11

Chapter 01Chapter 01Introduction to WaveletsIntroduction to WaveletsChapter 01Chapter 01Introduction to WaveletsIntroduction to Wavelets

22

Wavelets is a relative new mathematical methodwith many interesting applications.

Wavelets = New mathematical methodWavelets = New mathematical methodWavelets = New mathematical methodWavelets = New mathematical method

33

FunctionFunctionTransformedTransformedFunctionFunction

We want a suitable representation of a function

- Mathematical operation of a function- Draw new information from a function

Mathematical operation - New informationMathematical operation - New informationMathematical operation - New informationMathematical operation - New information

44

Wavelets = Small Waves

Wavelets = Small WavesWavelets = Small WavesWavelets = Small WavesWavelets = Small Waves

55

Wavelets are building blocksthat can quickly decorrelate data.

At the present day it is almost impossible to give a precise definition of wavelets. The research field is growing so fast and novel contributionsare made at such a rate that even if one manages to give a definition today,it might be obsolute tomorrow.

One, very vague, way of thinking about wavelets could be:

Wavelets = Building blocksWavelets = Building blocksWavelets = Building blocksWavelets = Building blocks

•• Wavelets are Wavelets are building blocksbuilding blocks for general functions. for general functions.• • Wavelets have Wavelets have space-frequency localizationspace-frequency localization..• • Wavelets have Wavelets have fast transform algorithmsfast transform algorithms..

66

•• Wavelets are mathematical functionsthat can cut up data into different frequency components, and then study each componentwith a resolution matched to its scale.

• • Wavelets have advantages over traditionalFourier methods in analyzing physical situation where the signal is transientor contains discontinuities and sharp spikes.

Frequency / Transient signals / DiscontinuityFrequency / Transient signals / DiscontinuityFrequency / Transient signals / DiscontinuityFrequency / Transient signals / Discontinuity

Adopting a whole Adopting a whole new mindsetnew mindset or perspective in prosessing data or perspective in prosessing data

DataData

77

Wavelets - Different scalesWavelets - Different scalesWavelets - Different scalesWavelets - Different scales

88

•• Wavelet transform has been perhaps the most exciting development in the last decade to bring together researchers in several different fields:

Seismic GeologySignal processing (frequency study, compression, …)Image processing (image compression, video compression, ...)Denoising dataCommunicationsComputer scienceComputer scienceMathematicsMathematicsElectrical EngineeringQuantum PhysicsMagnetic resonanceMusical tonesDiagnostic of cancerEconomics…

Interesting applicationsInteresting applicationsThe subject of Wavelets is expanding at a tremendous rateThe subject of Wavelets is expanding at a tremendous rateInteresting applicationsInteresting applicationsThe subject of Wavelets is expanding at a tremendous rateThe subject of Wavelets is expanding at a tremendous rate

99

•• Before 1930: The main branch of mathematics leading to wavelets began withJoseph Fourier (1807) with his theories of frequency analysis.

•• 1930: Several groups working independently researced the representationof functions using scale-varying basis functions.Physicists Paul Levy was studying small complicated detailsin Brownian motion using Haar basis function.Paley and Stein discovered a scale-varying function that conservethe energy of the function. This function was used by David Marrin numerical image processing in early 1980.

•• 1980- : S. Mallat discovered som relationships between quadrature mirror filters,pyramid algorithms, and orthonormal wavelet bases.Y. Meyer constructed the first non-trivial wavelets.Meyer wavelets are continuously differentiable, but do not have compact

support.I. Daubechies constructed orthonormal wavelet basis funcionsthat has become the comberstone of wavelet applications today.

•• 1995: A new philosophy in biorthogonal Wavelet construction: The Lifting Scheme.

HistoryHistoryHistoryHistory

1010

•• New technology - Rediscovered by I. Daubechies in 1987.

•• Signal analysis - Weighted sum of basis functions.

•• Infinitely many possible sets of wavelets.

•• Wavelet-coefficients contain information about the signal.

•• Basis functions containing information about both the time and frequency.

(Heisenberg inequality: Resolution in time and frequency cannotboth be made arbitrarily small.)

PropertiesPropertiesPropertiesProperties

1111

•• Wavelab at Standford University: Matlab library.

•• Wavelet Workbench from Research Systems, Inc.

•• Liftpack from Gabriel Fernandez, Senthil Periaswamy, and Wim Sweldens: C-routins.

•• Mathematica Lifting Notebook by Paul Abbott.

•• …..

SoftwareSoftwareSoftwareSoftware

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Analysis - SynthesisAnalysis - SynthesisAnalysis - SynthesisAnalysis - Synthesis

AnalysisAnalysis

SynthesisSynthesis

1313

ComponentsComponentsBreadBreadComponentsComponentsBreadBread

Brød = 1 kg Hvetemel+ 1/2 kg Grovt mel+ 1 1/2 ts Salt+ 50 g Gjær+ 100 g Margarin+ 1 1/2 l Vann/Melk

Koeffisienter

Basisfunksjoner

AnalysisSynthesis

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ComponentsComponentsBloodBloodComponentsComponentsBloodBlood

Blod = 0.45 % Blodlegemer= +0.55 % Blodplasma

BlodPlasma = 7 %Proteiner

+0.9 % Salter+0.1 % Glukose+…

Koeffisienter

Basisfunksjoner

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Components - Components / PositionsComponents - Components / PositionsComponents - Components / PositionsComponents - Components / Positions

Interested in components,but not in the positions.

Interested in components,and in the positions.

