Second-generation wavelets on finite
intervals
Stephan Houben
Eindhoven University of Technology
January 17, 2002
1
Overview of the talk
1. Quick review of some concepts
2. The Lifting scheme
3. Numerical examples of Lifting, and some problems
4. Wavelets on finite intervals
(second-generation wavelets)
5. Numerical examples of interval wavelets
6. Other second-generation wavelets
2
Quick review of some concepts
• Riesz bases and their dual
• Multi-resolution analysis
• Wavelets and scaling functions
• Discrete Wavelet Transform (DWT)
3
Riesz bases
Suppose we have a Hilbert space H and a (countably
infinite) sequence of vectors {φn} ⊂ H. Define the
operator Φ : `2 → H as
Φ({αn }) :=
∞
X
αn φn .
(1)
n=0
Then {φn} is a Riesz basis of H if and only if:
1. The series (1) converges for all {αn} ∈ `2.
2. The operator Φ is bounded.
3. The inverse Φ−1 : H → `2 exists.
4. The inverse Φ−1 is also bounded.
4
What does it mean?
This means that a Riesz basis gives us a way to map
the abstract space H one-to-one on the concrete space
`2. Convergence of a sequence in `2 implies convergence of the Φ-transformed sequence in H, and vice
versa.
Note that if we have a bounded operator Ψ : `2 → H
with bounded inverse Ψ−1, then we can find back the
accompanying Riesz basis {ψ n} by
ψ n := Ψ(en),
where en(m) = δnm.
(2)
5
Dual basis
The dual basis {φ̃n} of a Riesz basis {φ}n satisfies
the following equation.
(φi, φ̃j ) = δij .
(3)
The corresponding operators Φ and Φ̃ satisfy the relation
−1 ∗
Φ̃ = Φ
(4)
.
If {φn} = {φ̃n} (which is true iff Φ = Φ̃), then we
say that {φn} is an orthonormal basis.
Note that Φ is in that case an orthogonal operator,
since
Φ∗ = Φ−1.
(5)
6
Multi-resolution analysis
Consider the function space L2(R).
A Multi-resolution analysis consists of a sequence of
linear subspaces {Vn}, n ∈ Z, with the following properties.
1. Vn = Vn−1 ⊕ Wn−1, for some {Wn}.
2.
3.
S
n∈Z
T
n∈Z
Vn = L2(R).
Vn = {0}.
4. f (t) ∈ Vn ⇔ f (2t) ∈ Vn+1.
5. f (t) ∈ V0 ⇔ f (t + 1) ∈ V0.
7
Analysis and synthesis
Analysis
V3
V2
V1
V0
W2
W1
W0
V2
V1
V0
W2
W1
W0
Synthesis
V3
8
Wavelets and scaling functions
1. A Riesz basis of Vi is formed by the scaling functions {φi,n(t)}n .
2. A Riesz basis of the detail space Wi is formed by
the wavelets {ψi,n(t)}n .
3. All scaling functions are scaled, translated versions of the father wavelet φ(t).
4. All wavelet functions are scaled, translated versions of the mother wavelet ψ(t).
5. Scaling functions and wavelets are localised, i.e.
lim φi,n(t), ψi,n(t) = 0.
t→±∞
6.
∞
R
−∞
ψi,n(t) = 0, i.e. wavelets have zero average.
9
Discrete Wavelet Transform
We have a f ∈ Vi, represented by coefficients {vi(n)}n.
∞
X
f (t) =
(6)
vi(n)φi,n(t).
n=−∞
Since Vi = Vi−1 ⊕Wi−1, we can write f (t) = g(t)+
h(t), with g ∈ Vi−1 and h ∈ Wi−1.
g(t) =
h(t) =
∞
X
n=−∞
∞
X
vi−1(n)φi−1,n(t),
(7)
wi−1(n)ψi−1,n(t).
(8)
n=−∞
The linear operation that produces {vi−1(n)}n and
{wi−1(n)}n from {vi(n)} is called the Discrete Wavelet
Transform (DWT).
Its inverse is called the Inverse Discrete Wavelet Transform (IDWT).
10
The Lifting Scheme
The lifting scheme is a method for designing wavelet
transforms.
First we design a Discrete Wavelet Transform (DWT)
directly, without any reference to scaling functions φi,n
or wavelets ψi,n.
Then we find the scaling functions and wavelets belonging to this DWT. Note that we don’t need these for
most applications.
Exactly the opposite of what you might expect!
