the behavior of an actor with the motion characteristics of the approaching object. Information about the time to contact (TTC) of the approaching object with the actor is a critical aspect of such coordination. Lee (1976) has shown that a first-order approximation to TTC is specified by the ratio of the visual angle of an approaching object to its rate of change. The theoretical importance of this ratio, which Lee termed tau (τ), is that information about TTC is available directly from optic variables without the need for intermediate calculations of distance or speed. 
Although many TTC tasks can be reliably modeled on the τ hypothesis (e.g., Lee, Lishman, & Thomson, 1982; Lee & Reddish, 1981; Lee, Young, & Rewt, 1992; Savelsbergh, Whiting, & Bootsma, 1991), recent evidence suggests that there are multiple sources of information that may be used as cues for TTC (e.g., see DeLucia, 2004a; DeLucia, Kaiser, Bush, Meyer, & Sweet, 2003; Tresilian, 1995; van der Kamp, Savelsbergh, & Smeets, 1997). These cues mirror those used for static depth perception and include both monocular information sources such as relative size, height in field, occlusion, and motion parallax (e.g., DeLucia et al., 2003) and binocular information sources such as changing disparity (e.g., Regan & Beverley, 1979; Rushton & Wann, 1999). Observers' TTC judgments have also been shown to be influenced by the known size of approaching objects. When approaching objects are familiar to the observer and have a standard and known size, TTC errors are either reduced for standard-sized familiar objects (DeLucia, 2005) or increased when such objects are presented off-sized (Hosking & Crassini, 2003, 2010). López-Moliner, Field, and Wann (2007) propose that observers make use of information about the known size of approaching objects to determine the threshold optic expansion rates (
Θ ˙
) at which interceptive actions should be initiated. 
Recent analyses of interceptive actions appear to be consistent with the proposition that TTC responses are initiated when
Θ ˙
reaches a critical threshold value (Caljouw, van der Kamp, & Savelsbergh, 2004a, 2004b; Michaels, Zeinstra, & Oudejans, 2001; Smith, Flach, Dittman, & Stanard, 2001). For example, Michaels et al. (2001) found that flexion of the elbow to punch a falling ball was initiated when
Θ ˙
reached a critical value of between 0.01 and 0.04 rad/s. However, findings that interceptions of single approaching objects are based on threshold values of
Θ ˙
may not extend to the case when judging the relative TTC of two simultaneously approaching objects. This is because relative TTC tasks only require participants to make categorical judgments of which of two approaching objects will arrive first; there is no requirement for observers to judge the actual TTC of the two objects (see Tresilian, 1995). Given this absence of a timing component when judging relative TTC, it is unlikely that a critical value of
Θ ˙
would be useful. The following experiments aimed to test the proposal that relative TTC judgments are not based on a threshold value of
Θ ˙
but rather are based on a comparison of the instantaneous
Θ ˙
of the two approaching objects such that the object with the larger
Θ ˙
will be judged to arrive first. 
The hypothesis that relative TTC judgments are based on
Θ ˙
also provides an alternative explanation for previously documented effects of object size on TTC judgments. The “size-arrival effect” occurs when a larger object is judged as having an earlier TTC than that of a smaller object (e.g., Caird & Hancock, 1994; DeLucia, 1991; DeLucia & Warren, 1994; Smith et al., 2001). DeLucia (2004a) has proposed that the size-arrival effect is due to observers' reliance on the visual angle (Θ) of approaching objects to infer their distance from the viewpoint. However, relative TTC judgments based on
Θ ˙
would also produce size-dependent TTC errors because under many conditions a larger object will have a greater
Θ ˙
than a smaller object (but see below). DeLucia and her colleagues (e.g., DeLucia, 2004a, 2004b; DeLucia & Novak, 1997; DeLucia & Warren, 1994) have rejected the role of the average rate of change in visual angle (ΔΘ) as a potential determinant of the size-arrival effect. However, it should be noted that average ΔΘ is not the same as instantaneous
Θ ˙
because the former does not take into account the non-linear expansion in visual angle that occurs for approaching objects. 
Consider a task where observers are asked to judge the relative TTC of two balls that are approaching the viewpoint at identical speeds. The critical temporal separation between the two balls in the median plane required to elicit judgments that their TTCs are equivalent will vary depending on whether such judgments are based on τ, Θ, or
Θ ˙
. If observers were able to use a reliable source of TTC information (e.g., τ), then the two objects would be judged as having equal TTCs when there was no separation (i.e., 0 ms) between them. However, the magnitude of the separation between the two objects will increase if observers are assumed to base their relative TTC judgments on Θ or
Θ ˙
. If T c and T s are taken as the absolute TTC of a comparison object of size S c and standard object of size S s, then it can be shown that the critical temporal separation δ (where δ = T cT s) between two approaching objects required for them to subtend equivalent Θ is when 
δ = ( S c S s 1 ) T s ,
(1)
(see 1). 1 also shows that the critical temporal separation between two approaching objects necessary for them to have equal values of
Θ ˙
is when 
δ = ( S c S s 1 ) T s .
(2)
Equations 1 and 2 define the functions shown in Figures 1A and 1B, respectively (substituting the sizes of a tennis ball and football). Note that positive values represent the situation where the larger object is further away from the viewpoint in space/time than the standard object; negative separation values represent the reverse relationship. The functions can be considered as threshold separations that determine which of the two approaching objects will be judged as arriving sooner: separation values above the dotted lines in Figures 1A and 1B would produce judgments that the smaller object will arrive first; separation values below the dotted lines would produce judgments that the larger object will arrive first. 
Figure 1
 
(A) Function showing the critical temporal separation between two approaching objects (a standard, smaller object that is in front and a comparison, larger object that is behind) required to produce equal values of Θ. (B) A similar function showing the critical temporal separation between two approaching objects required to produce equal values of Θ ˙ . The object separations that would result in either “Smaller first” or “Larger first” relative TTC judgements are also shown in (A) and (B).
Figure 1
 
