Optimal Smoothing Splines for Detecting Extrema from Observational Data

In this paper, we present a method for detecting and computing the extrema from observational data by using spline curves. First, we approximate a given set of discrete data by designing optimal smoothing spline curves using normalized uniform B-splines as the basis functions. Then, we show the method for detecting and computing all the extrema of designed splines and/or of its first derivative. Here, utilizing the fact that splines are continuously differentiable piecewise polynomials, we only need to detect and compute the extrema of the polynomial and its first derivative in turn for each interval between the knot points. This process is easily carried out since the polynomial in each interval is characterized by a few control points. Finally we verified the validities by numerical experiments. In particular, the detection and computation of the extrema of the first derivative are used for edge detection in digital images.