GIT - GVU - 07 - 06 April 10 , 2007

Both the four-point and the uniform cubic B-spline refinement (i.e. subdivision) schemes double the number of vertices of a closed-loop polygonal curve P and produce sequences of vertices fj and bj respectively. We proposed to analyze a subdivision scheme Js that blends the rules of these two refinement methods to produce vertices of the form vj=(1–s)fj+sbj. Iterative applications of Js yield a family of curves parameterized by s that includes curves produced by four-point subdivision (J0), uniform cubic B-spline curves (J8/8), and uniform quintic B-spline curves (J12/8). We analyze the smoothness of curves produced by this subdivision for different parameters and show that the curves are at least C (-1.7 ≤ s ≤ 5.8), C (0 < s < 4), C (1 < s ≤ 2.8) and even C (s=3/2) despite the fact that the four-point scheme produces C curves and cubic B-spline subdivision produces C curves. Using this parameterized subdivision scheme, we then explore optimizing this parameter in both a control polygondependent and independent fashion to match B-splines of various degrees, interpolate the vertices of the control polygon, interpolate the midpoints of the edges on the control polygon, preserve the enclosed area of the curve and reduce popping artifacts for multi-resolution rendering. We also extend Js subdivision to open curves and show how to generalize this curve subdivision scheme to create a smooth surface subdivision for arbitrary topology surfaces. To reduce memory requirements when evaluating the final refined curve or surface, we introduce a new evaluation technique, called ringing, which requires storage space linear in the subdivision level as opposed to traditional methods that require storage exponential in the subdivision level without introducing any redundant calculations. 1. Curve refinements Subdivision has become a standard tool for modeling curves in CAD and Graphics both for its simplicity and expressive power. Subdivision is a set of rules that refine polygonal curves by increasing their number of vertices. Applying a subdivision operator repeatedly to a curve produces a sequence of curves that, if the subdivision rules are chosen correctly, converge to a smooth shape in the limit. There are many ways of choosing subdivision rules and refinement operations. We concentrate on the simplest class of subdivision methods called linear subdivision schemes where the new vertices are specified using linear combinations of old vertices. Furthermore, we restrict ourselves to the class of binary subdivision schemes (the number of control points double at each refinement) Ja,b whose rules are of the form P2j = (a Pj–1 + (8–2a) Pj + a Pj+1)/8 P2j+1 = ((b–1) Pj–1 + (9–b) Pj + (9–b) Pj+1 + (b–1) Pj+2)/16 where Pj represents the j control point at the k level of subdivision. Notice that several popular curve subdivision schemes fall under this class. Specifically, J0,0 corresponds to the four-point scheme [NDG87, DD89], which produces C curves, and J1,1 corresponds to uniform cubic B-splines [Sab02, LR80], which produces C curves. Recently, Rossignac [Ros04] created a new curve subdivision scheme entitled Jarek Subdivision by averaging the subdivision rules for cubic B-spline curves and the 4-point subdivision scheme. These rules correspond to J1/2,1/2. Despite the fact that the four-point rule produces curves that are only C, averaging these subdivision rules surprisingly creates a subdivision scheme that generates C curves. Inspired by this work, we investigate a generalized form of Jarek subdivision using an arbitrary affine combination of the rules of these two schemes. The result is a subdivision scheme Js,s, which we refer to simply as Js for convenience, parameterized by a single parameter s. When s=0, Js reproduces the four-point scheme. For s=1/2, Js produces curves generated by Jarek Subdivision. And for s=1, Js produces uniform, cubic b-spline curves (Fig. 1).