# $\ell_1$-minimization method for link flow correction

## TL;DR

This paper introduces an $ ext{l}_1$-minimization approach for correcting inconsistent link flow data in road networks, effectively identifying and fixing corrupted sensors under certain recoverability conditions, even with measurement noise.

## Contribution

It develops a novel $ ext{l}_1$-minimization method with a recoverability condition to accurately correct corrupted traffic flow data, including an analytical framework and algorithm for robustness.

## Key findings

- Method accurately corrects corrupted link flows in synthetic and real data.
- Recoverability condition ensures robustness to sensor miscounts.
- Provides bounds on correction errors under measurement noise.

## Abstract

A computational method, based on $\ell_1$-minimization, is proposed for the problem of link flow correction, when the available traffic flow data on many links in a road network are inconsistent with respect to the flow conservation law. Without extra information, the problem is generally ill-posed when a large portion of the link sensors are unhealthy. It is possible, however, to correct the corrupted link flows \textit{accurately} with the proposed method under a recoverability condition if there are only a few bad sensors which are located at certain links. We analytically identify the links that are robust to miscounts and relate them to the geometric structure of the traffic network by introducing the recoverability concept and an algorithm for computing it. The recoverability condition for corrupted links is simply the associated recoverability being greater than 1. In a more realistic setting, besides the unhealthy link sensors, small measurement noises may be present at the other sensors. Under the same recoverability condition, our method guarantees to give an estimated traffic flow fairly close to the ground-truth data and leads to a bound for the correction error. Both synthetic and real-world examples are provided to demonstrate the effectiveness of the proposed method.