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Frequency - Frequency / TimeFrequency - Frequency / TimeMusicMusicFrequency - Frequency / TimeFrequency - Frequency / TimeMusicMusic

Tools for analysis / synthesis:Tools for analysis / synthesis:-- Fourier transformationFourier transformation (frequence)(frequence)-- Wavelet transformationWavelet transformation (frequence / time)(frequence / time)-- ……

AnalysisAnalysis SynthesisSynthesis

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Components / PostitionsComponents / PostitionsFourier / WaveletsFourier / WaveletsComponents / PostitionsComponents / PostitionsFourier / WaveletsFourier / Wavelets

Fourier

Components = Freqyency

Wavelets

Components = Freqyency

Positions = Place or Time

1818

Potential of Wavelet AnalysisPotential of Wavelet AnalysisPotential of Wavelet AnalysisPotential of Wavelet Analysis

Engineers, physicists, astronomers, geologists, medical researchers, and others have begun exploring the extraordinary array of potential applicationsof wavelet analysis, ranging from signal and image processing to data analysis.

Wavelet analysis, in contrast to Fourier analysis, uses approximating functions that are localized in both time and frequency space.

1919

Seismic traceSeismic traceSeismic traceSeismic trace

2020

Fingerprints

Without wavelet technology, digitizing the FBI's constantly growing database of over 200 million fingerprint records (originally stored as inked impressions on paper cards) would have required an unmanageable 2,000 terabytes (1 Tb = 1000 Mb) of storage and filled over a billion 3.5-inch high-density floppy disks. Faced with this digital storage dilemma, the FBI researched a variety of image compression techniques before finally settling on one robust enough to preserve vital fine-scale fingerprint image details--a breakthrough wavelet-based image coding algorithm developed in cooperation with Los Alamos National Laboratory researchers answered the call.

Original

Reconstructed from 26:1 compression

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FingerprintOriginal - JPEG - Wavelet

Original

JPEG

Wavelet

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FingerprintOriginal

2323

FingerprintJPEG

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FingerprintWavelet

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Compression

JPEG WaveletOriginal

4 kb1.7 Mb 4 kb

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Wavelet transformation

•• From a signal processing standpoint, one may view an image as a signal that has

•• high-frequency (high-spatial detail) and•• low-frequency (smooth) components.

The algorithm filters the signal and then iterates the process.

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Compression 1:50

JPEG Wavelet

Originalt

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Wavelets and TelemedicineWavelets and TelemedicineWavelets and TelemedicineWavelets and Telemedicine

•• Massachusetts General Hospital:No clinically significant image degradation was identifiedin radiologi images up to 30:1.

•• Wavelet-based compression technology is superior toall other compression technologies(keep details, high compression ratio).

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Denoising Noisy Data

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Sea Surface Temperature

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CommunicationCompression

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WavesWavesConstruction of boatsConstruction of boatsWavesWavesConstruction of boatsConstruction of boats

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Medical imageMedical imageUltrasound / ECGUltrasound / ECGMedical imageMedical imageUltrasound / ECGUltrasound / ECG

ECG

Ultrasound

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Medical imageMedical imageThresholding - SegmentationThresholding - SegmentationMedical imageMedical imageThresholding - SegmentationThresholding - Segmentation

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Medical imageMedical imageUltrasound - Operation in the brainUltrasound - Operation in the brainMedical imageMedical imageUltrasound - Operation in the brainUltrasound - Operation in the brain

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DNRDNR

Bildebhandling

Lineærakselerator

3737

Stråleterapi - PasientposisjonStråleterapi - PasientposisjonStråleterapi - PasientposisjonStråleterapi - Pasientposisjon

Referansebilde Kontrollbilde

3838

Bildebehandling - HistogramBildebehandling - Histogram

3939

Bildebehandling - GråtoneskalaerBildebehandling - Gråtoneskalaer

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Bildebehandling - ConvolutionBildebehandling - Convolution

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Bildebehandling - Fourier transformasjon IBildebehandling - Fourier transformasjon I

4242

Bildebehandling - Fourier transformasjon IIBildebehandling - Fourier transformasjon II

4343

BilderepresentasjonBilderepresentasjonBilderepresentasjonBilderepresentasjon

Pixel

Bilderepresentasjonvha pixel-verdieri intervallet [0,255]

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Fourier-transformation of a square waveFourier-transformation of a square waveFourier-transformation of a square waveFourier-transformation of a square wave

f(x) square wave (T=2)

N=2

N=10

1

1

0

])12sin[(12

14

2sin

2cos

2)(

n

nnn

xnn

T

xnb

T

xna

axf

N

n

xnn

xf1

])12sin[(12

14)(

N=1

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FrequenceFrequenceFrequenceFrequence

Sinuswave with frequence f1 = 1

Sinuswave with frequence f2 = 2

f1 < f2

4646

Signals and FTSignals and FTSignals and FTSignals and FT

)sin( 11 ty

)sin( 22 ty

)sin()sin( 213 tty

FT

FT

FT

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Stationary / Non-stationary signalsStationary / Non-stationary signalsStationary / Non-stationary signalsStationary / Non-stationary signals

60 hvis )sin(

60 hvis )sin(

2

14 tt

tty

)sin()sin( 213 tty

FT

FT

Stationary

Non stationary

The stationary and the non-stationary signal both have the same FT.FT is not suitable to take care of non-stationary signals to give information about time.

4848

WaveletsWaveletsLocalization both in frequency and timeLocalization both in frequency and timeWaveletsWaveletsLocalization both in frequency and timeLocalization both in frequency and time

WT is suitable to take care of non-stationary signals to give information about time.

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Dissimilarities of Fourier and Wavelet TransformsDissimilarities of Fourier and Wavelet TransformsDissimilarities of Fourier and Wavelet TransformsDissimilarities of Fourier and Wavelet Transforms

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EndEnd