advertisement

Simple Linear Interpolation Limit Curve k00 m k10 kk21 Element km31 Element Unchanged k42 Interpolating Subdivision Schemes , find filters • Given a set of data such that: • e.g. two point (linear) scheme k0 m0 k1 four point (cubic) scheme k-1 k0 m0 k1 k2 • Generalizes easily to multiple dimensions, non-uniformly spaced points, boundaries, etc. Interpolating Subdivision Schemes • Limit curve is an interpolating function Wavelets From Subdivision • Limit curves can be used to interpolate data. On coarse grid k0 k1 k2 k0 k1 k2 k3 k4 On fine grid Suppose that is coarsened by subsampling and remaining data is predicted using subdivision k0 m0 k1 m1 k2 Wavelets From Subdivision • Does this fit the wavelet framework? fine approximation coarse approximation If we set strategy gives So the “wavelets” are details , our coarsening/prediction Wavelets From Subdivision • Similarly, setting produces the refinement equation: Wavelets From Subdivision • So subdivision schemes naturally lead to hierarchical bases + + Wavelets From Subdivision • The coarsening strategy is generally less than ideal – some smoothing (antialiasing) desirable k0 m0 k1 m1 k2 Accomplished by forcing the wavelet to have one or more vanishing moments Larger means smaller coefficients in wavelet series ~ Wavelets From Subdivision • How to improve wavelets using lifting as before tunable parameters Choose to make the moments zero. • Regardless of the choice for , and are orthogonal to the dual functions from which we obtain an improved coarsening strategy: Predict as before Then update Butterfly Subdivision − 1 8 1 16 1 16 1 16 1 2 1 2 − − 1 8 − 1 16 Loop Subdivision 1 8 3 8 3 8 1 8 − − 1 16 − 1 16 5 8 1 16 − 1 16 − − 1 16 1 16 Finite Elements From Subdivision • Key difference: subdivision mask is varied so that prediction operation is confined within an element k0 m0 k1 m1 k2 m2 k3 m3 k4 Element m4 k5 m5 k6 Element • Limit functions are finite element shape functions Finite Elements From Subdivision Scalar subdivision Finite Element generated from vector subdivision piecewise polynomial, but lacks smoothness at element boundaries Smoother vector subdivision schemes also possible Vector Refinement • e.g. vector refinement relation for Hermite interpolation functions ��ϕ uj +1,m (x)�� ��ϕ uj,k (x) �� ��ϕ uj +1,k (x) �� � θ �=� θ � + H j [k, m]� θ � (x) (x) ϕ ϕ �� j,k �� �� j +1,k �� m∈n( j,k ) ��ϕ j +1,m (x) �� �ϕ u (x ) H j [k, m] = � k m � θ �ϕ k (xm ) dϕ ku ( xm ) � dx � dϕ θk ( xm ) � � dx Cubic subdivision for displacements and rotations • Wavelets ��ϕ uj,k (x)�� ��wuj,m (x)�� ��ϕ uj+1,m (x)�� T �=� θ � − S j [k, m]� θ � � θ ϕ w (x) (x) �� j,m �� �� j+1,m �� k∈A( j,m) ��ϕ j,k (x)�� � i u � θ i u θ x ϕ ϕ dx (x) (x) �� � S j [k, m] = � x ϕ j+1,m (x) ϕ j +1,m (x) dx � j,k j,k k∈A( j,m) � S S � { } { }