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8/10/2019 Introduction to Wavelet Analysis
1/27
27/9/2005 1
DEIS
University of Bologna
Italy
AN INTRODUCTION TO
N INTRODUCTION TO
WAVELETAVELET
TRANSFORMS
RANSFORMS
Luca De Marchi
8/10/2019 Introduction to Wavelet Analysis
2/27
27/9/2005 2
DEIS
University of Bologna
Italy
OUTLINEUTLINE
Time-Frequency Analysis Introduction on Wavelet Operators
Examples of applications: Radar/Sonar Experimental results
Conclusions
8/10/2019 Introduction to Wavelet Analysis
3/27
27/9/2005 3
DEIS
University of Bologna
Italy
Fourier
ourier
Analysis
nalysis
=
=
deFtf
dtetfF
tj
tj
)(2
1)(
)()(
Fast Discrete Algorithm (FFT)
FFT: a rotation in function space
New basis functions sines and cosines Not localized in time
8/10/2019 Introduction to Wavelet Analysis
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DEIS
University of Bologna
Italy
Signal Analysis
ignal Analysis
f(t) = f1(t) + f
2(t) + f
3(t)
2
1230
1
1
302sin)(
= T
t
eT
ttf
2
28.1
100
2
2
1002sin)(
=
T
t
eT
ttf
2
32.3
155
3
3
1552sin)(
=
T
t
e
T
ttf
T1=28
T2 = 14
T3
= 7
8/10/2019 Introduction to Wavelet Analysis
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27/9/2005 5
DEIS
University of Bologna
Italy
Fast
ast
Fourier
ourier
Transform
ransform
8/10/2019 Introduction to Wavelet Analysis
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27/9/2005 6
DEIS
University of Bologna
Italy
Time
ime
Frequency
requency
Analysis
nalysis
:
A Well
ell
Known
nown
Example
xample
Freq
Time
DEIS
8/10/2019 Introduction to Wavelet Analysis
7/27
27/9/2005 7
DEIS
University of Bologna
Italy
Wavelet Transforms
avelet Transforms
( ) ( ) RbRadt
a
bttf
a
bac
= ++
,
1,
Continuous WT, () finite energy
c(a,b) is a resemblance index between () and ()located at a position b and scale a representing how
closely correlated is the wavelet with a portion of the
signal () is localized in frequency and in time
DEIS
8/10/2019 Introduction to Wavelet Analysis
8/27
27/9/2005 8
DEIS
University of Bologna
Italy
Wavelet
avelet
Analysis
nalysis
( ) ( )xeCx
x
5cos2
2
=
DEIS
8/10/2019 Introduction to Wavelet Analysis
9/27
27/9/2005 9
CWT
WT
Analysis
nalysis
DEIS
University of Bologna
Italy
DEIS
8/10/2019 Introduction to Wavelet Analysis
10/27
27/9/2005 10
DEIS
University of Bologna
Italy
Fourier
ourier
Analysis
nalysis
1 21 2 , ,( ) sin(2 ) sin(2 ) [ ]n n n nf n f n f n = + + +
f1= 500Hz
f2=1 KHz
=1/8000 s
=1.5n1=250
n2=282
DEIS
8/10/2019 Introduction to Wavelet Analysis
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DEIS
University of Bologna
Italy
Wavelet
avelet
Analysis
nalysis
22 22
4( )t i tt Ce e e
=
DEIS
8/10/2019 Introduction to Wavelet Analysis
12/27
27/9/2005 12
University of Bologna
Italy
Radar/Sonar
adar/Sonar
Appplications
ppplications
Radar Signal: fc=64Mhz, Tr=50us, =6us, fcarrier=1Mhz
Tx Tx
Tr
Rx
s
T
APPLICATIONS: airport Radar, metal detector, medical
application (tissue imaging, velocity blood measurements)
DEIS
8/10/2019 Introduction to Wavelet Analysis
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27/9/2005 13
University of Bologna
Italy
DENOISINGENOISING
Problem: Radar/Sonar pulses detection andfiltering in presence of strong noise andjamming signals
Solution: using a thresholding procedureperformed on coefficients resulting from a
Wavelet Transform analysis
DEIS
8/10/2019 Introduction to Wavelet Analysis
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27/9/2005 14
University of Bologna
Italy
Experimental results
xperimental results
System description Signal used to tune the
filter
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DEIS
i i f l
