multi-resolution analysis
DESCRIPTION
Multi-Resolution Analysis. Non-stationary Property of Natural Image. Pyramidal Image Structure. Z-transform. The z-transform is the discrete time version of Laplace transform. Given a sequence {x(n)}, its z-transform is: In particular,. Z-Transform and Fourier Transform. - PowerPoint PPT PresentationTRANSCRIPT
© 2002-2003 by Yu Hen Hu1ECE533 Digital Image Processing
Multi-Resolution Analysis
© 2002-2003 by Yu Hen Hu2ECE533 Digital Image Processing
Non-stationary Property of Natural Image
© 2002-2003 by Yu Hen Hu3ECE533 Digital Image Processing
Pyramidal Image Structure
© 2002-2003 by Yu Hen Hu4ECE533 Digital Image Processing
Z-transform
The z-transform is the discrete time version of Laplace transform.
Given a sequence {x(n)}, its z-transform is:
In particular, ( ) ( ) ( ) n
n
X z x n x n z
Z
1 ( ) ( ) 1
( ) 1 ( ) ( )
n n n
n
n nn
n n
x n x n z
x n z x n z X z
Z
© 2002-2003 by Yu Hen Hu5ECE533 Digital Image Processing
Z-Transform and Fourier Transform
Discrete time Fourier transform (DTFT):
Discrete Fourier Transform:
( ) ( ) ( ) ( ) j
j j n
z en
X e X j x n e X z
2 /
12 /
2 /0
1( ) ( ) ( ) ( )j kn N
Nj kn N j
z e k Nn
X k x n e X z X eN
© 2002-2003 by Yu Hen Hu6ECE533 Digital Image Processing
Frequency Domain Representation
Discrete time Fourier transform (DTFT)
Discrete Fourier Transform (DFT)
( ) ( ) ( )jj j n
z en
X e X z x n e
2 /
12 /
0
( ) ( ) ( )j k N
Nj kn N
z en
X k X z x n e
Z-plane
Re{z}
Im{z}
© 2002-2003 by Yu Hen Hu7ECE533 Digital Image Processing
Sub-sequence of a Finite Sequence
Let x(n) = x(0) x(1) x(2) x(3) x(4) x(5) x(6) …
x0(n) = x(0) 0 x(2) 0 x(4) 0 x(6) …
x1(n) = 0 x(1) 0 x(3) 0 x(5) 0 …
Then, clearly, x0(n) + x1(n) = x(n), and
x0(n) = [x(n) + (1)nx(n)]/2, x1(n) = [x(n) (1)nx(n)]/2
Denote X(z) to be the Z-transform of x(n), then
0
1
( ) 1 ( ) 1( ) ( ) ( )
2 2
( ) 1 ( ) 1( ) ( ) ( )
2 2
n
n
x n x nX z X z X z
x n x nX z X z X z
Z
Z
© 2002-2003 by Yu Hen Hu8ECE533 Digital Image Processing
Z transform of a Sub-sequence
Define
Let WM =exp(j2/M), then
One may write
( )( )
0 .kx n n Mn k
x notherwise
1( )
0
1( ) ( )
Mm n k
k Mm
x n W x nM
1( )
0
,
0
Mm n kM
m
M n kW
otherwise
1( )
0
1
0
1
0
1
0
1( ) ( )
1( )
1( )
1
Mm n k
k Mm
Mmk mn
M Mm
Mmk mn n
M Mm n
Mmk m
M Mm
X z W x nM
W W x nM
W W x n zM
W X zWM
Z
Z
© 2002-2003 by Yu Hen Hu9ECE533 Digital Image Processing
Decimation (down-sample)
M-fold decimator
yk(n) = x(Mn+k) = xk(Mn+k) , 0 k M1
Example. M = 2. y0(n) = x(2n), y1(n) = x(2n+1),
( ) /
/ 1/ 1/ 1/
0
( ) ( ) ( )
( ) ( )
Mn kn k M
k k kn
k M Mk M M mk M m
k M Mm
Y z x Mn k z x z
zz x z W X z W
M
11/ 2 1/ 2 1/ 2
0 20
1/ 2 1/ 211/ 2 1/ 2
1 2 20
1 1( ) ( ) ( ) ( )
2 2
( ) ( ) ( ) ( )2 2
m
m
m m
m
Y z X z W X z X z
z zY z W X zW X z X z
© 2002-2003 by Yu Hen Hu10ECE533 Digital Image Processing
Interpolation (up-sample)
L-fold Expander
Example. L = 2. {zL(n)} ={x(0), 0, x(1), 0, x(2), 0, …} and
( / ) / : integer,( )
0 .Lx n L n L
z nOtherwise
/( ) ( ) ( ) ( / )
( )
n Ln LL L L
n n
mL L
m
Z z z n z n z x n L z
x m z X z
Z
22 ( )z n X zZ
© 2002-2003 by Yu Hen Hu11ECE533 Digital Image Processing
Frequency Scaling
2 44 2 0
X(j)
2 44 2 0
X(j)
2 44 2 0
X(j2)
© 2002-2003 by Yu Hen Hu12ECE533 Digital Image Processing
Frequency domain Interpretation
For M = 2, with decimation
Note that
For L = 2, with interpolation,
In general, M-fold down-samples will stretch the spectrum M-times followed by a weighted sum. This may cause the aliasing effect.