11
Lifting scheme: DWT
{vi (n)}n
SPLIT
{vi (2n)}n
{vi (2n + 1)}n
-
PREDICT
UPDATE
{wi−1 (n)}n
+
{vi−1 (n)}n
12
Lifting scheme: IDWT
{vi−1 (n)}n
{wi−1 (n)}n
UPDATE
+
-
PREDICT
{vi (2n)}n
{vi (2n + 1)}n
MERGE
{vi (n)}n
13
Discrete Wavelet Transform:
{vi(2n)}n , {vi(2n + 1)}n = SPLIT({vi(n)}n),
{wi−1(n)}n = {vi(2n + 1)}n − P({vi(2n)}n),
{vi−1(n)}n = {vi(2n)}n + U({wi−1(n)}n).
Inverse Discrete Wavelet Transform:
{vi(2n)}n = {vi−1(n)}n − U({wi−1(n)}n).
{vi(2n + 1)}n = {wi−1(n)}n + P({vi(2n)}n ),
{vi(n)}n = MERGE({vi(2n)}n , {vi(2n + 1)}n).
14
Choosing P and U
• P is selected so that certain classes of functions
can be represented exactly (lead to 0 detail coefficients) in any space Vi . E.g. constant functions
or polynoms up to some degree k.
• U is selected so that the contribution of the detail
coefficients {wi−1(n)}n to the signal {vi(n)}n
has zero average.
More generally, one requires that the first k moments are 0.
∞
X
vi(n)np = 0,
for p = 0, . . . , k − 1.
n=−∞
(9)
15
Designing the predict step
{vi (n)}n
SPLIT
{vi (2n)}n
{vi (2n + 1)}n
-
PREDICT
{. . . , 0, . . . }
We design the operator P so that for certain classes
of inputs {vi(n)}n, the detail signal becomes exactly
0.
16
Designing the update step
{. . . , 0, 1, 0, . . . }
UPDATE
+
-
PREDICT
{vi (2n)}n
{vi (2n + 1)}n
MERGE
{vi (n)}n
We design the operator U so that if a δ-pulse is put
on the detail wire and 0 on the other wire, we obtain a
{vi(n)}n with 0 average.
17
Example: Haar wavelet
• P({vi(2n)}n) = {vi(2n)}n,
i.e. just the identity.
• U({wi−1(n)}n) = { 1
2 wi−1(n))}n ,
i.e. multiplication by 1
2.
18
Haar wavelet update step
{. . . , 0, 1, 0, . . . }
UPDATE
+
{. . . 0,
-
PREDICT
{. . . 0, − 12 , 0 . . . }
1
,0...}
2
MERGE
{. . . 0, − 12 , 12 , 0 . . . }
19
Example:
Cohen-Daubechies-Feauveau
biorthogonal (2, 2) wavelet
• P({vi(2n)}n) = { 1
2 (vi (2n) + vi(2n + 2))}n ,
• U({wi−1(n)}n) = { 1
4 (wi−1(n−1)+wi−1(n))}n .
20
Computing the scaling function
In the space V0, the basic scaling function φ0,0 (the
father wavelet) is represented by the coefficients
{v0(n)}n = {. . . , 0, 1, 0 . . . } = δn.
Using the Inverse DWT (with all detail coefficients wi(n) =
0), we can compute the representation of the scaling
function in any space Vi, i > 0.
For i → ∞, the coefficients {vi(n)}n approach the
sampling {φ0,0(n2−i)}n .
21
The cascade algorithm
{. . . 0, 1, 0 . . . }
PREDICT
IDWT
IDWT
MERGE
IDWT
{vi (n)}n ≈ {φ0,0 (n2−i)}n
22
Example: CDF-(2, 2) scaling
function
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-2
-1.5
Iteration 0
-1
-0.5
0
0.5
1
1.5
2
1
1.5
2
Iteration 1
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-2
-1.5
-1
-0.5
0
0.5
Iteration 2
1
1.5
2
-2
-1.5
-1
-0.5
0
0.5
Iteration 8
23
Computing the wavelet
In the detail space W0, the basic wavelet ψ0,0 (the
mother wavelet) is represented by the coefficients
{w0(n)}n = {. . . , 0, 1, 0 . . . } = δn.
Using the Inverse DWT (with all coefficients v0(n) =
0), we can compute the representation of the scaling
function in V1.
Then we can again use the cascade algorithm to find
the representation in any space Vi , i > 1.
For i → ∞, the coefficients {vi(n)}n approach the
sampling {ψ0,0(n2−i)}n.
