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27/9/2005 1 DEIS University of Bologna Italy AN INTRODUCTION TO AN INTRODUCTION TO WAVELET WAVELET TRANSFORMS TRANSFORMS Luca De Marchi

AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

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Page 1: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 1

DEISUniversity of Bologna

Italy

AN INTRODUCTION TO AN INTRODUCTION TO WAVELET WAVELET

TRANSFORMSTRANSFORMS

Luca De Marchi

Page 2: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 2

DEISUniversity of Bologna

Italy

OUTLINEOUTLINE

Time-Frequency AnalysisIntroduction on Wavelet Operators Examples of applications: Radar/SonarExperimental results Conclusions

Page 3: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 3

DEISUniversity of Bologna

ItalyFourierFourier AnalysisAnalysis

∫∞

∞−

∞−

Π=

=

ωω

ω

ω

ω

deFtf

dtetfF

tj

tj

)(21)(

)()(

• Fast Discrete Algorithm (FFT)• FFT: a rotation in function space• New basis functions sines and cosines• Not localized in time

Page 4: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 4

DEISUniversity of Bologna

ItalySignal Analysis Signal Analysis f(t) = f1(t) + f2(t) + f3(t)

2

1230

11

302sin)(

−−

−= T

t

eT

ttf π

2

28.1100

22

1002sin)(

−−

−= T

t

eT

ttf π

2

32.3155

33

1552sin)(

−−

−= T

t

eT

ttf π

T1=28

T2 = 14

T3 = 7

Page 5: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 5

DEISUniversity of Bologna

ItalyFast Fast FourierFourier TransformTransform

Page 6: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 6

DEISUniversity of Bologna

Italy

TimeTime--FrequencyFrequency AnalysisAnalysis::A A WellWell--KnownKnown ExampleExample

Freq

Time

Page 7: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 7

DEISUniversity of Bologna

Italy

Wavelet TransformsWavelet Transforms

( ) ( ) RbRadta

bttfa

bac ∈∈

⋅= +∞+

∞−

∗∫ ,1, ψ

Continuous WT, ƒ(τ) finite energyc(a,b) is a resemblance index between ƒ(τ) and ψ(τ)located at a position b and scale a representing how closely correlated is the wavelet with a portion of the signalψ(τ) is localized in frequency and in time

Page 8: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 8

DEISUniversity of Bologna

Italy

Wavelet Wavelet AnalysisAnalysis

( ) ( )xeCxx

5cos2

2−

⋅=ψ

Page 9: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 9

CWT CWT AnalysisAnalysisDEIS

University of Bologna Italy

Page 10: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 10

DEISUniversity of Bologna

ItalyFourierFourier AnalysisAnalysis

1 21 2 , ,( ) sin(2 ) sin(2 ) [ ]n n n nf n f n f nτ π τ π τ α δ δ= + + +

f1= 500Hzf2=1 KHzτ=1/8000 sα=1.5n1=250n2=282

Page 11: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 11

DEISUniversity of Bologna

ItalyWavelet Wavelet AnalysisAnalysis2 2 2

2 4( )t

i tt Ce e eπ απαψ

− − = −

Page 12: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 12

DEISUniversity of Bologna

Italy

Radar/Sonar Radar/Sonar AppplicationsAppplications

Radar Signal: fc=64Mhz, Tr=50us, τ=6us, fcarrier=1Mhz

Tx Tx

Tr

τ τ

Rx

T

APPLICATIONS: airport Radar, metal detector, medical application (tissue imaging, velocity blood measurements)

Page 13: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 13

DEISUniversity of Bologna

Italy

DENOISINGDENOISING

Problem: Radar/Sonar pulses detection and filtering in presence of strong noise and jamming signals

Solution: using a thresholding procedure performed on coefficients resulting from a Wavelet Transform analysis

