1. The Story of Wavelets Theory and Engineering Applications
2D-DWT using MATLAB (review)
Implementation issues
Advanced Topics: Wavelet Packets
Other Applications
Density Estimation

2. Recall: 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)

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

4. 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 rows along the columns: 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

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

6. 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

7. 2-D DWT ON MATLAB Load Image Choose (must be wavelet type .mat file)
Hit
Analyze
Choose
display
options

8. Recall 1-D DWT g[n] h[n] x[n]
B: 0 ~ 
g[n]
h[n] 2
d1: Level 1 DWT
Coeff.
B: 0 ~ /2 Hz
d2: Level 2 DWT
d3: Level 3 DWT
…….
B: 0 ~ /4 Hz
B: 0 ~ /8 Hz
In DWT, only approximation coefficients are decomposed.
Each decomposition allows dyadic dichotomization of the frequency spectrum
What if we were decompose the detail coefficients as well…?
Frequency
Time

9. Wavelet Packets : H H H G G H G H H G H G H G H G 2 x[n] A(1) D(1)
B: 0 ~ 
A(1)
D(1) H G
0 ~ /2 G
/4 ~ /2 H
0 ~ /4 G
3/4 ~  H
/2 ~ 3/4
/2 ~ 
AA(2)
DA(2)
AD(2)
DD(2) H
0 ~ /8 G
/8 ~ /4 H
/4 ~ 3/8 G
3/8 ~ /2 H
/2 ~ 5/8 G
5/8 ~ 3/4 H
3/4 ~ 7/8 G
7/8 ~ 
AAA(3)
DAA(3)
ADA(3)
DDA(3)
AAD(3)
DAD(3)
ADD(3)
DDD(3)

10. Wavelet Packets
Frequency
Time

11. Wavelet Packets on MATLAB

12. What About Scaling and Wavelet Functions ???
You Ask…
In DWT, we used scaling functions to generate lowpass filters, and wavelet functions to generate highpass filters.
In WP analysis, filters are generated by related, but different analysis functions.
Two-scale (dilation) equations
where
2N: Filter length

13. How Many Decompositions Are Too Many???
For a signal of length N=2L, we can have L levels of 1D-DWT.
For the same signal, we can have a maximum of 2N levels of decompositions
For a 512 sample signal 
x10123

14. Choosing the Best Tree
The best tree is the one that gives the most information.
What is “the most information”…you ask….
Entropy based definitions
Normalized Shannon entropy
Norm based entropy
Energy based entropy
Threshold based entropy
If at any level, splitting a branch results in less sntropy, the splitting provides more information.
Matlab Demo: noisychirp

15. Density Estimation Density:???
HISTOGRAM
Density
function

16. Density Estimation
Density estimation allows us to infer statistical characteristics of data
From what distribution is the data coming
Reliability, life cycle
Average quality, etc.
mean
Number of
“60W” bulbs
Watts 60

17. Density Estimation How do we estimate density???
Matlab demo….Load ex1cusp2.mat from wavelet toolbox
Plot the data…What do you observe? Can you infer any information from this data?
Plot as “points”… What can you say now?
Plot the histogram of the data …>>hist(ex1cusp2)
Histogram can be used as a rough estimate of the density
Too noisy
Takes “every sample” into account, regardless how irrelevant (noisy) it may be
Better way? What else, but of course,…wavelets

18. Wavelets to the Rescue (again)
If the histogram is a noisy rough estimate of the density
Denoise histogram using wavelet shrinkage denoising
Select
wavelet
Choose
denoising
thresholds
Select
number of
bins
Select
thresholding
scheme