Even degree B-spline curves and surfaces: A note on the paper “B-spline Curves and Surfaces Viewed as Digital Filters” by A. Goshtasby, F. Cheng, and B. Barsky

✍ Scribed by Christophe Rabut

• Publisher: Elsevier Science
• Year: 1992
• Weight: 510 KB
• Volume: 54
• Category: Article
• ISSN: 1049-9652
• DOI: 10.1016/1049-9652(92)90082-9

✦ Synopsis

This paper compares two different ways of building B-spline curves and surfaces for even degree polynomial splines. Viewing these problems as digital filters, we find that one way gives better results than the other one. This paper is intended to be a reflection and a continuation of the paper "B-spline Curves and Surfaces Viewed as Digital Filters," by A. Goshtasby, F. Cheng, and B. Barsky (Comput. Vision Graphics Image Process. 52, 1990, 264 275). 0 19% Academic Press, he.

1. INTRODUCTION Let us define Bk as the O-centered, degree k -1 polynomial B-spline, i.e., the function defined for any x E Iw by Bk(X) = (k-l)! )=a --!-i (-l)(f) (x + k/2 -1)";'

(1) (here, as usual, U+ denotes 0 if u is negative, u if u is nonnegative). Note that here Bk is an even function (i.e., Vx E Iw, BL(-X) = By) and that the joints of Bk (i.e., the points where Bk is not k -1 times differentiable) are at integer values if k is even, and at half integer values ifk is odd.

Now, given some equispaced data (j, yj)jEz let us define the degree k -1 B-spline curve associated to these data. We define the function Sk by Sk = 2 YjBk(. -j).

(2)

We call Sk the "symmetric B-spline curve associated to the data (j, yj)jEz ." Note that the joints of Sk are at integer values if k is even, and at half integer values if k is odd. Some authors prefer having the joints of the B-spline curve at the same abscissae as the data points, and they define the even degree B-spline curve associated to the data (j, yj)jez by (here k is odd) flk =zYjBk(' -j + k).