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
�
{
}
{
}