Piecewise convex estimation for signal processing

Traditional methods of nonparametric function estimation (splines, kernels and especially wavelet lters) usually produce arti cial features/spurious oscillations. Piecewise convex function estimation seeks to reliably estimate the geometric shape of the unknown function. We outline how piecewise convex tting may be applied to signal recovery, instantaneous frequency estimation, surface reconstruction, image segmentation, spectral estimation and multivariate adaptive regression. Two distinct methodologies for shape-correct estimation are given. First, we propose a piecewise convex information criterion that strongly penalizes additional in ection points and \e ciently" penalizes additional degrees of freedom. Second, a two-stage adaptive (pilot) estimator is described. In the rst stage, the number and location of the change points are estimated using strong smoothing. In the second stage, a constrained smoothing spline t is performed with the smoothing level chosen to minimize the MSE. The imposed constraint is that a single second-stage change point occurs in a region about each empirical change point of the rst-stage estimate. This constraint is equivalent to requiring that the third derivative of the second-stage estimate has a single sign in a small neighborhood about each rst-stage change point.