Locally monotonic regression

The concept of local monotonicity appears in the study of the set of root signals of the median filter and provides a measure of the smoothness of the signal. The median filter is a suboptimal smoother under this measure of smoothness, since a filter pass does necessarily yield a locally monotonic output; even if a locally monotonic output does result, there is no guarantee that it will possess other desirable properties such as optimal similarity to the original signal. Locally monotonic regression is a technique for the optimal smoothing of finite-length discrete real signals under such a criterion. A theoretical framework in which the existence of locally monotonic regression is proved and algorithms for their computation are given. Regression is considered as an approximation problem in R_n , the criterion of approximation is derived from a semimetric, and the approximating set is the collection of signals sharing the property of being locally monotonic