Various equations that describe how observers could recover the trajectory of an approaching object have been put forward. Many are relatively complex formulations that recover the veridical trajectory by scaling retinal cues, such as looming and changing disparity. However, these equations do not seem to describe human perception as observers typically misjudge trajectory angles. Thus, we examine whether a simpler formulation—one that does not predict veridical judgments—may better explain performance. We test the hypothesis that perceived trajectory is based on a speed ratio: the ratio of lateral angular speed to the sum of looming and changing disparity signals. To discriminate between this and alternative proposals, we examined the effect of object size on trajectory perception: The speed ratio hypothesis predicts that perceived trajectory will become less eccentric with increasing object size, while the alternatives predict that perceived trajectory will be independent of object size. Observers performed a trajectory judgment task in which they compared the trajectory direction of two approaching objects, of the same or different size, seen in separate intervals. We estimated perceptually parallel trajectories from their responses. In Experiment 1, objects differed in horizontal and vertical size, and in Experiment 2, they differed only in vertical size. In both experiments, observers' data showed a clear effect of object size and were close to predictions of the speed ratio hypothesis. We conclude that the alternate proposals we tested were not supported and that the speed ratio account is a sufficient account of the data.

*S*at a distance

*D*directly in front of the observer and on an oblique trajectory as in Figure 1 (left panel). The sphere is seen by two eyes that are separated by a distance

*I*.

*X*

_{c}is the “crossing distance” at which the ball will pass the observer (measured in the frontal plane containing the two eyes). The sphere in the left panel has trajectory angle

*β*with respect to the median plane of the head. Lateral visual direction is given by

*α*(illustrated in the right panel). The rate of change of lateral visual direction, or lateral angular speed, is

*α*and

*α*

_{L}and

_{L}and

*α*

_{R}and

_{R}.

*θ*and its rate of change

*θ*is the angle at the eye subtended by the horizontal extent of the object, and we can define a similar angle,

*ϕ,*which is the angle at the center of the object that is subtended by the two eyes.

*ϕ*is the object's absolute (or optic array) disparity. This quantity is approximately equal to the retinal disparity,

*α*

_{R}−

*α*

_{L}, when disparities are measured relative to a distant point,

*P*(see Sousa, Brenner, & Smeets, 2010). The rate of change of

*ϕ*is

*α*, then Equation 1 provides sufficient information to determine which will pass furthest from the head. Equation 1 provides ambiguous information about an object's trajectory. Objects traveling on the same trajectory produce different values when their widths are different, and objects on parallel trajectories produce different values when at different distances from the observer.

*ϕ*is geometrically similar to

*θ*. The similarity is evident from Figure 2. If an object moves toward the observer, angles

*ϕ*and

*θ*will both increase and so both provide motion-in-depth information. Indeed, if the object's width is equal to the observer's interocular distance, then angles

*θ*and

*ϕ*are equal and so are their rates of change,

*β*, but they do not support accurate perception and they are not sufficient to guide action. However, scaling these quantities appropriately provides the crossing distance,

*X*

_{c}(see left panel of Figure 1), which gives the object's position at some future time, and so is potentially useful for guiding interception or avoiding collisions. For example, Peper, Bootsma, Mestre, and Bakker (1994) proposed that interception could be achieved by comparing the crossing distance of an approaching object with the current position of the hand and executing hand movements to close the gap within the remaining time before arrival; the latter quantity is given by the ratios

*X*

_{c}, is obtained by scaling Equation 1 by the object's width,

*S*(see Bootsma, 1991 for a full derivation):

*X*

_{c}determined in this way expresses the distance that the object will pass the midpoint of the eyes, in units of the object's width.

*X*

_{c}can also be obtained by scaling Equations 2 and 3 by

*I*to obtain

*D*= 0. This is the frontal plane at the eyes when the object is straight ahead (Figure 1, left panel). To obtain crossing distance in the plane of the eyes when the object is not straight ahead, the results of Equations 4–6 need to be scaled by cos

*α*.

