Biorthogonal Wavelets

Ref: Rao & Bopardikar, Ch.4

Jyun-Ming Chen

Spring 2001

Why is orthogonality useful

• Orthonormal bases further simplify the computation

2211 aax

T12a1

T21a 2 T11x

5/3a ,a/a x, 1111 5/1a ,a/a x, 2222

Ortho v. Non-Ortho Basis

Sum of projection vectors !?

Dual Basis

1b ,a 11 1b ,a 22 0b ,a 21 0b ,a 12

2211 aax T11x

3/1b ,a/b x, 1111

3/1b ,a/b x, 2222

T3/13/2b1 T3/23/1b2

Dual Bases

a1-a2 and b1-b2 are biorthogonal

T12a1

T21a 2

Dual Basis (cont)

• Dual basis may generate different spaces– Here: a1-a2 and b1-b2 generate two different 2D subspaces in Euclidean

3space.

• Semiorthogonal:– For dual basis that generates the same subspace

• Orthogonal:– Primal and dual are the same bases

T211a1

T010a 2 T011b2 T001b1

Verify duality !

Extend to Function Space

• MRA types: – orthogonal, semiorthogonal, biorthognal

• Extend the concept to using biorthogonal MRA– More flexible design– Lifting scheme: a general design method for bio

rthogonal wavelets

Alternative Wavelets: Biorthogonal Wavelets

Proposed by Cohen (1992)

Characteristics of Orthogonal Basis

• Decomposition and reconstruction filters are FIR and have the same length

• Generally do not have closed-form expressions

• Usually not symmetric (linear phase)

• Haar wavelet is the only real-valued wavelet that is compactly supported, symmetric and orthogonal

• Higher-order filters (with more coefficients) have poor time-frequency localization

• Desired property: perfect reconstruction FIR symmetric (linear-phase) filters– Not available in orthogonal

bases

The Need for Biorthogonal Basis

• delegate the responsibilities of analysis and synthesis to two different functions (in the biorthogonal case) as opposed to a single function in the orthonormal case– more design freedom

• compactly supported symmetric analyzing and synthesis wavelets and scaling functions

Biorthogonal Scaling Functions

• Two sequences serve as impulse response of FIR filters

• Two sets of scaling functions generate subspaces respectively

• The basis are orthogonal; the two MRAs are said to be biorthogonal to each other

n

ntnht )2()(2)(

n

ntnht )2(~

)(~

2)(~

)()(~

),( kktt

)(~

and )( nhnh

)(~

and )( tt kk VV

~ and

dual

)(2)2(~

),2( nntt kkk

Dual MRA (cont)

• Basis of – Translated copy of appropriate dilation of

3210 VVVV

3210

~~~~VVVV

)(~

and )( tt

kk VV~

and

Function approximation in subspaces

n

ntnatf )(),0()(0

)(~

),(),0( nttfna

n

ntnatf )2(),1()(1

)2(~

),(2),1( nttfna

0

~on )( projecting

by obtained tscoefficien

Vtf

Coarser approx

Finer approx

2

1),1(

)2(~

),2(),1()2(~

),(

na

ntntnanttf

Relation between Finer and Coarser Coefficients

m

mntmhnt )22(~

)(~

2)(~ )(

~),(),0( nttfna

)22(~

),()(~

2),0( mnttfmhnam

)2(~

),(2),1( nttfna

m

mnamhna )2,1()(~

),0(

)2(~

),1(),0( nmhmanam

)2(~

),1(),0( nmhmanam

)22(~

),(2)2,1()2( mnttfmnamnn

Biorthogonal Wavelets

0)( dtt

0)(~ dtt

fns scaling dualwavelet0)(~

),( ntt fns scaling waveletdual0)(),(~ ntt

)()(~),( kktt Dual

• Two sets of wavelets generate subspaces respectively

• The basis are orthogonal; the two MRAs are said to be biorthogonal to each other

0 spans :)( WZkkt

0

~ spans :)(~ WZkkt

)(~ and )( tt kk WW

~ and

Require:

Two-scale relations of wavelet: primal and dual

n

ntngt )2(~

)(~2)(~

n

ntngt )2()(2)(

n

ntnbtftftg )(),0( )()()( :fn detail 010

)(~),(

)(~),()(~),(

)(~),()()(~),(),0(

1

01

010

nttf

nttfnttf

nttftfnttgnb

m

nmgmanb )2(~),1(),0( m

nmgmanb )2(~),1(),0(

and

)(),0()(0 n

ntnatf

0)(),(~ ntt

m

mtmatf )2(),1()(1

l

lntlgnt )22(~

)(~2)(~

Function Projection

m=2n+l

Function Reconstruction

m lm l

ll

mltmglbmltmhla

ltlbltla

tgtftf

)22()(),0(2)22()(),0(2

)(),0()(),0(

)()()( 001

n ln l

ntlnglbntlnhlatf

mln

)2()2(),0(2)2()2(),0( 2)(

2 ngSubstituti

1

l l

n

lnglblnhlana

ntnatf

)2(),0(2)2(),0(2),1(

Hence

)2(),1( )(1

Filter Bank

Primal and Dual MRA (biorthogonal)

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

NV~

1-NV~

2-NV~

3-NV~

3-NW~

2-NW~

1-NW~

kkkk WVWV ~~kkkk WVWV ~~

Filter Relations (between primal and dual)

m

p q

p q

mhnmh

qnpqhph

qntptqhph

nntt

)(~

)2(2

)2()(~

)(2

)22(~

),2()(~

)(4

)()(~

),(

2

)()(

~)2(

nmhnmh

m

2

)()(

~)2(

nmhnmh

m

Similarly,

2

)()(~)2(

nmgnmg

m

2

)()(~)2(

nmgnmg

m

qnp

qnp

2

02

:left only term

0)(~),( ntt

Filter Relations (cont)

0)(~)2( m

mgnmh 0)(~)2( m

mgnmh

mp q

p q

mgnmhqnpqgph

qntptqgph

ntt

)(~)2(2)2()(~)(2

)22(~

),2()(~)(4

0)(~),(

Similarly,

0)()2(~

m

mgnmh 0)()2(~

m

mgnmh0)(~

),( ntt

Design of Biorthogonal Wavelets

• because there is quite a bit of freedom in designing the biorthogonal wavelets, there are no set steps in the design procedure. …

• Lifting (Sweldens 94): a scheme for custom-design biorthogonal wavelets

Special Cases: orthogonal and semiorthogonal

• Common property:

• Differences: – if orthogonal: scaling functi

ons (and wavelets) of the same level are orthogonal to each other

– If semiorthogonal, wavelets of different levels are orthogonal (from nested space)

kkkkk WVVWV 1

VN

VN-1 WN-1

VN-2 WN-2

VN-3 WN-3

kkkk WWWV~

and ~

Dual and primal are the

same