Wavelets - Fagside fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htm

Hovedprojekt våren 2004Hovedprojekt våren 2004Bruk av Bruk av WaveletsWavelets (en relativt ny matematisk metode) (en relativt ny matematisk metode)innen innen medisinsk bildebehandlingmedisinsk bildebehandling

Hovedprojekt våren 2004Hovedprojekt våren 2004Bruk av Bruk av WaveletsWavelets (en relativt ny matematisk metode) (en relativt ny matematisk metode)innen innen medisinsk bildebehandlingmedisinsk bildebehandling

Wavelets - FagsideWavelets - Fagsidehttp://fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htmhttp://fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htmWavelets - FagsideWavelets - Fagsidehttp://fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htmhttp://fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htm

Prosjekter våren 2004Prosjekter våren 2004Matematisk behandling av medisinsk bilde-informasjonMatematisk behandling av medisinsk bilde-informasjonUbegrenset sett av oppgaver av alle vanskelighetsgrader (fra ingen matematikk til avansert matematikk)Ubegrenset sett av oppgaver av alle vanskelighetsgrader (fra ingen matematikk til avansert matematikk)

Prosjekter våren 2004Prosjekter våren 2004Matematisk behandling av medisinsk bilde-informasjonMatematisk behandling av medisinsk bilde-informasjonUbegrenset sett av oppgaver av alle vanskelighetsgrader (fra ingen matematikk til avansert matematikk)Ubegrenset sett av oppgaver av alle vanskelighetsgrader (fra ingen matematikk til avansert matematikk)

- Bruk av Wavelets til å bestemme blodårekanter fra ultralydbilder.Oppdragsgiver: SINTEF Unimed Ultralyd i Trondheim.

- Bruk av Wavelets til å bestemme brystkreftsvulster på tidlig stadium.Oppdragsgiver: Det Norske Radiumhospital i Oslo (DNR).

- Bruk av Wavelets til å bestemme benmasse i kroppen.Oppdragsgiver: Sørlandet Sykehus i Kristiansand.

- Bruk av Wavelets til å bestemme blodåretykkelse i lever.Oppdragsgiver: Sørlandet Sykehus i Kristiansand.

- Bruk av bl.a. Wavelets innen diagnostikk vha IR.Oppdragsgiver: Sørlandet Sykehus i Arendal.

Tittel - Eksempel på oppgavedefinisjon DNRTittel - Eksempel på oppgavedefinisjon DNR

Internasjonalt samarbeidInternasjonalt samarbeid

Resterende sider (på engelsk) er utarbeidet i forbindelse med en invitasjon jeg fikk til en internasjonal matematikk-konferanse i Baltikum høsten 2003 for å holde foredrag om mitt arbeid med bruk av Wavelets (en relativt ny matematisk metode) innen matematisk behandling av medisinsk bildeinformsjon.

IntroductionIntroductionIntroductionIntroduction

Per Henrik Hogstad

Associate Professor

Agder University CollegeFaculty of Engineering and ScienceDept of Computer ScienceGrooseveien 36, N-4876 Grimstad, NorwayTelephone: +47 37253285 Email: [email protected]

1 3

2

4

IntroductionIntroductionIntroductionIntroduction

Per Henrik Hogstad

- Mathematics- Statistics- Physics (Main subject: Theoretical Nuclear Physics)- Computer Science

- Programming / Objectorienting- Algorithms and Datastructures- Databases- Digital Image Processing- Supervisor Master Thesis

- Research- PHH : Mathem of Wavelets + Computer Application Wavelets/Medicine- Students : Application + Test Wavelets/Medicine

ResearchResearchResearchResearch

SINTEF Unimed Ultrasound in Trondheim

The Norwegian Radiumhospital in Oslo

Sørlandet hospital in Kristiansand / Arendal

Mathematics - Computer Science - Medicine

Mathematical Image OperationMathematical Image Operation - - ApplicationApplicationMathematical Image OperationMathematical Image Operation - - ApplicationApplication