8/10/2019 Introduction to Wavelet Analysis
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University of Bologna
Italy
Experimental results
xperimental results
DEIS
U i it f B l
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University of Bologna
Italy
Denoising images (1)
Algorithm Performance on a echografic image
sensors
samples100 200 300 400 500 600 700 800 900 1000 1100
5
10
15
20
25
30
sensors
samples100 200 300 400 500 600 700 800 900 1000
5
10
15
20
25
30
DEIS
University of Bologna
8/10/2019 Introduction to Wavelet Analysis
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University of Bologna
Italy
Denoising Images (2)
Enhancement of attenuation effects
8/10/2019 Introduction to Wavelet Analysis
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DEIS
University of Bologna
8/10/2019 Introduction to Wavelet Analysis
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University of Bologna
Italy
Dataata compressionompression
Fast Discrete algorithms
WT renders sparse large
classes of functionsi.e. few noticeable coefficients
many negligible
Ex. Standard JPEG 2000
DEIS
University of Bologna
8/10/2019 Introduction to Wavelet Analysis
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University of Bologna
Italy
Research topics:
esearch topics:
Music Signal Analysisusic Signal Analysis
Wavelet Spectrogram
Midi Scores Source:http://hil.t.u-tokyo.ac.jp
DEIS
University of Bologna
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y g
Italy
Research topics:
esearch topics:
Music Signal Analysisusic Signal Analysis
Definition of algorithms
Hardware implementations on FPGA board,
on DSP, or Full Custom Design.
Applications: Music Information Retrieval,Sound Synthesis and Analysis
La musique est une mathmatique mystrieuse dontles lment partecipent de linfini C.Debussy
DEIS
University of Bologna
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y g
Italy
Research topics:
esearch topics:
Device Simulationevice Simulation
DEIS
University of Bologna
8/10/2019 Introduction to Wavelet Analysis
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Italy
Research topics:
esearch topics:
Device Simulationevice Simulation
Definition of numerical algorithms
Physical relevances analysis
Computational Grid Automatic Design Software Engineering
Entia non sunt multiplicanda praeter necessitatem
Occam
DEIS
University of Bologna
8/10/2019 Introduction to Wavelet Analysis
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Conclusions
onclusions
Italy
Wavelet Transform: a tool for time -frequencyanalysis
Easy to implement: fast algorithms
Well suited for many applications: such as
non-stationary analysis or data compression
DEIS
University of Bologna
I l
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Italy
Wavelet Research Group
avelet Research Group
Professors: Guido Masetti, Sergio Graffi,Nicol Speciale.
(Sistemi Integrati per lAnalisi Spettrale LS)
PhD Students: Emanuele Baravelli, Luca De
Marchi, Matteo Montani, Nicola Testoni.
Fellows: Salvatore Caporale, Francesco Franz,
Simona Maggio, Marco Messina, AlessandroPalladini.
DEIS
University of Bologna
It l
8/10/2019 Introduction to Wavelet Analysis
27/27
27/9/2005 27
Italy
Students Publications
tudents Publications
FPGA Implementation of QCWT Based Algorithmfor filtering Low SNR Signals, A.Marcianesi,R.Padovani, N.Speciale, N.Testoni, G. Masetti,2003.
Wavelet-based Algorithms for Speckle Removalfrom B-Mode Images, S. Caporale,A. Palladini, L.De Marchi, N. Speciale, G. Masetti, 2004.
Wavelet-based Deconvolution Algorithms Appliedto Ultrasound Images, S. Maggio, N. Testoni, L. DeMarchi, N. Speciale, G. Masetti, 2005.