(A) Function showing the critical temporal separation between two approaching objects (a standard, smaller object that is in front and a comparison, larger object that is behind) required to produce equal values of Θ. (B) A similar function showing the critical temporal separation between two approaching objects required to produce equal values of Θ ˙ . The object separations that would result in either “Smaller first” or “Larger first” relative TTC judgements are also shown in (A) and (B).
Experiment 1
Experiment 1 measured relative TTC responses to two approaching objects that were simulated as either ambiguous in identity/familiarity (black textureless spheres) or unambiguous in identity/familiarity (familiar-sized sports balls). The predicted performance for relative TTC judgments based on familiar size information is similar to that predicted by the τ hypothesis; that is, relative TTC judgments for familiar-sized objects should be veridical. The point of difference between the two hypotheses is that while the familiar-size hypothesis predicts relative TTC judgments to be less veridical when objects are of unknown size, the τ hypothesis predicts equally accurate TTC performance irrespective of whether or not the size of an object is known. The Θ and
Θ ˙
hypotheses predict inaccurate TTC judgments irrespective of object familiarity. 
Methods
Participants
Eight students and staff members (5 males and 3 females) from Deakin University with a mean age of 23.5 years took part in the experiment and received $10 for their participation. All participants had normal or corrected-to-normal visual acuity and were naive to the purpose of the experiment. 
Apparatus and stimuli
Data collection took place in a quiet, dimly lit, laboratory on the Waurn Ponds campus of Deakin University. The main source of ambient light came from a lamp located in a corner of the room and also from the computer monitor. Participants made judgments while seated comfortably with their head positioned in a forehead and chin rest at one end of a semicircular viewing tunnel. The viewing tunnel was constructed of black lightproof cloth that excluded all sources of visual information available to participants except that provided by the computer monitor. The monitor was positioned at the other end of the viewing tunnel at a viewing distance of 43.5 cm. At this viewing distance, the computer monitor subtended a visual angle of approximately 49° horizontally and 38° vertically. 
Displays consisted of a textureless blue background with two spherical objects of different size that traveled toward the participants' viewpoint on a linear trajectory. The smaller object had a diameter of 7.0 cm, and the larger object had a diameter of 22.0 cm. The centers of the two objects were in the same horizontal plane as the participants' viewpoint and were presented so that the centers of the objects were 15.0 cm to the left and right of this viewpoint, which coincided with the center of the monitor screen. During data collection, participants viewed the displays monocularly with an eye patch fitted to the non-dominant eye. Displays were generated, display presentation was controlled, and participants' responses were collected, using a Silicon Graphics 230 workstation. The workstation drove a 53-cm Silicon Graphics monitor with a resolution of 1280 pixels (vertically) by 1024 pixels (horizontally) and a refresh rate of 60 Hz. The viewing geometry of the display was defined such that the screen coordinates of the simulations appeared in true perspective. 
Objects were displayed in two different texture-defined object familiarity conditions. In the no-texture condition, both objects appeared as black disks with no surface texture. In the familiar-object condition, the smaller object appeared as a yellow/green tennis ball, and the larger object appeared as a black-and-white football (see Figure 2). Both objects were simulated to approach participants' viewpoint at a constant velocity of 14.4 m/s. In both texture-defined object familiarity conditions, the objects appeared to travel toward the participants' eye plane for 1000 ms, and then both objects disappeared from view. In a further experimental manipulation, the TTC of the smaller standard object when it disappeared from view (hereafter referred to as TTC at disappearance) was systematically varied across six values of 50, 100, 150, 200, 300, and 400 ms. For each of these six TTC at disappearance conditions, the separation between the smaller standard object and the larger comparison object was manipulated by varying the position of the larger object in 11 equal steps ranging from a minimum separation of 0 ms, which occurred for all six TTC at disappearance conditions, to a maximum separation, which varied for each of the six TTC at disappearance conditions and was chosen such that both the smaller standard and larger comparison objects subtended the same Θ. This was achieved by repositioning the larger object in space/time such that it was further from the viewpoint than the smaller object. Table 1 sets out the 11 separation values used for each of the six TTC at disappearance conditions. It should be noted that the minima and maxima of these separation values were selected on the basis of pilot studies that showed a strong bias in observers to judge the larger objects as having an earlier TTC than the smaller objects. That is, observers responded with 0% “Smaller object will arrive first” responses when the separation between the objects was 0 ms (i.e., when there was no separation between the objects) and 100% “Smaller object will arrive first” responses when the maximum separation was used (i.e., when both objects subtended equal Θ). 1 Therefore, the temporal separations shown in Table 1 were chosen to accommodate a strong bias for larger objects and produce responses that could be fit to a 0–100% psychometric curve (see Figure 3A for sample data and fitted curves). Importantly, it was within this range of separations that observers' responses changed from “smaller first” to “larger first” (see bold numbers in Table 1) and, therefore, could be used to provide a measure of the threshold temporal separation. Table 2 shows that the
Θ ˙
of the smaller and larger objects at TTC at disappearance for the minimum and maximum object separation displays were above threshold, which has been taken to be approximately 0.08 deg/s (see Cavallo & Laurent, 1988; Lee, 1976). 
Figure 2
 
Screen shots taken from the last frame in the displays before the two objects disappeared from view in Experiment 1. The left-hand panel shows the black objects in the no-texture condition; the right-hand panel shows the yellow/green tennis ball and black-and-white football in the familiar-object condition. In both conditions, the objects appeared to approach participants against a textureless blue background.
Figure 2
 
Screen shots taken from the last frame in the displays before the two objects disappeared from view in Experiment 1. The left-hand panel shows the black objects in the no-texture condition; the right-hand panel shows the yellow/green tennis ball and black-and-white football in the familiar-object condition. In both conditions, the objects appeared to approach participants against a textureless blue background.
Table 1
 
Actual separations between the smaller standard object and larger comparison object for each of the six TTC at disappearance conditions used in Experiment 1.
Table 1
 
Actual separations between the smaller standard object and larger comparison object for each of the six TTC at disappearance conditions used in Experiment 1.
TTC at disappearance (ms) Actual object separations (ms)
50 0 11 22 33 44 55 66 77 88 99 110
100 0 22 43 66 86 108 129 151 172 194 220
150 0 32 64 96 128 160 192 224 256 288 320
200 0 42 85 127 169 212 254 297 339 381 420
300 0 64 128 192 256 319 383 447 511 575 640
400 0 85 171 256 342 427 513 598 683 769 850
 

Note: Numbers in bold indicate the temporal separations between which it was predicted that participants would switch their judgments from “larger object will arrive first” to “smaller object will arrive first” on the assumption that such judgments are based on Θ ˙ .

