Measure fields for function approximation

The computation of a piecewise smooth function that approximates a finite set of data points may be decomposed into two decoupled tasks: 1) the computation of the locally smooth models, and hence, the segmentation of the data into classes that consist of the sets of points best approximated by each model; 2) the computation of the normalized discriminant functions for each induced class (which maybe interpreted as relative probabilities). The approximating function may then be computed as the optimal estimator with respect to this measure field. For the first step, we propose a scheme that involves both robust regression and spatial localization using Gaussian windows. The discriminant functions are obtained fitting Gaussian mixture models for the data distribution inside each class. We give an efficient procedure for effecting both computations and for the determination of the optimal number of components. Examples of the application of this scheme to image filtering, surface reconstruction and time series prediction are presented.