Introduction to Wavelet Analysis

8/10/2019 Introduction to Wavelet Analysis

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University of Bologna

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AN INTRODUCTION TO

N INTRODUCTION TO

WAVELETAVELET

TRANSFORMS

RANSFORMS

Luca De Marchi


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OUTLINEUTLINE

Time-Frequency Analysis Introduction on Wavelet Operators

Examples of applications: Radar/Sonar Experimental results

Conclusions


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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


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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


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Fast

ast

Fourier

ourier

Transform

ransform


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Time

ime

Frequency

requency

Analysis

nalysis

:

A Well

ell

Known

nown

Example

xample

Freq

Time

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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

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Wavelet

avelet

Analysis

nalysis

( ) ( )xeCx

x

5cos2

2

=

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CWT

WT

Analysis

nalysis

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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

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Wavelet

avelet

Analysis

nalysis

22 22

4( )t i tt Ce e e

=

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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)

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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

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Experimental results

xperimental results

System description Signal used to tune the

filter


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Experimental results

xperimental results

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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

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Denoising Images (2)

Enhancement of attenuation effects


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Dataata compressionompression

Fast Discrete algorithms

WT renders sparse large

classes of functionsi.e. few noticeable coefficients

many negligible

Ex. Standard JPEG 2000

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Research topics:

esearch topics:

Music Signal Analysisusic Signal Analysis

Wavelet Spectrogram

Midi Scores Source:http://hil.t.u-tokyo.ac.jp

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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

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Research topics:

esearch topics:

Device Simulationevice Simulation

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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

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Conclusions

onclusions

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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

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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.

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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.

Documents

Introduction to Wavelet Analysis