Table 2
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 1.
Table 2
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 1.
TTC at disappearance (ms) Smaller object Larger object
Mina Maxb
50 111.41 350.14 34.19
100 27.85 87.54 8.55
150 12.40 38.91 3.96
200 6.96 21.88 2.28
300 3.10 9.73 0.99
400 1.74 5.47 0.56
 

aMinimum temporal separation condition.

 

bMaximum temporal separation condition.

Figure 3
 
(A) Examples of data and fitted curves for a representative participant in the familiar-object condition of Experiment 1. The data show the percentage of “Smaller object will arrive first” responses for each of the six levels of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations for smaller and larger objects in the familiar-object (open circles) and no-texture (filled circles) object familiarity conditions as a function of the TTC at disappearance of the smaller object. Error bars represent one standard error of the mean. The dashed lines represent the predicted separations between the smaller (standard) object and larger (comparison) object used in Experiment 1 required to produce equal τ, Θ, and Θ ˙ .
Figure 3
 
(A) Examples of data and fitted curves for a representative participant in the familiar-object condition of Experiment 1. The data show the percentage of “Smaller object will arrive first” responses for each of the six levels of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations for smaller and larger objects in the familiar-object (open circles) and no-texture (filled circles) object familiarity conditions as a function of the TTC at disappearance of the smaller object. Error bars represent one standard error of the mean. The dashed lines represent the predicted separations between the smaller (standard) object and larger (comparison) object used in Experiment 1 required to produce equal τ, Θ, and Θ ˙ .
Procedure and experimental design
Participants completed one block of 30 practice trials with the no-texture displays, and then completed 12 blocks each of 88 experimental trials in a single session. Each block of experimental trials consisted of eight randomly presented repetitions of the 11 displays consisting of the same TTC at disappearance condition together with the 11 separations of the larger object. On half the trials, the smaller object was presented to the left of participants' viewpoint and the larger object on the right; on the remaining trials, the reverse size–location pairing occurred. This size–location pairing was randomized within blocks. The texture condition of the displays was kept constant throughout each block and the order of blocks was randomized for each participant. 
On each trial, the two objects appeared on the display and immediately began their simulated approach toward the viewpoint. Participants were instructed to look at the display until the two approaching spheres disappeared from view and judge which object would pass their viewpoint first. Participants indicated their relative TTC judgment by pressing a mouse button. Participants could take as long as required to respond on each trial, with a new trial beginning 2 s after a response was made. No feedback was provided. Participants took a 5-min rest period between each block. In summary, the design of Experiment 1 was a two-factor repeated-measures factorial design, with the two factors being TTC at disappearance (six levels: 50, 100, 150, 200, 300, and 400 ms) and texture-defined object identity/familiarity (two levels: no texture and familiar object). 
Results and discussion
For each of the participants, the percentage of “Smaller object will arrive first” responses was calculated for each of the 11 separations between the larger and smaller spheres for each of the 12 experimental conditions. Cumulative Gaussian functions were fitted to each participant's relative TTC judgment data using Psignifit (version 2.5.41) curve fitting software. This implements the maximum likelihood method described by Wichman and Hill (2001a, 2001b). Two measures of performance were derived from these functions: the critical temporal separation between smaller and larger spheres corresponding to the point on the Gaussian functions where 50% of “Smaller object will arrive first” responses were produced and the jnd, defined as
1 2
(T 75T 25), where T 75 and T 25 were the points on the Gaussian functions that produced 75% and 25% of “Smaller object will arrive first” responses, respectively (cf., Regan & Hamstra, 1993). 
The three alternate hypothetical “equivalence” functions are shown in Figure 3B for the smaller standard object and larger comparison object. Inspection of Figure 3B shows that as the TTC at disappearance of the smaller object increases, a greater (positive) temporal separation between the larger and smaller objects is required for them to be judged as arriving at participants' viewpoint at the same time. Inspection of Figure 3B shows that this occurs equally for the no-texture objects and the familiar-object condition. Furthermore, the mean temporal separations between the smaller and larger objects required for them to be judged as having equal relative TTC matched very closely those separations predicted by the
Θ ˙
hypothesis. These critical separation data were analyzed using a linear regression (see Table 3), which revealed that the slope and intercept of the fitted line was not significantly different to that predicted by the
Θ ˙
hypothesis (R 2 = 0.716). 
Table 3
 
Slopes and intercepts of the regression line fitted to the obtained critical temporal separations in Experiments 1 and 2. The predicted slopes according to the τ, Θ, and Θ ˙ hypotheses are also presented.
Table 3
 
Slopes and intercepts of the regression line fitted to the obtained critical temporal separations in Experiments 1 and 2. The predicted slopes according to the τ, Θ, and Θ ˙ hypotheses are also presented.
Obtained Predicted slope
Intercept Slope
M SE M SE τ Θ ˙ Θ
Experiment 1 0.034 0.015 0.705 0.064 0* 2.142* 0.773
Experiment 2 −0.023 0.066 0.702 0.035 0* 2.143* 0.773
 

Note: * denotes a significant difference between predicted and obtained slopes in a two-tailed test with an alpha of 0.001.

