Lifting and Biorthogonality

Ref: SIGGRAPH 95

Projection Operator

• Def: The approximated function in the subspace is obtained by the “projection operator”

• As j↑, the approximation gets finer, and

•

2Levery for lim

fffPjj

kjk

kjj

jkj

ffP

V

,,

,

,then

for basis lorthonormaan form If

k

kjkjj xfP )( .,,

Projection Operator (cont)

• In general, it is hard to construct orthonormal scaling functions

• In the more general biorthogonal settings,

kjk

kjj

kj

ffP

dual

,,

,

~,then

~ functions scaling ofset second a have we

1)(~

purpose,ion normalizatfor and ~

,

,

',',,

dxxkj

kkkjkj

Ex: Linear Interpolating

functionhat the )(

pulse Dirac the)()(~

,

,,

x

xxx

kj

kjkj

biorthogonal! kjkjkjk

kjj ffP ,,,,

~, , kjkjkj

kkjj ffP ,,,,

~, ,

Ex: Constant Average-Interpolating

support of

width toalproportioninversely

height with functionsbox :~

,kj

kjkjkjk

kjj ffP ,,,,

~, ,

)1dth intervalWi(ht) average(

)]1,[between under area(

)1()(~

,1

,,

kkf

dxxffk

kkjkj

)(, xkj

j,k

Think …

• What does Pj+1 look like in linear interpolating and constant AI?

• What does Pj look like in other lifting schemes? (cubic interpolating, quadratic AI, …)

Polynomial Reproduction

• If the order of MRA is N, then any polynomial of degree less than N can be reproduced by the scaling functions

• That is,

NpxxP ppj 0for NpxxP pp

j 0for

This is true for all j

Ex: MRA of Order 4

• as in the case of cubic predictor in lifting …– Pj can reproduce x0, x1, x2, and x3 (and any linea

r combination of them)

• …

Interchange the roles of primal and dual …

• Define the dual projection operator w.r.t. the dual scaling functions

• Dual order of MRA:– Any polynomial of degree less than is

reproducible by the dual projection operator

kjk

kjjjkj ffPV ,,,

~,

~ then ,

~for basis a form

~ If

N~

N~

jP~

jP~

NpxxP ppj 0for

~ NpxxP ppj 0for

~

fxPxf

xf

fxfPx

Npf

pj

kkjkj

p

kjp

kkj

kkjkj

pj

p

,~~

,,

,~

,

~,,,

~0for and arbitrary For

,,

,,

,,

dxxgxffNote )()( represents g, space,function In :

,~

, fxPfPx pjj

p ,~

, fxPfPx pjj

p

kjk

kjj ffP ,,

~,

~

fxPfxP pj

pj ,

~,

~1

• From property of :jP~

! moments ~

toup preserve

,,~

,~

,

11

NP

fPxfxP

fxPfPx

j

jpp

j

pjj

p

j+1: one level finerin MRA

This means: “The pth moment of finer and coarsened approximations are the same.”

This means: “The pth moment of finer and coarsened approximations are the same.”

Summary•

• If the dual order of MRA is – Any polynomial of degree less than is reproducible by t

he dual projection operator– Pj preserves up to moments

• If the order of MRA is N– Any polynomial of degree less than N is reproducible by t

he projection operator Pj

– preserves up to N moments

N~

fn scaling dual oforder :~

fn scaling primal oforder :

N

N

N~ jP

~

jP~

N~

~

Subdivision

• Assume

• The same function can be written in the finer space:

• The coefficients are related by subdivision:

kjkjk

kjkj ff ,,,,

~, ,

kjkjk

kjkj ff ,1,1,1,1

~, ,

k

kjlkjlj h ,,,,1 k

kjlkjlj h ,,,,1

Recall “lifting-2.ppt”, p.16, 18

Coarsening

• On the other hand, to get the coarsened signal from finer ones: substitute the dual refinement relation

• into

• Recall

ljl

lkjkj h ,1,,,

~~~

kjkj f ,1,1

~,

l

ljlkjl

ljlkjkjkj fhhff ,1,,,1,,,,

~,

~~~,

~,

l

ljlkjkj h ,1,,,

~ l

ljlkjkj h ,1,,,

~

Ex: Coarsening for Linear Interpolation

kllkj

kjkj

h 2,,

2,1,

~ sequencefilter

Wavelets …

• form a basis for the difference between two successive approximations

• Wavelet coefficients: encode the difference of DOF between Pj and Pj+1

Pj

PjPj+1

Pj+1- Pj

This implies …

0,

then

any for 0,,or

~0 ,,

:moments ~

toup preserves if that Note

,

,,1

1

,,1

pkj

p

kkjkj

pjj

pj

pj

j

kkjkjjj

x

fxxfPfP

NpxfPxfP

NP

fPfP

(primal) wavelet has vanishing

momentsN~

MRA

1

0

12

0,,0

1

00

Then n

j mmjmjn

j

n

jn

j

fPfP

WVV

ljl

lmjmjjj

jjj

gVW

WVV

,1,,,1

1

, Since

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

• Wj depends on …

– how Pj is calculated from Pj+1

• Hence, related to the dual scaling function

orthogonal are waveletsand functions scaling

0~

, is,That

}0{ therefore

and Since

,,

11

Dual

WP

WVVVVP

kjmj

jj

jjjjjj

kj ,

~

kjkj

kkjkj

f

f

,,

,,

~,

0~

,0

~,

~

, ~

,

as drepresente becan in function Any

,,

,,,,

,,,,,,

,,

kjmjjj

kjk j

kjmjmj

kjk

kjmjj

mjkjk

kjj

mjl

mj

j

WP

ggP

g

W

Details

Dual Wavelets

• To find the wavelet coefficients j,m

mjmj

j mmjmj

f

fPfj

,,

0

12

0,,0

~,

PrimalScaling Fns

mj ,

DualScaling Fns

mj ,

~

basis of

coeff. obtained by

Primal Wavelets

mj ,

moments vanishing~N

jP~

DualProjection

N~

Order

jPPrimalProjection

NOrder

DualWavelets

mj ,~moments vanishingN

basis of

complement(refinement relation)

complement(refinement relation)

Lifting (Basic Idea)

• Idea: taken an old wavelet (e.g., lazy wavelet) and build a new, more performant one by adding in scaling functions of the same level

old waveletsscaling fns at level j

scaling fns at level j+1combine old wavelet with 2 scalingfns at level j to form new wavelet

Lifting changes …

• Changes propagate as follows:

Primalwavelet

DualScaling fn

Pj: Computing Coarser rep.

Dualwavelet

Inside Lifting

• From above figure, we know P determines the primal scaling function (by sending in delta sequence)

• Different U determines different primal wavelets (make changes on top of the old wavelet)

Inside Lifting (cont)

• U affects how sj-1 to be computed (has to do with ). Scaling fns are already set by P.

• ? From the same two-scale relations with (same )

• Visualizing the dual scaling functions and wavelets by cascading

kj ,

~

kj ,

~kj ,~

g~