by dr. rajeev srivastava cse, iit(bhu)
DESCRIPTION
Its Understanding Wavelet means ‘small wave’. So wavelet analysis is about analyzing signal with short duration finite energy functions. They transform the signal under investigation in to another representation which presents the signal in more useful form. This transformation of the signal is called Wavelet Transform Dr. Rajeev SrivastavaTRANSCRIPT
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Wavelet TransformBy
Dr. Rajeev SrivastavaCSE, IIT(BHU)
Dr. Rajeev Srivastava
Dr. Rajeev Srivastava 2
Its Understanding
Wavelet means ‘small wave’.
So wavelet analysis is about analyzing signal with short duration finite energy functions. They transform the signal
under investigation in to another representation which presents the signal in more useful form.
This transformation of the signal is called Wavelet Transform
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Wavelet Analysis and Synthesis
Wavelets :
The wavelets are a family of functions generated from a single function by translation and dilation.
Have a zero mean.
Used for analyzing and representing signals or other functions.
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Wavelet analysis:
A windowing technique with variable-sized regions.
Allows the use of long time intervals where we want
more precise low-frequency information, and shorter regions where we want
high-frequency information.
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Wavelets………
Time –Frequency plane of Discrete Wavelet Transform Fourier Transform
Translation Dilations(scaling)
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Wavelets……..
•In the time domain we have full time resolution, but no frequency localization or separation. •In the Fourier domain we have full frequency resolution but no time separation. •In the wavelet domain we have some time localization and some frequency localization.
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Wavelets…….• A set of dilations and translations ψ τ,s (t) of a chosen mother wavelet ψ (t) is used for analysis of a signal. The general form of wavelets :
st
stS
1)(,
•Where s is the scaling (dilations) factor and τ is the translation (location) factor. •Manipulating wavelets by translation ( change the central position of the wavelet along the time axis) and scaling ( change the locations or levels). •The forward wavelet transform (Analysis Part), calculates the contribution (wavelet coefficients, denoted as C τ,s )of each dilated and translated version of the mother wavelet in the original data set.Wavelet transform is defined as
dtttfC ss )()( ,,
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Wavelets……•Inverse wavelet transform (Synthesis Part) uses the computed wavelet coefficients and superimposes them to calculate the original data set. )()( ,
,, tCtf s
ss
•Discrete Wavelet Transform(DWT)The scale and translate parameters are chosen such that the resulting wavelet set forms an orthogonal set. Dilation factors are chosen to be powers of 2. A common choice for τ and s is τ =2m , s=n.2 m where n, m εZ i.e.)2(2)( 2/
, ntt mmnm
Where m is the scaling factor and n is the translation factor .
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Discrete Wavelet Transform (DWT)
Forward (DWT) and inverse transforms (IDWT) are then calculated using the following equations:
dtttfC
tCtf
nmm
nm
nmnm
nm
)().(2
)(.)(
,2/
,
,,
,
f(t)=original signal, =Wavelet coefficient, =mother wavelet.
Advantages of Using Wavelets:
Good de-correlating behavior Easily detect local features in a signal
Based on multi-resolution analysis.
Fast and stable algorithms are available to calculate the Discrete
Wavelet Transform and its inverse
𝜑𝑚 ,𝑛
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Wavelets and its relation to Sub-band coding
h0(n)LPF
h1(n)HPF
2↓
2↓
2↑
2↑
g0(n)
g1(n)+ x(t) x’(t)
NOTE: y0(n) is approximation part of x(n) and y1(n) is detail part of x(n)
y0(n)
y1(n)
A two-band filter bank for 1D sub-band coding and decoding
|H0(ω)| |H1(ω)|
LOW BAND HIGH BAND
0 π/2 π ω Spectrum splitting properties of sub-band coding and decoding
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Splitting the signal spectrum with an iterated filter bank.
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Wavelets…..•The Z-Transform of sequence x(n) for n=0,1,2,3,…. is
nznxZX )()(
•Where z is a complex variable .If , above equation becomes DFT. Basic advantage of using Z-Transform is that it easily handles the sampling rate changes .
jez
•Down Sampling by a factor of 2 in the time domain corresponds to the simple Z-domain operation:
)()(21)()2()( 2/12/1 zXzXzXnxnx downdown
•Up Sampling by a factor of 2 is defined as:
0
)2/()(
nxnxup
for n=0,2,4,…..Otherwise
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Perfect Reconstruction Filter Bank
The following figure illustrates the decomposition and reconstruction process.
The filter bank is said to be a perfect reconstruction filter bank when a2 = a0 . If, additionally, h1 = h2 and g1 = g2, the filters are called conjugate mirror filters
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A 2D ,four –band filter bank for sub-band Image Coding (Analysis Part)
h0(m)
h1 (m)
2↓
2↓
h0(n)
h1 (n)
h0(n)
h1 (n)
2↓
2↓
2↓
2↓
x(m,n)
Rows (along m)
a(m,n)
dV(m,n)
dH(m,n)
dD (m,n)
Columns (along n)
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Representation of Spatial and Frequency
Hierarchies
Spatial Hierarchy for 2D Image
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Representation of frequency hierarchies
Frequency hierarchy for a two level 2D DWT decomposition
Frequency hierarchy for a two level full 2D WPT decomposition
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Example of an 128x128 image at different levels of decompositions
by 2D DWT
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Applications
Image Compression
Image De-noising
Image Enahancement
Image Segmentation
Etc.
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