1. Multi-resolution image processing & Wavelet
Multi-resolution Analysis (MRA)
- An image can be structured by a series images with different resolutions
Image pyramids
-- collection of decreasing resolution images arranged in the shape of a pyramid
N/2*N/2
Level j-1
Level j (base)
N*N

2. Multi-resolution analysis
Coarse to fine analysis
- Low –resolution levels in a pyramid structure can be used to analyze the overall image context or large structures
- High-resolution levels can be used for analyzing individual object characteristics
Approximate pyramid (AP) & prediction residual pyramids (PRP)
- The approximation images in multiple levels construct an “AP”
- Prediction residual image is the difference of neighbor levels of images
- Encoding the PRP is more efficient than encoding the AP.

3. Sub-band representation
Sub-band coding
A filter band can be used for a multi-band representation of images.
E.g., a low-pass filter will produce an approximation of the original image, a high-pass filter will keep the high frequency details of the image.
An image can be constructed by combining a few of band-limited components (called sub-band)
E.g., a two-band filter bank
|H0(w)|
|H1(w)|
Low-band
High-band w
/2 

4. Harr transform Harr transform matrix T = H F H F: N by N image
H: N*N transform matrix (whose basis is called orthonormal wavelet)
E.g.,

5. Harr transform (cont’d)
Harr basis function:
( - can be used for discrete wavelet transform)

6. Multi-resolution representation
Scaling function – series of approximations of image
Wavelets – encode the difference in information between adjacent approximation
Series expansion concept:
Expansion coefficient expansion function (basis)
Dual function of
e.g., Fourier transform, DCT transform, etc

7. Multi-resolution representation (cont’d)
Scaling function
Determines the width Position
Nested function spaces spanned by
a scaling function
Function space: V0
V0 V1V2
j increases  size of Vj increases

8. Multi-resolution representation (cont’d)
Example of Scaling function V0
V0 V1

9. Multi-resolution representation (cont’d)
Example of expansion function:
Expansion function of subspace Vj can be represented by expansion function of subspace Vj+1
Define:
Scaling function coefficient

10. Multi-resolution representation (cont’d)
Example of expansion function: (cont’d)
In Harr function:
Therefore:

11. Wavelet function
Relationship between scaling and wavelet function spaces
V2 = V1W1= V0 W0 W1 W1 W0 V0
V1 = V0 W0
Wavelet function (x) spans the difference between any two adjacent scaling subspaces Vj and Vj+1

12. Wavelet function (cont’d)
Wj space:

13. Wavelet function (cont’d)
If f(x) Wj, then:
Vj+1 = VjWj
Union of spaces
Vj and Wj are orthogonal each other:
wavelet
Scaling function
Function space = V0W0 W1 …
= V1W1 W2 …

14. Wavelet function (cont’d)
If f(x) V1, then:
F(x) = approximation of f(x) + details of f(x)
fa(x)
fd(x)
V0 scaling function
Wavelets from W0
Note:
- wavelets encode the difference between the approximation and the actual function
- wavelet function can be expressed as a weighted sum of shifted, double-resolution scaling functions.

15. Wavelet function (cont’d)
Example: Haar wavelet function:
Haar scaling vector:
Wavelet vector:
f(x) = fa(x) + fd(x)
Where:

16. Wavelet transform Definition: Scaling coefficient Wavelet coefficient
(approximation)
Wavelet coefficient
(details)
Where:

17. Wavelet transform (cont’d)
1-D discrete wavelet transform (DWT)
- Definition:
Where: jj0
Where: M = 2J
x=0,1,…, M-1
j=0,1, …, J-1
k=0,1, …, 2j-1

18. Wavelet transform (cont’d)
Example of DWT using Haar scaling and wavelet functions
Given f(x), x=0,1,2,3; M=4 = 2J; J=2; j0=0
f(x)
k=0 for j=0
k=0,1 for j=1 4 1 x -3
Based on Haar transform matrix H4and f(x), we obtain:

19. Wavelet transform (cont’d)
2-D discrete wavelet transform of f(x,y) (size: M*N)
- Definition:
where i={H,V,D}
Normally: j0 = 0; N=M = 2J
j=0,1, …, J-1
m,n = 0,1, …, 2j-1

20. Wavelet transform (cont’d)
2-D discrete wavelet transform of f(x,y) (size: M*N)
- Definition:
where i={H,V,D}
- 2D scaling function:
-2D wavelets:
-2D functions:

21. Wavelet transform (cont’d)
2-D discrete wavelet transform of f(x,y) (cont’d)
- Decomposition of 2D wavelet transform:

22. Wavelet transform (cont’d)
2-D discrete wavelet transform of f(x,y) (cont’d)
- analysis filter bank for the decomposition of 2D wavelet transform:
Columns
(along n)
Rows
(along m)

23. Wavelet transform (cont’d)
2-D discrete wavelet transform of f(x,y) (cont’d)
- synthesis filter bank of 2D wavelet transform:
Columns
(along n)
Rows
(along m)

24. Wavelet transform (cont’d)
Other wavelets family:
- Example:
Fohen-Daubechies-Feauveau biorthogonal wavelet family