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Article  |   January 2016
Sources of bias in the perception of heading in the presence of moving objects: Object-based and border-based discrepancies
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Journal of Vision January 2016, Vol.16, 9. doi:10.1167/16.1.9
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      Oliver W. Layton, Brett R. Fajen; Sources of bias in the perception of heading in the presence of moving objects: Object-based and border-based discrepancies. Journal of Vision 2016;16(1):9. doi: 10.1167/16.1.9.

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      © ARVO (1962-2015); The Authors (2016-present)

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Abstract

The focus of expansion (FoE) specifies the heading direction of an observer during self-motion, and experiments show that humans can accurately perceive their heading from optic flow. However, when the environment contains an independently moving object, heading judgments may be biased. When objects approach the observer in depth, the heading bias may be due to discrepant optic flow within the contours of the object that radiates from a secondary FoE (object-based discrepancy) or by motion contrast at the borders of the object (border-based discrepancy). In Experiments 1 and 2, we manipulated the object's path angle and distance from the observer to test whether the heading bias induced by moving objects is entirely due to object-based discrepancies. The results showed consistent bias even at large path angles and when the object moved far in depth, which is difficult to reconcile with the influence of discrepant optic flow within the object. In Experiment 3, we found strong evidence that the misperception of heading can also result from a specific border-based discrepancy (“pseudo FoE”) that emerges from the relative motion between the object and background at the trailing edge of the object. Taken together, the results from the present study support the idea that when moving objects are present, heading perception is biased in some conditions by discrepant optic flow within the contours of the object and in other conditions by motion contrast at the border (the pseudo FoE). Center-weighted spatial pooling mechanisms in MSTd may account for both effects.

Introduction
Self-motion of an observer through an otherwise static environment produces a radial spatiotemporal pattern on the retina (Figure 1a). Gibson (1950) first proposed that a sighted animal can use the singularity, known as the focus of expansion (FoE), which appears at the center of the radially expanding optic flow when traveling forward to “see where he is going.” Research indicates that humans use optic flow to guide locomotion when walking through environments with rich visual texture (Warren, Kay, Zosh, & Duchon, 2001). Even in minimal dot-defined visual environments, humans are capable of accurately perceiving their direction of travel (heading) from optic flow when moving along a straight path, within <1° (Cuturi & MacNeilage, 2013; Van Den Berg, 1992; Warren & Hannon, 1988). These findings support the idea that humans use optic flow to perceive heading during self-motion. 
Figure 1
 
(a) The self-motion of an observer along a straight path toward a frontoparallel back plane yields a radial optic flow field (blue arrowheads). We refer to the point of the outflow as the background focus of expansion (FoE). The angular difference between the actual and perceived heading direction is called heading error. (b) When the observer moves in the presence of an independently moving object (red arrowheads) that approaches in depth, the object produces a secondary FoE (object FoE). The object may cross the observer's path and occlude the background FoE and create uncertainty about the heading direction. Heading error is systematically biased toward the object FoE.
Figure 1
 
(a) The self-motion of an observer along a straight path toward a frontoparallel back plane yields a radial optic flow field (blue arrowheads). We refer to the point of the outflow as the background focus of expansion (FoE). The angular difference between the actual and perceived heading direction is called heading error. (b) When the observer moves in the presence of an independently moving object (red arrowheads) that approaches in depth, the object produces a secondary FoE (object FoE). The object may cross the observer's path and occlude the background FoE and create uncertainty about the heading direction. Heading error is systematically biased toward the object FoE.
Mechanisms in the visual system that may use the FoE to determine heading face a challenging problem when objects that move independently of the observer occupy the environment during self-motion. Moving objects produce patterns of optic flow distinct from those experienced during self-motion in a static environment. Whereas the optical motion from a stationary background radiates from the singularity in the direction of heading (which we refer to as the background FoE), an object that moves in depth to approach the observer generates optic flow that emanates from a FoE in a different location (object FoE; Figure 1b). Therefore, the optic flow field may contain different regions of flow radiating from different FoE, some of which are not informative about self-motion. To add to the complexity of the problem, an opaque object that crosses the observer's future path completely occludes the background FoE and neighboring regions (Figure 1b), which are known to be the most informative regions of the optic flow field for heading perception (Crowell & Banks, 1993, 1996). 
Human heading perception in the presence of moving objects
Heading estimation performance is generally good in the presence of moving objects, but humans do make systematic errors in judged heading under some circumstances. Royden and Hildreth (1996) studied human heading perception in the presence of objects that move laterally with respect to the heading direction while also maintaining a fixed distance from the observer. Subjects viewed simulated self-motion in a visual environment that contained moving dots in one of two frontoparallel depth planes. One experiment assessed heading judgments to a horizontally moving object as a function of initial object position. When the object occluded the background FoE for more than half of the trial, Royden and Hildreth found that heading biases were in the direction of object motion by ∼1°. Conditions in which the moving object did not obscure the background FoE resulted in virtually no heading bias. 
Warren and Saunders (1995) assessed judgments of heading in the presence of an object that moved laterally to the path of a forwardly translating observer but also approached in depth (Figure 1b). The visual displays contained randomly positioned dots on two planar surfaces, where each dot moved in a manner consistent either with the background or the independently moving object. The object was always initially located 6° to either side of the center of the display and optically expanded as the trial progressed due to the decreasing distance between the observer and the object. The object moved along a 0° (parallel) or ±6° straight course relative to the observer (path angle). When the object moved on the side of the screen opposite of the observer's heading (i.e., the background FoE was visible), the accuracy of heading judgments was good, consistent with the findings of Royden and Hildreth (1996). However, Warren and Saunders (1995) found that when the object moved along an oblique trajectory and obscured the background FoE for much of the trial, heading judgments were biased in the direction of the object FoE (i.e., where the object came from; see Figure 1b). On average, the magnitude of the bias was 3.7° when the object was opaque and 2.3° when it was transparent. 
Models of heading perception
Warren and Saunders (1995) introduced a simple template model that explains human heading bias garnered in the presence of approaching moving objects. The model computes fits between a vector field representation of the optic flow and templates tuned to radial expansion. Each template responds preferentially to optic flow corresponding to heading in a particular direction (i.e., emanating from a particular FoE location). As such, optic flow emanating from a location other than the background FoE will contribute to the activation of templates associated with directions that are not aligned with the observer's heading direction. The heading estimate derived from the model reflects the heading sensitivity of the most active template. In the case of self-motion in the presence of an object that approaches the observer in depth and occludes the background FoE, the Warren and Saunders model yields a heading estimate that is biased toward the object FoE, which is consistent with human judgments. However, the direction of the bias garnered by the model is inconsistent with human judgments when the object maintains a fixed depth with respect to the observer (Royden, 2002). Royden (1997, 2002) developed a differential motion model to explain the bias both when the object approaches the observer and when it maintains a fixed depth. The differential motion model works by subtracting the background motion signals from others at the border of the object. Lines drawn through the resulting difference vectors intersect on the side of the object that coincides with the direction of the heading bias. 
Object-based and border-based discrepancies
Whereas spatial pooling and differential motion models focus on the mechanisms underlying the heading bias induced by moving objects, it is also useful to consider the locus of the discrepancy in the optic flow field that gives rise to the heading bias. Our focus in the present study is on the bias induced by moving objects that approach the observer in depth. Objects that maintain a fixed depth (as in Royden & Hildreth, 1996) or retreat from the observer generate a qualitatively different optic flow devoid of expansion and may yield a heading bias for different reasons (see The pseudo FoE and the influence of fixed-depth objects). 
One possibility is that approaching objects induce a heading bias because the optic flow within the object radiates from a location (the object FoE) that is offset from the background FoE (Figure 1b). If the visual system pools motion across the visual field, as suggested by Warren and Saunders (1995), such discrepant optic flow may lead to a bias in heading perception. Because the locus of the discrepancy is the optic flow within the object, we refer to this source of bias as an object-based discrepancy
Alternatively, moving objects may bias heading perception because of contrasting motion at the borders of the object, where the motion on the object side of the border differs from the motion on the background side of the border. Such border-based motion contrast is responsible for generating bias in Royden's (2002) differential motion model. However, spatial pooling models may also yield a heading bias due to border-based discrepancies. As we show below, motion contrast at the trailing edge of a moving object can, under certain conditions, appear sufficiently radial-like to attract the radial templates in spatial pooling models. 
We contend that there is already sufficient evidence that object-based discrepancies can induce a heading bias under some conditions. For example, when an object approaches the observer and occupies a large part of the visual field, as in Warren and Saunders (1995), the discrepant optic flow within the object is extremely salient. By comparison, the discrepant differential motion exists only at the borders of the object, which are in the periphery when the object is sufficiently close, and is therefore unlikely to account for the heading bias under such conditions. 
However, it is not yet known whether object-based discrepancies are the sole cause of the bias induced by approaching objects or if border-based motion contrast also contributes. As such, the aim of the present study is to test the sufficiency of the object-based account as an explanation for the bias induced by moving objects. Our approach is to delineate the conditions in which heading perception should and should not be biased if object-based discrepancies are the sole source of the heading bias. If heading perception is also affected under other conditions, that would indicate that the heading perception can be biased by other factors, possibly including border-based discrepancies. To this end, we identified the following five predictions concerning the conditions in which heading perception should and should not be biased due to object-based discrepancies.1 
Predictions of the object-based discrepancy account
First, if heading perception is biased due to optic flow within the object (an object-based discrepancy), moving objects should not induce a bias when the object moves parallel to the observer, even if the object occludes the background FoE (i.e., the object is on a collision course with the observer). In this case, the optic flow within the object radiates from the same location as the background FoE and is therefore identical in direction to the background optic flow. This prediction is supported by the findings of Warren and Saunders (1995), who demonstrated that heading is not biased by objects that move parallel to the observer. 
Second, objects that move along an oblique path should bias heading perception only when they occlude or move adjacent to the background FoE. An object should not affect heading perception when the background FoE and neighboring region are visible. This prediction follows from the observation that the region of the optic flow field that is most informative and influential for heading estimation is that which is near the FoE (Crowell & Banks, 1993, 1996). In the Warren and Saunders (1995) model, this is captured by Gaussian spatial weighting that more heavily weights input vectors near the center of the receptive field of each heading template. Several previous studies (Layton & Fajen, in press; Royden & Hildreth, 1996; Warren & Saunders, 1995) provide support for this prediction. 
Third, the magnitude of the bias should increase with the spatial offset between the object FoE and the background FoE. This is not to imply that the visual system identifies the object and background FoEs and then computes a weighted average. In the Warren and Saunders (1995) model, the effect of spatial offset occurs because the heading template that is maximally active is farther away from the background FoE when the offset is greater. The offset depends on the path angle of the moving object relative to the observer's path (larger path angles result in larger offsets; see Figure 2a). Warren and Saunders (1995) tested only a single nonzero path angle (6°), so the relationship between the object FoE/background FoE offset and the bias cannot be determined from their experiments. In Experiment 1, we systematically varied the path angle to test the prediction that the heading bias increases with the offset. 
Figure 2
 
