Transcript Chapter 4
Chapter 7
Wavelets and Multi-resolution
Processing
Background
Image Pyramids
Total number of elements in a P+1 level pyramid for P>0 is
1 1
1
4 2
N (1 2 ... P ) N
4 4
4
3
2
Example
Subband Coding
An image is decomposed into a set of band-limited
components, called subbands, which can be
reassembled to reconstruct the original image without
error.
Z-Transform
The Z-transform of sequence x(n) for n=0,1,2
is:
n
X ( z)
x(n) z
n
Down-sampling by a factor of 2:
1
x ( n ) x ( 2n ) X ( z ) X ( z ) X ( z )
2
1/ 2
down
1/ 2
down
Up-sampling by a factor of 2:
x(n / 2), n 0,2,4...
x (n)
X up ( z ) X ( z 2 )
otherwise
0
up
Z-Transform (cont’d)
If the sequence x(n) is down-sampled
and then up-sampled to yield x^(n),
then:
1
ˆ
X ( z ) X ( z ) X ( z )
2
From Figure 7.4(a), we have:
1
Xˆ ( z ) G0 ( z )H 0 ( z ) X ( z ) H 0 ( z ) X ( z )
2
1
G1 ( z )H1 ( z ) X ( z ) H1 ( z ) X ( z )
2
1
Xˆ ( z ) H 0 ( z )G0 ( z ) H1 ( z )G1 ( z )X ( z )
2
1
H 0 ( z )G0 ( z ) H1 ( z )G1 ( z )X ( z )
2
Error-Free Reconstruction
H 0 ( z )G0 ( z ) H1 ( z )G1 ( z ) 0
H 0 ( z )G0 ( z ) H1 ( z )G1 ( z ) 2
•
Matrix expression
[G0 ( z) G1( z)]Hm ( z) [2 0]
•
Analysis modulation matrix Hm(z):
H 0 ( z ) H 0 ( z )
Hm ( z )
H
(
z
)
H
(
z
)
1
1
H1 ( z )
G0 ( z )
2
G ( z ) det(H ( z )) H ( z )
0
1
m
FIR Filters
• For finite impulse response (FIR) filters, the
determinate of Hm is a pure delay, i.e.,
det(Hm ( z)) az ( 2k 1)
• Let a=2
• Let a=-2
g 0 ( n ) ( 1) n h1 ( n )
g1 ( n ) ( 1) n 1 h0 ( n )
g 0 ( n ) ( 1) n 1 h1 ( n )
g1 ( n ) ( 1) n h0 ( n )
Bi-orthogonality
Let P(z) be defined as:
2
P( z ) G0 ( z ) H 0 ( z )
H 0 ( z ) H1 ( z )
det(Hm ( z ))
2
G1 ( z ) H1 ( z )
H 0 ( z ) H1 ( z ) P( z )
det(Hm ( z ))
Thus, G0 ( z)H0 ( z) G0 ( z)H0 ( z) 2
Taking inverse z-transform:
g (k )h (n k ) (1) g (k )h (n k ) 2 (n)
n
0
0
0
k
Or,
0
k
g (k )h (2n k )
0
k
0
g0 (k ), h0 (2n k ) (n)
Bi-orthogonality (Cont’d)
It can be shown that:
g1(k ), h1(2n k ) (n)
g0 (k ), h1(2n k ) 0
g1(k ), h0 (2n k ) 0
Or, hi (2n k ), g j (k ) (i j ) (n ), i, j {0,1}
Examples: Table 7.1
Table 7.1
2-D Case
Daubechies Orthonormal Filters
Example 7.2
The Haar Transform
Oldest and simplest known orthonormal
wavelets.
T=HFH where
F: NXN image matrix,
H: NxN transformation matrix.
Haar basis functions hk(z) are defined over
the continuous, closed interval [0,1] for
k=0,1,..N-1 where N=2n.
Haar Basis Functions
k 2 p q 1 where0 p n 1,
q 0 or 1 for p 0,1 q 2 p for p 0
h0 ( z ) h00 ( z )
1
, z [0,1]
N
2 p / 2 ( q 1) / 2 p z ( q 0.5) / 2 p
1 p/2
hk ( z ) h pq ( z )
( q 0.5) / 2 p z q / 2 p
2
N
otherwise,z [0,1]
0
Example
Multiresolution Expansions
Multiresolution analysis (MRA)
A scaling function is used to create a
series of approximations of a function
or image, each differing by a factor of 2.
Additional functions, called wavelets,
are used to encode the difference in
information between adjacent
approximations.
Series Expansions
A signal f(x) can be expressed as a linear
combination of expansion functions:
f ( x)
a ( x)
k
k
k
Case 1: orthonormal basis: j ( x),k ( x) jk
Case 2: orthogonal basis: j ( x),k ( x) 0 j k
2
2
2
A
f
(
x
)
(
x
),
f
(
x
)
B
f
(
x
)
Case 3: frame:
k
k
Scaling Functions
Consider the set of expansion functions
composed of integer translations and
binary scaling of the real, square-integrable
function, (x),i.e.,
j,k ( x) 2 j / 2 (2 j x k )
By choosing wisely, {j,k(x)} can be made
to span L2(R)
Haar Scaling Function
MRA Requirements
Requirement 1: The scaling function is
orthogonal to its integer translates.
Requirement 2:The subspaces spanned by
the scaling function at low scales are nested
within those spanned at higher resolutions.
Requirement 3:The only function that is
common to all Vj is f(x)=0
Requirement 4: Any function can be
represented with arbitrary precision.
Wavelet Functions
Wavelet Functions
A wavelet function, y(x), together with its
integer translates and binary scalings,
spans the difference between any two
adjacent scaling subspace, Vj and Vj+1.
y j ,k ( x ) 2 y ( 2 x k )
j/2
y ( x)
h (n)
y
n
2 (2 x n)
j
hy (n) (1)n h (1 n)
Haar Wavelet Functions
Wavelet Series Expansion
f ( x)
c
j0
(k ) j0 ,k ( x )
k
c j0 ( k )
d j (k )
d (k )y
j
j j0
k
f ( x ) j0 ,k ( x )dx
f ( x)y
j ,k
( x )dx
j ,k
( x)
Harr Wavelet Series Expansion of y=x2
Discrete Wavelet Transform
W ( j0 , k )
f ( x )
M
1
j0 ,k
( x)
x
Wy ( j, k )
f ( x )y
M
1
j ,k
( x)
x
f ( x)
W ( j , k )
M
1
k
0
j0 ,k
( x)
W ( j, k )y
M
1
y
j j0
k
j ,k
( x)
The Continuous Wavelet Transform
Wy ( s, )
1
f ( x)
Cy
f ( x)y s , ( x)dx
Wy ( s, )
0
y s , ( x )
s2
dds
x
y s, ( x) y (
)
s
s
1
Cy
(u )
u
2
du
Misc. Topics
The Fast Wavelet Transform
Wavelet Transform in Two Dimensions
Wavelet Packets