L-fold up-sample will compress the spectrum L times
/ 2 / 20
/ 2/ 2 / 2
1
1( ) ( ) ( )
2
( ) ( ) ( )2
j j j
jj j j
Y e X e X e
eY e X e X e
/ 1/
0
12 / 2 /
0
( ) ( )
1( )
j k M Mj mk j M m
k M Mm
M kj m M j m M
m
eY e W X e W
M
e X eM
( )j j LLZ e X e
2 20 1( ) ( ) ( )j j j jX e Y e e Y e
22 ( ) ( )j jZ e X e
© 2002-2003 by Yu Hen Hu13ECE533 Digital Image Processing
Two-band Sub-band Filter
© 2002-2003 by Yu Hen Hu14ECE533 Digital Image Processing
Filter-banks
10 0 1 1
1/ 2 1/ 20 0 0
1/ 21/ 2 1/ 2
1 1 1
20 0 0 0
21 1 1 1
0 0 1 1
0 0 0
( ) ( ) ( ), ( ) ( ) ( )
1( ) ( ) ( )
2
( ) ( ) ( )2
( ) ( ) ( ) ( ) / 2
( ) ( ) ( ) ( ) / 2
ˆ ( ) ( ) ( ) ( ) ( )
1( ) ( ) ( ) (
2
Y z H z X z Y z z H z X z
V z Y z Y z
zV z Y z Y z
U z V z Y z Y z
U z V z z Y z Y z
X z G z U z G z U z
G z H z X z H
1 11 1 1
0 0 1 1
0 0 1 1
) ( )
( ) ( ) ( ) ( ) ( )2( )
( ) ( ) ( ) ( )2( )
( ) ( ) ( ) ( ) .(7.1.8)2
z X z
zG z z H z X z z H z X z
X zG z H z G z H z
X zG z H z G z H z Eq
G0(z)
G1(z)
v0(n)
v1(n)
u0(n)
u1(n)
+ˆ( )x n
H0(z)
H1(z)
z
x(n) v0(n)
v1(n)
y0(n)
y1(n)
© 2002-2003 by Yu Hen Hu15ECE533 Digital Image Processing
Frequency Response
1/ 2 1/ 20 0 0
1/ 2 1/ 2 1/ 2 1/ 20 0
/ 2 / 2/ 2 / 20 0 0
1/ 21/ 2 1/ 2
1 1 1
1/ 21/ 2 1/ 2 1/ 2 1/ 2 1/ 2
1 1
1( ) ( ) ( )
21
( ) ( ) ( ) ( )2
1( ) ( ) ( ) ( ) ( )
2
( ) ( ) ( )2
( ) ( ) 1 ( )2
j jj j j
V z Y z Y z
H z X z H z X z
V e H e X e H e X e
zV z Y z Y z
zz H z X z z H z
1/ 2
1/ 2 1/ 2 1/ 2 1/ 21 1
/ 2 / 2/ 2 / 21 1 1
( )
1( ) ( ) ( ) ( )
21
( ) ( ) ( ) ( ) ( )2
j jj j j
X z
H z X z H z X z
V e H e X e H e X e
H0(z)
H1(z)
z
x(n) v0(n)
v1(n)
y0(n)
y1(n)
© 2002-2003 by Yu Hen Hu16ECE533 Digital Image Processing
Frequency Domain Interpretation
22 0
|X(j)|= |X(ej)|
22 0
|X(j)Ho(j)|=|X(j()Ho(j()|=|Y0(j)|
22 0
|X(j)Ho(j)|=|X(j()Ho(j()|=|V0(j)|
© 2002-2003 by Yu Hen Hu17ECE533 Digital Image Processing
Frequency Domain Interpretation
22 0
|X(j)|= |X(ej)|
22 0
|X(j)H1(j)|=|X(j()H1(j()|=|Y1(j)|
22 0
|X(j)H1(j)|=|X(j()H1(j()|=|V1(j)|
© 2002-2003 by Yu Hen Hu18ECE533 Digital Image Processing
Perfect Reconstruction
Desired PR (perfect reconstruction) condition:
Implies:
It can be shown that
H0(z): low pass filter, H1(z): high pass filter
Usually, both are chosen to be FIR filters
ˆ( ) ( )x n x n D
0 0 1 1
0 0 1 1
( ) ( ) ( ) ( ) 2
( ) ( ) ( ) ( ) 0
DG z H z G z H z z
G z H z G z H z
0 1 1 0( ) ( ), ( ) ( )G z H z G z H z
© 2002-2003 by Yu Hen Hu19ECE533 Digital Image Processing
Perfect Reconstruction Filter Families
© 2002-2003 by Yu Hen Hu20ECE533 Digital Image Processing
2D Sub-band Filter
2-D four-band filter bank for sub-band image coding
© 2002-2003 by Yu Hen Hu21ECE533 Digital Image Processing
Daubechie’s Orthogonal Filters
© 2002-2003 by Yu Hen Hu22ECE533 Digital Image Processing
Sub-band Decomposition Example
A 4-band split of the vase in fig.7.1 using sub-band coding system of Fig. 7.5
© 2002-2003 by Yu Hen Hu23ECE533 Digital Image Processing
3-stage Forward DWT