J-splines

doi:10.1016/j.cad.2008.09.002

J-splines

2008-10-01

15

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AI

DOI：10.1016/j.cad.2008.09.002

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Abstract

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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 f(j) and b(j) respectively. We study the J-spline subdivision scheme J, introduced by Maillot and Stam, which blends these two methods to produce vertices of the form v(j) = (1 - s)f(j) + sb(j). Iterative applications of J(s) yield a family of limit curves, the shape of which is parameterized by s. They include four-point subdivision curves (J(0)), uniform cubic B-spline curves (J(1)), and uniform quintic B-spline curves (J(1.5)). We show that the limit curve is at least C(1) when. - 1.7 <= s <= 5.8, C(2) when 0 < s < 4, C(3) when 1 < s < 2.8, and C(4) when s = 3/2, even though a 4-point yields only C(1) curves and a cubic B-spline yields only C(2) curves. We generalize the J, scheme to a two-parameter family J(a,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. (c) 2008 Published by Elsevier Ltd.

Keywords:

Subdivision

Continuity

B-splines

Four-point

Interpolation

Fitting

Area preservation

Curves

Surfaces

J-splines

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Journal

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