Table 4 shows that mean jnds also increased as a function of increasing values of TTC at disappearance and that participants' ability to discriminate relative TTC was not affected by the addition of object familiarity information in the displays. Table 4 also shows that when jnd data were converted to Weber fractions, participants could discriminate approximately 13–18% differences in the relative TTC of the approaching objects. Some of these Weber fractions are larger than the ≤13% Weber fractions for discriminating TTC reported by López-Moliner and Bonnet (2002) and Regan and Hamstra (1993). This finding will be discussed in more detail following the results of the jnd analysis in Experiment 2
Table 4
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
Table 4
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
TTC at disappearance (ms)
50 100 150 200 300 400
jnd (ms) 20 (3) 31 (4) 40 (6) 49 (7) 67 (8) 88 (12)
Weber (%) 17.8 (2.5) 15.9 (1.7) 14.3 (2.2) 14.0 (2.4) 13.0 (1.5) 13.1 (2.0)
 

Note: Numbers in brackets show one standard error of the mean.

The results of Experiment 1 demonstrate that the
Θ ˙
hypothesis is robust for relative TTC judgments under various conditions of TTC at disappearance. Furthermore, TTC errors were not attenuated by object identity/familiarity information. However, there may be necessary conditions for an effect of familiar size/object identity on relative TTC judgments that were not included in Experiment 1. The aim of Experiment 2 was to investigate such conditions. 
Experiment 2
It has been proposed that sources of information other than τ may be used when the TTC of approaching objects is greater than 500 ms (Tresilian, 1995). If this proposition is correct, then it is possible that TTC at disappearance values of longer duration than those used in Experiment 1 are required in order for object familiarity to influence TTC judgments. This proposal was investigated in Experiment 2 by measuring the critical object separations required to produce relative TTC judgments under conditions that replicated two of the TTC at disappearance conditions used in Experiment 1 and two additional TTC at disappearance conditions that were longer than the 500 ms proposed by Tresilian (1995). Furthermore, Experiment 1 did not provide a preview of a real tennis ball and football because it was assumed that participants would have had prior experience with these objects and would, therefore, be familiar with their standard sizes. This assumption was tested in Experiment 2 by showing half of the participants a real tennis and football and telling them that the objects in the familiar-object displays accurately simulated the sizes of the real objects. Experiment 2 also included a check-texture condition to test whether the separations producing equal relative TTC judgments for familiar-sized tennis balls and footballs are only different to check-texture objects that have a moderate level of object familiarity (cf., DeLucia, 2005). If the results of Experiment 1 are robust across the experimental conditions outlined above, then it would be expected that temporal separations producing equal relative TTC judgments would equate to the separations predicted by the
Θ ˙
hypothesis for all levels of object familiarity. 
Methods
Participants
Twenty new participants (8 males and 12 females) were recruited from Deakin University students and staff (mean age = 20.4) and received $20 for taking part. All had normal or corrected-to-normal visual acuity and were naive to the purpose of the experiment. 
Apparatus and stimuli
Data collection was conducted in the same laboratory used for Experiment 1, and the same apparatus was used. Displays consisted of the same objects and backgrounds used in the Experiment 1 displays. However, TTC at disappearance was one of four values; these being: 150, 400, 1000 and 3600 ms. As in Experiment 1, the object separations were varied in equal steps ranging from a minimum separation of 0 ms to a maximum separation, which varied for each of the sets of displays and was chosen such that both smaller and larger objects had the same value of Θ when the objects disappeared. Table 5 sets out the 10 separation values used for each of the four TTC at disappearance conditions used in Experiment 2. Table 6 shows the
Θ ˙
of the smaller and larger objects at TTC at disappearance for the minimum and maximum object separation displays. Examples of response data and fitted curves for each TTC at disappearance condition are shown in Figure 5A
Table 5
 
Actual separations (in ms) between the smaller and larger objects used to produce the displays for each of the four TTC at disappearance conditions used in Experiment 2.
Table 5
 
Actual separations (in ms) between the smaller and larger objects used to produce the displays for each of the four TTC at disappearance conditions used in Experiment 2.
TTC at disappearance (ms) Actual object separations (ms)
150 0 32 64 96 128 160 192 224 256 288
400 0 85 170 256 341 426 511 596 681 767
1000 0 213 426 639 852 1065 1278 1491 1704 1917
3600 0 767 1534 2300 3067 3833 4600 5367 6133 6900
 

Note: Numbers in bold indicate the temporal separations between which it was predicted that participants would switch their judgments from “larger object will arrive first” to “smaller object will arrive first” on the assumption that such judgments are based on Θ ˙ .

Table 6
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 2.
Table 6
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 2.
TTC at disappearance (ms) Smaller object Larger object
Mina Maxb
150 12.38 38.90 4.56
400 4.64 14.59 1.71
1000 1.90 5.94 0.68
3600 0.52 1.62 0.19
 

aMinimum temporal separation condition.

 

bMaximum temporal separation condition.

Objects were displayed with the same texture-defined object familiarity conditions as those used in Experiment 1. An additional check-texture condition that replicated that used by DeLucia (2005) was also used (see Figure 4). In order to replicate the TTC at disappearance conditions used in Experiment 1 and the larger TTC at disappearance values chosen for Experiment 2, it was necessary to have the objects disappear from view at a constant 2.16 m distance from the viewpoint and vary the velocities of the approaching objects to generate the four TTC at disappearance values. These separate velocities were 14.4, 5.4, 2.2, and 0.6 m/s. 
Figure 4
 
Displays and backgrounds used in Experiment 2 for each of the texture-defined object familiarity conditions. (Left) No-texture condition. (Center) Check-texture condition. (Right) Familiar-object condition.
Figure 4
 
