We discuss surface reconstruction preserving discontinuities. Given a set of function values, we first detect discontinuities. Then we construct a surface that preserves discontinuities. The discontinuity detection is based on a residual analysis in high dimensions, and the surface fitting uses multivariate smoothing splines with discontinuities.
David T. Lee,
Jyh-Jen Horng Shiau,
"Surface reconstruction with discontinuities", Proc. SPIE 1383, Sensor Fusion III: 3D Perception and Recognition, (1 April 1991); doi: 10.1117/12.25266; https://doi.org/10.1117/12.25266