Discontinuity detection by null rules for adaptive surface reconstruction

We present a discontinuity detection method based on the so-called null rules, computed as a vector in the null space of certain collocation matrices. These rules are used as weights in a linear combination of function evaluations to indicate the local behavior of the function itself. By analyzing the asymptotic properties of the rules, we introduce two indicators (one for discontinuities of the function and one for discontinuities of its gradient) by locally computing just one rule. This leads to an efficient and reliable scheme, which allows us to effectively detect and classify points close to discontinuities. We then show how this information can be suitably combined with adaptive approximation methods based on hierarchical spline spaces in the reconstruction process of surfaces with discontinuities. The considered adaptive methods exploit the ability of the hierarchical spaces to be locally refined, and fault detection is a natural way to guide the refinement with low computational cost. A selection of test cases is presented to show the effectiveness of our approach.