1. Multi-Resolution Analysis

2. Non-stationary Property of Natural Image

3. Pyramidal Image Structure

4. Z-transform
The z-transform is the discrete time version of Laplace transform.
Given a sequence {x(n)}, its z-transform is:
In particular,

5. Z-Transform and Fourier Transform
Discrete time Fourier transform (DTFT):
Discrete Fourier Transform:

6. Frequency Domain Representation
Discrete time Fourier transform (DTFT)
Discrete Fourier Transform (DFT)
Im{z}
Re{z}
Z-plane

7. 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) x(2) x(4) x(6) …
x1(n) = x(1) x(3) x(5) …
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

8. Z transform of a Sub-sequence
Define
Let WM =exp(j2/M), then
One may write

9. Decimation (down-sample)
M-fold decimator
yk(n) = x(Mn+k) = xk(Mn+k) , 0  k  M-1
Example. M = y0(n) = x(2n), y1(n) = x(2n+1),

10. Interpolation (up-sample)
L-fold Expander
Example. L = 2. {zL(n)} ={x(0), 0, x(1), 0, x(2), 0, …} and

11. Frequency Scaling X(jw) X(jw/2) X(j2w) -4p -2p 2p 4p -4p -2p 2p 4p -4p2p 4p
X(jw/2)
-4p
-2p 2p 4p
X(j2w)
-4p
-2p 2p 4p

12. Frequency domain Interpretation
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
For M = 2, with decimation
Note that
For L = 2, with interpolation,

13. Two-band Sub-band Filter

14. Filter-banks + H0(z) H1(z) 2 x(n) v0(n) v1(n) G0(z) G1(z) 2 v0(n)
y0(n)
y1(n)
G0(z)
G1(z) 2
v0(n)
v1(n)
u0(n)
u1(n) +

15. Frequency Response
H0(z)
H1(z)
z-1 2
x(n)
v0(n)
v1(n)
y0(n)
y1(n)

16. Frequency Domain Interpretation
|X(jw)|= |X(ejw)| w
-2p -p p 2p
|X(jw)Ho(jw)|=|X(j(w+2p))Ho(j(w+2p))|=|Y0(jw)|
-2p -p p 2p w
|X(jw/2)Ho(jw/2)|=|X(j(w+p)/2)Ho(j(w+p)/2)|=|V0(jw)|
-2p -p p 2p w

17. Frequency Domain Interpretation
|X(jw)|= |X(ejw)| w
-2p -p p 2p
|X(jw)H1(jw)|=|X(j(w+2p))H1(j(w+2p))|=|Y1(jw)|
-2p -p p 2p w
|X(jw/2)H1(jw/2)|=|X(j(w+p)/2)H1(j(w+p)/2)|=|V1(jw)|
-2p -p p 2p w

18. 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

19. Perfect Reconstruction Filter Families

20. 2D Sub-band Filter
2-D four-band filter bank for sub-band image coding

21. Daubechie’s Orthogonal Filters

22. Sub-band Decomposition Example
A 4-band split of the vase in fig.7.1 using sub-band coding system of Fig. 7.5

23. 3-stage Forward DWT