Schematic depiction of the conditions of Experiment 1 that test whether the discrepant optic flow within the object suffices in accounting for the heading bias that arises during self-motion in the presence of approaching objects. The object-based account predicts that the heading bias should increase with the spatial offset between the object FoE and the background FoE (see predictions 3 and 4). (a) Optic flow fields for several different path angles (i.e., angle between the object trajectory and observer's heading). Optic flow due to the object is depicted in red and background flow is depicted in blue. Manipulating the path angle moves the position of the object FoE. (b) Top-down view of the experimental setup of Experiment 1 and the path angles used. The blue line indicates the observer's future path and the red arrow indicates the object trajectory. (c) The five different positions of the object at the end of the trial are shown relative to the observer's heading. In the Cross condition, the object was centered on the future path at the end of the trial. The conditions to the left indicate that the object crossed and cleared the path by the end of the trial, and conditions to the right indicate that the object did not cross the path.
Figure 2
 
Schematic depiction of the conditions of Experiment 1 that test whether the discrepant optic flow within the object suffices in accounting for the heading bias that arises during self-motion in the presence of approaching objects. The object-based account predicts that the heading bias should increase with the spatial offset between the object FoE and the background FoE (see predictions 3 and 4). (a) Optic flow fields for several different path angles (i.e., angle between the object trajectory and observer's heading). Optic flow due to the object is depicted in red and background flow is depicted in blue. Manipulating the path angle moves the position of the object FoE. (b) Top-down view of the experimental setup of Experiment 1 and the path angles used. The blue line indicates the observer's future path and the red arrow indicates the object trajectory. (c) The five different positions of the object at the end of the trial are shown relative to the observer's heading. In the Cross condition, the object was centered on the future path at the end of the trial. The conditions to the left indicate that the object crossed and cleared the path by the end of the trial, and conditions to the right indicate that the object did not cross the path.
Fourth, the influence of the moving object should be mediated by the structure of the optic flow within the contours of the object. When the object FoE offset is sufficiently small and the object occupies a sufficiently large part of the visual field, as was the case in the experiments of Warren and Saunders (1995), the structure of the optic flow within the object is more radial, as illustrated in the left and middle panels of Figure 2a. However, for larger path angles, where the object FoE offset is greater, the optic flow within the object will be less radial (i.e., more laminar), as illustrated in the right panel of Figure 2a. Because the templates in the Warren and Saunders (1995) model are spatially weighted toward the heading direction to which they are tuned (i.e., center-weighted), laminar flow contributes less than radial flow to the activity of the template associated with the object FoE location. As such, the influence of moving objects should diminish when the structure of the optic flow within the contours of the object is more laminar, which occurs when the object's path angle is large.2 Experiment 1 also provided a test of this prediction. 
Fifth, the influence of moving objects should diminish to zero as object distance increases. This prediction follows from the fact that the visual angle subtended by the moving object, and hence the spatial extent of the visual field with discrepant optic flow, decreases with distance. Therefore, the heading bias should attenuate to zero as the object moves farther away from the observer. We tested this prediction in Experiment 2
To summarize, heading perception may be biased by approaching objects because of discrepant optic flow within the contours of the object, which radiates from a location (the object FoE) other than the background FoE. This hypothesis, which we refer to as the object-based discrepancy hypothesis, is compatible with the spatial pooling model of Warren and Saunders (1995) and leads to five predictions about heading perception in the presence of approaching objects. The first two predictions are supported by existing data, but the third, fourth, and fifth predictions have not yet been tested. The aim of Experiments 1 and 2 of the present study was to test the latter three predictions. To anticipate, we found that heading judgments were biased when the object-based account predicts a bias but that judgments were also biased under conditions in which no such bias was predicted. In Experiment 3, we considered the possibility that under certain conditions, the bias in heading perception can emerge from a specific type of border-based discrepancy, which we call the “pseudo FoE.” The pseudo FoE results from the relative motion between the object and background around the trailing edge of the object, which under some conditions exhibits a radial FoE-like appearance. The results from Experiment 3 demonstrate that heading judgments can be biased by the pseudo FoE, which expands the range of conditions in which moving objects can influence the perception of heading. 
Experiment 1: Path angle
In Experiment 1, we tested the third and fourth predictions.3 Subjects viewed simulated self-motion in the presence of an independently moving object that moved laterally and approached the observer's future path over a wide range of possible path angles (Figure 2b). The third prediction states that the heading bias should increase with the spatial offset between the object FoE and the background FoE. Because the offset increases with path angle, the heading bias should also increase with path angle. 
Note that the heading bias is not expected to increase linearly with path angle because the relation between path angle and object FoE offset is nonlinear. As illustrated in Figure 3a, object FoE offset increases rapidly with path angle when path angle is small but less rapidly as path angle increases (see Appendix B for details). As such, the heading bias should be more sensitive to changes in path angle at smaller path angles compared with larger path angles. 
Figure 3
 
(a) The object FoE plotted as a function of the path angle (black). The gray circles represent the path angles used in Experiment 1, and the gray dashed line shows the linear mapping between path angle and object FoE. (b) Results for Experiment 1. The mean heading bias is plotted in the Before Cross Far (light orange), Before Cross Near (dark orange), Cross (black), After Cross Near (dark blue), and After Cross Far (light blue) conditions for each path angle. Error bars show ±1 SEM. Positive heading bias indicates heading error in the direction of the object FoE. The object FoE angle is shown in (a) for comparison.
Figure 3
 
