Signal Processing and Piecewise Convex Estimation
This method addresses signal recovery and related tasks like image segmentation, but it appears incremental as it builds on existing smoothing and spline techniques.
The authors tackled the problem of nonparametric function estimation in signal processing by proposing piecewise convex fitting (PCF), a two-stage adaptive method that estimates change points and fits constrained smoothing splines to minimize mean squared error.
Many problems on signal processing reduce to nonparametric function estimation. We propose a new methodology, piecewise convex fitting (PCF), and give a two-stage adaptive estimate. In the first stage, the number and location of the change points is 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 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. We sketch how PCF may be applied to signal recovery, instantaneous frequency estimation, surface reconstruction, image segmentation, spectral estimation and multivariate adaptive regression.