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Maximal Overlap DWT: I
- MODWT is like DWT in many ways,
but differs in certain key properties
- unlike DWT, MODWT is not orthonormal
(in fact MODWT is highly redundant)
- like DWT, can do MRA & analysis of variance
- unlike DWT, MODWT works for all samples sizes N
(i.e., power of 2 assumption is not required)
- if N is power of 2, can compute MODWT
using O(N log 2
N ) operations
(i.e., same as FFT algorithm)
- contrast to DWT, which uses O(N ) operations
- MODWT additive decomposition (MRA)
- details & smooths shift along with X:
X has detail
˜
D j
=⇒ T
m X has detail T
m ˜
D j
Maximal Overlap DWT: II
- MODWT analysis of variance
- based on MODWT wavelet coefficients,
but not details & smooths
- MODWT discrete wavelet power spectrum same
for X & T
m
X
- MODWT also appears under these names:
- undecimated DWT (or nondecimated DWT)
- stationary DWT
- translation invariant DWT
- time invariant DWT
- basic idea: use values downsampled out of DWT
Maximal Overlap DWT: IV
h l
≡ h l
˜
W 1 ,t
L− 1 ∑
l=
h l
X
t−l mod N
˜
W 1 ,t
L− 1 ∑
l=
h l
X
t−l mod N
- t = 1, 3 ,... of above yield W 1 of usual DWT:
W
1
[
˜
W 1 , 1
˜
W 1 , 3
˜
W 1 ,N − 1
] T
(recall that W 1 ,t
˜
W 1 , 2 t+
- t = 0, 2 ,... yield W T , 1
of DWT of T X:
W
T , 1
[
˜
W 1 , 0
˜
W 1 , 2
˜
W 1 ,N − 2
] T
note: these values usually subsampled out of DWT
1 √
2
W
1
1 √
2
W
T , 1 to obtain N × 1 vector
˜
W 1
[
˜
W 1 , 0
˜
W 1 , 1
˜
W 1 ,N − 1
] T
˜
W 1
has scale 1 MODWT wavelet coefficients
- likewise, can define N × 1 vector
˜
V 1
with scale 2 MODWT scaling coefficients
Maximal Overlap DWT: V
˜
W 1
via
˜
W 1
˜
B 1
X, where
˜
B 1
˜ h 0
h 3
h 2
h 1
h 1
h 0
h 3
h 2
h 2
h 1
h 0
h 3
h 3
h 2
h 1
h 0
h 3
h 2
h 1
h 0
h 3
h 2
h 1
h 0
˜
B 1 is N × N matrix
˜
B 1 is rescaled row from B T , 1
˜
B 1 is rescaled row from B 1
˜
W 1
˜
B 1
X is matrix formulation of
˜
W 1 ,t
L− 1 ∑
l=
h l
X
t−l mod N
, t = 0, 1 ,... , N − 1
˜
V 1
˜
A 1
X as matrix formulation of
˜
V 1 ,t
L− 1 ∑
l=
g ˜ l
X
t−l mod N
, t = 0, 1 ,... , N − 1
MODWT Analysis of Variance
• P
T
1
P
1
= I
N
& T
T T = I N
imply
P
T
T , 1
P
T , 1
= T
T
P
T
1
P
1
T = I
N
i.e., P T , 1
orthonormal
‖X‖
2
= ‖W 1
2
+‖V 1
2
and ‖X‖
2
= ‖W T , 1
2
+‖V T , 1
2
2 ‖X‖
2
= ‖W 1
2
2
2
2
W
1
[
˜
W 1 , 1
˜
W 1 , 3
˜
W 1 ,N − 1
] T
W
T , 1
[
˜
W 1 , 0
˜
W 1 , 2
˜
W 1 ,N − 2
] T
˜
W 1 ,t ’s form elements of
˜
W 1 , have
‖W
1
2
2
= 2‖
˜
W 1
2
‖V
1
2
2
= 2‖
˜
V 1
2
2 = ‖
˜
W 1
2
˜
V 1
2