(a) The object FoE plotted as a function of the path angle (black). The gray circles represent the path angles used in Experiment 1, and the gray dashed line shows the linear mapping between path angle and object FoE. (b) Results for Experiment 1. The mean heading bias is plotted in the Before Cross Far (light orange), Before Cross Near (dark orange), Cross (black), After Cross Near (dark blue), and After Cross Far (light blue) conditions for each path angle. Error bars show ±1 SEM. Positive heading bias indicates heading error in the direction of the object FoE. The object FoE angle is shown in (a) for comparison.
The fourth prediction is that the influence of moving objects should be mediated by the structure of the optic flow within the contours of the object. Because of the assumed Gaussian weighting of heading templates (e.g., Warren & Saunders, 1995), the heading bias should be greater when the structure of the flow within the object is more radial compared to when it is more laminar. Because the optic flow within the object becomes more laminar at larger path angles (Figure 2a), the heading bias should flatten out and possibly even begin to diminish in the largest path angle conditions. 
Method
Participants
Twelve naïve subjects (nine men, three women) from Rensselaer Polytechnic Institute between the ages of 18 and 21 years participated in the study for course credit. All subjects had a valid driver's license and normal or corrected-to-normal vision. 
Visual displays
Subjects viewed displays (100° W × 80° H) of simulated self-motion (heading angle θ = ±5°, ±15°) along a straight path on a dot-defined ground plane (5,000 dots; 50-m depth). Negative and positive heading angles correspond to simulated self-motion into the left and right sides of the screen, respectively. The simulated observer had an average human eye height of 1.8 m. Translation of the observer was simulated at 5 m/s. The cylindrical moving object (1-m radius; 3-m height) initially subtended 11° vertically, consisted of 500 dots superimposed on a plane that matched the black sky, and started 21 m away from the observer in the Cross condition. 
The object moved to five positions relative to the observer's heading (Figure 2c): (a) the Before Cross Far condition, (b) the Before Cross Near condition, (c) the Cross condition, (d) the After Cross Near condition, and (e) the After Cross Far condition. In each condition, the object moved along the same relative trajectory to the observer but covered a different range of positions. Each condition was defined relative to the Cross condition, when the center of the object crossed the observer's path at the end of the trial, 2 m away from the observer. The starting and ending object positions were determined by temporally shifting the range of positions covered by the object relative to the Cross condition (τ = 0 s). The shifts were set to τ = −1.0, −0.5, 0.5, and 1.0 s in the Before Cross Far, Before Cross Near, After Cross Near, and After Cross Far conditions, respectively. For example, the Before Cross Far condition was tantamount to the Cross condition, except the object's starting and ending positions retreated to what they would be in the Cross condition, had the trial started earlier (see Appendix A). 
We manipulated the path angle (Figure 2b), the angular deviation between the object and observer trajectories (δ = ±5°, ±10°, ±20°, ±35, ±60°, ±90°). Positive and negative path angles indicate that the object approached the heading direction symmetrically from the right and left, respectively. To test this wide range of trajectories, we fixed the amount of time that the object occluded the observer's future path. We imposed this condition because if the object were to approach the locomotor path at a right angle (δ = ±90°) and move as quickly as it does when the path angle is small (e.g., δ = ± 5°), the object would cross the path for only a very short amount of time at the end of the trial. This may result in a reduced bias, not because of the object's trajectory but because of the limited occlusion time. Conversely, the object would cross the path very gradually for smaller path angles. Therefore, we covaried object speed with path angle to ensure that the object maintained contact with the observer's future path for a fixed amount of time, irrespective of the angle of approach. Specifically, the object moved at 22.94, 11.52, 5.85, 3.49, 2.31, or 2 m/s in the ±5°, ±10°, ±20°, ±35°, ±60°, and ±90° path angle conditions, respectively. Irrespective of the path angle, we fixed the final position of the object within each position condition relative to the heading direction (e.g., Cross Near). In the Before Cross Far, Before Cross Near, Cross, After Cross Near, and After Cross Far conditions, the center of the object was visually offset from the background FoE by 8°, 11°, 15°, 23°, and 26° at the beginning of the trial and offset by 45°, 32°, 0, 32°, and 45° at the end, respectively. 
The visual displays were generated in the WorldViz Vizard 3.0 environment on an Alienware Area 51 desktop computer equipped with two NVIDIA GeForce GTX 480 graphics cards, a 3.2-GHz Intel Core i7 processor, 6 GB of memory running Microsoft Windows 7 x64. The displays were projected onto a large rear-projection screen using a Barco Ciné 8 projector (1280 × 1024 resolution; 60-Hz refresh rate). Subjects sat in a chair approximately 1 m away from the rear-projection screen (100° W × 80° H) and viewed the visual displays binocularly in a dark room. Subjects were allowed to freely move their eyes during the experiment. 
The experimental protocol was approved by the Institutional Review Board at Rensselaer Polytechnic Institute and is in compliance with the Declaration of Helsinki. All subjects gave informed consent in writing before participating in the experiment. 
Procedure
At the beginning of the experiment, subjects completed a short basic heading judgment task that consisted of simulated travel along a straight path toward a backplane defined by dots. The practice block contained no independently moving objects and allowed subjects to become familiar with the experimental instructions. In both the practice block and the actual experiment, the first frame of each trial appeared for 0.5 s at the outset of each trial. At the end of each 1.5-s trial of simulated self-motion, a blue rectangular postmotion probe (4° V) appeared on the screen at a random horizontal position along with the final frame of the optic flow sequence. Subjects aligned the horizontal position of the probe with their perceived heading by manipulating a steering wheel. Turning the wheel left or right moved the probe in the corresponding direction, and subjects confirmed their probe placement by pressing a button located on the front of the steering wheel. Prior to the main experiment, subjects were notified about the presence of a moving cluster of dots that may appear similar to a moving object. Subjects were instructed to ignore the object as much as possible and base their heading judgment on the direction they are moving through the environment. No feedback was given during the practice or experimental trials. 
Subjects completed 720 trials, blocked by three repetitions of experimental conditions in a fully crossed design. For each repetition, the heading angle was randomly jittered around the specified angle (θ + X, X ∼ U(−2°, 2°). Each block consisted of 240 trials (5 object positions τ × 4 headings θ × 12 path angles δ). Experimental blocks were counterbalanced across subjects, and the whole experiment lasted less than 90 min. 
Bias correction
Consistent with other heading experiments (Royden & Hildreth, 1996; Warren & Hannon, 1988; Warren & Saunders, 1995), subject responses tended to exhibit a constant bias toward the center of the screen. The postmotion probe appeared in a random position at the end of each trial, so the center screen bias did not occur because of the initial position of the probe. Unless noted otherwise in the text, we report heading bias after correcting for the center screen bias. We calculated a center screen bias correction for each subject and environment by averaging all subject judgments garnered for a particular heading and subtracted that value from the data. That is, we computed the corrected response  where R is the response of subject s for heading θ, path angle δ, object position τ, and repetition r. The E[ ] symbol corresponds to the mean, taken with respect to δ, τ, and r.  
Results and discussion
Given symmetric performance for left and right headings and positive and negative path angles, we collapsed across heading and path angle sign. Figure 3b plots the mean heading bias as a function of path angle in the Before Cross Far (light orange), Before Cross Near (dark orange), Cross (black), After Cross Near (dark blue), and After Cross Far (light blue) conditions. Positive heading bias indicates that the error in human heading judgments is in the direction of the object FoE. Note that Figure 3b shows the heading bias after correcting for the center-screen bias (see the Bias Correction section). As such, zero heading bias indicates veridical performance or judgments that are globally offset by a fixed center-screen bias but not by the moving object. We verified sphericity in the data in all analyses using Mauchly's test. 
Recall that the object-based discrepancy account predicts that when the moving object occludes the heading direction and the object FoE position is spatially offset from that of the background FoE, heading perception should be biased toward the object FoE. Furthermore, the magnitude of the bias should increase with path angle because of the increasing discrepancy between the optic flow within the object and the background optic flow, as depicted in Figure 3a. The findings were largely consistent with these predictions. As depicted in Figure 3b, when the moving object influenced heading perception, the bias was positive, that is, in the direction of the object FoE, consistent with the findings of Warren and Saunders (1995). The heading bias was greater when the object occluded the background FoE at the end of the trial (Cross condition) or shortly before the end of the trial (After Cross conditions) compared with when the object did not occlude the background FoE at all (Before Cross conditions). This resulted in a significant main effect of object position, F(4, 44) = 3.74, p < 0.01, Image not available = 0.25. In addition, the magnitude of the heading bias increased with path angle in a manner that is consistent with the nonlinear increase in object FoE offset with path angle (Figure 3a). As expected, the effect of path angle was stronger in conditions in which the object crossed the future path (Cross, After Cross Near, After Cross Far) compared with when it did not (Before Cross Near, Before Cross Far), yielding a significant Object Position × Path Angle interaction, F(20, 220) = 6.39, p ≪ 0.001, Image not available = 0.37. The simple main effect of path angle was significant in the Cross, After Cross Near, and After Cross Far conditions but not in the Before Cross Near or Before Cross Far conditions.  
Although the third prediction was supported, we found no evidence to support the fourth prediction, which is that the heading bias should flatten out and possibly even begin to diminish at large path angles because of the more laminar structure of the optic flow within the object. To the contrary, the bias in the 90° path angle condition was consistently larger than (or at least as large as) it was in the other path angle conditions. This finding is difficult to reconcile with the assumed Gaussian weighting of heading templates (e.g., Warren & Saunders, 1995), which leads to reduced sensitivity to the kind of laminar flow that was present within the contours of the moving object in the larger path angle conditions. It could be that the flow within the object was still sufficiently radial to influence the activation of heading templates more closely aligned with the object FoE. In other words, the increase in heading bias with path angle captured by the third prediction may have offset the decrease in heading bias with path angle captured by the fourth prediction. Alternatively, heading perception may have been affected for some other reason in the larger path angle conditions; that is, there may be a cause for the influence of moving objects on heading perception other than discrepant optic flow within the object. We will return to this issue in Experiment 3
Before moving on to Experiment 2, it is worth briefly discussing the fact that the heading bias and the effect of path angle were observed even in the After Cross conditions, in which the background FoE was occluded by the object during the trial but was revealed before the trial once the object cleared the future path (see Figure 2c). This suggests that the bias can persist, at least for a brief period, even after the background FoE reappears. The finding is somewhat surprising in light of the conclusion from previous research that approaching objects influence heading perception only when they obscure the background FoE for a significant portion of the trial (Royden & Hildreth, 1996; Warren & Saunders, 1995). We interpret this finding as a reflection of a dynamic heading perception process that evolves over time rather than being based on the instantaneous optic flow field (Layton & Fajen, manuscript under review; Layton, Mingolla, & Browning, 2012). 
In summary, heading perception was biased when the object crossed the future path at or slightly before the end of the trial. The bias was in the direction of the object FoE and increased with path angle, consistent with the second and third predictions of the object-based discrepancy account. However, the bias was also present at large path angles when the optic flow within the object's contours was more laminar, which is inconsistent with the fourth prediction. 
Experiment 2: Final distance
In Experiment 2, we tested the fifth prediction of the object-based discrepancy account, which is that the influence of the moving object on heading perception should diminish to zero as the distance to the object increases. This prediction follows from the fact that as object distance increases, the size of the region of the visual field with discrepant optic flow decreases. To test this prediction, we manipulated the final distance in depth between the object and the observer (Figure 4). 
Figure 4
 
