This preview shows page 1. Sign up to view the full content.
Unformatted text preview: Course 18.327 and 1.130
Course
Wavelets and Filter Banks
Mallat pyramid algorithm Pyramid Algorithm for Computing
Pyramid
Wavelet Coefficients
Wavelet
Goal: Given the series expansion for a function fj(t) in Vj
Goal:
(t)
fj(t) = ∑ aj[k]φj,k(t)
(t)
k how do we find the series
fj1(t) = ∑ aj1[k]φj1,k(t)
(t)
k in Vj1 and the series
in
gj1(t) = ∑ bj1[k]wj1,k(t)
(t)
k in Wj1 such that
such
in
fj(t) = fj1(t) + gj1(t) ?
(t)
(t)
(t)
2 Example: suppose that φ(t) = box on [0,1]. Then
Example:
(t)
functions in V1 can be written either as a combination of
can
functions L φ(2t) 1 0 1
, ½ φ(2t1) 0 L
, ½1 or as a combination of L φ(t) 1 0 φ(t1) 1 L
1 , 0 1 2 ,
3 plus a combination of w(t) 1 w(t1) 1 L L
0 1 , Easy to see because
Easy 0 φ(2t)
(2t)
φ(2t –1)
(2t 1) =
= 1 ½[φ(t)
½[φ(t)
(t) +
 2 , w(t)]
w(t)]
4 • Suppose that f(t) is a function in L2(R). What are the
(R).
coefficients, aj[k], of the projection of f(t) on to Vj?
[k],
Call the projection fj(t),
Call
fj(t) = ∑ aj[k]φj,k(t)
(t)
k aj[k] must minimize the distance between f(t) and fj(t)
[k]
∞ ∂
∫
∂aj[k] ∞ {f(t) – fj(t)}2 dt = 0
dt
{f(t) ∞ ∫ 2 {f(t)  ∑aj[l]φj,l(t)} φj,k(t)dt = 0
(t)}
dt
{f(t) ∞ l f(t) aj[k] = ∫f(t)φj,k(t)dt
[k]
dt
fj(t) 5 • How does φj,k(t) relate to φj1,k(t), wj1,k(t)?
How
(t)
(t),
N φ(t) = 2 ∑ h0[l]φ(2t  l)
(2t
(t)
l=0 refinement equation φj1,k(t) = 2(j1)/2 φ(2j1tk)
N = 2(j1)/2. 2 ∑ h0[l]φ (2jt – 2k l)
l=0 N φj1,k(t) = √2 ∑h0[l] φjj,2k + l(t)
(t)
,2k
l=0 Similarly, using the wavelet equation, we have
N wj1,k(t) = √2 ∑h1[l]φjj,2k + l(t)
(t)
,2k
l=0 6 Multiresolution decomposition equations
Multiresolution decomposition
∞ a j1[n] = ∫ f(t)φj1,n(t) dt
[n]
(t) dt
∞ ∞ = √2 ∑ h0[l] ∫ f(t)φjj,2n + l(t) dt
(t) dt
,2n
l ∞ = √2 ∑ h0[l] aj[2n + l]
[2n
l So
aj1[n] = √2 ∑ h0[k2n]aj[k]
[n]
k → Convolution with h0[n] followed by downsampling
n]
downsampling 7 Similarly
∞
bj1[n] = ∫ f(t) wj1,n(t) dt
[n]
f(t)
(t) dt
∞ which leads to
bj1[n] = √2 ∑ h1[k – 2n] aj[k]
[n]
[k 2n]
k 8 Multiresolution reconstruction equation
Multiresolution reconstruction
Start with
fj(t) = fj1(t) + gj1(t)
(t)
(t)
Multiply by φj,n(t) and integrate
Multiply
∞ ∞ ∞ ∞ ∞ ∞ ∫ fj(t) φj,n(t) dt = ∫ fj1(t)φj,n(t)dt + ∫ gj1(t)φj,n(t) dt
(t)
(t)
dt
(t) dt So ∞ aj[n] = ∑ aj1[k] ∫ φj1,k(t) φj,n(t) dt +
[n]
[k]
(t)
(t)
k ∞
∞ ∑ bj1[k] ∫ wj1,k(t) φj,n(t) dt
[k]
(t)
(t) dt
k ∞ 9 ∞ ∞ ∫ φj1,k(t) φj,n(t) dt = √2 ∑ h0[l] ∫ φj,2k+l(t) φj,n(t) dt
(t)
(t)
(t) dt ∞ l ∞ = √2 ∑ h0[l] δ[2k + l  n]
[2k
l = √2 h0[n – 2k]
[n 2k]
Similarly ∞ ∫ wj1,k(t)φj,n(t) dt = √2 h1[n –2k]
(t)
[n ∞ Result:
aj[n] = √2 ∑ aj1[k]h0[n  2k] +
[n]
[n
k √2 ∑ bj1[k]h1[n – 2k]
[n
k 10
10 Filter Bank Representation aj[n] u0[n]
aj1[n]
v [n]
~
↑2 0 √2h [n]
↓2
√2h0[n]
0 Synthesis Analysis v1[n]
bj1[n]
u1[n]
~
↑2
↓2
√2h1[n]
√2h1[n] ~
h0[n] = h0[n]
~
h1[n] = h1[n] ⊕ aj[n] time reversal Verify that filter bank implements MRA equations:
~
u0[n] = √2 ∑ h0[n  k]aj[k]
[n]
[n
k = √2 ∑ h0[k – n]aj[k]
[k
k 11
11 aj1[n] = u0[2n]
[2n] downsample by 2
by = √2 ∑ h0[k – 2n]aj[k]
[k
k bj1[n] = u1[2n]
= √2 ∑ h1[k – 2n]aj[k]
[k
k 678 aj[n] = √2 ∑ h0[n  l]v0[l] + √2 ∑ h1[n  l]v1[l]
[n]
[n
[n
l
l
aj1[0] aj1[1] v0[n]
L
aj1[l/2] ; l even
/2]
L
v0[l] =
0
; otherwise
n
1 0 1 2
upsample by 2
So
aj[n] = √2 ∑ h0[n  l]aj1[l/2] + √2 ∑ h1[n  l]bj1[l/2]
[n]
[n
/2]
[n
l even = √2 ∑ h0[n –2k]aj1[k] + √2 ∑ h1[n – 2k]bj1[k]
[n
[k]
[n
k k 12
12 ...
View
Full
Document
This note was uploaded on 12/04/2011 for the course ESD 18.327 taught by Professor Gilbertstrang during the Spring '03 term at MIT.
 Spring '03
 GilbertStrang

Click to edit the document details