Displays and backgrounds used in Experiment 2 for each of the texture-defined object familiarity conditions. (Left) No-texture condition. (Center) Check-texture condition. (Right) Familiar-object condition.
Procedure and experimental design
In a single session, participants completed one block of 30 practice trials selected randomly from the no-texture condition, followed by 12 blocks of 60 experimental trials. Each block of experimental trials consisted of six repetitions of the 10 displays in which object separation was varied. For each block, TTC at disappearance was held constant, and all displays involved the one texture-defined object familiarity condition. The horizontal position of the smaller and larger objects relative to the participants' viewpoint was varied using the same procedure as in Experiment 1. All participants completed the following sequence of groups of four blocks based on texture-defined object familiarity: four blocks involving no-texture displays, four blocks involving check-texture displays, four blocks involving familiar-object displays. The four blocks in each group varied in terms of the TTC at disappearance, and presentation of the four blocks within groups was varied randomly for each participant. 
Before completing the group of four blocks with familiar-object displays, participants in the preview group were shown the real tennis ball and football and told that these were the objects being simulated in the displays. The instructions to participants regarding the procedure for making relative TTC judgments, the time between a participant's response and the onset of a new trial, and rest periods between blocks were the same as in Experiment 1. No feedback was provided. In summary, the design of Experiment 2 was a three-way mixed factorial design. The between-subjects factor was object preview (two levels: preview and no preview), and the two within-subjects factors were TTC at disappearance (four levels: 150, 400, 1000, and 3600 ms) and texture-defined object familiarity (three levels: no texture, check texture, and familiar object). 
Results and discussion
As in Experiment 1, the initial stage of data analysis involved the fitting of cumulative Gaussian functions to each participant's mean percentages of “Smaller object will arrive first” responses. From these individual cumulative Gaussian functions, the critical object separations producing equal relative TTC judgments were calculated. Figure 5B shows that the mean object separations required for them to be judged as having equal relative TTC increased as a function of increasing TTC at disappearance in the displays. This relationship between object separation and TTC at disappearance was the same irrespective of texture-defined object familiarity information. Inspection of Figure 5B shows that the four data points fall on the function representing object separations predicted on the assumption that relative TTC judgments are based on
Θ ˙
. This was confirmed by a linear regression analysis (see Table 3), which found that the fitted line was not significantly different to that expected by the
Θ ˙
hypothesis (R 2 = 0.836). 
Figure 5
 
(A) Examples of data and fitted curves for a representative participant in the familiar-size condition of Experiment 2. The data show the percentage of “Smaller object will arrive first responses” for each level of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations between the smaller (standard) object and larger (comparison) object used in Experiment 2 as a function of texture-defined object familiarity and TTC at disappearance. Standard error bars represent one standard error of the mean. The dashed lines represent the predicted separations required to produce equal relative TTC judgements based on the τ, Θ, and Θ ˙ hypotheses.
Figure 5
 
(A) Examples of data and fitted curves for a representative participant in the familiar-size condition of Experiment 2. The data show the percentage of “Smaller object will arrive first responses” for each level of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations between the smaller (standard) object and larger (comparison) object used in Experiment 2 as a function of texture-defined object familiarity and TTC at disappearance. Standard error bars represent one standard error of the mean. The dashed lines represent the predicted separations required to produce equal relative TTC judgements based on the τ, Θ, and Θ ˙ hypotheses.
Inspection of Table 7 shows that as the TTC at disappearance of the smaller object increases, a greater critical temporal separation between the two objects is required for them to be judged as having different TTCs. This occurs equally for no-texture, check-texture, and familiar-object displays when TTC at disappearance is 150, 400, or 1000 ms. However, when TTC at disappearance was at 3600 ms, the mean jnd for the familiar object was significantly less than that of the no-texture object (p < 0.05). When jnd data were converted to Weber fractions, Table 7 shows that participants could discriminate approximately 11–14% differences in the relative TTC of the approaching objects. To our knowledge, there have not been any studies determining the minimum jnd for discriminating the relative TTC of two simultaneously approaching objects. Todd (1981) measured TTC judgments for two simultaneously approaching stimuli and reported that participants were 90% accurate in detecting TTC differences equal to or greater than 150 ms and showed performance above chance for TTC differences greater than 10 ms. However, Todd did not use a formal psychophysical procedure for obtaining discrimination thresholds. Furthermore, the 6–13% Weber fractions reported by López-Moliner and Bonnet (2002) and Regan and Hamstra (1993) were derived from TTC judgments comparing two objects that were presented successively, while the stimuli in Experiments 1 and 2 were presented simultaneously. If TTC judgments are like other psychophysical judgments (e.g., discrimination of orientation; see Heeley & Buchanan-Smith, 1992), then the discrimination thresholds reported by López-Moliner and Bonnet and Regan and Hamstra were likely to be underestimations of participants' ability to discriminate relative TTC with simultaneous displays. However, the results of Experiments 1 and 2 showed that observers could reliably discriminate differences in TTC of 11–18%. These slightly larger discrimination thresholds are probably due to the different TTC conditions used in the respective studies. That is, since the TTC at disappearance values used in Experiments 1 and 2 were considerably larger than those used by López-Moliner and Bonnet and Regan and Hamstra, the precision of TTC judgments may be reduced due to additional strategies coming into play (cf., Tresilian, 1995). 
Table 7
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
Table 7
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
TTC at disappearance (ms)
150 400 1000 3600
jnd (ms)
  No texture 37 (3) 89 (8) 199 (16) 924 (86)
  Check texture 35 (3) 91 (12) 178 (21) 829 (71)
  Familiar object 42 (5) 74 (7) 184 (26) 727 (93)
Weber (%) 14.2 (0.7) 12.9 (0.9) 11.5 (0.7) 13.0 (0.7)
 

Note: Numbers in brackets show one standard error of the mean.