The projected size of the object in its final position compared with the dimensions of the visual display. The positions of the object are horizontally offset for visual clarity. The dashed lines in the 2-m condition indicate that part of the object was not visible in the visual display.
Figure 4
 
The projected size of the object in its final position compared with the dimensions of the visual display. The positions of the object are horizontally offset for visual clarity. The dashed lines in the 2-m condition indicate that part of the object was not visible in the visual display.
Method
Participants
Thirteen naïve subjects (eight men, five women) from the Rensselaer Polytechnic Institute between the ages of 18 and 25 years participated in the study for course credit. All subjects had a valid driver's license and normal or corrected-to-normal vision. 
Visual displays
The displays were similar to those used in Experiment 1 with the following exceptions. First, when the trial ended, the object was either directly ahead of the observer (identical to the Cross condition in Experiment 1) or its leading edge was aligned with the observer's future path, which we refer to as the Before Cross condition. Because our main focus in Experiment 2 was on the effect of distance, it was not necessary to vary the final object position as widely as we did in Experiment 1. However, it was also necessary to include more than one final object position so as not to encourage a strategy of aligning the probe with the object on every trial. Thus, the object's position at the end of the trial was on the future path in some trials and before the future path on other trials. Second, the object moved along one of four path angles (δ = ±15°, ±35°), where positive and negative values indicate symmetric trajectories that approach the observer's future path from the right and left, respectively. Third, the object trajectory was rigidly displaced in depth such that the object's distance at the end of the trial (d) was 2, 4, 8, 16, or 32 m. The object in its final position subtended 52°, 32°, 19°, 10°, and 5°, respectively (Figure 4). 
Procedure
Subjects completed 480 trials, blocked by three repetitions of experimental conditions in a fully crossed design. For each repetition, the heading angle was randomly jittered around the specified angle (θ + X, X ∼ U(−2°, 2°)). Each block consisted of 160 trials (5 object final distances d × 4 headings θ × 4 path angles δ × 2 object final positions τ). Experimental blocks were counterbalanced across subjects, and the whole experiment lasted less than 60 min. 
Results and discussion
As in Experiment 1, judgments were symmetric in heading and path angle sign, so we collapsed across these variables. Figure 5 shows the heading bias as a function of final object distance averaged across the 13 subjects, divided into two plots to focus separately on Cross (Figure 5a) and Before Cross (Figure 5b) conditions. As in Experiment 1, the mean heading bias in all conditions was positive, which indicates bias in the direction of the object FoE. 
Figure 5
 
Results from Experiment 2 in the Cross (a) and Before Cross (b) conditions. The heading bias, averaged across subjects, is plotted for each final distance between the observer and the object in depth. The green curve shows the bias for the 35° path angle and the orange curve shows the bias for the 15°. Error bars show ±1 SEM.
Figure 5
 
Results from Experiment 2 in the Cross (a) and Before Cross (b) conditions. The heading bias, averaged across subjects, is plotted for each final distance between the observer and the object in depth. The green curve shows the bias for the 35° path angle and the orange curve shows the bias for the 15°. Error bars show ±1 SEM.
Given previous results (including those of Experiment 1) indicating that the heading bias is strongest when the object crosses the future path, we will focus on judgments in the Cross condition. Perhaps the most striking result is that the heading bias was significantly greater than zero (p < 0.05) in all 10 conditions and in fact remained quite large (∼2°), even in the 32-m final distance condition. This is difficult to reconcile with the object-based account, which predicts accurate, unbiased heading perception when the object is far away. The Warren and Saunders (1995) model, which captures the influence of discrepant optic flow within the object, pools motion over a large area that mostly includes background optic flow when the object is far away. Such pooling serves to reduce the influence of small discrepant regions of optic flow. Thus, the fact that a small disturbance in the optic flow field resulted in a bias in heading perception is problematic for the object-based account. 
One could argue that the persistence of the effect in the larger distance conditions could be accommodated in the Warren and Saunders (1995) model by adjusting the parameters to make the templates more center weighted. This would increase the effect of the small region of discrepant optic flow from the moving object, which could account for the influence of moving objects at a distance. However, making the templates more center weighted would also decrease their sensitivity to laminar flow, which would be inconsistent with the bias induced by objects at larger path angles in Experiment 1. Thus, the parameter on the Gaussian filter that determines the degree of center weighting would have to change in opposite directions to account for the results of Experiments 1 and 2
The effect of final distance interacted with path angle, F(4, 48) = 2.95, p < 0.03, Image not available = 0.10. When final distance was small (2–4 m), the heading bias was significantly affected by path angle. This replicates the effect of path angle reported in Experiment 1 and is consistent with the object-based account. However, path angle did not affect heading bias in the 8-, 16-, and 32-m conditions. This is an important result because if heading perception in the 8-, 16-, and 32-m conditions was biased by the discrepant optic flow within the object, one would expect the bias to be larger in the 35° path angle condition because of the larger discrepancy. The fact that the effect of path angle was present in the 2- and 4-m conditions but not in the 8-, 16-, and 32-m conditions strongly suggests that the cause of the bias in the latter conditions is something other than the discrepant optic flow within the object. Taken together, the results from the Cross conditions were consistent with the object-based account when final distance was small (2–4 m). However, the findings suggest that heading may be biased for some other reason when the object is farther away.  
Our main focus was on conditions in which the object crossed the future path (i.e., the Cross condition), but we will briefly discuss judgments in the Before Cross condition. The overall mean heading bias was slightly smaller in the Before Cross condition than it was in the Cross condition, but the difference was not statistically significant, F(1, 12) = 1.13, p = 0.3. However, the bias was significantly greater than zero in all conditions, which is again inconsistent with the object-based account and provides additional evidence that heading is biased for some other reason when the object is farther away. The effect of path angle is consistent with the object-based account in the 4- and 8-m conditions. However, for reasons that we do not understand, there was no effect of path angle in the 2-m condition. 
In summary, Experiment 2 revealed that moving objects that cross or approach the future path influence heading perception, independent of the final distance between the observer and the object. The results were consistent with the object-based account when the object moved close to the observer in depth but suggest that heading perception is affected by some other factor when objects are farther away. 
Experiment 3: The pseudo FoE
The findings of Experiments 1 and 2 provide evidence that heading perception can be affected by discrepant optic flow within the object emanating from a location (the object FoE) other than the background FoE. However, we also found that heading perception was biased under conditions in which the optic flow within the object was more laminar and when the object was far away and hence occupied a small part of the visual field. This suggests that moving objects can influence heading perception for reasons other than the pooling of motion that includes discrepant optic flow radiating from a different location. Here, we consider the possibility that moving objects may influence heading perception because of discrepancies arising at the borders of the moving object rather than within the object. 
Figure 6a shows an object moving from right to left and crossing the future path at a large path angle. As such, the optic flow within its contours is less radial and more laminar. However, all of the optical motion on the interior of the object is in the leftward direction, and all of the optical motion from the background on the outside of the trailing edge is in the rightward direction. Furthermore, the motion from both the object and the background is upward above the horizon and downward below the horizon. Although the vectors from the object and background do not radiate from the same point, the pattern of motion near the trailing edge of the object may be sufficiently radial-like to influence heading perception. We refer to the point that lies at the intersection of the horizon and the trailing edge of the object (blue circle) as the “pseudo FoE.” 
Figure 6
 
The setup for Experiment 3 that tests whether flow discrepancies arising at the borders rather than within the moving object can influence heading perception. Under certain circumstances, one such discrepancy creates radial-like motion contrast between the object and the background at the trailing edge of the moving object (pseudo FoE). (a) In the Object Condition, there is motion contrast between the trailing edge of the object (red arrowheads) and the background (blue arrowheads). In the example depicted, the object occludes the background FoE and the pseudo FoE effect is strong because of the radial-like motion around the trailing edge of the object. (b) In the Blank Object condition, a gap is introduced between the trailing edge of the object and the background (black) to eliminate the motion contrast. The blank Object traveled with the object during the trial. (c) The trajectory of the object was rigidly displaced horizontally in eight different locations relative to the observer's future path (lateral offset). When the lateral offset was zero, the trailing edge made tangential contact with the future path at the end of the trial. Positive and negative lateral offsets correspond to rightward or leftward shifts in the trajectory.
Figure 6
 
