Analysis of emergent patterns in crossing flows of pedestrians reveals an invariant of `stripe' formation in human data

TL;DR

This study investigates how pedestrian crossing patterns form and remain consistent across different crossing angles, revealing an invariant of stripe orientation relative to the crossing angle bisector through experimental and computational analysis.

Contribution

The paper introduces two novel computational methods for analyzing pedestrian stripe patterns and demonstrates that stripe orientation is invariant across various crossing angles, supporting the bisector hypothesis.

Findings

Stripes are parallel and perpendicular to the bisector at all angles.

Stripe spacing and size decrease as crossing angle increases.

Number of stripes increases with crossing angle.

Abstract

When two streams of pedestrians cross at an angle, striped patterns spontaneously emerge as a result of local pedestrian interactions. This clear case of self-organized pattern formation remains to be elucidated. In counterflows, with a crossing angle of 180{\deg}, alternating lanes of traffic are commonly observed moving in opposite directions, whereas in crossing flows at an angle of 90{\deg} diagonal stripes have been reported. Naka (1977) hypothesized that stripe orientation is perpendicular to the bisector of the crossing angle. However, studies of crossing flows at acute and obtuse angles remain underdeveloped. We tested the bisector hypothesis in experiments on small groups (18-19 participants each) crossing at seven angles (30{\deg} intervals), and analyzed the geometric properties of stripes. We present two novel computational methods for analyzing striped patterns in pedestrian data: (i) an edge-cutting algorithm, which detects the dynamic formation of stripes and allows us to measure local properties of individual stripes; and (ii) a pattern-matching technique, based on the Gabor function, which allows us to estimate global properties (orientation and wavelength) of the striped pattern at a time T. We find an invariant property: stripes in the two groups are parallel and perpendicular to the bisector at all crossing angles. In contrast, other properties depend on the crossing angle: stripe spacing (wavelength), stripe size (number of pedestrians per stripe), and crossing time all decrease as the crossing angle increases from 30{\deg} to 180{\deg}, whereas the number of stripes increases with crossing angle. We also observe that the width of individual stripes is dynamically squeezed as the two groups cross each other. The findings thus support the bisector hypothesis at a wide range of crossing angles, although the theoretical reasons for this invariant remain unclear.