General discussion
The two experiments reported above were designed to test precise predictions of the critical separations between two approaching objects that would produce equal relative TTC judgments under the assumption that such judgments are based on τ, Θ, or
Θ ˙
. These predictions were tested by varying the relative TTC of smaller and larger objects and measuring the temporal separation between the objects that resulted in observers effectively judging that the two objects would arrive at the same time. Consistent with the size-arrival effect, the results found that two approaching objects were judged to have equivalent TTCs when the TTC of the smaller object was in fact less than that of the larger object. Moreover, the magnitude of these TTC errors closely fit that predicted by the hypothesis that relative TTC judgments are based on the
Θ ˙
. In Experiment 1, this reliance on
Θ ˙
was found to occur across a range of TTC at disappearance values and occurred independently of whether the approaching objects were ambiguously sized, or alternatively, when the objects were familiar to the observers and had known sizes. Experiment 2 found that the effect of
Θ ˙
was robust under a larger range of TTC at disappearance conditions than those used in Experiment 1 and also showed that the absence of an effect of familiar size on relative TTC judgments could not be explained by the possibility that observers required pre-calibration to the sizes of familiar objects via a preview, nor on the possibility that differences would only be produced when comparing familiar objects with ambiguously sized check-textured objects. Taken together, the results suggest that relative TTC judgments are based solely on
Θ ˙
, even when other variables such as relative size and familiar size are present in the displays. This finding contradicts previously reported effects of familiar size on TTC judgments and the following discussion will attempt to address these seemingly contradictory findings in terms of their differing methodologies and task restraints. 
Although López-Moliner et al. (2007) showed that TTC responses are initiated more accurately when the size of an approaching object is known, the two experiments reported in this paper did not find an effect of familiar size on relative TTC judgments. An important difference between the two studies is that they used different classes of TTC tasks. While López-Moliner et al. measured TTC performance for a single approaching object, the present study measured relative TTC judgments for two approaching objects. As noted earlier, relative TTC tasks do not require a timed response (Tresilian, 1995). Therefore, familiar size information may not be used for relative TTC judgments because the additional calculations required to disambiguate
Θ ˙
would be of no benefit in terms of satisfactory completion of the task. Hence, when making categorical judgments of which of two approaching objects will arrive sooner, observers may converge on simple looming information. 2 It should be noted that López-Moliner et al. reported that TTC judgments for familiar-sized objects were only more accurate in binocular viewing conditions. In monocular conditions, participants converged on a single threshold value of
Θ ˙
for all familiar-sized balls. These results are more consistent with the findings of Experiments 1 and 2 but raise the question of why observers would converge on different sources of information in binocular and monocular conditions. López-Moliner et al. seem to suggest that familiar size may be allocated more weighting in binocular conditions, a proposal that is consistent with DeLucia's (2005) finding that use of disparity information for TTC judgments is moderated by the availability of familiar size. 
It is perhaps more difficult to explain why we were not able to reproduce the effects of familiar size reported by DeLucia (2005) given that both studies used a relative TTC task. However, there were a number of methodological differences between the two studies that were not addressed in our experiments, and it is possible that the conflicting findings are a reflection of a flexible visual system that can select from an array of information sources for TTC judgments depending on the constraints of the task and the saliency/reliability of the available cues (DeLucia, 2004a; DeLucia et al., 2003; Tresilian, 1995; van der Kamp et al., 1997). Interestingly, we did find differences in jnd values for familiar-sized objects and no-texture objects when TTC at disappearance approximated the relatively large value used by DeLucia. Objects with such large TTCs have
Θ ˙
values that are closer to threshold (see Table 6), and therefore, this latter finding may indicate that familiar size is a variable of importance for discriminating TTC when other information is less salient. 
The
Θ ˙
of an approaching object is dependent on the size of the object, its distance from the observer, and its velocity. Therefore,
Θ ˙
on its own is an ambiguous source of TTC information. Despite this ambiguity, many studies have provided evidence for observers' use of
Θ ˙
when hitting and catching real objects and when making purely perceptual TTC judgments of virtual stimuli (e.g., Caird & Hancock, 1994; Caljouw et al., 2004b; Gray & Regan, 2000; Michaels et al., 2001; Smith et al., 2001). Indeed, humans appear to be very sensitive to looming from a very young age (Bower, Broughton, & Moore, 1970; Kayed & van der Meer, 2000; Schmuckler & Li, 1998) and recent neurological studies have located areas of the visual cortex that respond to
Θ ˙
(e.g., Billington, Wilkie, Field, & Wann, 2010; Sun & Frost, 1998). However, the reliance on
Θ ˙
for TTC judgments is problematic for theories of TTC because it fails to account for the precise timing required of many interceptive actions. One explanation is that the use of different sources of TTC information varies as a function of the time/distance between the observer and an object (DeLucia, 2004a, 2008; Tresilian, 1995). According to DeLucia (2004a), when judging objects with relatively large TTCs, heuristic source of TTC information such as
Θ ˙
is used to guide TTC judgments. When TTC is reduced to a more critical region such that precise timing is necessary, observers converge on more accurate sources of TTC information. Many ball sports require a level of precision in judging TTC that only the expert athlete can achieve. However, there is mounting evidence that τ is unlikely to be the source of accurate TTC judgments, not least because the assumptions that must be met for τ to be reliable are very constrained (e.g., see Bootsma & Oudejans, 1993; Tresilian, 1990, 1995). Over years of ball-playing experience, the athlete develops knowledge about the physical and dynamic characteristics of the ball including its familiar size (Gray, 2002). While the present study did not find that familiar size was used in relative TTC tasks, a plausible alternative to τ for precise TTC judgments is that observers learn to scale
Θ ˙
on the basis of familiar size (López-Moliner et al., 2007). 
In conclusion, we have demonstrated that
Θ ˙
is used for relative TTC judgments when approaching objects (1) are either ambiguous in their identity/size or are familiar object with standard sizes, (2) have relatively brief actual TTCs or relatively long actual TTCs, and (3) travel at a range of speeds and between a range of viewing distances. The finding that relative TTC judgments are based on comparisons of the instantaneous
Θ ˙
of approaching objects, in combination with previous reports that absolute TTC judgments are initiated when
Θ ˙
reaches a threshold value, suggests that
Θ ˙
is a critical variable for judging TTC. 
Appendix A
Equivalence functions
The formal analysis of the determinants of relative TTC that follows is designed to answer the following question: What separation between two approaching objects is required for them to have the same Θ or
Θ ˙
at any time t? The analysis is based on the assumptions that (i) the two objects are approaching a viewpoint at the same constant velocity and (ii) that the two objects are located on the observers' line of sight. 3  
The Θ of each of the approaching objects can be calculated from tan−1(
S D
), where S is the size of the object and D is its distance from the viewpoint. Let T c and T s be the absolute TTC of the approaching objects (where the subscripts c and s denote the comparison object and standard object, respectively). The critical temporal separation (δ; defined as T cT s) between these two objects with diameters of S c and S s required for them to have equal Θ at any time t is when 
S c D c = S s D s .
(A1)
Since D = VT, where V is the velocity of the approaching object, Equation A1 can be rewritten as 
T c V c = S c S s · T s V s .
(A2)
Therefore, the critical temporal separation necessary for two approaching objects to subtend the equivalent Θ is when 
δ = ( S c S 0 s 1 ) T s .
(A3)
 
