Wavelets and filterbanks

Mallat 2009, Chapter 7

Outline

•  Wavelets and Filterbanks

•  Biorthogonal bases

•  The dual perspective: from FB to wavelet bases –  Biorthogonal FB –  Perfect reconstruction conditions

•  Separable bases (2D)

•  Overcomplete bases –  Wavelet frames (algorithme à trous, DDWF) –  Curvelets

Wavelets and Filterbanks

Wavelet side

•  Scaling function –  Design (from multiresolution

priors) –  Signal approximation –  Corresponding filtering operation

§  Condition on the filter h[n] → Conjugate Mirror Filter (CMF)

•  Corresponding wavelet families

Filterbank side

•  Perfect reconstruction conditions (PR) –  Reversibility of the transform

•  Equivalence with the conditions on the wavelet filters –  Special case: CMFs →

Orhogonal wavelets –  General case → Biorthogonal

wavelets

Wavelets and filterbanks

•  The decomposition coefficients in a wavelet orthogonal basis are computed with a fast algorithm that cascades discrete convolutions with h and g, and subsample the output

•  Fast orthogonal WT

{ }

0 0

* *0

( ) [ ] ( )

[ ] ( ), ( ) ( ) ( ) ( ) ( ) * ( )

( ) ( )

Since (t-n) is an orthonormal basisn

n

f t a n t n V

a n f t t n f t t n dt f t n t dt f n

t t

ϕ

ϕ

ϕ ϕ ϕ ϕ

ϕ ϕ

∈Ζ

+∞ +∞

−∞ −∞

= − ∈

= − = − = − =

= −

∑

∫ ∫

Linking the domains

)()(ˆ)(ˆ)()(ˆ)(ˆ

)()(ˆ)(ˆ)(ˆ)()(ˆ)(ˆ

1*1

)(

−

−−

+

↔−=↔=−

−↔−==+↔=

=

zfffzfeff

zfefeffzfeff

ez

jjj

jj

ωωωπω

ω

ω

ωπω

ω

ω

f [n]↔ f (z) = f [k]z−kk=−∞

+∞

∑

f [n−1]↔ z−1 f (z) unit delay

f [−n]↔ f z−1( ) reverse the order of the coefficients

(−1)n f [n]↔ f (−z) negate odd terms

Switching between the Fourier and the z-domain

Switching between the time and the z-domain

Fast orthogonal wavelet transform

•  Fast FB algorithm that computes the orthogonal wavelet coefficients of a discrete signal a0[n]. Let us define

Since is orthonormal, then

•  A fast wavelet transform decomposes successively each approximation PVjf in the coarser approximation PVj+1f plus the wavelet coefficients carried by PWj+1f.

•  In the reconstruction, PVjf is recovered from PVj+1f and PWj+1f for decreasing values of j starting from J (decomposition depth)

,,

,

[ ] ,

[ ] ,j nj j n

j j n

a n f

d n f

ϕϕ

ψ

=

=

jsince is an orthonormal basis for V

( )0 0( ) [ ]n

f t a n t n Vϕ= − ∈∑

( ){ }nt nϕ∈Ζ

−

0[ ] ( ), ( ) ( )a n f t t n f nϕ ϕ= − = ∗

Fast wavelet transform

•  Theorem 7.7 –  At the decomposition

–  At the reconstruction

∑

∑∞+

−∞=+

+∞

−∞=+

∗=−=

∗=−=

njjj

njjj

pganapngpd

phanapnhpa

]2[][]2[][

]2[][]2[][

1

1

⎩⎨⎧

+=

==

∗+∗=−+−= ++

+∞

−∞=

+∞

−∞=++∑ ∑

1202][

][

][][][]2[][]2[][ 1111

pnpnpx

nx

ngdnhandnpgnanphpa jjn n

jjj

(1)

(2)

(4)

x[p]

x n!" #$

Proof: decomposition (1) 1 1 1 1

1*

1 11

11 1

1

1

1

[ ] [ ] [ ], [ ] [ ]

