We suggest a three-stage procedure to recover discontinuous regression surfaces when noisy data are present. In the first stage, jump candidate points are detected using a jump detection criterion. A local principal component line is then fitted through these points in a neighborhood of a design point. This line provides a first-order approximation to the true jump location curve in that neighborhood. In the third stage, observations on the same side of the line as the given point are combined using a weighted average procedure to fit the surface at that point. If there are no jump candidate points in the neighborhood, then all observations in that neighborhood are used in the surface fitting. If, however, the center of the neighborhood is on a jump location curve, only those observations on one side of the line are used. Thus blurring is automatically avoided around the jump locations. This methodology requires $O(N(k^*)^2)$ computation, where $N$ is the sample size and $k^*$ is the window width. Its assumptions on the model are flexible. Some numerical results are presented to evaluate the surface fit and to discuss the selection of the window widths.
"Discontinuous regression surfaces fitting." Ann. Statist. 26 (6) 2218 - 2245, December 1998. https://doi.org/10.1214/aos/1024691468