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Interpolating subdivision schemes and their connection to wavelets. The concept of limit curves, coarsening and prediction strategies, hierarchical bases, and improving wavelets using lifting. It also mentions subdivision masks and finite elements.
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Limit�Curve
Element (^) Element
such�that:��
four�point�(cubic)�scheme
spaced�points,�boundaries,�etc.
1
On�coarse�grid
On�fine�grid
Suppose�that������������is�coarsened�by�subsampling
and�remaining�data�is�predicted�using�subdivision
2
Choose���������������to�make�the�moments�zero.
are�orthogonal�to�the�dual�functions
from�which�we�obtain�an�improved�coarsening�strategy:
tunable�parameters
Predict�as�before
as�before
Then�update
5
Scalar�subdivision
Finite�Element�generated
from�vector�subdivision�
piecewise�polynomial,�but�
lacks�smoothness�at�element�
boundaries
Smoother�vector�subdivision�schemes�also�possible
�
k m
m
u k
ϕ
ϕ
�
jk
u jk
kAjm
j j m
u j m
jm
u jm θ θ θ
x { x x } dx km x { x x } dx
S
j m
u j m
i j k Ajm S
jk
u jk
i � (^) � � + + ∈
� = �
� � �
� ( ) () [ , ] 1 , ( ) 1 , () (, )
ϕ ϕ S ϕ ϕ
7
