Bus bunching is a perennial phenomenon that not only diminishes the efficiency of a bus system, but also prevents transit authorities from keeping buses on schedule. We present a physical theory of buses serving a loop of bus stops as a ring of coupled self-oscillators, analogous to the Kuramoto model. Sustained bunching is a repercussion of the process of phase synchronisation whereby the phases of the oscillators are locked to each other. This emerges when demand exceeds a critical threshold. Buses also bunch at low demand, albeit temporarily, due to frequency detuning arising from different human drivers’ distinct natural speeds. We calculate the critical transition when complete phase locking (full synchronisation) occurs for the bus system, and posit the critical transition to completely no phase locking (zero synchronisation). The intermediate regime is the phase where clusters of partially phase locked buses exist. Intriguingly, these theoretical results are in close correspondence to real buses in a university’s shuttle bus system.

a) (b) (c) (d) (e) Figure 1: (a) The input low lighting video frame I. (b) The inverted video frame R: R is obtained by inverting the input low-lighting video frame I. (c) The marked video frame: pixels with low intensity in at least one color (RGB) channel are marked in green. (d) The de-haze video frame J: J is obtained by applying the adapted de-haze algorithm on the inverted video frame R. (e) The final output video frame E: E is obtained by inverting the de-haze video frame J. AbstractWe describe a novel and effective video enhancement algorithm for low lighting video. The algorithm works by first inverting the input low-lighting video and then applying an image de-haze algorithm on the inverted input. To facilitate faster computation and improve temporal consistency, correlations between temporally neighboring frames are utilized. Simulations using naive implementations of the algorithm show good enhancement results and 2x speed-up as compared with frame-wise enhancement algorithms, with further improvements in both quality and speed possible.

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