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~