The setup for Experiment 3 that tests whether flow discrepancies arising at the borders rather than within the moving object can influence heading perception. Under certain circumstances, one such discrepancy creates radial-like motion contrast between the object and the background at the trailing edge of the moving object (pseudo FoE). (a) In the Object Condition, there is motion contrast between the trailing edge of the object (red arrowheads) and the background (blue arrowheads). In the example depicted, the object occludes the background FoE and the pseudo FoE effect is strong because of the radial-like motion around the trailing edge of the object. (b) In the Blank Object condition, a gap is introduced between the trailing edge of the object and the background (black) to eliminate the motion contrast. The blank Object traveled with the object during the trial. (c) The trajectory of the object was rigidly displaced horizontally in eight different locations relative to the observer's future path (lateral offset). When the lateral offset was zero, the trailing edge made tangential contact with the future path at the end of the trial. Positive and negative lateral offsets correspond to rightward or leftward shifts in the trajectory.
The pseudo FoE could account for the fact that heading judgments in Experiment 1 were biased even in the large path angle conditions. When path angle was sufficiently large that the optic flow within the object was more laminar, the pseudo FoE was well defined (see the right panel of Figure 2a). Thus, the presence of the bias in the large path angle conditions in Experiment 1 can be attributed to the pseudo FoE. Likewise, in Experiment 2, as final object distance increased and the object occupied a smaller visual angle, the optic flow within the object appeared more laminar, giving rise to a pseudo FoE at the trailing edge. This could explain why heading judgments were biased even when the object was far away. Lastly, the absence of a path angle effect in the 8-, 16-, and 32-m conditions is consistent with an influence of the pseudo FoE, because the position of the pseudo FoE was very similar (trailing edge of the object) toward the end of the trial in both the 15° and 35° path angle conditions. 
Although the results of Experiments 1 and 2 are consistent with the hypothesis that heading perception can be influenced by the pseudo FoE, a direct test is needed. This was the aim of Experiment 3. Figure 6a depicts one of the two conditions (the Object condition), wherein radial-like motion contrast exists at the border between the object and background (i.e., the pseudo FoE is present). If the pseudo FoE influences heading perception, judgments should be biased even in conditions in which the optic flow within the object is more laminar. Furthermore, reducing the motion contrast between the object and the background should decrease the effect. As illustrated in Figure 6b (Blank Object condition), the introduction of an opaque blank object the same color as the background that trails behind the moving object should decrease the heading bias. (Note that the contours of the object are shown in Figure 6b for the purposes of illustration but were not visible in the actual stimuli.) This is a strong prediction because by introducing a blank “gap” between the background and the object, we effectively remove a region of the visual field that would otherwise contain background optic flow and hence provide reliable information about heading direction. As such, one might expect heading perception to be less accurate in the Blank Object condition. However, if heading perception is influenced by the pseudo FoE, then the blank object should actually reduce the bias and make heading perception more accurate. 
In Experiment 3, we used a single path angle and laterally shifted the trajectory (lateral offset; Figure 6c) of the object. Conditions in which the lateral offset was small yielded radial-like motion contrast between the object and background, giving rise to the pseudo FoE. It is under this set of conditions that the main prediction (i.e., weaker bias in the Blank Object condition) should hold. In conditions with a large positive lateral offset, the optic flow within the object was radial. As such, heading judgments should be similar in the Object and Blank Object conditions. Lastly, there should be no heading bias when the background FoE is visible and when the trailing edge of the object is not near the background FoE (large negative lateral offset). 
Method
Participants
Twelve naïve subjects (eight men, four women) from the Rensselaer Polytechnic Institute between the ages of 18 and 22 years participated in the study for course credit. All subjects had a valid driver's license and normal or corrected-to-normal vision. 
Visual displays
The displays were identical to those used in the Cross condition of Experiment 1, except for the following modifications. The object approached the observer's trajectory symmetrically from either side at a fixed angle (δ = ±45°) and stopped 4 m in depth. The object trajectory was rigidly displaced laterally (lateral offset l) such that the object stopped at one of eight horizontal positions relative to the observer's future path (l = −1.75, −1.25, −0.75, −0.25, 0.0, 0.25, 0.75, 1.25 m; Figure 6c). The reference lateral offset (0 m) was similar to the Cross condition, except the trailing edge of the object was tangent to the observer's future path (τ = 0.5s). Positive lateral offsets indicate that the object trajectory was displaced rightward, and negative offsets indicate a leftward shift in the trajectory. 
In the Blank Object condition, a 0.6-m radius black cylinder devoid of dots was positioned 1.5 m behind the moving object and rotated 70° about the center of the moving object relative to the z-axis. During the trial, the black cylinder moved with the same velocity and for the same amount of time as the moving object. 
Procedure
Subjects completed 384 trials, blocked by three repetitions of experimental conditions in a fully crossed design. For each repetition, the heading angle was randomly jittered around the specified angle (θ + X, X ∼ U(−2°, 2°)). Each block consisted of 128 trials (8 lateral offsets l × 4 headings θ × 2 path angles δ × 2 object conditions). Experimental blocks were counterbalanced across subjects, and the whole experiment lasted less than 60 min. 
Results and discussion
Performance was symmetric across heading and path angle sign, which allowed us to collapse across these variables. Figure 7 shows heading bias as a function of object lateral offset for the Object (blue) and Blank Object (black) conditions. Positive heading bias indicates heading errors in the direction of the object FoE. A two-way (Object Condition × Lateral Offset) repeated-measures analysis of variance revealed significant main effects of both variables, F(1, 11) = 10.30, p < 0.01, Image not available = 0.48, for object condition, and F(7, 77) = 4.67, p < 0.001, Image not available = 0.30, for lateral offset. The interaction did not reach significance, F(7, 77) = 1.88, p = 0.08, Image not available = 0.15.  
Figure 7
 
The heading bias, averaged across subjects, from Experiment 3 in the Object (blue) and Blank Object condition (black) as a function of the lateral offset of the object. Error bars show ±1 SEM, and stars indicate a significant difference between heading bias garnered in the Object and Blank Object conditions (p < 0.05). Data points are partitioned into three groups and labeled depending on whether the optic flow within the object is more radial (“Radial flow inside object”), the pseudo FoE effect is strong (“Pseudo FoE”), or neither applies (“Neither”). Sample optic flow for conditions in each group is shown above the curves.
Figure 7
 
