Unformatted text preview: Course 18.327 and 1.130
Course
Wavelets and Filter Banks
Multiresolution Analysis (MRA):
Requirements for MRA;
Requirements
Nested Spaces and
Complementary Spaces;
Scaling Functions and Wavelets
Scaling Scaling Functions and Wavelets
φ(t) Box function Continuous time:
1 0
1 0 1 φ(2t) Scaling
(2t) Scaling 1/2 t t
φ(2t  1)
(2t
Scaling +
Scaling
Shifting
Shifting 1 0 1/2 1 t
2 For this example:
φ(t) = φ(2t) + φ(2t – 1)
(t)
(2t)
(2t
More generally:
Refinement equation
φ(t) = 2∑ h0[k]φ(2t – k)
(2t
or
k=0
Twoscale difference
equation
N φ(t) is called a scaling function
The refinement equation couples the representations
The
of a continuoustime function at two time scales. The
of
time
continuoustime function is determined by a discretecontinuous
time filter, h0[n]! For the above (Haar) example:
h0[0] = h0[1] = ½
[1] (a lowpass filter)
filter)
3 Note: (i) Solution to refinement equation may not
Note:
always exist. If it does…
always
(ii) φ(t) has compact support i.e.
(ii)
φ(t) = 0 outside 0 ≤ t < N
(t)
(comes from the FIR filter, h0[n])
(comes
(iii) φ(t) often has no closed form solution.
(iii)
(iv) φ(t) is unlikely to be smooth.
(iv)
Constraint on h0[n]:
N ∫ φ(t)dt = 2 ∑ h0[k] ∫ φ(2t – k)dt
dt
[k] (2t
k=0
N = 2 ∑ h0[k] • ½ ∫ φ(τ)dτ
[k]
k=0 So N ∑ h0[k] = 1
[k] k=0 Assumes ∫ φ(t)dt ≠ 0
dt
4 1 Square wave
of finite length of
Haar wavelet w(t) Now consider:
1
0 1 t 1/2 φ(2t)
Scaled
1/2 0 1/2 t 0 w(t) = φ(2t)  φ(2t – 1)
(2t)
(2t φ(2t – 1)
(2t
Scaled + shifted
Scaled
+ sign flipped
sign
1
t 5 More generally:
N w(t) = 2∑ h1[k] φ(2t – k)
[k] (2t Wavelet equation k=0 For the Haar wavelet example:
For
wavelet
h1[0] = ½
[0] h1[1] = ½
[1] (a highpass filter)
(a
filter) 6 Some observations for Haar scaling function and wavelet
Some
scaling
1. Orthogonality of integer shifts (translates):
Orthogonality of
1 0 φ(t) 1 φ(t  1)
(t 1 t 0 1 2 t 1 if k = 0
∫ φ(t) φ(t – k)dt = 0 otherwise
(t) (t
dt = δ[k]
Similarly
∫ w(t) w(t – k)dt = δ[k]
w(t)
dt
Reason: no overlap 7 2. Scaling function is orthogonal to wavelet:
1 φ(t) 1 w(t)
+ + 1
0 1 t 0 t 1/2
 ∫φ(t) w(t)dt = 0
dt
Reason: +ve and –ve areas cancel each other.
ve
ve areas 8 3. Wavelet is orthogonal across scales:
1 w(2t) w(t) w(2t  1)
w(2t + + 1/2 1
0 1/2  + t 0 ∫ w(t) w(2t)dt = 0 ,
dt  t  t ∫ w(t) w(2t – 1)dt = 0
w(t)
dt Reason: finer scale versions change sign while
Reason:
coarse scale version remains constant.
coarse 9 Wavelet Bases
Our goal is to use w(t), its scaled versions (dilations)
Our
and their shifts (translates) as building blocks for
continuoustime functions, f(t). Specifically, we are
continuous time
interested in the class of functions for which we can
define the inner product:
define
∞
<f(t) , g(t)> = ∫ f(t) g*(t)dt < ∞
<f(t)
dt
∞ Such functions f(t) must have finite energy:
∞ f(t) = ∫f(t)2 dt
∫
dt
2 ∞ <∞ and they are said to belong to the Hilbert space, L2(ℜ). 10
10 Consider all dilations and translates of the Haar wavelet:
Consider
wavelet:
/2
wj,k(t) = 2jj/2 w(2jt – k) ; ∞ ≤ j ≤ ∞
k) ∞ ≤ k ≤ ∞
Normalization factor so that wj,k(t) = 1
Normalization

