Extrema Detection of Optimal Smoothing Splines with Application to Edge Detection Problem

In this paper, we develop a method for detecting and computing the global extrema from a given set of discrete data by using spline curves. First, we approximate the set of discrete data by designing optimal smoothing spline curves using normalized uniform B-splines as the basis functions. Then, we show a method for detecting and computing all the extrema of designed splines and of its derivatives. 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 derivatives 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 small number of control points. Also, when we use the cubic spline, the simplified method can be developed by considering the property of smoothing splines. 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