By Dr Rajeev Srivastava CSE IITBHU Dr Rajeev Srivastava 1 Its Understanding Dr Rajeev Srivastava 2 3 Wavelet Analysis and Synthesis Dr Rajeev Srivastava Dr Rajeev Srivastava ID: 441434
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Slide1
Wavelet Transform
ByDr. Rajeev SrivastavaCSE, IIT(BHU)
Dr. Rajeev Srivastava
1Slide2
Its Understanding
Dr. Rajeev Srivastava2Slide3
3Wavelet Analysis and Synthesis
Dr. Rajeev SrivastavaSlide4
Dr. Rajeev Srivastava
4Slide5
5Wavelets………
Time –Frequency plane of Discrete Wavelet Transform
Fourier Transform
Translation Dilations(scaling
)
Dr. Rajeev SrivastavaSlide6
6Wavelets……..
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.
Dr. Rajeev SrivastavaSlide7
7Wavelets…….
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 :
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
Dr. Rajeev SrivastavaSlide8
8Wavelets……
Inverse wavelet transform (Synthesis Part)
uses the computed wavelet coefficients and superimposes them to calculate the original data set.
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
=2
m
,
s
=n.2
m
where n, m
ε
Z
i.e.
Where m
is the scaling factor and
n
is the translation factor
.
Dr. Rajeev SrivastavaSlide9
9
.
Dr. Rajeev Srivastava
Slide10
10
h
0
(n)
LPF
h
1
(n)
HPF
2
↓
2
↓
2
↑
2
↑
g
0
(n)
g
1
(n)
+
x(t)
x’(t)
NOTE: y0(n) is approximation part of x(n) and y1(n) is detail part of x(n)
y
0
(n)
y
1
(n)
A two-band filter bank for 1D sub-band coding and decoding
|H
0
(
ω
)|
|H
1
(
ω
)|
LOW BAND
HIGH BAND
0 π/2 π
ω
Spectrum splitting properties of sub-band coding and decoding
Dr. Rajeev SrivastavaSlide11
11Splitting the signal spectrum with an iterated filter bank.
Dr. Rajeev SrivastavaSlide12
12Wavelets…..
The Z-Transform of sequence x(n) for n=0,1,2,3,…. is
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 .
Down Sampling
by a factor of 2 in the time domain corresponds to the simple Z-domain operation:
Up Sampling
by a factor of 2 is defined as:
for n=0,2,4,…..
Otherwise
Dr. Rajeev SrivastavaSlide13
13
The filter bank is said to be a
perfect reconstruction filter bank
when
a
2
= a
0
. If, additionally,
h1 = h
2
and
g1 = g
2
, the filters are called
conjugate mirror filters
Dr. Rajeev SrivastavaSlide14
14
h0(m)h1
(m)
2↓
2↓
h
0
(n)
h
1
(n)
h
0
(n)
h
1
(n)
2↓
2↓
2↓
2↓
x(m,n)
Rows
(along m)
a(m,n)
d
V
(m,n)
d
H
(
m,n
)
d
D
(m,n)
Columns (along n)
Dr. Rajeev SrivastavaSlide15
15
Spatial
Hierarchy for 2D Image
Dr. Rajeev SrivastavaSlide16
16
Frequency hierarchy for a two level 2D DWT decomposition
Frequency hierarchy for a two level full 2D WPT decomposition
Dr. Rajeev SrivastavaSlide17
17Example of an 128x128 image at different levels of decompositions by 2D DWT
Dr. Rajeev SrivastavaSlide18
Dr. Rajeev Srivastava
18Slide19
Dr. Rajeev Srivastava19