Both the 4-point and the uniform cubic B-spline subdivisions double the number of vertices of a closed-loop polygon k P and produce sequences of vertices fj and bj respectively. We study the J-spline subdivision scheme Js, introduced by Maillot and Stam, which blends these two methods to produce vertices of the form vj = (1 - s) fj + s bj. Iterative applications of Js yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves (J0), uniform cubic B-spline curves (J1), and uniform quintic B-spline curves (J1.5). We show that the limit curve is at least C1 when - 1.7 s 5.8, C2 when 0 < s < 4, C3 when 1 < s 2.8, and C4 when s = 3 / 2, even though a 4-point yields only C1 curves and a cubic B-spline yields only C2 curves. We generalize the Js scheme to a two-parameter family Ja, b and propose data-dependent and data-independent solutions for computing values of parameters a and b that minimize various objective functions (distance to the control vertices, deviation from the control polygon, change in surface area, and popping when switching levels of subdivision in multi-resolution rendering). We extend the J-spline subdivision to open curves and to a smooth surface subdivision scheme for quad-meshes with arbitrary connectivity. 2008.