J/2
∫ wj,k(t) wJ,K(t) dt = ∫ 2j/2 w(2jt – k) . 2J/2 w(2Jt – K)dt
(t)
(t) 1 if j = J and k = K
=
0 otherwise = δ[ j – J ] δ[ k – K ] 11
11 M 1
v2
L L w1,k(t) 4 2
1 3 t 1
L 1 4 3 2 L w0,k(t)
t v2
L 1 2 3 4 L w1,k(t)
t
12
12 wjk(t) form an orthonormal basis for L2(ℜ).
(t)
basis
f(t) = ∑ bjk wjk(t) ;
f(t)
(t)
j,k
∞ /2 w(2
wjk(t) = 2jj/2 w(2jt – k) bjjk = ∞ f(t) wjk(t) dt
∫ f(t)
(t) dt
k 13
13 Multiresolution Analysis
Multiresolution Analysis
Key ingredients:
1. A sequence of embedded subspaces:
{0} ⊂ … ⊂ V1 ⊂ V0 ⊂ V1 ⊂ … ⊂ Vj ⊂ Vj+1 ⊂ … ⊂ L2(ℜ)
{0}
L2(ℜ) = all functions with finite energy
∞
= {ƒ(t): ∫ ƒ(t) 2 dt < ∞}
Hilbert
(t):
(t)
dt

space
space
Requirements:
• Completeness as j → ∞ . If ƒ(t) belongs to
Completeness
If (t)
L2(ℜ) and ƒj(t) is the portion of ƒ(t) that lies in
and (t)
(t)
Vj, then lim∞ ƒj(t) = ƒ(t)
then j→
(t)
(t)
14
14 Restated as a condition on the subspaces:
∞ ∪ j=∞ • Vj = L2 (ℜ) Emptiness as j →  ∞
Emptiness
lim 
j →  ∞  fj(t)  = 0 Restated as a condition on the subspaces:
∞ ∩ Vj = {0} j=∞ 15
15 2. A sequence of complementary subspaces, Wj,
sequence
such that Vj + Wj = Vj+1
and
and Vj ∩ Wj = {0}
{0} (no overlap) This is written as
Vj ⊕ Wj = Vj+1 (Direct sum)
Note: An orthogonal multiresolution will have Wj
Note:
orthogonal to Vj : Wj ? Vj .
orthogonal
So orthogonality will ensure that Vj ∩ Wj = {0}
So 16
16 We thus have
V1 = V 0 ⊕ W 0
V2 = V 1 ⊕ W 1 = V 0 ⊕ W 0 ⊕ W 1
V3 = V 2 ⊕ W 2 = V 0 ⊕ W 0 ⊕ W 1 ⊕ W 2
M
J1
VJ = VJ1 ⊕ WJ1 = V0 ⊕ ∑ Wj
j=0
M
∞
2(ℜ) = V ⊕ ∑ W
L
0
j
j=0 We can also write the recursion for j < 0
We
V0 = V1 ⊕ W1
= V2 ⊕ W2 ⊕ W1
M 1
= Vk ⊕ ∑ Wj
j=k
M
∞
1
= ∑ Wj
⇒ L2(ℜ) = ∑ Wj
j = ∞ j = ∞ 17
17 3. A scaling (dilation) law:
If ƒ(t) ∈ Vj then ƒ(2t) ∈ Vj+1
If (t)
then (2t)
4. A shift (translation) law:
If ƒ(t) ∈ Vj then ƒ(tk) ∈ Vj
k integer
If (t)
then
k)
integer
5. V0 has a shiftinvariant basis, {φ(tk) :  ∞ ≤ k ≤ ∞}
k)
W0 has a shiftinvariant basis, {w(tk) :  ∞ ≤ k ≤ ∞}
k)
We expect that V1 = V0 + W0 will have twice as
will
many basis functions as V0 alone.
many
First possibility: {φ(tk) , w(tk) :  ∞ ≤ k ≤ ∞}
k)
Second possibility: use the scaling law i.e.
if φ(t k) ∈ V0 , then φ(2t k) ∈ V1
if
k)
k) 18
18 So V1 has a shiftinvariant basis, {v 2 φ(2tk):  ∞ ≤ k ≤ ∞}
k): Can we relate this basis for V1 to the basis for V0?
We know that
V0 ⊂ V1
So any function in V0 can be written as a combination
can
of the basic functions for V1.
of
In particular, since φ(t) ∈ V0, we can write
In
(t) φ(t) = 2∑ h0[k] φ(2t – k)
[k] (2t
k This is the Refinement Equation (a.k.a. the TwoScale Difference Equation or the Dilation Equation).
19
19 We also know that
W0 = V1 – V0
So
W0 ⊂ V1
This means that any function in W0 can also be written
can
as a combination of the basic functions for V1.
as
Since w(t) ∈ W0, we can write
w(t) = 2∑ h1[k] φ(2t – k)
[k] (2t
k Wavelet
Equation 20
20 Multiresolution Representations
Functions: L2 (ℜ) = V0 ⊕ W0 ⊕ W1 ⊕ W2 ⊕ ...
Level 2 detail
Level 1 detail
Level 0 detail Finite energy
Finite
functions
functions Coarse
Coarse
approximation
approximation Images:
V0 W0 + V1 W1 V2 +
21
21 Multiresolution Representations
Geometry: Mesh courtesy of Igor Guskov (Caltech) 22
22 ...
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Full Document
 Spring '03
 GilbertStrang
 Wj, dt, Hilbert space, Wavelet, Discrete wavelet transform, Haar wavelet

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