The heading bias, averaged across subjects, from Experiment 3 in the Object (blue) and Blank Object condition (black) as a function of the lateral offset of the object. Error bars show ±1 SEM, and stars indicate a significant difference between heading bias garnered in the Object and Blank Object conditions (p < 0.05). Data points are partitioned into three groups and labeled depending on whether the optic flow within the object is more radial (“Radial flow inside object”), the pseudo FoE effect is strong (“Pseudo FoE”), or neither applies (“Neither”). Sample optic flow for conditions in each group is shown above the curves.
If the pseudo FoE affects heading perception, then the heading bias should be greater than zero in conditions in which the pseudo FoE is present (i.e., the middle set of conditions in Figure 7). Furthermore, the heading bias should be smaller in the Blank Object condition because the Blank Object occluded the region of the visual field that contains the pseudo FoE. As indicated in Figure 7, the data were consistent with both predictions: The heading bias was greater than zero in the −0.75-, −0.25-, 0-, and 0.25-m lateral offset condition and was significantly smaller in the Blank Object condition (p < 0.05). 
To rule out the possibility that the aforementioned difference between the Object and Blank Object conditions is due to something other than the pseudo FoE, we can compare the heading bias in these two conditions when the pseudo FoE is not present. In that case, the heading bias in the Object and Blank Object conditions should be similar. The findings from the −1.75, −1.25, 0.75, and 1.25 conditions support this prediction. Heading judgments were not significantly biased and were unaffected by the presence of the blank object in the −1.75 and −1.25 conditions (left side of Figure 7), when the optic flow within the object was more laminar and the pseudo FoE was not defined. In the 0.75 and 1.25 conditions, heading judgments were biased in the direction of the object FoE, which was expected given that the optic flow within the object was more radial (right side of Figure 7). However, there were no significant differences between the Object and Blank Object conditions: l = 0.75, t(23) = −0.47, p > 0.32; l = 1.25, t(23) = −0.38, p > 0.35. 
Together, the results from Experiment 3 provide compelling evidence that humans' heading judgments may be biased because of two factors: (a) optic flow within the object radiating from the object FoE and (b) the pseudo FoE near the border of the object. When the optic flow within the interior of the object is more radial, such motion influences heading perception due to center-weighted spatial pooling. On the other hand, heading judgments are also influenced by the pseudo FoE when the movement of the object against the background results in a radial-like pattern of motion. 
General discussion
We conducted three experiments to test the hypothesis that heading perception in the presence of moving objects that approach in depth is biased due to discrepant optic flow radiating from a location (object FoE) other than the background FoE. We manipulated the relative trajectory between the observer and object (path angle δ), which in turn varied the object FoE offset and the structure of the flow within the object and the final distance between observer and the object, which affected the spatial extent of the region with discrepant optic flow. The object-based account was not sufficient to explain the pattern of heading bias in Experiments 1 and 2. The results were consistent with an influence of optic flow within the object only for small path angles and when the object moved close to the observer. In Experiment 1, we observed the largest heading bias (∼4°) when the optic flow within the object was more laminar. This is contrary to the prediction that the bias should attenuate under such conditions (although we cannot rule out the possibility that such attenuation was counteracted by the increase in spatial offset between the object FoE and background FoE). Despite the diminished optical size of the object at far final distances in Experiment 2, the bias did not decrease compared to when the object moved near the observer. These findings are difficult to reconcile with the hypothesis that discrepant optic flow within the object is the sole source of the heading bias induced by moving objects. 
In Experiment 3, we considered the possibility that heading perception can also be affected by a specific type of border-based discrepancy that we call the pseudo FoE. The pseudo FoE emerges when the relative motion near the border of the object and the background appears radial enough to give the impression of an illusory or false FoE. We tested whether the pseudo FoE accounts for the bias by introducing a gap between the trailing edge of the object and the background to eliminate the motion contrast. In support of our prediction, the heading bias that was predicted by the pseudo FoE effect was significantly reduced when the motion contrast at the trailing edge was occluded by the gap. 
An important implication of the pseudo FoE effect is that moving objects can influence heading perception across a wider range of conditions than previously believed. The object-based account predicts that objects influence heading perception only when they occupy a large portion of the visual field and when the optic flow within the object is more radial. However, objects that occupy a smaller visual angle and generate more laminar optic flow may give rise to a pseudo FoE in the optic flow field that can also influence heading perception. In other words, moving objects can influence heading perception even when they are farther away and moving at a larger path angle. 
Mechanisms underlying the pseudo FoE effect
Although the two sources of bias that were highlighted in the present study are qualitatively different in terms of the stimulus information, the underlying neural mechanisms may be the same. Like discrepant optic flow within the object radiating from the object FoE, the pseudo FoE may activate neurons in the primate visual area MSTd that ordinarily respond to radial patterns associated with heading during self-motion in a rigid environment (Duffy & Wurtz, 1991, 1995; Gu, DeAngelis, & Angelaki, 2012). The activation of heading-sensitive cells may occur because the optic flow around the pseudo FoE may appear sufficiently radial. It is clear from the emerging physiology that the neurons in MSTd have far more complex receptive fields than the iconic radial structure that is often assumed (Mineault, Khawaja, Butts, & Pack, 2012). Across the MSTd population, the response of neurons may well tolerate deviations from the radial flow field. Thus, regardless of whether the flow discrepancy is object based or border based, the diverse population of MSTd cells may generate a biased heading signal so long as flow produced during self-motion in the presence of independently moving objects appears sufficiently radial. 
Because the Warren and Saunders (1995) model contains templates with characteristics similar to MSTd cells, it may also yield heading estimates that are influenced by the pseudo FoE. Under conditions in which the pseudo FoE was present, the template centered on the radial-like motion contrast near the trailing edge may become the most active template, especially when the object occludes the background FoE. However, whether the model captures the effect of the blank object in Experiment 3 depends on its parameterization. In our simulations of the Warren and Saunders model, the model is weakly affected by the blank object when the units are parameterized to integrate motion over a large spatial extent. If the model units are configured to pool motion over a region that is smaller than the extent of the gap introduced by the blank object, the bias attenuates in the Blank Object condition. In this case, the response diminishes in those center-weighted units whose receptive fields are centered near the border that induces the bias. This shifts the maximal response toward the object, closer to the heading direction, which reduces the bias. However, although a template model configured in this manner may better account for the pseudo FoE effect, limiting the extent of spatial integration is at odds with the well-documented properties of heading-sensitive cells in primate MSTd, which often have receptive fields that span much of the visual field and would likely be weakly affected or unaffected by the blank object manipulation (Duffy & Wurtz, 1991). 
There are two additional features of the pseudo FoE that are problematic for the Warren and Saunders (1995) model. The first is the fact that its position relative to the background FoE changes as the object sweeps across the observer's visual field. Nonetheless, one's impression when viewing stimuli containing the pseudo FoE is that the heading is stable rather than continually changing, as if moving along a curved path. That is, one does not perceive that heading is drifting with the motion of the pseudo FoE at the trailing edge. 
The second observation that deserves consideration is that the pseudo FoE influenced heading perception even in trials in which the directions of the motion vectors at the trailing edge of the object did not contrast at the end of the trial (i.e., in the −0.75 and −0.25 conditions of Experiment 3). In these conditions, the motion at the trailing edge was in opposing directions, forming a pseudo FoE, during the early part of the trial. However, once the trailing edge crossed the observer's future path, the background motion was in the same direction as the object motion. Furthermore, the background FoE was no longer occluded by the object. Nonetheless, the bias in heading perception persisted in these conditions. 
We believe that these two phenomena may be related and may reflect temporal hysteresis in the visual system. An implicit assumption of most computational models of heading perception is that heading estimates are based on the instantaneous optic flow field. Recently, Layton and colleagues (2012) designed a dynamical neural model that exhibits temporal hysteresis and yields heading estimates that are more stable than estimates based on the instantaneous optic flow field. Model units simulate the dynamics of neurons in the dorsal medial superior temporal area (MSTd) of the primate cortex that are sensitive to radial expansion (Layton & Browning, 2012), and the responses exhibit a similar pattern of heading bias as humans in the presence of independently moving objects (Layton et al., 2012). Heading estimates produced by the model would be biased by the pseudo FoE because units would respond most strongly to the illusory radial-like motion contrast between the trailing edge of the object and the background when the background FoE is not visible and optic flow within the object is more laminar. 
Temporal dynamics may allow the model to explain the strong pseudo FoE effect, even after the trailing edge of the object disoccludes the background FoE. As a dynamical system, the model's estimates reflect the evolution of the optic flow field over time rather than at a single instant. If the model produces a heading estimate in the direction of the pseudo FoE at the trailing edge of the object, and the object crosses the path to reveal the background FoE, units in the network would require time to respond to the background FoE. In other words, the bias due to pseudo FoE would persist for some time until the response to the background FoE begins to dominate. The model prediction that the visual system develops a heading estimate by integrating optic flow over time has been supported by recent findings showing that human heading judgments depend on the time history of the optic flow field, rather than any instant (Layton & Fajen, manuscript under review). Whether the model of Layton et al. can explain the large heading bias at extreme path angles (Experiment 1) and the bias at far distances (Experiment 2) remains an open question and will be the topic of future investigation. 
The pseudo FoE effect and differential motion
As mentioned in the Introduction, Royden's (2002) differential motion model also attributes the heading bias induced by moving objects to discrepancies arising at the border between the object and the background. In this section, we consider how the differential motion model may account for the border-based effects reported in the present study. The defining characteristic of the pseudo FoE is motion in opposing directions, which would activate the model's local opponent motion operators with receptive fields near the trailing edge of the object. However, differential motion is also found at the leading edge of the object (see Figure 6a), activating opponent motion operators with receptive fields in that region as well. A key feature of the opponent motion operators in the Royden model is that their response is multiplicatively related to the difference in speeds within the receptive field (see equation on p. 3048 of Royden, 2002). If the object is close to crossing the path, which is the case when the pseudo FoE effect is strongest, the speed difference between the object and background is considerably greater at the leading edge of the object than it is at the trailing edge (Figure 6a). This discrepancy in the speed differences means that heading estimates in this scenario are driven by difference vectors at the leading rather than the trailing edge of the object. As such, manipulation of the difference vectors at the trailing edge, such as that which occurred in the Blank Object condition, would have a negligible effect on the heading estimate. In other words, the differential motion model would not capture the effect of the blank object in Experiment 3. Our simulations of the Royden model confirm that this is indeed the case—model heading estimates did not differ in the Object and Blank Object conditions (Layton & Fajen, in press). 
The pseudo FoE and the influence of fixed-depth objects
In the present study, the focus was on heading perception in the presence of objects that approach the observer in depth, which create localized patterns of radial expansion. However, as mentioned in Experiments 1 and 2, at extreme path angles or at far distances, the flow can appear more laminar than radial, making the pattern in some cases resemble that which is generated when objects remain at a fixed depth, as in Royden and Hildreth (1996). The flow patterns are nevertheless distinct because the objects in Royden and Hildreth have only horizontal flow on their interiors, whereas the objects in the present study also have a vertical motion component.4 The difference in the flow patterns created by the objects is apparently enough to induce qualitatively different influences on heading perception. The pseudo FoE effect clearly does not occur in the stimuli of Royden and Hildreth, otherwise, the direction of the bias in their study would not be in the direction of object motion. The differing direction of the bias and ∼2× greater magnitude of the pseudo FoE effect suggest different mechanisms are responsible for the bias or the same mechanisms process the objects in qualitatively different ways. Layton et al. (2012) showed how a single MSTd network with recurrent competition can account for the different patterns of human heading bias caused by the objects of Warren and Saunders (1995) and Royden and Hildreth (1996). The approaching objects of Warren and Saunders (1995) created a broad distribution of activity across the template-matching MSTd layer, biased toward the object FoE, whereas the object of Royden and Hildreth (1996) created a sharp peak in the direction of object motion. Future investigation will elucidate whether heading estimates produced by the Layton model are influenced by the pseudo FoE or if separate mechanisms are needed. 
Conclusions
Discrepant optic flow within an approaching object and at the borders of the object represent potential sources of heading bias that arise under different circumstances during self-motion in dynamic environments. When an object approaches the observer at a small angle and occupies a sufficiently large part of the visual field that the optic flow within the object is more radial, heading perception is biased toward the object FoE. At larger path angles and when the object is farther away, the optic flow within the object is more laminar but heading perception is biased by the pseudo FoE at the trailing edge of the object. Thus, moving objects can bias heading perception due to both object-based and border-based discrepancies. 
Acknowledgments
The authors thank Ennio Mingolla and Arash Yazdanbakhsh for generously supporting Oliver Layton as a visiting researcher at Rensselaer Polytechnic Institute during part of the period in which this research was conducted. This work was supported by grants from the Office of Naval Research (ONR N00014-11-1-0535 and N00014-14-1-0359). 
Commercial relationships: none. 
owl@bu.edu 
Corresponding author: Oliver W. Layton. 
Email: owl@bu.edu. 
Address: Department of Cognitive Science, Rensselaer Polytechnic Institute, Troy, NY, USA. 
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Footnotes
1  Some of the predictions are also consistent with an influence of border-based motion contrast. However, the predictions for this account are more complicated because there is more than one way in which border-based discrepancies can influence heading perception. Furthermore, our aim in Experiments 1 and 2 is to determine whether the heading bias due to approaching objects can be entirely attributed to object-based discrepancies. As such, it is not necessary to identify the predictions of the border-based account at this point.
Footnotes
2  The retreating object of Royden and Hildreth (1996) could be thought to move with a more extreme path angle than those considered in the present study (>90°), and indeed, there was no evidence in that study that the discrepant optic flow activated heading templates in the direction from which the object came. In fact, the heading bias was smaller (≤1°) than it was for the approaching objects (2–4°) in Warren and Saunders (1995) and in the opposite direction.
Footnotes
3  Recall that the first two predictions have already been tested and are supported by existing data.
Footnotes
4  The objects used in the present study always have a finite object FoE position, while that of the objects in Royden and Hildreth (1996) is infinite.
Appendix A
Analytic Derivations
Object position
In this section, we derive the initial and final positions of the moving object in the object position conditions (Before Cross Far, Before Cross Near, Cross, After Cross Near, and After Cross Far). Calculations are first performed for the Cross condition. Let us first consider the movement of an observer along a heading θ through the xz plane (Figure A1a). We assume that the observer begins at the origin, the observer's central axis remains parallel to the z-axis throughout self-motion (no rotation), and θ = 0 defines movement along the z-axis. Points (x, z) along the observer's path are given by    
Figure 1
 
