Post on 29-Mar-2015
Seminar Report ’03 Wavelet Transforms
INTRODUCTION
Wavelet transforms have been one of the important signal
processing developments in the last decade, especially for the applications
such as time-frequency analysis, data compression, segmentation and
vision. During the past decade, several efficient implementations of
wavelet transforms have been derived. The theory of wavelets has roots in
quantum mechanics and the theory of functions though a unifying
framework is a recent occurrence. Wavelet analysis is performed using a
prototype function called a wavelet. Wavelets are functions defined over
a finite interval and having an average value of zero. The basic idea of the
wavelet transform is to represent any arbitrary function f (t) as a
superposition of a set of such wavelets or basis functions. These basis
functions or baby wavelets are obtained from a single prototype wavelet
called the mother wavelet, by dilations or contractions (scaling) and
translations (shifts). Efficient implementation of the wavelet transforms
has been derived based on the Fast Fourier transform and short-length
‘fast-running FIR algorithms’ in order to reduce the computational
complexity per computed coefficient.
Dept. of EEE MESCE Kuttippuram1
Seminar Report ’03 Wavelet Transforms
TRANS…WHAT?
First of all, why do we need a transform, or what is a transform
anyway?
Mathematical transformations are applied to signals to obtain
further information from that signal that is not readily available in the raw
signal. Now, a time-domain signal is assumed as a raw signal, and a
signal that has been transformed by any available transformations as a
processed signal.
There are a number of transformations that can be applied such as
the Hilbert transform, short-time Fourier transform, Wigner transform,
the Radon transform, among which the Fourier transform is probably the
most popular transform. These mentioned transforms constitute only a
small portion of a huge list of transforms that are available at engineers
and mathematicians disposal. Each transformation technique has its own
area of application, with advantages and disadvantages.
Dept. of EEE MESCE Kuttippuram2
Seminar Report ’03 Wavelet Transforms
IMPORTANCE OF THE FREQUENCY
INFORMATION
Often times, the information that cannot be readily seen in the
time-domain can be seen in the frequency domain. Most of the signals in
practice are time-domain signals in their raw format. That is, whatever
that signal is measuring, is a function of time. In other words, when we
plot the signal one of the axis is time (independent variable) and the other
(dependent variable) is usually the amplitude. When we plot time-domain
signals, we obtain a time-amplitude representation of the signal. This
representation is not always the best representation of the signal for most
signal processing related applications. In many cases, the most
distinguished information is hidden in the frequency content of the signal.
The frequency spectrum of a signal is basically the frequency components
(spectral components) of that signal. The frequency spectrum of a signal
shows what frequencies exist in the signal.
Let’s give an example from the biological signals. Suppose we
are looking at an ECG signal (graphical recording of heart’s activity). The
typical shape of a healthy ECG signal is well known to cardiologists. Any
significant deviation from that shape is usually considered to be a
symptom of a pathological condition. This pathological condition,
Dept. of EEE MESCE Kuttippuram3
Seminar Report ’03 Wavelet Transforms
however, may not always be quite obvious in the time-domain signal.
Cardiologists usually use the time-domain ECG signals, which are
recorded on strip-charts to analyze ECG signals. Recently, the new
computerized ECG recorders/analyzers also utilize the frequency
information to decide whether a pathological condition exists. A
pathological condition can sometimes be diagnosed more easily when the
frequency content of the signal is analyzed.
Today Fourier transform is the most widely used transformation
technique to obtain the frequency representation.
Dept. of EEE MESCE Kuttippuram4
Seminar Report ’03 Wavelet Transforms
FOURIER TRANSFORMS
The Fourier transforms are used in many areas, in applications to
obtain the frequency representation of the signal. If the Fourier transform
of a signal in time-domain is taken, the frequency – amplitude
representation of that signal is obtained.
The Fourier transform is defined by the following two equations:
X (f) = -∞ ∫∞ x (t). e (- 2 j ∏ f t) dt. ...(1)
x (t) = -∞ ∫∞ X (f). e (2 j ∏ f t) df. …(2)
In the above equation, t stands for time, f stands for frequency,
and x denotes the signal in time domain and the X denotes the signal in
frequency domain. This convention is used to distinguish the two
representations of the signal.
