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AN INTRODUCTION TO AN INTRODUCTION TO WAVELET WAVELET
TRANSFORMSTRANSFORMS
Luca De Marchi
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OUTLINEOUTLINE
Time-Frequency AnalysisIntroduction on Wavelet Operators Examples of applications: Radar/SonarExperimental results Conclusions
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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
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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
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TimeTime--FrequencyFrequency AnalysisAnalysis::A A WellWell--KnownKnown ExampleExample
Freq
Time
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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
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Wavelet Wavelet AnalysisAnalysis
( ) ( )xeCxx
5cos2
2−
⋅=ψ
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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
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DEISUniversity of Bologna
ItalyWavelet Wavelet AnalysisAnalysis2 2 2
2 4( )t
i tt Ce e eπ απαψ
− − = −
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Radar/Sonar Radar/Sonar AppplicationsAppplications
Radar Signal: fc=64Mhz, Tr=50us, τ=6us, fcarrier=1Mhz
Tx Tx
Tr
τ τ
Rx
sτ
T
APPLICATIONS: airport Radar, metal detector, medical application (tissue imaging, velocity blood measurements)
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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
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Experimental resultsExperimental results
System description Signal used to tune the filter
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Experimental resultsExperimental results
Signal corrupted by colored noise
Signal spectrum
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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
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Denoising Images (2)
Enhancement of attenuation effects
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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
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Data Data compressioncompression
•Fast Discrete algorithms
• WT renders sparse largeclasses of functionsi.e. few noticeable coefficientsmany negligible
• Ex. Standard JPEG 2000
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Research topics: Research topics: Music Signal AnalysisMusic Signal Analysis
Wavelet Spectrogram
Midi Scores Source:http://hil.t.u-tokyo.ac.jp
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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
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Research topics: Research topics: Device SimulationDevice Simulation
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Research topics: Research topics: Device SimulationDevice Simulation
Definition of numerical algorithmsPhysical relevances analysisComputational Grid Automatic DesignSoftware Engineering
“Entia non sunt multiplicanda praeter necessitatem” Occam
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ConclusionsConclusions
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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
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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.
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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.