In applications like image analysis, signal detection and ecological spatial data smooth and non-smooth parts for curves and surfaces occur simultaneously. In the higher dimensional case, isolated discontinuities occurring in otherwise smooth surfaces correspond to boundaries or "break curves". Methods for analyzing break curves and to estimate smooth-non-smooth surfaces and regression functions will be developed, using curve estimation, least squares and (local) maximum likelihood techniques. Applications include confidence regions for break curves and isoklines, edge detection in image analysis, the Poisson forest problem, and dimension reduction techniques for high dimensional regression. Many phenomena in biological, environmental and economic sciences can be described by the occurrence of sharp changes when either time or location varies. Statistical tools will be developed for the analysis of data describing such changes. These will be of particular interest for the detection of edges in images and for parsimonious descriptions of high-dimensional data. In this context, confidence regions will be constructed which allow for instance to find "safe" areas where pollutant levels are below critical levels. The proposed methods will work under minimal assumptions on the nature of the data.