21 1 2[ ], [ ]2 22 2

2 '' 2 2 2 ' 2 2 2 '22

2 '22

j j j j j j jn

j j

j j j jj j

jj j j j j

j

j

j

p V V p p n n

butt p t np n dt

t p tt t p t t p t p t

t p t

ϕ ϕ ϕ ϕ ϕ

ϕ ϕ ϕ ϕ

ϕ ϕ

+ + + +

+

+ ++

+− + +

+

+

+

∈ ⊂ → =

⎛ ⎞ ⎛ ⎞− −= ⎜ ⎟ ⎜ ⎟

⎝ ⎠⎝ ⎠

−= − → = + → − = → =

⎛ ⎞− ⎛ ⎞=⎜ ⎟ ⎜

⎝ ⎠⎝ ⎠

∑

∫

(b)

(a)

let

then

( )

( ) ( )

* *

*1

1

2 ' 22

1 ' 1[ ], [ ] ' 2 ' , 2 [ 2 ]2 22 2

[ ] [ 2 ] [ ]

j

j

j j

j jn

t n t p n

t tp n t p n dt t p n h n p

p h n p n

ϕ ϕ

ϕ ϕ ϕ ϕ ϕ ϕ

ϕ ϕ

+

+

⎟

⎛ ⎞−= + −⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞= + − = + − = −⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

= −

∫

∑

replacing into (a)

thus (b) becomes

(3)

1 1' ' ' 22 22 2 2j j jt t t t tp p t p+ += − → = + → = +

Proof: decomposition (2) qui

•  Coming back to the projection coefficients

•  Similarly, one can prove the relations for both the details and the reconstruction formula

a j+1[ p]= f ,ϕ j+1,p = f , h[n− 2p]n∑ ϕ j ,n = f

−∞

+∞

∫ h[n− 2p]n∑ ϕ *j ,ndt =

= h[n− 2p]n∑ f (t)

−∞

+∞

∫ ϕ *j ,n (t)dt = h[n− 2p]n∑ f ,ϕ j ,n = h[n− 2p]

n∑ a j[n]→

a j+1[ p]= a j ∗h[2p]

Proof: decomposition (3)

•  Details

( ) [ ]

[ ]

[ ]

[ ] [ ] [ ]

1, 1 1, 1, , ,

'

1, ,

1, ,

1, ,

1

,

2 21, , 2 2

222

, 2 ,

2

j p j j j p j n j n j nn

j

j n j n

j p j nn

j n j nn

j jn

W V

t t pt t n p g n p

g n p

f g n p f

d p g n p a n

ψ ψ ψ ϕ ϕ

ψ ϕ ψ ϕ

ψ ϕ

ψ ϕ

+ + + +

−

+

+

+

+

∈ ⊂ → =

= − →

⎛ ⎞= − + = − →⎜ ⎟⎝ ⎠

= − →

= − →

= −

∑

∑

∑

∑

(3bis)

Proof: Reconstruction

1 1 , , 1, 1, , 1, 1,, ,j j j j p j p j n j n j p j n j nn n

V V W ϕ ϕ ϕ ϕ ϕ ψ ψ+ + + + + += ⊕ → = +∑ ∑

but , 1,

, 1,

, [ 2 ]

, [ 2 ]j p j n

j p j n

h p n

g p n

ϕ ϕ

ϕ ψ+

+

= −

= −

(see (3) and (3bis), the analogous one for g)

thus

, 1, 1,[ 2 ] [ 2 ]j p j n j nn nh p n g p nϕ ϕ ψ+ += − + −∑ ∑

Taking the scalar product with f at both sides:

⎩⎨⎧

+=

==

∗+∗=−+−= ++

+∞

−∞=

+∞

−∞=++∑ ∑

1202][

][

][][][]2[][]2[][ 1111

pnpnpx

nx

ngdnhandnpgnanphpa jjn n

jjj

CVD

Since Wj+1 is the orthonormal complement of Vj+1 in Vj, the union of the two respective basis is a basis for Vj. Hence

Graphically

aj p[ ] = h p− 2n[ ]n∑ aj+1 n[ ] = aj+1 n[ ]h p− 2n[ ]

n∑

aj 0[ ] = h −2n[ ]n∑ aj+1 n[ ] = aj+1 n[ ]h −2n[ ]

n∑

Let's assume that h is symmetric

aj 0[ ] = aj+1 n[ ]h 2n[ ]n∑

n0 1 2

n0 1 2

aj+1[n]

h[n]

Graphically

n0 1 2

n0 1 2

h[n]

aj+1 n[ ]

aj 0[ ] = aj+1 n[ ]h 2n[ ]n∑ =

aj+1 n[ ]h n[ ]n∑

Summary

•  The coefficients aj+1 and dj+1 are computed by taking every other sample of the convolution of aj with and respectively.

