Handout18

Handout18 - Simple�Linear�Interpolation Limit�Curve...

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Unformatted text preview: Simple�Linear�Interpolation Limit�Curve m1 k0 k0 0 m k31 k k21 Element k4 2 Element Unchanged Interpolating�Subdivision�Schemes • Given�a�set�of�data������������������������������������,�find�filters����������������������������� 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. 1 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 k 0 m0 k 1 m1 k 2 2 Wavelets�From�Subdivision • Does�this�fit�the�wavelet�framework? fine�approximation coarse�approximation details If�we�set��������������������������������������,�our�coarsening/prediction strategy�gives So�the�“wavelets”�are Wavelets�From�Subdivision • Similarly,�setting produces�the�refinement�equation: 3 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 k 0 m0 k 1 m1 k 2 Accomplished�by�forcing�the�wavelet�to�have�one�or�more� vanishing�moments Larger�����means�smaller�coefficients���������in�wavelet�series ~ 4 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 5 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 k 0 m0 k 1 m1 k 2 m2 k 3 m3 k 4 Element m4 k 5 m5 k 6 Element • Limit�functions�are�finite�element�shape�functions 6 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 �u � �u � �u � �ϕ j , k ( x) � �ϕ j +1, k ( x) � �ϕ j +1, m ( x) � �θ �=� θ � + � H j [ k , m]� θ � �ϕ j , k ( x) � �ϕ j +1, k ( x) � m∈n ( j , k ) �ϕ j +1, m ( x) � � �� � � � �ϕ u ( x ) H j [ k , m] = � k m �θ �ϕ k ( xm ) u dϕ k ( xm ) � dx � dϕ θ ( xm ) k � � dx Cubic�subdivision�for�displacements�and�rotations • Wavelets � � �� x {ϕ � k∈ A ( j , m ) i S � � �u � �u �u �ϕ j ,k ( x)� �w j ,m ( x)� �ϕ j +1,m ( x)� T � � − � S j [k , m]� θ �=� θ �θ �ϕ j ,k ( x)� �w j ,m ( x)� �ϕ j +1,m ( x)� k∈A( j ,m) � � � �� � u j ,k � ( x) ϕ θ,k ( x) dx � S j [k , m] = � x i ϕ u+1, m ( x) ϕ θ+1,m ( x) dx j j j S � } { } 7 ...
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This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.

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