Narrow Results

We extend some well known algorithms for planar Bezier and B-spline curves, including the de Casteljau subdivision algorithm for Bezier curves and several standard knot insertion procedures (Boehm's algorithm, the Oslo algorithm, and Schaefer's algorithm) for B-splines, from the real numbers to the complex domain. We then show how to apply these polynomial and piecewise polynomial algorithms in a complex variable to generate many well known fractal shapes such as the Sierpinski gasket, the Koch curve, and the C-curve. Thus these fractals also have Bezier and B-spline representations, albeit in the complex domain. These representations allow us to change the shape of a fractal in a natural manner by adjusting their complex Bezier and B-spline control points. We also construct natural parameterizations for these fractal shapes from their Bezier and B-spline representations.

This paper investigates how the mean envelope, the subtrahend in the sifting procedure for the Empirical Mode Decomposition (EMD) algorithm, represents as an expansion in terms of basis. To this end, a novel approach that gives an alternative analytical expression using B-spline functions is presented. The basic concept lies mainly on the idea that B-spline functions form a basis for the space of splines and have refined-node representations by knot insertion. This newly-developed expression is essentially equivalent to the conventional one, but gives a more explicit formulation on this issue. For the purpose of establishing the mathematical foundation of the EMD methodology, this study may afford a favorable opportunity in this direction.