Plan view of the placement of the moving object relative to the observer's path in the Cross condition. See Appendix A for variable definitions. (a) Overview of the xz coordinate system that is centered on the observer's initial position. The gray and blue dashed lines indicate the trajectories of the observer and cylindrical object, respectively, if movement continued beyond the end of the trial. Arrows indicate the movement of the observer and object from their initial to final positions during the trial. (b) The moment when the object makes tangential contact with the observer's future path in the x′z′ coordinate system centered on the object's final position. (c) Trigonometric relationship between the final position of the object and the moment when the object makes tangential contact with the observer's future path.
Figure 1
 
Plan view of the placement of the moving object relative to the observer's path in the Cross condition. See Appendix A for variable definitions. (a) Overview of the xz coordinate system that is centered on the observer's initial position. The gray and blue dashed lines indicate the trajectories of the observer and cylindrical object, respectively, if movement continued beyond the end of the trial. Arrows indicate the movement of the observer and object from their initial to final positions during the trial. (b) The moment when the object makes tangential contact with the observer's future path in the x′z′ coordinate system centered on the object's final position. (c) Trigonometric relationship between the final position of the object and the moment when the object makes tangential contact with the observer's future path.
If the observer moves at a constant speed (sobs) for the duration of the trial (ttrial), then the observer's final position is    
The first constraint on the placement of the object is that the object is centered at a depth (d) from the observer's final position along the future path. As Figure A1a shows, the final position of the object (xf, zf) can be determined from trigonometry as    
The second constraint on the placement of the object is that the object's leading edge makes tangential contact with the observer's future path when one third of the trial remains. To show when this occurs, we must determine the conditions whereby a point around the perimeter of the cylindrical object first makes tangential contact with the observer's future path (Figure A1b). For convenience, let us define the coordinate axes x′z′ centered on (xf, zf), the point at which the center of the object crosses the observer's future path, and assume that the object's path angle (δ) is defined relative to the heading (θ). For the present derivation, we focus on the case in which the object crosses the observer's path from the right; Similar results hold by symmetry if the object crosses from the left. The straight line trajectory of the object that coincides with its center can be parameterized as (u, ucot(δ + θ)), with respect to the nonnegative parameter u. To determine when the object of radius (r) makes tangential contact with the observer's future path, we compute the intersection between the observer's path (Equation A1) and the circle centered on the parameterized trajectory of the moving object:  The solution to this system of equations (x*, z*) is given by   The point at which the object is tangent to the observer's path is determined by values of u for which Equations A5A6 are well defined. This must occur when the expression inside the radical is nonnegative. Solving r2 csc2θu2(cotθ − cot(δ + θ))2 ≥ 0 for u yields  In general, both the positive and negative solutions must be considered when solving for u. However, because u is a nonnegative value and rcscδ sin(δ + θ) is always positive for the conditions used in our study, it is not necessary to consider the negative solution when taking the square root.  
The object first crosses the observer's path at the point of equality in Equation A7:    
The third constraint on the object placement is that the object speed should depend on the path angle such that the period of time (T) the object occludes the observer's future path is constant. Let us define T = ttrialtcrit, where tcrit is the point of time in the trial when the object makes tangential contact with the observer's future path (tcrit = 0.5 s). Given that x = u and the distance (d′) of the object's center from the origin in the xz′ coordinate system is u*csc(δ + θ) (Figure A1c), the speed of the object (sobj) is    
The initial position of the object (xi, zi) then is given by    
Recall that the preceding calculations (Equations A1A10) use constraints for the Cross condition and the other object position conditions are time-shifted versions. The object's path and speed in the other object position trials are identical to those in the Cross condition, except the object makes contact with the observer's path at different times. Let us define τ as the temporal offset relative to the Cross condition (τ = 0 s). For example, when (τ = 0.5 s), the object arrives at any point along its trajectory 0.5 s sooner than in the Cross condition. The following equation defines the position of the object at time t for each condition by substituting ttrialttrialtτ in Equation A10:  where the temporal offsets τ = −1.0, −0.5, 0, 0.5, and 1.0 s define the Before Cross Far, Before Cross Near, Cross, After Cross Near, and After Cross Far, respectively.  
Appendix B
Object FoE
Because both the observer and the object moved, and the object speed and path angle covaried in Experiment 1, the shift in the object's FoE was a nonlinear function of path angle (Figure 3a). Therefore, a large compression in the object FoE shift occurs for larger path angles. As indicated by the superimposed plot markers in Figure 3a, our selection of path angles in Experiment 1 samples the object FoE position space in approximately constant increments. 
The observer's self-motion (solid black arrow) and object motion (solid gray arrow) components are depicted in the left panel of Figure A2. In object coordinates the relative motion of the object (dashed arrow) is shown as the vector sum of the observer and object motion components. The angle between the observer's heading and the resultant object motion in object coordinates specifies the relative direction of the object's FoE. Note that as the observer's heading changes, the magnitude of the resultant object motion vector remains constant (just the direction rigidly rotates) because the object position and direction of motion also change to keep the object's path angle the same. 
Figure 2
 
The relationship between the object FoE position and the path angle of the object. (a, left panel) Depiction of the self-motion and object motion directions in world coordinates. (a, right panel) The object FoE can be expressed as the angle between the observer's motion vector and the vector representing the difference between the observer and object motion vector (“resultant” vector) in object coordinates.
Figure 2
 
The relationship between the object FoE position and the path angle of the object. (a, left panel) Depiction of the self-motion and object motion directions in world coordinates. (a, right panel) The object FoE can be expressed as the angle between the observer's motion vector and the vector representing the difference between the observer and object motion vector (“resultant” vector) in object coordinates.
To compute the position of the object FoE, consider the movement of the observer in object coordinates shown in Figure A2. The resultant speed sr and object FoE angle θobj are given by the law of cosines:     
The specific object FoE offsets used in Experiment 1 were ±4.1°, ±6.98°, ±10.79°, ±14.28°, ±18.0°, and ±21.8° in the ±5°, ±10°, ±20°, ±35°, ±60°, and ±90° path angle conditions. 
Figure 1
 
(a) The self-motion of an observer along a straight path toward a frontoparallel back plane yields a radial optic flow field (blue arrowheads). We refer to the point of the outflow as the background focus of expansion (FoE). The angular difference between the actual and perceived heading direction is called heading error. (b) When the observer moves in the presence of an independently moving object (red arrowheads) that approaches in depth, the object produces a secondary FoE (object FoE). The object may cross the observer's path and occlude the background FoE and create uncertainty about the heading direction. Heading error is systematically biased toward the object FoE.
Figure 1
 
