Piecewise convex estimation for signal processing

Additional methods of nonparametric function estimation (splines, kernels and especially wavelet filters) usually produce artificial features/spurious oscillations. Piecewise convex function estimation seeks to reliably estimate the geometric shape of the unknown function. We outline how piecewise convex fitting 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 inflection points and “efficiently” penalizes additional degrees of freedom. Second, a two-stage adaptive (pilot) estimator is described. In the first stage, the number and location of the change points are estimated using strong smoothing. In the second stage, a constrained smoothing spline fit 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 first-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 first-stage change point