- note: this provides rationale for pesky
MODWT MRA: I
& P
T , 1
are orthonormal, have
X = B
T
1
W
1
+A
T
1
V
1 and X = B
T
T , 1
W
T , 1
+A
T
T , 1
V
T , 1
- can thus define a (nonunique) MODWT synthesis as
X =
1
2
(
B
T
1
W
1
+ B
T
T , 1
W
T , 1
+ A
T
1
V
1
+ A
T
T , 1
V
T , 1
)
˜
B
T
1
˜
W 1
˜
A
T
1
˜
V 1
MODWT Detail & Smooth: I
˜
D 1
˜
B
T
1
˜
W 1
˜
S 1
˜
A
T
1
˜
V 1
so X =
˜
D 1
˜
S 1
˜
D 1
is MODWT detail of level j = 1
˜
S 1 is MODWT smooth of level j = 1
2 = ‖D 1
2
- Exer. [167a]: in general have
˜
D 1
2
=
1
2
(
˜
W 1
2
T
1
B
1
B
T
T , 1
W
T , 1
)
˜
W 1
2
- have assumed N to be even, but in fact, if N odd,
analysis of variance & MRA still hold
- Exer. [167b] argues this is true when N ≥ L
- will show later N ≥ L requirement not needed
- can argue
˜
D 1
is formed using zero phase filter
MODWT Detail & Smooth: II
˜
D 1
˜
B
T
1
˜
W 1
to get
˜
D 1 , 0
˜
D 1 , 1
˜
D 1 , 2
˜
D 1 ,N − 2
˜
D 1 ,N − 1
˜ h 0
h 1
h 2
h 3
h 0
h 1
h 2
h 3
h 0
h 1
h 2
h 3
h 3
h 0
h 1
h 2
h 2
h 3
h 0
h 1
h 1
h 2
h 3
h 0
˜
W 1 , 0
˜
W 1 , 1
˜
W 1 , 2
˜
W 1 ,N − 2
˜
W 1 ,N − 1
- in filtering notation, have
˜
D 1 ,t
L− 1 ∑
l=
h l
˜
W 1 ,t+l mod N
N − 1 ∑
l=
h
◦
l
˜
W 1 ,t+l mod N
where {
h
◦
l
} is periodized version of {
h l
{h l
} ←→ H(·) =⇒ {
h l
˜
H(·) ≡ H(·)/
h
◦
l
˜
H(
k
N
) : k = 0,... , N − 1 }
Summary of MODWT So Far
• J
0
= 1 MODWT maps X to
˜
W 1
˜
V 1
(all 3 are N × 1 vectors)
˜
W 1 denoted as {
˜
W 1 ,t
obtained by filtering {X t } with {
h l
˜
V 1 denoted as {
˜
V 1 ,t
obtained by filtering {X t } with {g˜ l
- Exer. [167b] shows, for N ≥ L,
X =
˜
B
T
1
˜
W 1
˜
A
T
1
˜
V 1
˜
D 1
˜
S 1
˜
D 1
˜
S 1
outputs from zero phase filters
- will be able to drop ‘N ≥ L’ restriction soon
- also have ‖X‖
2 = ‖
˜
W 1
2
˜
V 1
2
- Fig. 169 shows flow diagram (no downsampling)
MODWT Coefficients of Level j
˜
W j
˜
V j as N × 1 vectors with elements
˜
W j,t
L j − 1 ∑
l=
h j,l
X
t−l mod N
˜
V j,t
L j − 1 ∑
l=
g ˜ j,l
X
t−l mod N
h j,l ≡ h j,l
j/ 2 & ˜g j,l ≡ g j,l
j/ 2 , where
X −→
h j,l −→
↓ 2
j
W
j
& X −→
g j,l −→
↓ 2
j
V
j
- {h j,l } & {g j,l } & thus {
h j,l } & {g˜ j,l } have width
L
j
j
− 1)(L − 1) + 1
˜
H(f ) = H(f )/
˜
G(f ) = G(f )/
H
j
(f ) = H(
j− 1
f )
j− 2 ∏
l=
G(
l
f )
yield (since 2
j/ 2 is product of j copies of
˜
H j
(f ) ≡
˜
H(
j− 1
f )
j− 2 ∏
l=
˜
G(
l
f )
as transfer function for {
h j,l
} ≡ {h j,l
j/ 2 }
- likewise, transfer function for {g˜ j,l } given by
˜
G j
(f ) ≡
j− 1 ∏
l=
˜
G(
l
f ),
= L,
h 1 ,l
h l
˜
H 1 (f ) =
˜
H(f ) etc.
