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 fir

st

– Let be a piecewise-linear function with vertices at the integers

– be function at dyadic points

)(0 xf

)(xf j

ji 2

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)

jc0

j

ij

ij

i ccc 12

1

2

1jc2

jc3

jc1

jc5jc4

Equally Applicable to Parametric Curves

Controlpolygon

Refinement Mask

• Mask r determines important properties of the curve– Continuity, differentiab

ility, …

• 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