Equation (1) is called the Fourier Transform of x (t), and equation
(2) is called the inverse Fourier Transform of X (f), which is x (t).
Dept. of EEE MESCE Kuttippuram5
Seminar Report ’03 Wavelet Transforms
For the better understanding of wavelet transforms let’s look back
at the Fourier transforms more closely. Fourier transforms (as well as
Wavelet transforms) is a reversible transform, that is, it allows going back
and forth between the raw and processed signals. However, only either
one of them is available at any given time. That is, no frequency
information is available in time-domain signal, and no time information
in the Fourier transformed signal. The natural question that comes to our
mind is that is it necessary to have both the time and the frequency
information at the same time?
The answer depends on the particular application, and on the
nature of signal in hand. Recall that the Fourier transform 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 where
in time these frequency components exist. This information is not
required when the signal is so called stationary.
Let’s take a closer look at this stationary concept more closely,
since it is of paramount importance in signal analysis. Signals whose
frequency content does not change in time are called stationary signals. In
other words, the frequency content of stationary signals does not change
with time. In this case, one does not need to know at what times
Dept. of EEE MESCE Kuttippuram6
Seminar Report ’03 Wavelet Transforms
frequency components exist, since all frequency components exists at all
times!!!
For example, take the following signal…
x (t) = cos (2*∏*10t) + cos (2*∏*25t) + cos (2*∏*50t) + cos
(2*∏*100t)
The above signal is a stationary signal, because it has frequencies
of 10Hz, 25Hz, 50Hz,and 100Hz at any given time instant. This signal is
plotted below:
figure (1)
Dept. of EEE MESCE Kuttippuram7
Seminar Report ’03 Wavelet Transforms
And the following is its Fourier transform:
figure (2)
The top plot in figure (2) is the (half of the symmetric) frequency
spectrum of the signal in figure (1). The bottom plot is the zoomed
version of the top plot, showing only the range of the frequencies 10, 25,
50,100 Hz.
Contrary to the signal in the figure (1), the following is not
stationary. Figure(3) plots a signal whose frequency constantly in time.
This signal is known as the “chirp” signal. This is a non-stationary signal.
Dept. of EEE MESCE Kuttippuram8
Seminar Report ’03 Wavelet Transforms
figure (3)
Let’s look at another example. Figure (4) plots a signal with four
different frequency components at four different time intervals, hence a
non-stationary signal. The interval 0 to 300ms has a 100Hz sinusoid, the
interval 300 to 600ms has a 50Hz sinusoid, the interval 600 to 800ms has
a 25Hz sinusoid, and finally the interval 800 to1000ms has a 10Hz
sinusoid.
Dept. of EEE MESCE Kuttippuram9
Seminar Report ’03 Wavelet Transforms
figure (4)
And the following is its Fourier transform:
Dept. of EEE MESCE Kuttippuram10
Seminar Report ’03 Wavelet Transformsfigure (5)
The ripples present in the above figure are due to sudden changes
from one frequency component to another, which is neglected in our
analysis. Also, note that the amplitudes of higher frequency components
are higher than the low frequency ones. This is due to the fact that higher
frequencies last longer than the lower frequency components.
Now, compare the two figures (2) and (5). The similarity between
these two spectrums should be apparent. Both of them show four spectral
components at exactly the same frequencies, i.e., at 10, 25, 50 and 100Hz.
Other than the ripples, and the difference in the amplitudes (which can
always be normalized), the two spectrums are almost identical, although
the corresponding time-domain signals are not even close to each other.
Both of the signals involve the same frequency components, but the first
one has these frequencies at different intervals. So, how come the
spectrums of two entirely different signals look very much alike? Recall
that the Fourier transform gives the spectral component of the signal, but
it gives no information regarding where in time those spectral
components appear. Therefore, Fourier transform is not a suitable
technique for analyzing non-stationary signals, with one exception:
Dept. of EEE MESCE Kuttippuram11
Seminar Report ’03 Wavelet TransformsFourier transform can be used for non-stationary signals, if we
are only interested in what spectral components exist in the signal, but not
interested where these occur. However, if this information is required,
i.e., if we want to know, what spectral components occur at what time
(interval), then Fourier transform is not the right transform to use.
