Wavelets AND counterlets

Pattern
Recognition using
Wavelets &
Contourlets
PROJECT
Group No - 24

MOTIVATION
 Wavelet transform & Contourlet transform are relatively new
theories, they have enjoyed a tremendous attention and success
over the last decade.
 Almost all signals encountred in practice call for a time-frequency
analysis, and wavelets provide a very simple and efficient way to
perform such an analysis. While Contourlet tranform is a new two
dimensional transform method for image representation.
 Still, there’s a lot to discover in these new theories, due to the
infinite variety of non-stationary signals encountred in real life.

 Introduction
 About Fourier Transform
 Short-Time Fourier Transform
 Principle of Wavelet & Contourlet Transform
 Applications
 Time-Frame Distribution
 Conclusion
 References

Our main concern is to get the information from frequency domain which is not available in
Time Domain representation. There are different transform available for getting information
from frequency domain :
• FOURIER TRANSFORM
• SHORT TIME FOURIER TRANSFORM
• WAVELET TRANSFORM, etc.
Why do we need the frequency information ?
Often times, the information that cannot be readily seen in the time-domain can be seen in
the frequency domain.
So how do we measure the frequency content of a signal
FOURIER TRANSFORM (FT)
Fourier transform are applied to signals to obtain a further information from the signal that
is not readily available in the Time domain signal (raw signal).
Moreover, If the FT of a signal in time domain is taken, the frequency-amplitude
representation of that signal is obtained.

NOTHING MORE, NOTHING LESS ...
 Fourier Transform is a reversible transform, that is, it allows to go back and
forward between the raw and processed (transformed) signals. However, only either
of them is available at any given time.
 No frequency information is available in the time-domain signal, and no time
information is available in the Fourier transformed signal.
 FT gives the frequency information of the signal, which means that it tells us how
much of each frequency exists in the signal, but it does not tell us when in time
these frequency components exist.
STATIONARY AND NON-STATIONARY SIGNALS
FT identifies all spectral components present in the signal, however it does not
provide any information regarding the temporal (time) localization of these
components. Why?
Most of Transportation Signals are Non-stationary.
(We need to know whether and also when an incident was happened.)

Stationary signals consist of spectral components that do not change in time
 all spectral components exist at all times
 no need to know any time information
 FT works well for stationary signals
However,
Non-stationary signals consists of time varying spectral components
 How do we find out which spectral component appears when?
 FT only provides what spectral components exist , not where in time they are
located.
 Need some other ways to determine time localization of spectral components
)52cos()(1 ttx   )252cos()(2 ttx   )502cos()(3 ttx  
)502cos(
)252cos(
)52cos()(4
t
t
ttx






][)( 3215 xxxtx 
 Concatenation

X4(ω)
X5(ω)Perfect knowledge of what
frequencies exist, but no
information about where
these frequencies are
located in time

So We Need a local analysis scheme for a time-frequency representation of
non stationary signals
THE SHORT TIME FOURIER TRANSFORM
 Dennis Gabor (1946) Used STFT to analyze only
a small section of the signal at a time by a technique
called Windowing the Signal.
 The Segment of Signal is Assumed Stationary.
Steps Involved in STFT :
1. Choose a window function of finite length
2. Place the window on top of the signal at t=0
3. Truncate the signal using this window
4. Compute the FT of the truncated signal
5. Incrementally slide the window to the right
6. Go to step 3, until window reaches the end of the signal. For each time
location where the window is centered, we obtain a different FT
 Hence, each FT provides the spectral information of a separate time-slice
of the signal, providing simultaneous time and frequency information.

 STFT provides the time information by computing a different FTs for consecutive
time intervals, and then putting them together.
 Selection of width of STFT window:
Wide analysis window  Poor time resolution, Good frequency resolution
Narrow analysis window  Good time resolution, Poor frequency resolution

DRAWBACKS OF STFT
 Unchanged Window
 Dilemma of Resolution
Narrow window -> poor frequency resolution
Wide window -> poor time resolution
 Heisenberg Uncertainty Principle
Cannot know what frequency exists at what time intervals.
Heisenberg Principle
4
1
 ft
Time resolution: How well two spikes
in time can be separated from each
other in the transform domain
Frequency resolution: How well two
spectral components can be separated
from each other in the transform domain
Both time and frequency resolutions cannot be arbitrarily high!!!
We cannot precisely know at what time instance a frequency component is located.
We can only know what interval of frequencies are present in which time intervals.