•  The filter removes the higher frequencies of the inner product sequence aj , whereas is a high-pass filter that collects the remaining highest frequencies.

•  The reconstruction is an interpolation that inserts zeroes to expand aj+1 and dj+1 and filters these signals, as shown in Figure.

h gh g

a j+1[ p]= a j ∗h[2p]

Filterbank implementation

h

g

g

h

↓2

↓2

↑2

↑2

+

a0

_

_

a1

a1

d1

h

g

↓2

↓2

a2

_

_

d2

a2

d2

ă0

g

h ↑2

↑2

+

d1

Fast DWT

•  Theorem 7.10 proves that aj+1 and dj+1 are computed by taking every other sample of the convolution on aj with and respectively

•  The filter h removes the higher frequencies of the inner product and the filter g is a band-pass filter that collects such residual frequencies

•  An orthonormal wavelet representation is composed of wavelet coefficients at scales

plus the remaining approximation at scale 2J

h g

1 2 2j J≤ ≤

{ }1 ,j Jj Jd a

≤ ≤⎡ ⎤⎢ ⎥⎣ ⎦

Summary

∀ω ∈ , h ω( )2+ h ω +π( )

2= 2

andh(0) = 2

↓2 h

↓2 gja

1ja +

1jd +

Analysis or decomposition Synthesis or reconstruction

* 1ˆˆ( ) ( ) [ ] ( 1) [1 ]j ng e h g n h nωω ω π− −= + ↔ = − −

The fast orthogonal WT is implemented by a filterbank that is completely specified by the filter h, which is a CMF The filters are the same for every j

↑2 h

↑2 g ja

Teorem 7.2 (Mallat&Meyer) and Theorem 7.3 [Mallat&Meyer]

Filter bank perspective

h

g g

h ↓2

↓2

↑2

↑2

+ a0

_

ă0

_

Taking h[n] as reference (which amounts to choosing the synthesis low-pass filter) the following relations hold for an orthogonal filter bank:

1 1

(1 )

neglecting the unitary shift, as usually done in applications

[ ] ( 1) [ ] ( 1)

[ ] [ ] ( 1) [ ]

[ ] [ ]

[ ] ( 1) [1 ] ( 1) [ 1]

[ ] [ ] ( 1) [ (1 )]

[ ]n n

n

n n

n

g n h n

g n g n h n

h n h n

g n h n h n

g n g n h n

h n− −

− −

− −

= − − = −

= − = −

= −

= − − = − −

= − = − − −

Finite signals

•  Issue: signal extension at borders

•  Possible solutions: –  Periodic extension

§  Works with any kind of wavelet §  Generates large coefficients at the borders

–  Symmetryc/antisymmetric extension, depending on the wavelet symmetry §  More difficult implementation §  Haar filter is the only symmetric filter with compact support

–  Use different wavelets at boundary (boundary wavelets) –  Implementation by lifting steps

Wavelet graphs

Orthogonal wavelet representation

•  An orthogonal wavelet representation of aL=< f ,ϕL,n> is composed of wavelet coefficients of f at scales 2L<2j<=2J , plus the remaining approximation at the largest scale 2J :

•  Initialization –  Let b[n] be the discrete time input signal and let N-1 be the sampling period, such that the

corresponding scale is 2L=N-1

–  Then:

original continuous time signal discrete time signal interpolation function

N-1: discrete sample distance 2L= N-1 scale

Initialization

,

1 11/ 2

, ,1 1

1 222

1 1 122

L

L n LL

LL n L nL

t n

t N n t N nN N NN N N N

ϕ ϕ

ϕ ϕ ϕ ϕ− −

− −

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

⎛ ⎞ ⎛ ⎞− −= → = = → = → =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠

following the definition:

