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ECE 6258 Russell M. Mersereau
ECE6258 Lecture 23
Multiresolution Processing
ECE 6258 Russell M. Mersereau
Motivation
Fourier transforms
Basis functions are sinusoids of infinite duration
Wavelet transforms
Basis functions are signals (wavelets) of varying frequency and limited duration
.
Formalized in 1987
Allows for a representation and analysis of signals at more thanone resolution.
ECE 6258 Russell M. Mersereau
Local statistical variation in images
source: Gonzalez and Woods
ECE 6258 Russell M. Mersereau
Image pyramids
A collection of decreasingresolution images(Burt and Adelson)
Total number ofpixels in pyramid= 4/
N
ECE 6258 Russell M. Mersereau
Gaussian and Laplacian pyramids
Gaussian pyramid Gaussian pyramid
Laplacian pyramid Laplacian pyramid
ECE 6258 Russell M. Mersereau
Subband Decompositions
A (1-D) signal isdecomposed into alowpass and ahighpass component.
Each is
critically
sampled
, so no extra
samples areinvolved.
ECE 6258 Russell M. Mersereau
Upsampling and Downsampling
↓
2
↑
2
x
[
n
]
x
[
n
]
x
down
[
n
]
x
up
[
n
]
DOWNSAMPLER
UPSAMPLER
ECE 6258 Russell M. Mersereau
Analysis of System
Aliasing terms
ECE 6258 Russell M. Mersereau
Result of four-band split
LL
HL
LH
HH
ECE 6258 Russell M. Mersereau
Filter banks and Haar transforms
The only two-point FIR filters that satisfy the exactreconstruction conditions are the basis functions of the Haartransform.
When these are used to form a wavelet system, the result iscalled a
Haar wavelet
.
ECE 6258 Russell M. Mersereau
Haar wavelet decomposition
A filter bank isconstructed fromtwo-point FIRfilters.
The LL image isdecomposedusing the samefilter bank.
The LLLL imageis alsodecomposed.
Reconstructionsat severalresolutions arepossible.
ECE 6258 Russell M. Mersereau
Multiresolution expansions
In multiresolution analysis, a
scaling function
, is used to
create a series of approximation of an image, each differingby a factor of 2 (in size) from its nearest neighboringapproximations.
Additional functions, called
wavelets
, are then used to encode
the difference in information between adjacentapproximations.
ECE 6258 Russell M. Mersereau
Series Expansions
A signal or function
f
(
x
) can often be analyzed by expressing it as a linear
combination of expansion functions
If the set {
α
k
} is unique, the set of expansion functions form a
basis
.
The set of expressible functions form a function space
V
.
For any function space
V
and corresponding expansion set {
φ
i (
x
)}, there is a
set of dual function
, that can be used to compute the
α
i .
∑
=
k
k
k
x
x
f
)
(
)
(
ϕ
α
)}
(
{
x
span
V
k
ϕ
=
)}
(
~
{
x
k
ϕ
dx
x f x x f x
k
k
k
) ( ) ( ~ ) (
),
(
~
∫
=
=
ϕ
ϕ
α
ECE 6258 Russell M. Mersereau
Orthogonal Bases and Frames
Case 1:
If the expansion functions form an orthonormal basis, the basisand its dual are equivalent.
Case 2:
If the expansion functions are merely orthogonal, the basisfunctions and their duals are
biorthogonal
.
Case 3:
If the expansion set is not a basis, but supports the expansion,the expansion functions and their duals are
overcomplete
and
form a
frame
.
ECE 6258 Russell M. Mersereau
Scaling functions
Consider the set of expansion functions composed of integertranslations and binary scalings of the real, square-integrablefunction
.
By choosing
wisely,
can be made to span
L
(
R
).
If we restrict
j
, the resulting expansion set will span a subset
of
L
(
R
).
)
2
(
2
)
(
k
x
x
j
j
k
j
−
=
ϕ
ϕ
)}
(
{
x
span
V
k
j
k
j
ϕ
=
)
(
x
ϕ
)
(
x
ϕ
{
} )
(
x
k
j
ϕ
ECE 6258 Russell M. Mersereau
Haar scaling functions
) 1
(
)
( 1 , 0
−
=
x
x
ϕ
ϕ
0 , 1
x
x
Source: Gonzalez and Woods
)
(
)
( 0 , 0
x
x
ϕ
ϕ
=
1
)
(
V
x
f
∈
1
0 , 0
)
(
V
x
∈
ϕ
1 , 1
x
x
ECE 6258 Russell M. Mersereau
Wavelets and scaling functions
Since
W
j
V
j , the wavelet can be expressed in terms of the scaling
function.
It can be shown that
In terms of filter banks,
⊂
]
1 [
)
1
(
]
[
n
h
n
h
n
−
−
=
]
[
]
[ 0
n
h
n
h
φ
=
]
[
]
[ 1
n
h
n
h
=
)
2 ( 2 ] [ ) (
n
x
n
h
x
n
−
=
∑
ϕ
ψ
ECE 6258 Russell M. Mersereau
The Haar wavelet
)
(
)
(
0 , 0
x
x
ϕ
ϕ
=
0 , 1
x
x
2 , 0
x
x
0
1
W
V
V
x
f
0
)
(
W
x
f
d
∈
)
(
V
x
f
a
∈
ECE 6258 Russell M. Mersereau
1-D wavelet series expansion
Any function
can be expanded relative to the wavelet
and scaling function
.
Analogous to Fourier series
j
is an arbitrary starting scale
The
are normally called the
approximation or scaling
coefficients
.
The
are called the
detail or wavelet coefficients
.
)
(
)
(
R
L
x
f
∈
)
(
x
ϕ
)
(
x
ψ
∑
∑∑
=
k
j
j
k
k
j
k
k
j
j
x k d x k c x f
0
0
0
) ( ] [ ) ( ] [ ) (
ψ
ϕ
]
[
k
d
k
]
[ 0
k
c
j
ECE 6258 Russell M. Mersereau
Calculating the wavelet series coefficients
The wavelet series coefficients can be computed byperforming the following inner products
ECE 6258 Russell M. Mersereau
Example
Consider the simple example
ECE 6258 Russell M. Mersereau
Grinding away
ECE 6258 Russell M. Mersereau
The expansion
Substituting these values gives the wavelet series expansion
L 4 4 4 4 4 4 4 4 4 4
3 4 4 4 4 4 4 4 4 4 4
2
1
4
4
4
4
4
3 4 4 4 4 4
2 1 4 4 4 4
3
4
4
4
4
2
1
4
43
4 42
1
4
3
4
2
1
−
−
−
−
=
1 0 0 1 1 2
1
0
0
1
0
0
)
(
32
2
3
)
(
32
2
)
(
4 1
)
(
1 3
W W V W V V
W
W
V
V
W
V
x
x
x
x
y
ψ
ψ
ψ
ϕ
ECE 6258 Russell M. Mersereau
Example at scales 0, 1, and 2
