Extrema detection of bivariate spline functions Hiroyuki Kano a, * , Hiroyuki Fujioka a , Clyde F. Martin b a Division of Science, School of Science and Engineering, Tokyo Denki University, Hatoyama, Hiki-gun, Saitama 350-0394, Japan b Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409, USA Abstract In this paper, we develop a systematic method for detecting the extrema of bivariate spline functions and of their deriv- atives. It is assumed that the splines are constituted by employing normalized, uniform B-splines as the basis functions, and the detection procedure fully utilizes the spline properties. All the extrema can be found except those with singular Hessian matrix. By numerical examples, we demonstrate the effectiveness of the method. Ó 2007 Elsevier Inc. All rights reserved. Keywords: B-spline; Spline approximation; Bivariate spline; Extrema; Hessian; Resultant 1. Introduction Splines have been used for approximating functions in one or more variables since the seminal work of Schoenberg [1], in 1946. As the number of nodes increases the approximations in general become better and the rate of convergence is determined by both the maximal distance between adjacent nodes and the degree of the spline. Increasing the number of nodes increases computational cost and increasing the degree often introduces unwanted oscillations. To determine these oscillations it is necessary to find the zeros of the derivative of the spline. It follows from the general theory of splines that the derivative of a spline func- tion is itself a spline, albeit not a spline of odd dimension. The problem of determining the zeros has been considered in the literature off and on since the early 1970s beginning with the work of Schumaker [2] and continuing with representative papers [3–5]. This problem is important for other reasons. We would hope that if the function being approximated has a local maximum or minimum that the spline would search this out and we could find an approximated extrema. This problem remains open in the sense that the spline may have many more extrema than the original function. It is possible to construct spline approximations to mono- tone functions for which the spline is monotone. There does not seem to be any systematic way to construct spline approximations to functions with a single extrema that are guaranteed to have a single extrema. This problem is of acute interest in the statistical problem of constructing probability distribution functions. The extrema detection plays key roles in edge detection problems of digital images [6–8]. Also, in [9], its usefulness 0096-3003/$ - see front matter Ó 2007 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2007.10.054 * Corresponding author. E-mail addresses: kano@j.dendai.ac.jp (H. Kano), fujioka@j.dendai.ac.jp (H. Fujioka), clyde.f.martin@ttu.edu (C.F. Martin). Available online at www.sciencedirect.com Applied Mathematics and Computation 200 (2008) 58–69 www.elsevier.com/locate/amc