( ) [ ] [ ] ( )1

,11

L nn n

t N nf t b n b n tN N

ϕ ϕ−+∞ +∞

−=−∞ =−∞

⎛ ⎞−= =⎜ ⎟

⎝ ⎠∑ ∑

[ ] [ ]1

,11 1, , L n L

t N nb n f f a nN N N

ϕ ϕ−

−

⎛ ⎞−= = =⎜ ⎟

⎝ ⎠

but

since

[ ] ( )

[ ] ( )

1

1

1

by definition, thenL

L

t N na n f t N dtN

a n N f N n

ϕ+∞ −

−−∞

−

⎛ ⎞−= ⎜ ⎟

⎝ ⎠

≈

∫

[ ] ,,L L na n f ϕ=

if f is regular, the sampled values can be considered as a local average in the neighborhood of f(N-1n)

N-1: discrete sample distance 2L= N-1 scale

Basis for VL

The filter bank perspective

Perfect reconstruction FB

•  Dual perspective: given a filterbank consisting of 4 filters, we derive the perfect reconstruction conditions

•  Goal: determine the conditions on the filters ensuring that

a0 ≡ a0

↓2 h

↓2 g0a

1a

1d

↑2 h

↑2 g ~

~

a0

PR Filter banks

•  The decomposition of a discrete signal in a multirate filter bank is interpreted as an expansion in l2(Z)

[ ] [ ] [ ] [ ] [ ]1 0 0 0[ ] * 2 2 2n n

a l a h l a n h l n a n h n l= = − = −∑ ∑since

then

and the signal is recovered by the reconstruction filter

thus

dual family of vectors

points to biorthogonal

wavelets

The two families are biorthogonal

Thus, a PR FB projects a discrete time signals over a biorthogonal basis of l2(Z). If the dual basis is the same as the original basis than the projection is orthonormal.

Discrete Wavelet basis

•  Question: why bother with the construction of wavelet basis if a PR FB can do the same easily?

•  Answer: because conjugate mirror filters are most often used in filter banks that cascade several levels of filterings and subsamplings. Thus, it is necessary to understand the behavior of such a cascade

1

,11

L nt N nN N

ϕ ϕ−

−

⎛ ⎞−=⎜ ⎟

⎝ ⎠

[ ] ,,L L na n f ϕ=

N-1: discrete sample distance 2L= N-1 scale

discrete signal at scale 2L

for depth j>L

Discrete wavelet basis

Perfect reconstruction FB

•  Theorem 7.7 (Vetterli) The FB performs an exact reconstruction for any input signal iif

h* ω( ) h(ω)+ g* ω( ) g(ω) = 2h* ω +π( ) h(ω)+ g* ω +π( ) g(ω) = 0

)(ˆ)(ˆ)(ˆ)(ˆ)(

)(ˆ)(ˆ

)(2

)(~)(~

*

*

ωπωπωωω

πω

πω

ωω

ω

ghgh

hg

gh

+−+=Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+

Δ=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Matrix notations

(alias free)

When all the filters are FIR, the determinant can be evaluated, which yields simpler relations between the decomposition and the reconstruction filters.

Changing the sampling rate

•  Downsampling

•  Upsampling

y 2ω( ) = 12 x ω( )+ x ω +π( )( ) = x 2n!" #$n=−∞

+∞

∑ e− jnω

y ω( ) = x 2ω( ) = x n!" #$n=−∞

+∞

∑ e− j2nω

ê 2 x ω( ) xdown ω( ) = y ω( )

y ω( ) = 12 xω2!

"#

$

%&+ x

ω2+π

!

"#

$

%&

!

"#

$

%&

é 2 x ω( ) xup ω( ) = y ω( )

Subsampling: proof y ω( ) =… y 0!" #$+ y 1!" #$e− jω + y 2!" #$e− j2ω +…=

=…x 0!" #$+ x 2!" #$e− jω + x 4!" #$e

− j2ω +…→

thusy 2ω( ) =…x 0!" #$+ x 2!" #$e− j2ω + x 4!" #$e− j4ω +…but

x 1!" #$e− jω + x 1!" #$e

− j (ω+π ) = 0→ 12x 1!" #$e

− jω + x 1!" #$e− j (ω+π )( ) = 0

x 2!" #$e− j2ω =

12x 2!" #$e

− j2ω + x 2!" #$e− j2(ω+π )( )

thus

y 2ω( ) =…x 0!" #$+ 12 x 1!" #$e

− jω + x 1!" #$e− j (ω+π )( )+ 12 x 2

!" #$e− j2ω + x 2!" #$e

− j2(ω+π )( )+…=

y 2ω( ) = 12 x ω( )+ x ω +π( )( )