(a) The self-motion of an observer along a straight path toward a frontoparallel back plane yields a radial optic flow field (blue arrowheads). We refer to the point of the outflow as the background focus of expansion (FoE). The angular difference between the actual and perceived heading direction is called heading error. (b) When the observer moves in the presence of an independently moving object (red arrowheads) that approaches in depth, the object produces a secondary FoE (object FoE). The object may cross the observer's path and occlude the background FoE and create uncertainty about the heading direction. Heading error is systematically biased toward the object FoE.
Figure 2
 
Schematic depiction of the conditions of Experiment 1 that test whether the discrepant optic flow within the object suffices in accounting for the heading bias that arises during self-motion in the presence of approaching objects. The object-based account predicts that the heading bias should increase with the spatial offset between the object FoE and the background FoE (see predictions 3 and 4). (a) Optic flow fields for several different path angles (i.e., angle between the object trajectory and observer's heading). Optic flow due to the object is depicted in red and background flow is depicted in blue. Manipulating the path angle moves the position of the object FoE. (b) Top-down view of the experimental setup of Experiment 1 and the path angles used. The blue line indicates the observer's future path and the red arrow indicates the object trajectory. (c) The five different positions of the object at the end of the trial are shown relative to the observer's heading. In the Cross condition, the object was centered on the future path at the end of the trial. The conditions to the left indicate that the object crossed and cleared the path by the end of the trial, and conditions to the right indicate that the object did not cross the path.
Figure 2
 
Schematic depiction of the conditions of Experiment 1 that test whether the discrepant optic flow within the object suffices in accounting for the heading bias that arises during self-motion in the presence of approaching objects. The object-based account predicts that the heading bias should increase with the spatial offset between the object FoE and the background FoE (see predictions 3 and 4). (a) Optic flow fields for several different path angles (i.e., angle between the object trajectory and observer's heading). Optic flow due to the object is depicted in red and background flow is depicted in blue. Manipulating the path angle moves the position of the object FoE. (b) Top-down view of the experimental setup of Experiment 1 and the path angles used. The blue line indicates the observer's future path and the red arrow indicates the object trajectory. (c) The five different positions of the object at the end of the trial are shown relative to the observer's heading. In the Cross condition, the object was centered on the future path at the end of the trial. The conditions to the left indicate that the object crossed and cleared the path by the end of the trial, and conditions to the right indicate that the object did not cross the path.
Figure 3
 
(a) The object FoE plotted as a function of the path angle (black). The gray circles represent the path angles used in Experiment 1, and the gray dashed line shows the linear mapping between path angle and object FoE. (b) Results for Experiment 1. The mean heading bias is plotted in the Before Cross Far (light orange), Before Cross Near (dark orange), Cross (black), After Cross Near (dark blue), and After Cross Far (light blue) conditions for each path angle. Error bars show ±1 SEM. Positive heading bias indicates heading error in the direction of the object FoE. The object FoE angle is shown in (a) for comparison.
Figure 3
 
(a) The object FoE plotted as a function of the path angle (black). The gray circles represent the path angles used in Experiment 1, and the gray dashed line shows the linear mapping between path angle and object FoE. (b) Results for Experiment 1. The mean heading bias is plotted in the Before Cross Far (light orange), Before Cross Near (dark orange), Cross (black), After Cross Near (dark blue), and After Cross Far (light blue) conditions for each path angle. Error bars show ±1 SEM. Positive heading bias indicates heading error in the direction of the object FoE. The object FoE angle is shown in (a) for comparison.
Figure 4
 
The projected size of the object in its final position compared with the dimensions of the visual display. The positions of the object are horizontally offset for visual clarity. The dashed lines in the 2-m condition indicate that part of the object was not visible in the visual display.
Figure 4
 
The projected size of the object in its final position compared with the dimensions of the visual display. The positions of the object are horizontally offset for visual clarity. The dashed lines in the 2-m condition indicate that part of the object was not visible in the visual display.
Figure 5
 
Results from Experiment 2 in the Cross (a) and Before Cross (b) conditions. The heading bias, averaged across subjects, is plotted for each final distance between the observer and the object in depth. The green curve shows the bias for the 35° path angle and the orange curve shows the bias for the 15°. Error bars show ±1 SEM.
Figure 5
 
Results from Experiment 2 in the Cross (a) and Before Cross (b) conditions. The heading bias, averaged across subjects, is plotted for each final distance between the observer and the object in depth. The green curve shows the bias for the 35° path angle and the orange curve shows the bias for the 15°. Error bars show ±1 SEM.
Figure 6
 
The setup for Experiment 3 that tests whether flow discrepancies arising at the borders rather than within the moving object can influence heading perception. Under certain circumstances, one such discrepancy creates radial-like motion contrast between the object and the background at the trailing edge of the moving object (pseudo FoE). (a) In the Object Condition, there is motion contrast between the trailing edge of the object (red arrowheads) and the background (blue arrowheads). In the example depicted, the object occludes the background FoE and the pseudo FoE effect is strong because of the radial-like motion around the trailing edge of the object. (b) In the Blank Object condition, a gap is introduced between the trailing edge of the object and the background (black) to eliminate the motion contrast. The blank Object traveled with the object during the trial. (c) The trajectory of the object was rigidly displaced horizontally in eight different locations relative to the observer's future path (lateral offset). When the lateral offset was zero, the trailing edge made tangential contact with the future path at the end of the trial. Positive and negative lateral offsets correspond to rightward or leftward shifts in the trajectory.
Figure 6
 
The setup for Experiment 3 that tests whether flow discrepancies arising at the borders rather than within the moving object can influence heading perception. Under certain circumstances, one such discrepancy creates radial-like motion contrast between the object and the background at the trailing edge of the moving object (pseudo FoE). (a) In the Object Condition, there is motion contrast between the trailing edge of the object (red arrowheads) and the background (blue arrowheads). In the example depicted, the object occludes the background FoE and the pseudo FoE effect is strong because of the radial-like motion around the trailing edge of the object. (b) In the Blank Object condition, a gap is introduced between the trailing edge of the object and the background (black) to eliminate the motion contrast. The blank Object traveled with the object during the trial. (c) The trajectory of the object was rigidly displaced horizontally in eight different locations relative to the observer's future path (lateral offset). When the lateral offset was zero, the trailing edge made tangential contact with the future path at the end of the trial. Positive and negative lateral offsets correspond to rightward or leftward shifts in the trajectory.
Figure 7
 
The heading bias, averaged across subjects, from Experiment 3 in the Object (blue) and Blank Object condition (black) as a function of the lateral offset of the object. Error bars show ±1 SEM, and stars indicate a significant difference between heading bias garnered in the Object and Blank Object conditions (p < 0.05). Data points are partitioned into three groups and labeled depending on whether the optic flow within the object is more radial (“Radial flow inside object”), the pseudo FoE effect is strong (“Pseudo FoE”), or neither applies (“Neither”). Sample optic flow for conditions in each group is shown above the curves.
Figure 7
 
The heading bias, averaged across subjects, from Experiment 3 in the Object (blue) and Blank Object condition (black) as a function of the lateral offset of the object. Error bars show ±1 SEM, and stars indicate a significant difference between heading bias garnered in the Object and Blank Object conditions (p < 0.05). Data points are partitioned into three groups and labeled depending on whether the optic flow within the object is more radial (“Radial flow inside object”), the pseudo FoE effect is strong (“Pseudo FoE”), or neither applies (“Neither”). Sample optic flow for conditions in each group is shown above the curves.
Figure 1
 
Plan view of the placement of the moving object relative to the observer's path in the Cross condition. See Appendix A for variable definitions. (a) Overview of the xz coordinate system that is centered on the observer's initial position. The gray and blue dashed lines indicate the trajectories of the observer and cylindrical object, respectively, if movement continued beyond the end of the trial. Arrows indicate the movement of the observer and object from their initial to final positions during the trial. (b) The moment when the object makes tangential contact with the observer's future path in the x′z′ coordinate system centered on the object's final position. (c) Trigonometric relationship between the final position of the object and the moment when the object makes tangential contact with the observer's future path.
Figure 1
 
Plan view of the placement of the moving object relative to the observer's path in the Cross condition. See Appendix A for variable definitions. (a) Overview of the xz coordinate system that is centered on the observer's initial position. The gray and blue dashed lines indicate the trajectories of the observer and cylindrical object, respectively, if movement continued beyond the end of the trial. Arrows indicate the movement of the observer and object from their initial to final positions during the trial. (b) The moment when the object makes tangential contact with the observer's future path in the x′z′ coordinate system centered on the object's final position. (c) Trigonometric relationship between the final position of the object and the moment when the object makes tangential contact with the observer's future path.
Figure 2
 
The relationship between the object FoE position and the path angle of the object. (a, left panel) Depiction of the self-motion and object motion directions in world coordinates. (a, right panel) The object FoE can be expressed as the angle between the observer's motion vector and the vector representing the difference between the observer and object motion vector (“resultant” vector) in object coordinates.
Figure 2
 
The relationship between the object FoE position and the path angle of the object. (a, left panel) Depiction of the self-motion and object motion directions in world coordinates. (a, right panel) The object FoE can be expressed as the angle between the observer's motion vector and the vector representing the difference between the observer and object motion vector (“resultant” vector) in object coordinates.
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