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.