Perfect Reconstruction conditions

↓2 h

↓2 g

↑2

↑2 0a

1a

1da0

h

g

( ) ( ) ( ) ( )( )( )

( ) ( ) ( ) ( )( )

( ) ( )

1 0 0

*

* *1 0 0

*1 0

1 ˆ ˆ(2 )2

[ ]ˆ ˆ[ ] [ ] ( ) ( ) ( )

1 ˆ ˆ(2 )2

1 ˆ(2 )2

since h and g are real

thus, replacing in the first equation

Similarly, for the high-pass branch

a a h a h

h n h

h n h n h h h

a a h a h

d a g

ω ω ω ω π ω π

ω

ω ω ω

ω ω ω ω π ω π

ω ω ω

= + + +

→

− = → = − =

= + + +

= ( ) ( )( )*0 ˆa gω π ω π+ + +

a0 (ω) = a1(2ω) h(ω)+ d1(2ω) g(ω)

Perfect Reconstruction conditions

•  Putting all together

a0 (ω) = a1(2ω) h(ω)+ d1(2ω) g(ω) =

=12a0 ω( ) h* ω( )+ a0 ω +π( ) h* ω +π( )( ) h(ω)

+12a0 ω( ) g* ω( )+ a0 ω +π( ) g* ω +π( )( ) g(ω)

a0 (ω) =12h* ω( ) h(ω)+ g* ω( ) g(ω)!"#

$%&a0 ω( )+ 12 h

* ω +π( ) h(ω)+ g* ω +π( ) g(ω)!"#

$%&a0 ω +π( )

=1 =0 (alias-free)

h* ω( ) h(ω)+ g* ω( ) g(ω) = 2h* ω +π( ) h(ω)+ g* ω +π( ) g(ω) = 0

)(ˆ)(ˆ)(ˆ)(ˆ)(

)(ˆ)(ˆ

)(2

)(~)(~

*

*

ωπωπωωω

πω

πω

ωω

ω

ghgh

hg

gh

+−+=Δ

⎟⎟⎠

⎞⎜⎜⎝

⎛

+−

+

Δ=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Matrix notations

(alias free)

Perfect reconstruction biorhogonal filters

•  Theorem 7.8. Perfect reconstruction filters also satisfy

Furthermore, if the filters have a finite impulse response there exists a in R and l in Z such that

•  Conjugate Mirror Filters:

h* ω( ) h(ω)+ h* ω +π( ) h(ω +π ) = 2

g(ω) = e− jω h*(ω +π )g(ω) = e− jωh*(ω +π )

Correspondinglyg[n]= (−1)1−n h[1− n]g[n]= (−1)1−n h[1− n]

h = h→ h ω( )2+ h ω +π( )

2= 2

)(ˆ1)(~)(~)(ˆ

*)12(

*)12(

πωω

πωωω

ω

+=

+=+−

+−

hea

ghaeg

li

li

a=1, l=0

Perfect reconstruction biorthogonal filters qui

Given h and and setting a=1 and l=0 in (2) the remaining filters are given by the following relations

§  The filters h and are related to the scaling functions φ and ~φ via the corresponding two-scale relations, as was the case for the orthogonal filters (see eq. 1).

Switching to the z-domain

Signal domain

)(ˆ)(~)(~)(ˆ

*

*

πωω

πωωω

ω

+=

+=−

−

heg

hegi

i

]1[)1(][~]1[~)1(][

11

nhngnhng

nn

−−=−−=

−

−

)()(~)(~)(1111

−−

−−

−=−=zhzzgzhzzg

h~

h~

(3)

Biorthogonal filter banks

•  A 2-channel multirate filter bank convolves a signal a0 with

a low pass filter

and a high pass filter

and sub-samples the output by 2

A reconstructed signal ã0 is obtained by filtering the zero-expanded signals with a dual low-pass and high pass filter

Imposing the PR condition (output signal=input signal) one gets the relations that the different filters must satisfy (Theorem 7.7)