MODWT Analysis of Variance: I
} be the DFT of {X t
˜
W j,t
˜
H j
k
N
)X
k
˜
V j,t
˜
G j
k
N
)X
k
˜
W j
2
=
N
N − 1 ∑
k=
˜
H j
k
N
2
|X k
2
& ‖
˜
V j
2
=
N
N − 1 ∑
k=
˜
G j
k
N
2
|X k
2
˜
W j
2
+‖
˜
V j
2
=
N
N − 1 ∑
k=
(
˜
H j
k
N
2
˜
G j
k
N
2
)
|X
k
2
- when j ≥ 2, can reduce term in parentheses:
˜
H j
k
N
2
˜
G j
k
N
2
= |
˜
H(
j− 1 k
N
2
j− 2 ∏
l=
˜
G(
l k
N
2
j− 1 ∏
l=
˜
G(
l k
N
2
(
˜
H(
j− 1 k
N
2
˜
G(
j− 1 k
N
2
) j− 2 ∏
l=
˜
G(
l k
N
2
1
2
(
|H(
j− 1 k
N
2
j− 1 k
N
2
)
˜
G j− 1
k
N
2
˜
G j− 1
k
N
2
since |H(f )|
2
2 = H(f ) + G(f ) = 2
- thus have (after 2nd use of Parseval’s theorem)
˜
W j
2
+‖
˜
V j
2
=
N
N − 1 ∑
k=
˜
G j− 1
k
N
2
|X k
2
= ‖
˜
V j− 1
2
MODWT Analysis of Variance: II
- holds for j = 2,... , J 0
, so have
˜
V 1
2
=
J 0 ∑
j=
˜
W j
2
˜
V J 0
2
- analysis of variance holds if we can show
‖X‖
2
= ‖
˜
W 1
2
˜
V 1
2
- Exer. [167b]: above true when N ≥ L
- Exer. [171a]: above true when N ≥ L or N < L
- MODWT analysis of variance thus holds!
MODWT Multiresolution Analysis: II
˜
W
T
j
to get elements of
˜
D j
˜
S j
˜
D j,t
N − 1 ∑
l=
h
◦
j,l
˜
W j,t+l mod N
˜
S j,t
N − 1 ∑
l=
˜g
◦
j,l
˜
V j,t+l mod N
˜
D j
formed by filtering {
˜
W j,t
} with {
˜
H
∗
j
k
N
˜
S j formed by filtering {
˜
V j,t } with {
˜
G
∗
j
k
N
˜
W j,t
˜
V j,t
} formed by filtering
{X
t
} ←→ {X
k
} with {
h
◦
j,l
˜
H j
k
N
)} and
{X
t
} ←→ {X
k } with {g˜
◦
j,l
˜
G j
k
N
- implies the following for j ≥ 2:
˜
D j,t
˜
H j
k
N
˜
H
∗
j
k
N
)X
k
˜
H j
k
N
2
X k
˜
S j,t
˜
G j
k
N
˜
G
∗
j
k
N
)X
k
˜
G j
k
N
2
X k
˜
D j,t
˜
S j,t
(
˜
H j
k
N
2
˜
G j
k
N
2
)
X
k
˜
D j,t
˜
S j,t
˜
G j− 1
k
N
2
X k
˜
S j− 1 ,t
˜
G j− 1
k
N
2
X k
˜
S j− 1
˜
D j
˜
S j
for j ≥ 2, so have
˜
S 1
J 0 ∑
j=
˜
D j
˜
S J 0
& hence X =
J 0 ∑
j=
˜
D j
˜
S J 0
if we use Exer. [172]: X =
˜
S 1
˜
D 1 for all N & L
MODWT Pyramid Algorithm: I
˜
W j
˜
V j using
˜
V j− 1 rather than X
- can obtain all 3 by filtering X directly:
˜
V j , use {
˜
G j
k
N
˜
G j− 1
k
N
˜
G(
j− 1 k
N
˜
W j
, use {
˜
H j
k
N
˜
G j− 1
k
N
˜
H(
j− 1 k
N
˜
V j− 1
, use {
˜
G j− 1
k
N
˜
V j
˜
W j
using
˜
G(
j− 1 k
N
˜
H(
j− 1 k
N
h l
˜
H(f ), then
h 0
︸ ︷︷ ︸
2
j− 1 −1 zeros
h 1
︸ ︷︷ ︸
2
j− 1 −1 zeros
h L− 2
︸ ︷︷ ︸
2
j− 1 −1 zeros
h L− 1
˜
H(
j− 1
f )
˜
W j,t
L− 1 ∑
l=
h l
˜
V j− 1 ,t− 2
j− 1 l mod N
˜
V j,t
L− 1 ∑
l=
g ˜ l
˜
V j− 1 ,t− 2
j− 1 l mod N
≡ X
t
, can use to get
˜
W 1
˜
W J 0
˜
V J 0
