Signal Processing and Piecewise Convex Estimation

TL;DR

This paper introduces a novel piecewise convex fitting method for nonparametric signal estimation, combining adaptive change point detection with constrained smoothing splines to improve accuracy in various signal processing tasks.

Contribution

The paper presents a new two-stage adaptive estimation technique called piecewise convex fitting (PCF) that effectively detects change points and fits signals with convex constraints.

Findings

Effective change point detection using strong smoothing.

Improved signal estimation with constrained smoothing splines.

Versatile application to multiple signal processing problems.

Abstract

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.