][][ nhnh −=][][ ngng −=

]2[][]2[][

01

01ngandnhana

∗=

∗=

][~ nh][~ ng

⎩⎨⎧

+=

===

∗+∗=

1202][

][][

][~][~][~ 110

pnpnpx

nxny

ngdnhana

Revisiting the orthogonal case (CMF)

h

g g

h ↓2

↓2

↑2

↑2

+ a0

_

ă0

_

Taking as reference (which amounts to choosing the analysis low-pass filter) the following relations hold for an orthogonal filter bank:

1

[ ] [ ] [ ] [ ]

[ ] ( 1) [1 ]n

h n h n h n h n

g n h n−

= − ↔ = −

= − −

synthesis low-pass (interpolation) filter: reverse the order of the coefficients

negate every other sample

[ ] [ ]h n h n= −

Orthogonal vs biorthogonal PRFB

↓2 h

↓ 2 g

↑ 2

↑ 2 0a

1a

1da0

h

g

h* ω( ) h(ω)+ h* ω +π( ) h(ω +π ) = 2g(ω) = e− jω h*(ω +π )g(ω) = e− jωh*(ω +π )

In the signal domaing[n]= (−1)1−n h[1− n]g[n]= (−1)1−n h[1− n]

h ≠ h h = h

h ω( )2+ h ω +π( )

2= 2

g = g

Biorthogonal PRFB Orthogonal PRFB

Fast BWT

•  Two different sets of basis functions are used for analysis and synthesis

•  PR filterbank

][~][~][

]2[][]2[][

11

11

ngdnhana

ngandnhana

jjj

jjjj

∗+∗=

∗=∗=

++

+

+

][][ 00 nana =

h

g g

h ↓2

↓2

↑2

↑2

+ a0

_

_

~

~

ă0

]1[)1(][~]1[~)1(][

11

nhngnhng

nn

−−=−−=

−

−

Be careful with notations!

•  In the simplified notation where –  h[n] is the analysis low pass filter and g[n] is the analysis band pass filter, as it is the case in most of the

literature; –  the delay factor is not made explicit;

•  The relations among the filters modify as follows

][][ 00 nana =

h

g g

h ↓2

↓2

↑2

↑2

+ a0

~

~

ă0

][)1(][~][~)1(][

nhng

nhngn

n

−

−

−=

−= Slightly different formulation: the high pass filters are obtained by the low pass filters by negating the odd terms

Biorthogonal bases

Orthonormal basis

{en}n∈N: basis of Hilbert space

Ortogonality condition < en, ep>=0 ∀n≠p

∀y ∈ H,

There exists a sequence

|en|2=1 ortho-normal basis

Bi-orthogonal basis

{en}n∈N: linearly independent

∀y ∈ H, ∃A>0 and B>0 :

Biorthogonality condition:

A=B=1 ⇒ orthogonal basis

en

ep

∑==

nn

nenyeyn][:,][

λλ

∑

∑

≤≤

==

n

nn

n

Ay

nBy

enyeyn

22

2

][

~][:,][

λ

λλ

∑∑ ==

−=

nnn

nnn

pn

eefeefypnee

~,~,][~, δ

Biorthogonal bases

If h and h are FIR

Φ ω( ) =h 2− pω( )

2p=1

+∞

∏ Φ(0), Φ ω( ) =h 2− pω( )

2p=1

+∞

∏ Φ(0)

The functions φ and φ satisfy the biorthogonality relationϕ(t), ϕ(t − n) = δ[n]

The two wavelet families ψj,n{ }

( j ,n)∈Z 2 and ψ

j,n{ }( j ,n)∈Z 2

are Riesz bases of L2(R)

ψj ,n, ψ

j ' ,n'= δ[n− n ']δ[ j − j ']

Though, some other conditions must be imposed to guarantee that φ^ and φ^tilde are FT of finite energy functions. The theorem from Cohen, Daubechies and Feaveau provides sufficient conditions (Theorem 7.10 in M1999 and Theorem 7.13 in M2009)

Any f ∈ L2 R( ) has two possible decompositions in these bases

f = f ,ψ j ,nn, j∑ ψ j ,n = f , ψ j ,n

n, j∑ ψ j ,n

Reminder

Summary of Biorthogonality relations

•  An infinite cascade of PR filter banks yields two scaling functions and two wavelets whose Fourier transform satisfy