*X*

_{c}) define a right-angled triangle. Therefore, we can define

*β*as

*X*

_{c}in Equation 7 to obtain the following three formulations for trajectory angle:

*θ*

_{0}denotes initial angular size and

*θ*is current angular size:

*β*in Figure 1, are overestimated: Approaching objects appear to pass further from the head (e.g., Harris & Dean, 2003; Peper et al., 1994; Welchman et al., 2004). It may be that observers attempt veridical recovery of motion in depth, e.g., by implementing a scheme such as those above, but fail due to incorrectly estimating the necessary parameters, or that they do not attempt veridical recovery.

*β*′ is perceived trajectory angle):

*X*

_{c}, i.e., the lateral distance at which the sphere would intersect the frontal plane at the eyes. We also describe the trajectory in terms of its angle

*β*from the median plane (Figure 1, left panel), and we provide both quantities on our graphs in the Results and analysis section.

*Same size*—either both reference and test spheres were large, or both were small. We would not expect any systematic bias in trajectory judgments in these conditions. We included these conditions to examine the influence of size on 3D trajectory discrimination thresholds.

*Different size*—either the reference trajectory sphere was large and the test trajectory sphere was small (large reference, small test) or vice versa (small reference, large test). Using these data, we can examine the influence of object size on visual judgments of apparently parallel 3D trajectories.

*F*

_{3,9}= 10.522,

*p*= 0.003,

*η*

^{2}= 0.52) and no effect of the size of the spheres alone or in interaction with crossing distance. These data are provided in Supplementary Figure S1.

*F*

_{3,9}= 6.395,

*p*= 0.013,

*η*

^{2}= 0.50), as found in the same-size conditions, but no effect of same- vs. different-size condition alone or in interaction with crossing distance.

*F*

_{1,3}= 62.351,

*p*= 0.004,

*η*

^{2}= 0.15), crossing distance (

*F*

_{3,9}= 575.578,

*p*< 0.0005,

*η*

^{2}= 0.74), and size in interaction with crossing distance (

*F*

_{3,9}= 28.483,

*p*< 0.0005,

*η*

^{2}= 0.09).

*g*that determines the relative gain for looming rate vs. changing disparity (see Equation 15). The two signals are combined in the expected proportions when

*g*= 0.5:

*g*> 0.5) or the changing disparity signal (

*g*< 0.5).

*β*

_{test}′ =

*β*

_{reference}′) when

*g,*we can determine the physical test trajectory angle that would appear equal to a physical reference trajectory under this model, i.e., we find the physical test stimulus trajectory angle that satisfies Equation 16. We do so using a numerical method. Predictions of this model are given as follows:

- If observers made no use of changing disparity, and instead based their perception of trajectory angle on$ \alpha \xb7 / \theta \xb7 $, then
*g*= 1. In this case, expected performance on the apparently parallel judgment task is shown as fine dashed lines in Figure 4. The slopes of these lines are equal to the ratios of the physical size of the test and reference spheres. - If observers made no use of looming signals, then
*g*= 0. In this case, the size of the spheres would not influence judgments, so there will be no systematic error. Expected performance is shown as the diagonal line,*y*=*x*. Data lying on this line would tell us that observers do not use looming signals, but such data would not be informative about the extent to which observers use changing disparity signals (data on this line indicate that physically identical test and reference trajectories appear identical, but it does not mean perception is veridical because both may be misperceived by the same amount). - If observers used the sum of looming and changing disparity, consistent with the proposal of Regan and Beverley (1979), then
*g*= 0.5. Expected performance in this case is shown as bold dashed lines.

