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Wavelets are mathematical functions that cut up data into
different frequency components and then study each
component with a resolution matched to its scale.
Wavelets allow time and frequency domain analysis
simultaneously.
Wavelet algorithms
process data at
different scales
or resolutions.
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INTRODUCTIONINTRODUCTION
M.Tech Thesis-1 Presentation-1-2016
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WHY CHOOSEWHY CHOOSE
WAVELET TRANSFORM ?WAVELET TRANSFORM ?
One of the most useful features of wavelets is the ease
with which one can choose the defining co-efficient for a
given wavelet system to be adapted for a given problem.
Basis functions are localized in frequency, for example
power spectra.
Wavelet transform can vary in scale and can conserve
energy while computing functional energy.
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PROPERTIES OF WAVELETSPROPERTIES OF WAVELETS
Simultaneous localization in time and scale
• The location of the wavelet allows to explicitly represent
the location of events in time
• The shape of the wavelet allows to represent different
details or resolution.
Sparsity : many of the
coefficients in a wavelet
representation are either
zero or very small .
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PROPERTIES OF WAVELETS(CONTINUE..)PROPERTIES OF WAVELETS(CONTINUE..)
Adaptability : Can represent functions discontinuities or
corners more efficiently.
Wavelets are scaled and translated copies of a finite length or
fast-decaying oscillation waveform.
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ORTHOGONAL WAVELET FILTERORTHOGONAL WAVELET FILTER
BANKBANK
Coefficients of Orthogonal filters are real numbers.
The filters are of the same length and are not symmetric.
The relation between low pass and high pass filters are
given by the relation :
G0 = H0(-Z^-1)
For perfect reconstruction alternating flip is used.
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BIORTHOGONAL WAVELET FILTERBIORTHOGONAL WAVELET FILTER
BANKBANK
The low pass and high pass filters do not have the same
length.
The coefficients of the filters are either real numbers or
integers.
The low pass filter is always symmetric ,while the high
pass filter can be either symmetric or anti-symmetric.
For perfect re-construction bi-orthogonal filter has all odd
length or all even length filters.
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WAVELET TRANSFORMWAVELET TRANSFORM
Wavelet transforms have become the most useful tool of
signal representation.
It was developed to overcome the shortcomings for time-
frequency representation of non –stationary signals using
Short time Fourier transform(STFT) which gives a
constant resolution at all frequencies.
In 1-D DWT is applied in the rows first and then along
the columns, then we get the 2-D decomposition of the
image, in which we get the four components i.e.
Approximation, horizontal, vertical and diagonal
coefficients.
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WAVELET TRANSFORM(continue…)WAVELET TRANSFORM(continue…)
CONTINOUS WAVELET TRANSFORM :
It is the convolution of the input data sequence with a set
of functions generated by the mother wavelet .
Its advantageous while performing image compression as
it provides significant improvement in picture quality.
Examples are Meyer, Morlet, Mexican hat
Forward CWT:
Inverse CWT :
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WAVELET FAMILIESWAVELET FAMILIES
MEYER WAVELET :
It is an orthogonal wavelet which is infinitely
differentiable and defined in frequency domain.
-8 -6 -4 -2 0 2 4 6 8
-1
0
1
2
Meyer wavelet
-8 -6 -4 -2 0 2 4 6 8
-0.5
0
0.5
1
1.5
Meyer scalingfunction
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WAVELET FAMILIES(continue…)WAVELET FAMILIES(continue…)
MORLET WAVELET :
It is a wavelet composed of complex exponential(carrier)
multiplied with a Gaussian window(envelope) .
-4 -3 -2 -1 0 1 2 3 4
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Morlet wavelet
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WAVELET FAMILIES(continue…)WAVELET FAMILIES(continue…)
MEXICAN HAT WAVELET :
It is the negative normalized second derivative of a
Gaussian function.
-5 -4 -3 -2 -1 0 1 2 3 4 5
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
Mexican hat wavelet
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WAVELET FAMILIES(continue…)WAVELET FAMILIES(continue…)
DISCRETE WAVELET TRANSFORM :
In 1-D DWT is applied in the rows first and then along
the columns, then we get the 2-D decomposition of the
image, in which we get the four components i.e.
Approximation, horizontal, vertical and diagonal
coefficients.
In DWT based image fusion, DWT is first applied to
source images to obtain the wavelet coefficients and then
appropriate fusion rule is used.
Finally, for reconstruction of fused image inverse DWT is
used.
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WAVELET FAMILIES(continue…)WAVELET FAMILIES(continue…)
Commonly used DWTs are as:
(i)Haar wavelet :It is the first invented DWT. For an
input list of 2^n numbers, it is considered to pair up the
input values, stores the difference and passes the sum.
The process is repeated recursively, pairing up the sums
to provide the next scale which leads to 2^n-1 differences
and a final sum.
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WAVELET FAMILIES(continue…)WAVELET FAMILIES(continue…)
(i) Daubechies wavelet : The formulation is based on the use
of recurrence relation to generate progressively finite
discrete samplings of an implicit mother wavelet function,
each resolution is twice that of previous scale.
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ADVANTAGES OVER TRADITIONALADVANTAGES OVER TRADITIONAL
FOURIER TRANSFORMFOURIER TRANSFORM
Wavelets represent functions that have sharp peaks and
for accurately deconstructing and reconstructing finite,
non-periodic and non-stationary signals.
Fourier transform is not practical for computing spectral
information and cannot observe frequencies varying with
time . On the other hand, Wavelet transform are based on
wavelets which are varying frequency in limited duration.
In FFT OFDM it requires guard signal which is not
needed in Wavelet OFDM.
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APPLICATIONS OF WAVELET TRANSFORMAPPLICATIONS OF WAVELET TRANSFORM
1.DWT for data compression if signal is already sampled.
2.CWT for signal analysis.
3.Used for wavelet shrinkage
4.Used in communication as wavelet OFDM being the
modulation scheme used by Panasonic.
5.Noise filtering
6.Image fusion
7.Recognition
8.Image matching and retrieval
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CONCLUSIONCONCLUSION
Multiwavelets are better approach for high compression
ratio and to get better performance to medical imaging
applications and it may be found suitable for enhancing
the computability for compression of different areas of
applications.
TheThe great challenge in this field is to develop robust and
dynamic algorithm for compressing. In view of this, work
may be further extend to develop an universal wavelet
filter that may be suitable for types of images pertaining
to different application areas.
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BIBLIOGRAPHYBIBLIOGRAPHY
Paper on Multimodal image fusion and Robust object
tracking by CSE Department NIST Berhampur
Gonzalez Rafael C., Woods Richard E. Digital image
Processing .Upper Saddle river New Jersey: Prentice Hall,
Second Edition
Wikipedia
https://en.wikipedia.org/wiki/Wavelet_transform
Wavelet analysis for image processing Tzu-Heng Henry
Lee Graduate Institute of Communication
Engineering,
National Taiwan University, Taipei, Taiwan, ROC
An introduction to wavelets by A Graps