)~,~(),,( ghgh

Φ 2ω( ) = 12h ω( )Φ ω( ) ↔ ϕ

t2#

$%&

'(= h[n]ϕ t − n( )

n=−∞

+∞

∑ (i)

Φ 2ω( ) = 12h ω( ) Φ ω( ) ↔ ϕ t

2#

$%&

'(= h[n] ϕ t − n( )

n=−∞

+∞

∑ (ii)

Ψ 2ω( ) = 12g ω( )Φ ω( ) ↔ ψ

t2#

$%&

'(= g[n]ϕ t − n( )

n=−∞

+∞

∑ (iii)

Ψ 2ω( ) = 12g ω( ) Φ ω( ) ↔ ψ t

2#

$%&

'(= g[n] ϕ t − n( )

n=−∞

+∞

∑ (iv)

Properties of biorthogonal filters

Imposing the zero average condition to ψ in equations (iii) and (iv)

Ψ(0) = Ψ(0) = 0 → g(0) = g(0) = 0replacing into the relations (3) (also shown below)

g(ω) = e−iω h*(ω +π ) g(ω) = e−iω h*(ω +π )→ h*(π ) = h(π ) = 0Furthermore, replacing such values in the PR condition (1)

h*(ω) h(ω)+ g*(ω) g(ω) = 2→ h*(0) h(0) = 2It is common choice to set

h*(0) = h(0) = 2

Biorthogonal bases

•  If the decomposition and reconstruction filters are different, the resulting bases is non-orthogonal

•  The cascade of J levels is equivalent to a signal decomposition over a non-orthogonal basis

•  The dual bases is needed for reconstruction

{ } { }1 ,

2 , 2J jJ jn j J nk n k nϕ ψ

∈Ζ ≤ ≤ ∈Ζ

⎡ ⎤⎡ ⎤ ⎡ ⎤− −⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦

Example: bior3.5

Example: bior3.5

Biorthogonal bases

Biorthogonal bases

CMF : orhtogonal filters

•  PR filter banks decompose the signals in a basis of l2(Z). This basis is orthogonal for Conjugate Mirror Filters (CMF).

•  [Smith&Barnwell,1984]: Necessary and sufficient condition for PR orthogonal FIR filter banks, called CMFs

–  Imposing that the decomposition filter h is equal to the reconstruction filter h~, eq. (1) becomes

–  Correspondingly

h*(ω) h(ω)+ h*(ω +π ) h(ω +π ) = 2 (1) →h*(ω)h(ω)+ h*(ω +π )h(ω +π ) = 2→| h(ω) |2 + | h(ω +π ) |2= 2

]1[)1(][][~][][~

1 nhngngnhnh

n −−===

−

Summary

•  PR filter banks decompose the signals in a basis of l2(Z). This basis is orthogonal for Conjugate Mirror Filters (CMF).

•  [Smith&Barnwell,1984]: Necessary and sufficient condition for PR orthogonal FIR filter banks, called CMFs

–  Imposing that the decomposition filter h is equal to the reconstruction filter h~, eq. (1) becomes

–  Correspondingly 2|)(ˆ||)(ˆ|

2)(ˆ)(ˆ)(ˆ)(ˆ2)(~)(ˆ)(~)(ˆ

22

**

**

=++

→=+++

→=+++

πωω

πωπωωω

πωπωωω

hh

hhhh

hhhh

]1[)1(][][~][][~

1 nhngngnhnh

n −−===

−

Properties

•  Support –  h, are FIR → scaling functions and wavelets have compact support

•  Vanishing moments –  The number of vanishing moments of Ψ is equal to the order of zeros of in π. Similarly, the

number of vanishing moments of is equal to the order p of zeros of h in π.

•  Regularity –  One can show that the regularity of Ψ and φ increases with the number of vanishing moments

of , thus with the order p of zeros of h in π. –  Viceversa, the regularity of and increases with the number of vanishing moments of Ψ, thus with the order of zeros of in π.

•  Symmetry –  It is possible to construct both symmetric and anti-symmetric bases using linear phase filters

§  In the orthogonal case only the Haar filter is possible as FIR solution.

h~

p~ h~ψ~

ψ~ ϕ~ψ~

p~ h~