*g*> 0.5. Best fitting data for the group mean corresponded to a gain on looming of 0.64 in the small-reference, large-test condition and 0.51 in the large-reference, small-test condition. A paired-samples

*t*test revealed no significant difference. The average gain value across the two conditions was 0.58, indicating a slightly greater contribution of looming than changing disparity (0.42). As stated earlier, the crossing distance of the small spheres was found to be 69% of that for the large spheres. If looming and changing disparity signals had been summed with equal gains (i.e.,

*g*= 0.5), the crossing distance of the small spheres would have been 76% of that for the large spheres (the slope of the bold dashed light blue line in Figure 4). This prediction is close to the observed value.

Observer | Small reference, large test | Large reference, small test |
---|---|---|

PD | 0.72 | 0.50 |

PJ | 0.60 | 0.31 |

AN | 0.67 | 0.71 |

AA | 0.55 | 0.52 |

Group | 0.64 | 0.51 |

*n,*in units of the object's horizontal physical size and thus could provide a way to encode trajectories without involving distance information:

*n,*i.e., without scaling this quantity by object width. If observers perceive trajectories within the horizontal meridian plane in terms of crossing distance in units of horizontal physical size, as in Equation 17, then perceived trajectory should not depend on the object's vertical size. We tested this in Experiment 2. Observers compared the trajectories of a small sphere and a “tall” ellipsoid whose horizontal size was the same as the small sphere but whose vertical size was larger. Unlike a sphere, the ellipsoid stimulus provides a different looming signal on its horizontal and vertical axes. If only the horizontal size is important, then the vertical size manipulation should have no effect. Alternatively, if both dimensions are important, then apparently parallel trajectory judgments should reflect the use of a looming signal derived from retinal velocities measured in both horizontal and vertical directions. The latter prediction is supported by evidence from Gray and Regan (2000). They used stimuli with unequal rates of horizontal and vertical expansion in the retinal image and found that time-to-contact judgments, in monocular conditions, were influenced by the velocities in both horizontal and vertical directions.

*F*

_{1,3}= 11.698,

*p*= 0.042,

*η*

^{2}= 0.08), crossing distance (

*F*

_{3,9}= 214.997,

*p*< 0.0005,

*η*

^{2}= 0.85), and vertical size in interaction with crossing distance (

*F*

_{3,9}= 28.483,

*p*< 0.0005,

*η*

^{2}= 0.04).

*F*

_{3,9}= 5.035,

*p*= 0.026,

*η*

^{2}= 0.26) and no effect of the object's vertical size alone or in interaction with crossing distance.

*g*= 0, the motion-in-depth signal used in trajectory perception is not derived from vertical changing image size information; (2) where

*g*= 1, the motion-in-depth signal is derived from looming signals (where horizontal and vertical changing sizes contribute equally); and (3) where

*g*= 0.5, motion in depth is based on the sum of looming and changing disparity, consistent with the proposal of Regan and Beverley (1979).

*g*= 0 and

*g*= 1 are smaller than in Experiment 1, and therefore, parameter estimates will be more susceptible to noise.

Observer | Small reference, large test | Large reference, small test |
---|---|---|

PD | 0.90 | 0.62 |

PJ | 0.29 | 0.51 |

AN | 0.84 | 0.52 |

AA | 0.55 | 0.44 |

Group | 0.70 | 0.53 |

*t*test indicated no significant difference). The average gain value for looming was 0.61, indicating a greater contribution of looming than changing disparity (i.e., 0.39). This bias was larger than in Experiment 1 (looming gain was 0.54). Almost all observers in both of the reference-type conditions showed a deviation, corresponding to relatively greater use of looming than predicted, i.e.,

*g*> 0.5, as in Experiment 1.

*g*= 0.5), the crossing distance of the small spheres would have been 86% of that for the tall ellipsoids (the slope of the bold dashed blue line in Figure 5). This prediction is close to the observed value of 81%.

*β*′, is related to trajectory angle

*β*as shown in Equation A1, where

*D*and

*D*′ are the actual and perceived distances to the trajectory start point, respectively. A1 predicts that perceived trajectory angle is underestimated when perceived distance is overestimated: