1. Wavelet Transform A Presentation
By Subash Chandra Nayak
01EC3010
IIT Kharagpur INDIA

2. Introduction to the world of transform
What are transforms :-
A mathematical operation that takes a function or sequence and maps into another one
General Form :-
Examples :-
Laplace, Fourier, DTFT, DFT, FFT, z-transform

3. Fourier Transform Mathematical Form :- Notes :-
Fourier transform identifies all spectral components present in the signal, however it does not provide any information regarding the temporal (time) localization of the components

4. Fourier Transform :: Limitations
Signals are of two types
# Stationary
# Non – Stationary
Non stationary signals are those who have got time varying spectral components ... FT gives only provides the existence of the spectral components of the signal ... But does not provide any information on the time occurrence of spectral components
Explanation
The basis function e-jwt stretches to infinity , Hence only analyzes the signal globally
In order to obtain time-localization of spectral components , the signal need to be analyzed locally

5. Time – Frequency Representation
Instantaneous frequency :-
Group delay :-
Disadvantages of above expressions
These equations though have a huge theoretical significance but are not easy to implement easily

6. Short-time Fourier Transform
Also known as a STFT
given as
w(t) :- windowing function generally a Gaussian pulse is used, other choices are rectangular , elliptic etc..
Maps 1D function to 2D time-frequency domain
Advantages :-
# Gives us time-frequency description of the signal
# Overcomes the difficulties of Fourier transform by use of windowing functions

7. STFT :: Disadvantages Heisenberg Principle :- Trade offs :-
One can not get infinite time and frequency resolution beyond Heisenberg’s Limit
Trade offs :-
Wider window
Good frequency resolution , Poor time resolution
Narrower window
Good time resolution , Poor frequency resolution

8. Wavelet Transform Mathematical form:- where x(t) = given signal
Overcomes the shortcoming of STFT by using variable length windows :: i.e. Narrower window for high frequency thereby giving better time resolution and Wider window at low frequency thereby giving better frequency resolution
Heisenberg’s Principle still holds
Mathematical form:-
where x(t) = given signal
tau = translation parameter
s = scaling parameter = 1/f
phi(t) = Mother wavelet , All kernels are obtained by scaling and/or translating mother wavelet

9. Continuous Wavelet transform
The kernel functions used in wavelet transform are all obtained from one prototype function known as mother wavelet , by scaling and/or translating it
Here
a = scale parameter
b = translation parameter
Continuous Wavelet transform

10. CWT (Contd..)
In order to become a wavelet a function must satisfy the above two conditions

11. Inverse wavelet transform

12. Examples of wavelets

15. Constant Q-filtering CWT can be rewritten as
A special property of the above filter defined by the mother wavelet is that they are Constant-Q filters
Q factor = Center frequency/Bandwidth
Hence the filter defined by wavelet increases their Bandwidth as scale increases ( i.e. center frequency increases )
This boils down to filter bank implementation of discrete wavelet transform

16. Filter Banks :: General Structure
Condition for Perfect Reconstruction

17. Filter bank (Contd..) P(z) – P(-z) = 2 Product filter
Now let’s define product filter as :-
P0(z) = F0(z)H0(z)
And Normalized Product filter as
P(z) = zL P0(z) where L = delay in total process
So the PR condition boils down to this realationship
P(z) – P(-z) = 2

18. Harr Filter Bank
Note that f0(n) and f1(n) are non-causal ... Hence here
Unit delay is required to implement it hence here
L = 1

19. P(w) is said to be halfband filter because of its symmetry
Product filter
P(w) is said to be halfband filter because of its symmetry
Also
P(w) + P(w + pi) = 2

20. Product filter II
P(w) should be as flat as possible around 0 and pi . The more is the flatness of P(w) around 0 and pi the better the Product filter is . Hence P(w) is always tried to be designed as a MAXFLAT Filter
Order of filter :: p
p = (L+1)/2 ; L = number of delay elements
Methods of determination of P(z)
# Duabechies method
# Meyer methods
Both of the above methods give us P(z) for a given order “p” . The higher is the order the better is the filter but at the same time it will require more hardware complexity

21. Spectral factorization
The spectral factorization is the problem of finding h0(z) once P(z) is known
Linear Phase Factorization
H0(z) and F0(z) are of different degree. Gives filter with linear phase
Orthogonal Factorization
H0(z) and F0(z) are of same degree. Gives filter with non-linear phase. Daubechies family of filters belongs to this category. For orthogonal filter
For orthogonal filter

22. Discrete wavelet Transform
Discrete domain counterpart of CWT
Implemented using Filter banks satisfying PR condition
Represents the given signal by discrete coefficients {dk,n}
DWT is given by

23. Scaling Function Harr Scaling Function
These are functions used to approximate the signal up to a particular level of detail
For Harr System
Harr Scaling
Function

24. Refinement equation and wavelet Equation
Refinement equation is an equation relating to scaling function and filter coefficients
Wavelet equation is an equation relating to wavelet function and filter coefficients
By solving the above two we can obtain the scaling and wavelet function for a given filter bank structure
Refinement Equation
Wavelet Equation

25. DWT Implementation a(k,n) a(k,n) g`[n] h`[n] g[n] h[n] + g`[n] h`[n]
d(k+1,n)
g`[n]
h`[n] 2 2
g[n]
h[n] +
a(k+1,n)
d(k+2,n)
g`[n]
h`[n] 2 2
g[n]
h[n] +
a(k+1,n) 2
a(k+2,n)
Decomposition
Reconstruction
We have only shown the above implementation for the Haar Wavelet, however, as we will
see later, this implementation – subband coding – is applicable in general.

26. DWT Sub-band Decomposition
x[n]
Length: 512
B: 0 ~ 
|H(jw)| w
/2
-/2
g[n]
h[n]
Length: 256
B: 0 ~ /2 Hz
Length: 256
B: /2 ~  Hz 2 2
|G(jw)|
d1: Level 1 DWT
Coeff.
g[n]
h[n]
Length: 128
B: 0 ~  /4 Hz w
Length: 128
B: /4 ~ /2 Hz 2 2 -
-/2
/2 
d2: Level 2 DWT
Coeff.
g[n]
h[n] 2 2
Length: 64
B: 0 ~ /8 Hz
Length: 64
B: /8 ~ /4 Hz
…….
d3: Level 3 DWT
Coeff.

27. Sub-band coding

28. Some Important properties of wavelets
Compact Support :-
Finite duration wavelets are called compactly supported in time domain but are not band-limited in frequency. Can be implemented using FIR filters
Examples
Harr, Daubechies, Symlets , Coiflets
Narrow band wavelets are called compactly supported in frequency domain. Can be implemented using IIR filters
Meyer’s wavelet

29. Some Important properties of wavelets
Symmetry
Symmetric / Antisymetric wavelets have got liner-phase
Orthogonal wavelets are asymmetric and have a non-linear phase
Biorthogonal wavelets are asymmetric but have got linear phase can be implemented using FIR filters
Vanishing Moment
pth vanishing moment is defined as
The more the number of moments of a wavelets are zero the more is its compressive power

30. Some Important properties of wavelets
Smoothness
is roughly the number of times a function can be differentiated at any given point
Closely related to vanishing Moments
Smoothness provides better numerical stability
It also provides better reconstruction propertiy

31. 2D DWT Why would we want to take 2D-DWT of an image anyway?
Generalization of concept to 2D
2D functions  images f(x,y)  I[m,n] intensity function
Why would we want to take 2D-DWT of an image anyway?
Compression
Denoising
Feature extraction
Mathematical form

32. 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

33. 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

34. 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

35. DWT in Work

36. DWT at work

37. DWT at work

39. Applications of wavelets
There are a lots of uses of wavelets .... The most prominent application of wavelets are
Computer and Human Vision
FBI Finger Print compression
Image compression
Denoising Noisy data
Detecting self-similar behavior in noisy data
Musical Notes synthesis
Animations

40. Things that I didn’t Cover
Different algorithms for getting Product filter, max-flat filter realization, spectral factorization , solutions for refinement and wavelet equations and many more
Noble Identity, modulation matrix, Polyphase matrix forms
MRA , Mallat’s pyramidal algorithm
Lifting
Basic Vector algebra needed for wavelet analysis ( It’s too mathematical to present )
Orthogonality, biorthogonality, frames
Vector algebra approach for wavelets
Wavelet for denoising and many more

41. Thank You