For practical purposes it is difficult to make the separation, since
there are a lot of practical stationary signals, as well as non-stationary
ones. Almost all biological signals, for example are non-stationary. Some
of the famous ones are ECG (electrical activity of the heart,
electrocardiograph), EEG (electrical activity of the brain,
electroencephalograph), and EMG (electrical activity of the muscles,
electromyography).
When the time localization of the spectral components are
needed, a transform giving the time-frequency representation of the signal
is needed.
The Wavelet transform is a transform of this type.
Dept. of EEE MESCE Kuttippuram12
Seminar Report ’03 Wavelet Transforms
THE WAVELET TRANSFORM
The Wavelet transform provides the time-frequency
representation. (There are other types of transforms which give this
information too, such as short time Fourier transform, Wigner
distribution, etc.)
Often times a particular spectral component occurring at any
instant can be of particular interest. In these cases it may be very
beneficial to know that the time intervals these particular spectral
components occur. For example, in EEGs, the latency of an event-related
potential is of particular interest. (Event-related potential is the response
of the brain to a specific stimulus like flashlight, the latency of this
response is the amount of time elapsed between the onset of the stimulus
and the response).
Wavelet transform is capable of providing the time-frequency
information simultaneously, hence giving a time-frequency representation
of the signal.
Originally, the Wavelet transform was implemented as an
alternative to the short time Fourier transforms. The Wavelet transform
Dept. of EEE MESCE Kuttippuram13
Seminar Report ’03 Wavelet Transforms
analysis is similar to short time Fourier transform analysis, except
for two main differences:
1. The Fourier transforms of the windowed signals are not taken, and
therefore single peak will be seen corresponding to a sinusoid, i.e.,
negative frequencies are not computed.
2. The width of the window is changed as the transform is computed for
every single spectral component, which is probably the most significant
characteristic of the Wavelet transform.
The above points causes problem of resolution of the STFT. This
is due to the result of choosing a window function, once and for all, and
using that window in the entire analysis. The answer, of course, is
application dependent: if the frequency components are well separated
from each other
In the original signal, then we may sacrifice some frequency
resolution and go for good time resolution, since the spectral components
are already well separated from each other. However, if this is not the
case, then a good window function could be more difficult than finding a
good stock to invest in. Thus the Wavelet transform solves the dilemma
of resolution to a certain extent.
Dept. of EEE MESCE Kuttippuram14
Seminar Report ’03 Wavelet Transforms
WAVELETS – THEORY
The Wavelet analysis is performed using a prototype function
called a wavelet, which has the effect of a band pass filter. Wavelets are
functions defined over a finite interval and having an average value of
zero. The basic idea of the wavelet transform is to represent any arbitrary
function f (t) as a superposition of a set of such wavelets or basis
function. These basis functions are derived from a single prototype called
mother wavelet.
The term wavelet means a small wave. The smallness refers to
the condition that this window function is of finite length (compactly
supported). The ‘wave’ refers to the condition that this function is
oscillatory. The term ‘mother’ implies that the functions with different
region of support that are used in the transformation process are derived
from one main function, or the mother wavelet by dilations or
contractions (scaling) and translations (shifts).
There are many different implementations of the Wavelet
transforms, of which the continuous wavelet transform is the simplest to
start with.
Dept. of EEE MESCE Kuttippuram15
Seminar Report ’03 Wavelet Transforms
CONTINUOUS WAVELET TRANSFORM
The continuous wavelet transform was developed as an
alternative to short time Fourier transforms, to overcome the resolution
problem.
The continuous wavelet transform is defined as follows:
CWTxΨ (τ,s) = Ψx
Ψ (τ,s) = 1/√s ∫ x (t) Ψ*(t- τ /s) dt.
As seen in the above equation, the transformed signal is a
function of two variables, tau and s, the translation and scale parameter,
respectively. Psi(t) is the transforming function, and it is called the
mother wavelet.
The term translation is related to the location of the window, as
the window is shifted through the signal. This term obviously corresponds
to time information in time domain. However, we do not have a
frequency parameter, as we had before the STFT. Instead, we have a scale
parameter, which is defined as 1/ frequency. The parameter scale in the
Wavelet analysis is similar to the scale used in maps. As in the case of
maps, a high scale refers to a detailed view. Similarly, in
Dept. of EEE MESCE Kuttippuram16
Seminar Report ’03 Wavelet Transforms
terms of frequency, low frequencies (high scales) correspond to global
information of a signal (that usually spans the entire signal), whereas high
frequencies (low scales) correspond to detailed information of a hidden
pattern in the signal (that usually lasts a relatively short time). Scaling, as
a mathematical operation, either dilates or compresses a signal. Larger
scales correspond to dilated signals and small scales correspond to
compressed signals.