Differentiating Θ with respect to time gives
Θ ˙ = S V D 2
. If two objects have equal
Θ ˙
, then 
S c V c ( D c ) 2 = S s V s ( D s ) 2 .
(A4)
Therefore, the critical temporal separation between the two objects necessary for them to have equivalent
Θ ˙
is when 
δ = ( S c S s 1 ) T s .
(A5)
 
Acknowledgments
We thank two anonymous reviewers for their helpful comments on previous versions of this manuscript. Parts of this manuscript were prepared while the first author was a visiting researcher at NASA Ames Research Center. 
Commercial relationships: none. 
Corresponding author: Simon G. Hosking. 
Email: simon.hosking@gmail.com. 
Address: 506 Lorimer Street, Fishermans Bend, Victoria, Australia. 
Footnotes
Footnotes
1  These pilot data were consistent with previous findings that a larger object will be judged to arrive first even when it has a much larger actual TTC than a smaller object (e.g., DeLucia, 1991, 2005; DeLucia & Warren, 1994).
Footnotes
2  This distinction between absolute and categorical TTC judgments may also explain why the initiation of avoidance actions in response to approaching obstacles does not appear to rely on threshold values of
Θ ˙
(see Tresilian, Plooy, & Carroll, 2004). That is, if such avoidance actions do not require precise timing (there are no negative consequences for initiating an avoidance action too early) but rather are based on categorical judgments of when to “do nothing” and when to “get out of the way,” then it is unlikely that such actions would be dependent on threshold values for their initiation.
Footnotes
3  Kaiser and Mowafy (1993) have shown that for offset objects α = cos Φ · S/(D 2 + R 2)1/2, where α is the visual angle of the offset object, Φ is the angle between the medial edge of an object and the line of sight, S is the actual size of the object, D is its distance from the observers' viewpoint, and R is the offset of the object from the line of sight. As in previous derivations of the angular extent of offset objects (e.g., Caird & Hancock, 1994; DeLucia, 1991), the formula reported in the appendix assume that the approaching objects are located on the observers' line of sight.
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Figure 1
 
(A) Function showing the critical temporal separation between two approaching objects (a standard, smaller object that is in front and a comparison, larger object that is behind) required to produce equal values of Θ. (B) A similar function showing the critical temporal separation between two approaching objects required to produce equal values of Θ ˙ . The object separations that would result in either “Smaller first” or “Larger first” relative TTC judgements are also shown in (A) and (B).
Figure 1
 
(A) Function showing the critical temporal separation between two approaching objects (a standard, smaller object that is in front and a comparison, larger object that is behind) required to produce equal values of Θ. (B) A similar function showing the critical temporal separation between two approaching objects required to produce equal values of Θ ˙ . The object separations that would result in either “Smaller first” or “Larger first” relative TTC judgements are also shown in (A) and (B).
Figure 2
 
Screen shots taken from the last frame in the displays before the two objects disappeared from view in Experiment 1. The left-hand panel shows the black objects in the no-texture condition; the right-hand panel shows the yellow/green tennis ball and black-and-white football in the familiar-object condition. In both conditions, the objects appeared to approach participants against a textureless blue background.
Figure 2
 
Screen shots taken from the last frame in the displays before the two objects disappeared from view in Experiment 1. The left-hand panel shows the black objects in the no-texture condition; the right-hand panel shows the yellow/green tennis ball and black-and-white football in the familiar-object condition. In both conditions, the objects appeared to approach participants against a textureless blue background.
Figure 3
 
(A) Examples of data and fitted curves for a representative participant in the familiar-object condition of Experiment 1. The data show the percentage of “Smaller object will arrive first” responses for each of the six levels of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations for smaller and larger objects in the familiar-object (open circles) and no-texture (filled circles) object familiarity conditions as a function of the TTC at disappearance of the smaller object. Error bars represent one standard error of the mean. The dashed lines represent the predicted separations between the smaller (standard) object and larger (comparison) object used in Experiment 1 required to produce equal τ, Θ, and Θ ˙ .
Figure 3
 
(A) Examples of data and fitted curves for a representative participant in the familiar-object condition of Experiment 1. The data show the percentage of “Smaller object will arrive first” responses for each of the six levels of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations for smaller and larger objects in the familiar-object (open circles) and no-texture (filled circles) object familiarity conditions as a function of the TTC at disappearance of the smaller object. Error bars represent one standard error of the mean. The dashed lines represent the predicted separations between the smaller (standard) object and larger (comparison) object used in Experiment 1 required to produce equal τ, Θ, and Θ ˙ .
Figure 4
 
