Subdivision Curve (and its relations to wavelets) Jyun-Ming Chen Spring 2001

Road Map Introduce concepts of recursive subdivision Create uniform and non-uniform B-splines and Daubechies wavelet Use one-dimension curves (function and parametric curves) to motivate 1D wavelets Steer towards hierarchical function decomposition, nested spaces, MRA, …

Subdivision: Introduction Idea: repeatedly refining an initial piecewise-linear function to produce a sequence of increasing detailed functions that converge to the limit function

Subdivision Scheme History: Chaikin’s algorithm (1974) To simplify discussion –consider function curves first –Let be a piecewise- linear function with vertices at the integers – be function at dyadic points

Subdivision Scheme Averaging mask Chaikin’s scheme Uniform subdivision –Same scheme applied everywhere along the curve Stationary subdivision –Same scheme used in each iteration

Example: Chaikin’s Curve

Subdivision Steps Simplify the implementation, make it a two-step process –Splitting: introduce midpoints –Averaging: compute the weighted average Ignore the boundary conditions for now –assume periodicity (closed curve); or portions away from boundary Splitting & Averaging

This means…(Chaikin’s)

Equally Applicable to Parametric Curves Control polygon

Refinement Mask Mask r determines important properties of the curve –Continuity, differentiability, … Riesenfeld (1975) showed Chaikin’s algorithm produces uniform quadratic B- spline B-spline of any degree can be produced by the following mask (Lane and Riesenfeld) Ex: cubic B-spline

Daubechies Subdivision Scheme Daubechies scheme produces fractal-like function with the following mask Fractal-like