Page 14: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 14

DEISUniversity of Bologna

Italy

Experimental resultsExperimental results

System description Signal used to tune the filter

Page 15: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 15

DEISUniversity of Bologna

Italy

Experimental resultsExperimental results

Signal corrupted by colored noise

Signal spectrum

Page 16: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 16

DEISUniversity of Bologna

Italy

Experimental resultsExperimental results

Page 17: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 17

DEISUniversity of Bologna

Italy

Denoising images (1)

• Algorithm Performance on a echografic image

sens

ors

samples100 200 300 400 500 600 700 800 900 1000 1100

5

10

15

20

25

30

sens

ors

samples100 200 300 400 500 600 700 800 900 1000

5

10

15

20

25

30

Page 18: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 18

DEISUniversity of Bologna

Italy

Denoising Images (2)

Enhancement of attenuation effects

Page 19: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 19

DEISUniversity of Bologna

Italy

Research topics: Research topics: UltrasoundsUltrasounds

Definition of algorithmsHardware implementations on FPGA board, on DSP, or Full Custom Design. Applications: Biomedical Imaging Enhancement, Tissues properties investigation…

“ If you steal from one author it’s plagiarism, if you steal from many it’s research” W.Mizner

Page 20: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 20

DEISUniversity of Bologna

Italy

Data Data compressioncompression

•Fast Discrete algorithms

• WT renders sparse largeclasses of functionsi.e. few noticeable coefficientsmany negligible

• Ex. Standard JPEG 2000

Page 21: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 21

DEISUniversity of Bologna

Italy

Research topics: Research topics: Music Signal AnalysisMusic Signal Analysis

Wavelet Spectrogram

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

Page 22: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 22

DEISUniversity of Bologna

Italy

Research topics: Research topics: Music Signal AnalysisMusic Signal Analysis

Definition of algorithmsHardware implementations on FPGA board, on DSP, or Full Custom Design. Applications: Music Information Retrieval, Sound Synthesis and Analysis…

“ La musique est une mathématique mystérieuse dontles élément partecipent de l’infini” C.Debussy

Page 23: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 23

DEISUniversity of Bologna

Italy

Research topics: Research topics: Device SimulationDevice Simulation

Page 24: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 24

DEISUniversity of Bologna

Italy

Research topics: Research topics: Device SimulationDevice Simulation

Definition of numerical algorithmsPhysical relevances analysisComputational Grid Automatic DesignSoftware Engineering

“Entia non sunt multiplicanda praeter necessitatem” Occam

Page 25: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 25

ConclusionsConclusions

DEISUniversity of Bologna

Italy

Wavelet Transform: a tool for time -frequency analysis

Easy to implement: fast algorithms

Well suited for many applications: such as non-stationary analysis or data compression

Page 26: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 26

DEISUniversity of Bologna

Italy

Wavelet Research GroupWavelet Research Group

Professors: Guido Masetti, Sergio Graffi, Nicolò Speciale. (Sistemi Integrati per l’Analisi Spettrale LS)

PhD Students: Emanuele Baravelli, Luca De Marchi, Matteo Montani, Nicola Testoni.

Fellows: Salvatore Caporale, Francesco Franzè, Simona Maggio, Marco Messina, Alessandro Palladini.

Page 27: AN INTRODUCTION TO WAVELET TRANSFORMSmasetti/Dida01/intro_wavelet.pdfWavelet Transforms () dt a R b R a t b f t a c a b ∈ ∈ − = ⋅ + +∞ −∞ ∫ ∗, 1, ψ Continuous WT,

27/9/2005 27

DEISUniversity of Bologna

Italy

Students PublicationsStudents Publications

FPGA Implementation of QCWT Based Algorithm for filtering Low SNR Signals, A.Marcianesi, R.Padovani, N.Speciale, N.Testoni, G. Masetti, 2003.

Wavelet-based Algorithms for Speckle Removal from B-Mode Images, S. Caporale, A. Palladini, L. De Marchi, N. Speciale, G. Masetti, 2004.

Wavelet-based Deconvolution Algorithms Applied to Ultrasound Images, S. Maggio, N. Testoni, L. De Marchi, N. Speciale, G. Masetti, 2005.