In terms of mathematical functions, if f (t) is a given function, f
(st) corresponds to a contracted version of f (t) if s<1 and to an expanded
version of f (t) if s>1.
However, in the definition of the Wavelet transform, the scaling
term is used in the denominator, and therefore, the opposites of the above
statements holds, i.e., scales s<1 dilates the signals whereas scales s>1
compresses the signal.
Dept. of EEE MESCE Kuttippuram17
Seminar Report ’03 Wavelet Transforms
COMPUTATION OF THE CWT
1. The signal to be analyzed is taken.
2. The mother wavelet is chosen and the computation is begun with s = 1.
The CWT is computed for all values of s. the wavelet will dilate is s
increases and compresses when s is decreased.
3. The wavelet is placed in the beginning of the signal at the point which
corresponds to time = 0.
4. The wavelet is multiplied with the signal and integrated over all times.
The result is then multiplied by the constant 1/sqrt{s}.
5. The above step normalizes the energy so that the transformed signal
has same energy at every scale.
6. The wavelet at scale s =1 is then shifted to the right by τ and the above
steps are repeated until the wavelet reaches the end of the signal.
Dept. of EEE MESCE Kuttippuram18
Seminar Report ’03 Wavelet Transforms
CONCLUSION
The development of wavelets is an example where ideas from
many different fields combined to merge into a whole that is more than
the sum of its parts. Wavelet transforms have been widely employed in
signal processing application, particularly in image compression research.
It has been used extensively in multi-resolution analysis (MRA) for image
processing. In signal processing applications, different implementations
of the wavelet theory have been used for effective evaluation of
biological signals emanating from bio-medical devices.
Dept. of EEE MESCE Kuttippuram19
Seminar Report ’03 Wavelet Transforms
REFERENCES
IEEE Proceedings- Vis. Image Signal Processing, Vol. 144, No.6, December 1997.
IEEE Proceedings- Vol.84, No.4, April 1996
www.ieee.org/wavelettransform/
Wavelet transforms by Robi Polikar.
Digital Image Processing by Raphael.C.Gonzalez and Richard.C.Woods.
Dept. of EEE MESCE Kuttippuram20
Seminar Report ’03 Wavelet Transforms
ABSTRACT
Mathematical transformations are applied to signals to obtain
further information from the signal that is not readily available in the raw
signal. By applying the various transformations available today, the
frequency information in these signals is obtained. There are many
transforms that are used quite often by engineers and mathematician’s.
Hilbert transforms, short-time Fourier transforms, Radon transform, and
the Wavelet transform constitute only a small portion of a huge list of
transforms available at engineer’s and mathematician’s disposal where
each transformation technique has its own area of application, advantages
and disadvantages.
Dept. of EEE MESCE Kuttippuram21
Seminar Report ’03 Wavelet Transforms
ACKNOWLEDGEMENT
I express my sincere gratitude to Dr. P.M.S. Nambisan, Prof. and
Head, Department of Electrical and Electronics Engineering, MES College of
Engineering, Kuttippuram, for his cooperation and encouragement.
I would also like to thank my seminar guide and Staff in-charge
Asst. Prof. Gylson Thomas. (Department of EEE) for his invaluable advice and
wholehearted cooperation without which this seminar would not have seen the
light of day.
Gracious gratitude to all the faculty of the department of EEE and
friends for their valuable advice and encouragement.
Dept. of EEE MESCE Kuttippuram22
Seminar Report ’03 Wavelet Transforms
CONTENTS
1. Introduction 1
2. Trans…What? 2
3. Importance of frequency information 3
4. The Fourier transform 5
5. The Wavelet transform 13
6. Wavelets – Theory 15
7. Continuous wavelet transform 16
8. Computation of Wavelet transforms 18
9. Conclusion 19
10. References 20
Dept. of EEE MESCE Kuttippuram23