24
Example: CDF-(2, 2) wavelet
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-2
-1
0
1
2
3
-2
-1
Iteration 0
0
1
2
3
2
3
Iteration 1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
-0.2
-0.2
-0.4
-0.4
-2
-1
0
1
Iteration 2
2
3
-2
-1
0
1
Iteration 8
25
CDF-(2, 2)
1
Cohen-Daubechies-Feauveau (2,2) scaling function
0.8
0.6
0.4
0.2
0
-3
-2
-1
0
1
2
3
0.8
Cohen-Daubechies-Feauveau (2,2) wavelet
0.6
0.4
0.2
0
-0.2
-0.4
-3
-2
-1
0
1
2
3
26
CDF-(2, 2) for de-noising
4
4
Original signal
3
3
2
2
1
1
0
0
-1
Noisy signal
-1
0
1
2
3
4
5
6
0
1
Original signal
4
2
3
4
5
6
Noisy signal
4
Non-adaptive filter
3
3
2
2
1
1
0
0
-1
Adaptive filter
-1
0
1
2
3
4
5
Non-adaptive filter
6
0
1
2
3
4
5
Adaptive filter
27
6
CDF-(2, 2) for de-noising (2)
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
6
0
1
Original signal
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
Non-adaptive filter
3
4
5
6
5
6
Noisy signal
5
0
2
6
0
1
2
3
4
Adaptive filter
28
CDF-(2, 2), non-periodic signal,
adaptive filter
5
4
3
2
1
0
0
1
2
3
4
5
6
29
Wavelets on finite intervals
• Away from the boundary, wavelets and scaling
function remain the same.
• Near the boundary, we introduce special wavelets
and scaling functions.
• We again use the lifting scheme to find the DWT
first, and then compute the wavelets and scaling
functions by using the cascade algorithm.
• It is no longer true that all wavelets and scaling
functions are translated, dilated versions of the
mother and father wavelet. (Orphan wavelets)
• Interval wavelets are a special case of secondgeneration wavelets.
30
Predict for interval wavelets
Away from the boundary, P works as before.
P
P
Near the boundary, we have to do something special.
31
Update for interval wavelets
We design the operator U again by putting δ-pulses
on the detail wire, but now we have to consider the
boundary as a special case. E.g. for CDF-(2, 2),
{1, 0, 0, . . . }
UPDATE
+
{1 −
-
PREDICT
{−x, −y, 0, . . . }
x+y
, − 2y , 0, . . . }
2
MERGE
{−x, 1 −
x+y
, −y, − 2y , 0, . . . }
2
Take x = 3/4 and y = 1/8.
32
CDF-(2, 2) on the interval
1
1
0.8
0.8
0.6
0.6
0.4
0.4
0.2
0.2
0
0
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Scaling function in center
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Scaling function on boundary
0.8
0.6
0.4
0.6
0.2
0.4
0
0.2
-0.2
0
-0.4
-0.2
-0.6
-0.4
-0.8
0
0.5
1
1.5
2
2.5
3
3.5
Wavelet in center
4
4.5
5
0
0.5
1
1.5
2
2.5
3
3.5
4
Wavelet on boundary
33
4.5
5
Interval CDF-(2, 2) for de-noising
5
5
4
4
3
3
2
2
1
1
0
0
0
1
2
3
4
5
6
0
1
Original signal
5
4
4
3
3
2
2
1
1
0
0
1
2
3
4
5
Non-adaptive filter
3
4
5
6
5
6
Noisy signal
5
0
2
6
0
1
2
3
4
Adaptive filter
34
Other second-generation wavelets
1. Wavelets on non-equidistant grids
2. Wavelets with weighted inner products
3. Wavelets on a sphere (or even arbitrary surfaces)
35
Wavelets on non-equidistant grids
• The predict step generalises in a straightforward
way, as long as we have a subdivision scheme.
P
• The update step is complicated by the fact that
we need the averages (integrals) of the scaling
functions, which may all be different. We can
numerically approximate these integrals with the
cascade algorithm and a quadrature rule.
36
Wavelets with weighted inner
products
Instead of the standard L2 inner product
Z∞
(f, g) =
f (t)g(t) dt,
−∞
one may want to use a weighted inner product
Z∞
(f, g)w =
f (t)g(t)w(t) dt.
−∞
Again, the difficulty is with the update step.
We now want the weighted average and the first k
weighted moments of the wavelets to be 0.
Z∞
ψi,n(t)w(t)tp dt = 0,
p = 0, 1, . . . , k − 1.
−∞
The update step has to ensure all these conditions.
37
Wavelets on a sphere (or even
arbitrary surfaces)
“Butterfly” scaling function
“Butterfly” wavelet
Images by Wim Sweldens
38
THE END
39