1. The Story of Wavelets Theory and Engineering Applications
Stationary discrete wavelet transform
Two-dimensional wavelet transform
2D-DWT using MATLAB
Implementation issues
Image compressing using 2D-DWT

2. Stationary Wavelet Transfporm (SWT)
DWT is not time invariant… Not Good !
What makes DWT time varying? Decimation (down sampling)
DWT can be made time invariant, however, the transform must be redundant!!!
Stationary wavelet transform
-decimated DWT
-decimated DWT???
At any given level you can have two different DWT, due to choice in discarding even or odd indexed elements during subsampling
At J levels, you can have N=2J different DWTs. The particular DWT chosen can be denoted by =[1 2 … N], j=1, if odd indexed elements are chosen, j=0, if even indexed elements are chosen

3. SWT For 6 level DWT64 DWTs For 10 level DWT 1024 DWTs ….
SWT is defined as the average of all -decimated DWTs
For 6 level DWT64 DWTs
For 10 level DWT 1024 DWTs ….
An efficient algorithm: Hj Gj
cAj=a(j,n)
cAj+1=a(j+1,n)
cDj+1=d(j+1,n)
where
Hj-1 2 Hj
Gj-1 2 Gj
Note: No subsampling is involved!!!.

4. SWT
Does it work…? MATLAB DEMO

5. Applications of SWT Denoising… denoising…denoising
MATLAB demo: noisy doppler & noisy quadchirp
Interval dependent thresholds

6. 1D-DWT2D-DWT Recall fundamental concepts of 1D-DWT
Provides time-scale (frequency) representation of non-stationary signals
Based on multiresolution approximation (MRA)
Approximate a function at various resolutions using a scaling function, (t)
Keep track of details lost using wavelet functions, (t)
Reconstruct the original signal by adding approximation and detail coeff.
Implemented by using a series of lowpass and highpass filters
Lowpass filters are associated with the scaling function and provide approximation
Highpass filters are associated with the wavelet function and provide detail lost in approximating the signal

7. 2-D DWT Why would we want to take 2D-DWT of an image anyway?
How do we generalize these concepts to 2D?
2D functions  images f(x,y)  I[m,n] intensity function
What does it mean to take 2D-DWT of an image? How do we interpret?
How can we represent an image as a function?
How do we define low frequency / high frequency in an image?
How to we compute it?
Why would we want to take 2D-DWT of an image anyway?
Compression
Denoising
Feature extraction

8. 2D Scaling/Wavelet Functions
We start by defining a two-dimensional scaling and wavelet functions:
If (t) is orthogonal to its own translates,
is also orthogonal to its own translates. Then, if fo(x,y) is the projection of f(x,y) on the space Vo generated by s(x,y):

9. 2D-DWT
Just like in 1D we generated an approximation of the 2D function f(x,y). Now, how do we compute the detail lost in approximating this function?
Unlike 1D case there will be three functions representing the details lost:
Details lost along the horizontal direction
Details lost along the vertical direction
Details lost along the diagonal direction
1D  Two sets of coeff.; a(k,n) & d (k,n)
2D Four sets of coefficients: a(k,n), b(k, n), c(k, n) & d(k,n)

10. Four Faces of 2D-DWT One level of 2D DWT reconstruction:
Approximation coefficients
Detail coefficients along
the horizontal direction
Detail coefficients along
the vertical direction
Detail coefficients along
the diagonal direction

11. Implementation of 2D-DWTH ~
1 2 G
COLUMNS LL ~
ROWS H
2 1
COLUMNS …… LH
INPUT
IMAGE
COLUMNS
ROWS …… H ~
1 2 G HL ~ G
2 1
ROWS HH
COLUMNS
INPUT
IMAGE LH HL HH
LHH
LLH
LHL
LLL
LLH LL LH LH LL
LHL
LHH HL HH HL HH

12. Up and Down … Up and Down
Downsample columns along the rows: For each row, keep the even indexed columns, discard the odd indexed columns
2 1
Downsample columns along the rows: For each column, keep the even indexed rows, discard the odd indexed rows
1 2
Upsample columns along the rows: For each row, insert zeros at between every other sample (column)
2 1
Upsample rows along the columns: For each column, insert zeros at between every other sample (row)
1 2

13. Implementing 2D-DWT
Decomposition
COLUMN j
ROW i

14. Reconstruction LL H 1 2 2 1 H LH G 1 2 HL H 1 2 G 2 1 HH G 1 2
1 2 H
2 1 H LH
1 2 G
ORIGINAL
IMAGE HL
1 2 H
2 1 G HH
1 2 G

15. 2-D DWT ON MATLAB Load Image Choose (must be wavelet type .mat file)
Hit
Analyze
Choose
display
options