In recent years discontinuity meshing has become an important part of mesh-based solutions to the global illumination problem. Application of this technique accurately locates all radiance function discontinuities in the scene and is essential for limiting visual artifacts. In an environment containing $m$ edges there are $O(m^2)$ $D^1$ and $D^2$ discontinuities. With a typical scene this can result in many thousands of discontinuity lines being processed. We review existing methods for reducing these lines and introduce an improved, perception based metric for determining which discontinuities are important and which can be safely ignored. Our results show that a 50\% or more reduction in the number of discontinuity lines can be achieved with a corresponding reduction in general mesh complexity, with little or no perceptible change to the rendered result.