WAVELET TRANSFORM
Wavelet
A Wavelet is a wave-like oscillation with an amplitude that begins at zero, increases
and decreases back to zero.
Wavelet Transform
Wavelet Transformation is basically a mathematical technique in which a particular
signal is analyzed in the time domain by using different versions of a translated
and dilated basis function called the Wavelet Prototype or mother wavelet.
 It overcomes the preset resolution problem of the STFT by using a variable length
window
 Analysis windows of different lengths are used for different frequencies:
Analysis of high frequencies  Use narrower windows for better time resolution
Analysis of low frequencies  Use wider windows for better frequency resolution
This works well, if the signal to be analyzed mainly consists of slowly varying
characteristics with occasional short high frequency bursts.
 Heisenberg principle still holds!!!
The function used to window the signal is called THE WAVELET.
There are two type of wavelet Transform:
1) Continuous wavelet transform
2) Discrete wavelet transform

CONTINUOUS WAVELET TRANSFORM
The continuous transform is mainly done by using Fourier Transformation which uses sine and
cosine as basis functions for analyzing a signal. Because wavelet are generally infinites as well as
periodic in nature , so Fourier Transform is appropriate.
  




 
 
t
xx dt
s
t
tx
s
ssCWT

  1
),(),(
Continuous wavelet transform of the
signal x(t) using the analysis wavelet
(.)
Translation
parameter,
measure of time
Scale parameter,
measure of
frequency
The mother wavelet. All kernels are
obtained by translating (shifting) and/or
scaling the mother wavelet
A
normaliz
ation
constant
Signal to
be
analyzed
Scale = 1/frequency
Discrete Wavelet Transform
Discrete Wavelet transform are generally used for digital images when they need to be viewed or
processed at multiple resolutions.

Time
Frequency
Better time
resolution;
Poor
frequency
resolution
Better
frequency
resolution;
Poor time
resolution
• Each box represents a equal portion
• Resolution in STFT is selected once for entire analysis

Contourlet Transform
 The Contourlet transform which was proposed by Do and Vetterli in 2002,
is a new two-dimensional transform method for image representations
 Basic functions are multiscale & multidimensional
 Captures smooth contours and edges at any orientation
 Derived directly from discrete domain instead of extending from
continuous domain
 It consists of two filter bank stages, first is used to capture the point
discontinuities, which is then followed by a directional filter bank (DFB) to
link point discontinuities into linear structures
 Application : Graphic equalizer, Filters noise, etc.

APPLICATION
 Audio compression
 Speech recognition
 Image and video compression
 Denoising Signals

Time Frame Distribution
1st Semester
MSE:-
Literature Survey and Theoretical Comparison
ESE:-
(a). Problem Analysis
(b). Mathematical Formulation
2nd Semester
(a). Simulation and Verification
(b). Comparison of Simulation result

CONCLUSION
 FT works with Stationary signal.
 While STFT work with both stationary as well as non-stationary
signal, but having resolution problem.
 So to overcome this resolution problem we switched to Wavelet
transform using variable window size, but limited to 1-D
transform.
 Further for two-dimensional representation of image we proceed
to Contourlet transform.

References
 Robi Polikar, The Wavelet Tutorial,
http://users.rowan.edu/~polikar/WAVELETS/WTpart1html
 http://www.jitbm.com/12th%20volume/meaad%207.pdf
 The Contourlet Transform: An Efficient Directional Multiresolution Image
Representation
http://minhdo.ece.illinois.edu/publications/contourlet_txform.pdf
 Signals & Systems ~ By Ramesh Babu

Wavelets AND counterlets

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