# A Partial Differential Equation Obstacle Problem for the Level Set Approach to Visibility

## TL;DR

This paper introduces a simplified PDE-based method for computing visibility sets from a point with obstacles represented as level sets, offering an alternative to ray tracing with demonstrated convergence and numerical examples.

## Contribution

It proposes a new nonlinear obstacle PDE formulation for visibility problems, simplifying previous models and providing a convergent numerical scheme.

## Key findings

- The PDE approach effectively computes visibility sets in 2D and 3D.
- The numerical scheme converges under the Barles and Souganidis framework.
- The method offers advantages over traditional ray tracing.

## Abstract

In this article we consider the problem of finding the visibility set from a given point when the obstacles are represented as the level set of a given function. Although the visibility set can be computed efficiently by ray tracing, there are advantages to using a level set representation for the obstacles, and to characterizing the solution using a Partial Differential Equation (PDE). A nonlocal PDE formulation was proposed in Tsai et. al. (Journal of Computational Physics 199(1):260-290, 2004): in this article we propose a simpler PDE formulation, involving a nonlinear obstacle problem. We present a simple numerical scheme and show its convergence using the framework of Barles and Souganidis. Numerical examples in both two and three dimensions are presented.