Approximating Polygonal Curves in Two and Three Dimensions

We discuss theweighted minimum numberpolygonal approximation problem. Eu and Toussaint (1994, CVGIP: Graphical Models Image Process.56, 231–246) considered this problem subject to theparallel-striperror criterion inR2withLqdistance metrics, and they concluded that it can be solved inO(n2) time by using the Cone intersection method. In this note, we clarify part of their discussion and show that solving their problem correctly requiresO(n2logn) time. Also, we discuss theweighted minimum numberproblem subject to theline segmenterror criterion. When input curves are strictly monotone inR3, we demonstrate that if theL1orL∞metric is used, this problem also can be solved inO(n2) time.