Displays and backgrounds used in Experiment 2 for each of the texture-defined object familiarity conditions. (Left) No-texture condition. (Center) Check-texture condition. (Right) Familiar-object condition.
Figure 4
 
Displays and backgrounds used in Experiment 2 for each of the texture-defined object familiarity conditions. (Left) No-texture condition. (Center) Check-texture condition. (Right) Familiar-object condition.
Figure 5
 
(A) Examples of data and fitted curves for a representative participant in the familiar-size condition of Experiment 2. The data show the percentage of “Smaller object will arrive first responses” for each level of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations between the smaller (standard) object and larger (comparison) object used in Experiment 2 as a function of texture-defined object familiarity and TTC at disappearance. Standard error bars represent one standard error of the mean. The dashed lines represent the predicted separations required to produce equal relative TTC judgements based on the τ, Θ, and Θ ˙ hypotheses.
Figure 5
 
(A) Examples of data and fitted curves for a representative participant in the familiar-size condition of Experiment 2. The data show the percentage of “Smaller object will arrive first responses” for each level of TTC at disappearance as a function of the actual TTC of the larger object. (B) Mean critical temporal separations between the smaller (standard) object and larger (comparison) object used in Experiment 2 as a function of texture-defined object familiarity and TTC at disappearance. Standard error bars represent one standard error of the mean. The dashed lines represent the predicted separations required to produce equal relative TTC judgements based on the τ, Θ, and Θ ˙ hypotheses.
Table 1
 
Actual separations between the smaller standard object and larger comparison object for each of the six TTC at disappearance conditions used in Experiment 1.
Table 1
 
Actual separations between the smaller standard object and larger comparison object for each of the six TTC at disappearance conditions used in Experiment 1.
TTC at disappearance (ms) Actual object separations (ms)
50 0 11 22 33 44 55 66 77 88 99 110
100 0 22 43 66 86 108 129 151 172 194 220
150 0 32 64 96 128 160 192 224 256 288 320
200 0 42 85 127 169 212 254 297 339 381 420
300 0 64 128 192 256 319 383 447 511 575 640
400 0 85 171 256 342 427 513 598 683 769 850
 

Note: Numbers in bold indicate the temporal separations between which it was predicted that participants would switch their judgments from “larger object will arrive first” to “smaller object will arrive first” on the assumption that such judgments are based on Θ ˙ .

Table 2
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 1.
Table 2
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 1.
TTC at disappearance (ms) Smaller object Larger object
Mina Maxb
50 111.41 350.14 34.19
100 27.85 87.54 8.55
150 12.40 38.91 3.96
200 6.96 21.88 2.28
300 3.10 9.73 0.99
400 1.74 5.47 0.56
 

aMinimum temporal separation condition.

 

bMaximum temporal separation condition.

Table 3
 
Slopes and intercepts of the regression line fitted to the obtained critical temporal separations in Experiments 1 and 2. The predicted slopes according to the τ, Θ, and Θ ˙ hypotheses are also presented.
Table 3
 
Slopes and intercepts of the regression line fitted to the obtained critical temporal separations in Experiments 1 and 2. The predicted slopes according to the τ, Θ, and Θ ˙ hypotheses are also presented.
Obtained Predicted slope
Intercept Slope
M SE M SE τ Θ ˙ Θ
Experiment 1 0.034 0.015 0.705 0.064 0* 2.142* 0.773
Experiment 2 −0.023 0.066 0.702 0.035 0* 2.143* 0.773
 

Note: * denotes a significant difference between predicted and obtained slopes in a two-tailed test with an alpha of 0.001.

Table 4
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
Table 4
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
TTC at disappearance (ms)
50 100 150 200 300 400
jnd (ms) 20 (3) 31 (4) 40 (6) 49 (7) 67 (8) 88 (12)
Weber (%) 17.8 (2.5) 15.9 (1.7) 14.3 (2.2) 14.0 (2.4) 13.0 (1.5) 13.1 (2.0)
 

Note: Numbers in brackets show one standard error of the mean.

Table 5
 
Actual separations (in ms) between the smaller and larger objects used to produce the displays for each of the four TTC at disappearance conditions used in Experiment 2.
Table 5
 
Actual separations (in ms) between the smaller and larger objects used to produce the displays for each of the four TTC at disappearance conditions used in Experiment 2.
TTC at disappearance (ms) Actual object separations (ms)
150 0 32 64 96 128 160 192 224 256 288
400 0 85 170 256 341 426 511 596 681 767
1000 0 213 426 639 852 1065 1278 1491 1704 1917
3600 0 767 1534 2300 3067 3833 4600 5367 6133 6900
 

Note: Numbers in bold indicate the temporal separations between which it was predicted that participants would switch their judgments from “larger object will arrive first” to “smaller object will arrive first” on the assumption that such judgments are based on Θ ˙ .

Table 6
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 2.
Table 6
 
Visual angle expansion rates ( Θ ˙ , in deg/s) at TTC at disappearance for the smaller (standard) and larger (comparison) objects used in Experiment 2.
TTC at disappearance (ms) Smaller object Larger object
Mina Maxb
150 12.38 38.90 4.56
400 4.64 14.59 1.71
1000 1.90 5.94 0.68
3600 0.52 1.62 0.19
 

aMinimum temporal separation condition.

 

bMaximum temporal separation condition.

Table 7
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
Table 7
 
Mean jnds and Weber fractions for discriminating differences in the relative TTC of the smaller and larger objects as a function of the TTC at disappearance of the smaller object.
TTC at disappearance (ms)
150 400 1000 3600
jnd (ms)
  No texture 37 (3) 89 (8) 199 (16) 924 (86)
  Check texture 35 (3) 91 (12) 178 (21) 829 (71)
  Familiar object 42 (5) 74 (7) 184 (26) 727 (93)
Weber (%) 14.2 (0.7) 12.9 (0.9) 11.5 (0.7) 13.0 (0.7)
 

Note: Numbers in brackets show one standard error of the mean.

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