An Efficient Solution to the 2D Visibility Problem in Cartesian Grid Maps and its Application in Heuristic Path Planning

TL;DR

This paper presents a fast, memory-efficient method for solving the 2D visibility problem in grid maps using PDE-based transport, enabling rapid heuristic evaluations for path planning.

Contribution

It introduces a novel PDE-based approach for visibility computation that is faster and more memory-efficient than traditional ray-casting methods, with practical path planning applications.

Findings

Computes visibility in O(n) time with minimal operations per cell.

Provides an open-source implementation of the algorithm.

Enables real-time heuristic evaluation for path planning.

Abstract

This paper introduces a novel, lightweight method to solve the visibility problem for 2D grids. The proposed method evaluates the existence of lines-of-sight from a source point to all other grid cells in a single pass with no preprocessing and independently of the number and shape of obstacles. It has a compute and memory complexity of $\mathcal{O}(n)$, where $n = n_{x}\times{} n_{y}$ is the size of the grid, and requires at most ten arithmetic operations per grid cell. In the proposed approach, we use a linear first-order hyperbolic partial differential equation to transport the visibility quantity in all directions. In order to accomplish that, we use an entropy-satisfying upwind scheme that converges to the true visibility polygon as the step size goes to zero. This dynamic-programming approach allows the evaluation of visibility for an entire grid orders of magnitude faster than typical ray-casting algorithms. We provide a practical application of our proposed algorithm by posing the visibility quantity as a heuristic and implementing a deterministic, local-minima-free path planner, setting apart the proposed planner from traditional methods. Lastly, we provide necessary algorithms and an open-source implementation of the proposed methods.