The risk of pedestrian collisions with peripheral visual field loss

Abstract

Patients with peripheral field loss complain of colliding with other pedestrians in open-space environments such as shopping malls. Field expansion devices (e.g., prisms) can create artificial peripheral islands of vision. We investigated the visual angle at which these islands can be most effective for avoiding pedestrian collisions, by modeling the collision risk density as a function of bearing angle of pedestrians relative to the patient. Pedestrians at all possible locations were assumed to be moving in all directions with equal probability within a reasonable range of walking speeds. The risk density was found to be highly anisotropic. It peaked at ≈45° eccentricity. Increasing pedestrian speed range shifted the risk to higher eccentricities. The risk density is independent of time to collision. The model results were compared to the binocular residual peripheral island locations of 42 patients with forms of retinitis pigmentosa. The natural residual island prevalence also peaked nasally at about 45° but temporally at about 75°. This asymmetry resulted in a complementary coverage of the binocular field of view. Natural residual binocular island eccentricities seem well matched to the collision-risk density function, optimizing detection of other walking pedestrians (nasally) and of faster hazards (temporally). Field expansion prism devices will be most effective if they can create artificial peripheral islands at about 45° eccentricities. The collision risk and residual island findings raise interesting questions about normal visual development.

Figure 1

A Patient, P, walks in the y direction toward C for t seconds. A peDestrian, D, also walks toward C at a speed that will cause him to collide with the patient at C. Under the simplifying assumptions that both maintain constant speed and direction, D remains at a constant bearing angle β with respect to P as they approach collision.

Figure 2

The wedge diagram identifies (in red) the headings a pedestrian, D, starting at a given single location can take to collide with the patient, P, within assumed speed and time constraints. Headings that cannot result in a constrained collision are color coded to indicate the applicable constraint. Collisions in the blue area would require D to walk faster than the high speed limit, r_f, while those in the yellow area would require walking slower that the low speed limit, r_s. The gray striped area represents paths that require longer than the maximum time to collision (t_max), even though the speed constraints are met. The percentage of red area (without gray stripes) thus represents the relative risk of collisions from that location. Pedestrian speed decreases from infinite at α = 0 (heading directly at the patient start point), to equal r_P when α = β (not shown), reaching a minimum speed r_min at α = 90°, then increases to equal r_P again and collide at infinite time and distance on a parallel path. This example uses r_P = 1 m/s, r_f = 1.5 m/s, r_s = 0.7 m/s, t_max = 5 s, for pedestrians at β = 40° and d_PD = 1.3 m, resulting in a point risk of 0.31. The wedge diagram would be the same for any distance along that bearing angle, except for the α angle and collision point that t_max intercedes. (The wedges would also be unchanged if the speeds change but the ratios of r_f/r_P and r_s/r_P do not change.)

Figure 3

(A) The patient starts at (x, y) = (0, 0) and walks with increasing y. For each pedestrian position in the figure, extending from x = 0 to 10 m and y = −4 to 10 m, in 0.5-m steps, the angular range of collision hazards of interest is shown in red. In this example, the patient walks at 1 m/s and the pedestrians walk between 0.7 and 1.5 m/s. Collisions that would require times longer than 5 s are not shown. In the region below y = 0, the pedestrians start behind the patient, and remain behind the patient to collide. The patient would not be considered responsible for avoiding this type of collision. (B) The point risk (red wedge area) is calculated and shown with color representing the risk. To create this map, the wedge diagram shown in (A) is recalculated at a resolution of 0.02 m.

Figure 4

Collision risk as a function of bearing relative to the patient. Total area under the curve is normalized to 1, representing all risks in the quadrant of concern. The percentage area under the curve between any two bearings can thus represent the amount of risk that comes from a window of vision monitoring that range. (Same parameters as in Figure 3.)

Figure 5

Effect of the maximum time to collision, t_max. (A) With t_max = 2 s, calculated on a 0.5-m grid, the angular range of possible collisions (point risk) is shown in red wedges. (B) Same as Panel (A), but for t_max = 7 s. Collisions are possible with pedestrians that start much farther from the patient. (C) Same as Panel (A), but rendered on a high resolution grid of 0.02 m and the size of the collision wedges is represented by the color coding shown in the scale. (D) Same as Panel (C), but for t_max = 7 s. Despite different t_max values, the risk density as a function of the bearing angle β calculated from (C) and (D) is identical to the risk density function shown in Figure 4 for t_max = 5 s.

Figure 6

Fraction of risk monitored by islands of vision. (A) The collision risk density curve as a function of pedestrian bearing. The area under the curve represents the total collision risk posed by all pedestrians in any of the positions to the right of the patient (but not behind him). That is also the risk monitored by a normally sighted person, assuming a temporal field extent of 90°. The shaded area represents the fraction (31%) of that risk that would be monitored by a residual island of 20° centered at an eccentricity of 30° (achievable with a 57Δ prism). (B) The fraction of the total risk monitored by island windows of variable width as indicated, as a function of the island's center eccentricity (the shaded area of a sliding 20° window under the risk density curve of Panel [A], compared with corresponding curves for narrower windows [and the same risk density curve]).

Figure 7

The effect of pedestrian speed range on the risk as a function of bearing angle from a patient walking at a speed of 1 m/s. (A) The risk function for slower pedestrians walking in the speed range of 0.5 to 1.0 m/s (or 0.5 to 1.0 times the speed of the patient). The risk in this case is shifted centrally and peaks at 30°. (B) The risk function as a function of eccentricity calculated for faster pedestrians walking at speeds between 1.0 and 2.0 m/s is flatter, peaking at farther eccentricity (49°) and extending to eccentricities behind the patient. (C) The risk density function for the conditions of (A). (D) The risk density function for the conditions of (B). Note that these density functions are independent of t_max.

Figure 8

Histogram of field coverage of the 42 patients with residual islands. The height and the color coding mark the number of patients out of 42 that were seeing within each 1° × 1° cell. (A) The histogram for the LE. (C) The histogram for the RE. (B) The binocular histogram represents the union of the LE and RE panels. The horizontal axis represents the eccentricity (degrees) in the visual field along the horizontal meridian. The plane cutting through the binocular field marks a slice through the histogram used to calculate the visual field coverage along the horizontal meridian, corresponding to the lateral eccentricities that might be relevant to detecting a colliding pedestrian.

Figure 9

A slice through the data from Figure 8, showing the lateral angular extent of the residual islands with the calculated risk density function superimposed. The angular extents for the “either eye” data are marked at the 50% level, representing the minimum range of eccentricities monitored by half the patients. The peaks of the calculated risk density function (blue) closely match the locations of the nasal peaks of the patient peripheral islands data. Additional horizontal dotted lines mark the extent of the residual islands from both eyes at the lower and upper quartiles of the population.