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10/20/
ECE 6258 Russell M. Mersereau
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ECE6258 Lecture 25
2-D Wavelets
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ECE 6258 Russell M. Mersereau
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Announcement
Problem Set #5 has been posted.
Images “mandrill.gif”, “peppers.gif”, and “boats.gif”for PS #5 have also been posted.
10/20/
ECE 6258 Russell M. Mersereau
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Two-dimensional wavelet transforms
The wavelet concept can be easily extended separably to twodimensions.
In the separable case, a 2-D scaling function and three 2-D waveletsare required.
The function can be expanded in terms of the scaling function andwavelets.
] [ ] [ ] , [ n
m
n
m
=
] [ ] [ ] , [ n
m
n
m H
=
] [ ] [ ] , [ n
m
n
m V
=
] [ ] [ ] , [
n
m
n
m
D
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ECE 6258 Russell M. Mersereau
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2-D discrete wavelet expansions^
Any 2-D sequence can be expanded in any family of 2-D scalingand wavelet functions.
− =
− =
1 0
(^10)
(^2) /
(^2) /
]
[ ] , [ 1 ]
[
M m
N n
j
j
o^
l n k m n m x
MN
l k j
W
ϕ
− =
− =
1 0
1 0
(^2) /
(^2) /
]
2 [
]
[
]
[
M m
N n
j
j i
o i^
l n k m n m
x
MN
l k j
W
ψ
} , ,
{
D
V
H
i^
=
]
2 [
]
, , [ 1 ] , [
(^2) /
0
l n k m l k j W
MN
n m x
j
j
j
m
n
ϕ
]
[
2 ]
[
(^2) /
0
, ,^
0
l n k m l k j W
MN
j
j i
j
m
n
i
j j D V H i
∞ =
=
ϕ
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ECE 6258 Russell M. Mersereau
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Fast wavelet analysis (one stage)
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ECE 6258 Russell M. Mersereau
Fast wavelet synthesis (one stage)
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ECE 6258 Russell M. Mersereau
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Frequency decomposition
input
after first stage
after second stage
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ECE 6258 Russell M. Mersereau
Example: (computer generated image)
Source: Gonzalez and Woods
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ECE 6258 Russell M. Mersereau
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Wavelet-based noise removal 1.
Choose a wavelet and number of levels,
P
. Compute the
FWT of the noisy image.
Threshold the detail coefficients from scales
J
-1 to
J-P
using
either a hard or soft threshold.
Perform a wavelet reconstruction based on the originalapproximation coefficients at level
J-P
and the modified
detail coefficients for levels
J
-1 to
J-P
.
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ECE 6258 Russell M. Mersereau
Denoising example
Noisy original
Denoising with
P=2, hard global threshold
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ECE 6258 Russell M. Mersereau
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Denoising example (cont’d)
Result with highestresolution detail coefficientszeroed
Difference image
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ECE 6258 Russell M. Mersereau
Denoising example (cont’d)
Result with all detailcoefficients zeroed
Difference image
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ECE 6258 Russell M. Mersereau
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Wavelet Packets--Motivation
The FWT decomposes a signal into a series of logarithmicallyrelated frequency bands.
Constant-Q filter banks
To get greater control over the partitioning of the time-frequencyplane, we need a more flexible decomposition, called a
wavelet
packet
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ECE 6258 Russell M. Mersereau
Wavelet Decomposition
Filter bank^ Filter bank
Decompositionspace tree Decompositionspace tree
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ECE 6258 Russell M. Mersereau
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Three-scale wavelet packet analysis
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ECE 6258 Russell M. Mersereau
Filter bank for full wavelet packet analysis
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ECE 6258 Russell M. Mersereau
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Adaptive wavelet packets
Presumably subband signals with little energy are of littleimportance in the representation.
Define the easy to compute metric
One possibility: decompose only those bands for which
E
f ) passes
a threshold.
Compute the metric for the parent
E
P
and
E
A
,^
E
H
,^
E
V , and
E
D
If
E
A
E
H
E
V
E
D
E
P
include the offspring, else prune them.
∑
n m
n m f
f
E
,
| ] , [ | ) (
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ECE 6258 Russell M. Mersereau
Optimal tree: example
Note: method used by FBIFor compressing fingerprintImages.
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ECE 6258 Russell M. Mersereau
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Associated decomposition
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ECE 6258 Russell M. Mersereau
Cohen-Daubechies-Feauveau biorthogonal wavelets
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ECE 6258 Russell M. Mersereau
Corresponding analysis and synthesis filters