WaveletsWaveletsNew New mathematical methodmathematical method with many interesting with many interesting applicationsapplications

WaveletsWaveletsNew New mathematical methodmathematical method with many interesting with many interesting applicationsapplications

Divide a function into parts with frequency and time/position information

Signal Processing - Image Processing - Astronomy/Optics/Nuclear PhysicsImage/Speech recognition - Seismologi - Diff.equations/Discontinuity…

Definition of The Continuous Wavelet Transform Definition of The Continuous Wavelet Transform CWTCWTDefinition of The Continuous Wavelet Transform Definition of The Continuous Wavelet Transform CWTCWT

dxxfxfbafWbaW baba )()(),]([),( ,,

0 , )(, 2 aRbaRLf

The continuous-time wavelet transform (CWT)of f(x) with respect to a wavelet (x):

][ fW),]([ bafW

)(xf

)(xL2(R)

a

bxaxba

2/1, || )(

dadbxbaWaC

xf ba )(),(11

)( ,2

)(0,1 x )(0,2 x )(1,2 x

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

Fourier transformationFourier transformation




CWT - Time and frequency localizationCWT - Time and frequency localizationCWT - Time and frequency localizationCWT - Time and frequency localization

aatata

)()(0,

Time

Frequency

ˆ

1)()(

0,

aaa

a

Small a: CWT resolve events closely spaced in time.Large a: CWT resolve events closely spaced in frequency.

CWT provides better frequency resolution in the lower end of the frequency spectrum.

Wavelet a natural tool in the analysis of signals in which rapidlyvarying high-frequency components are superimposed on slowly varyinglow-frequency components (seismic signals, music compositions, pictures…).

Fourier - Wavelet Fourier - Wavelet Fourier - Wavelet Fourier - Wavelet

t

a=1/2

a=1

a=2

t

Signal

Time Inf

Fourier

Freq Inf

Wavelet

Time InfFreq Inf

Filtering / CompressionFiltering / CompressionFiltering / CompressionFiltering / Compression

)(xf ),]([ bafW

Data compression

Remove low W-values

Lowpass-filtering

Replace W-values by 0for low a-values

Highpass-filtering

Replace W-values by 0for high a-values

Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet


f

[f]Wψ

F[f]

[f]Wa

1ψ2

b)1,(a [f]Wψ

b)20,(a [f]Wψ

b)10,(a [f]Wψ

Fourier

Wavelet

xex x

2ln

2cos)(

2



Fourier

Wavelet

xex x

2ln

2cos)(

2

f

F[f]

[f]Wψ [f]W

a

1ψ2

Wavelet TransformWavelet TransformMorlet Wavelet - Visible OscillationMorlet Wavelet - Visible OscillationWavelet TransformWavelet TransformMorlet Wavelet - Visible OscillationMorlet Wavelet - Visible Oscillation

signal Original

f

[f]Wa

1ψ2

signal Modified f

[f]Wa

1ψ2

xex x

2ln

2cos)(

2

Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible OscillationMorlet Wavelet - Non-visible Oscillation [1/2] [1/2]Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible OscillationMorlet Wavelet - Non-visible Oscillation [1/2] [1/2]

][fWa

11ψ2

][fWa

12ψ2

xex x

2ln

2cos)(

2

210)0.01(x1 1000e(x)f

9,11 xif x)5sin(2)(

11,,9 xif (x)(x)f

1

12 xf

f

(x)f1

(x)f2

Scalogram

Scalogram

Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible OscillationMorlet Wavelet - Non-visible Oscillation [2/2] [2/2]Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible OscillationMorlet Wavelet - Non-visible Oscillation [2/2] [2/2]

xex x

2ln

2cos)(

2

][fW 1ψ

Scalogram

][fWa

11ψ2

(x)f2

][fW 2ψ

Scalogram

][fWa

12ψ2

(x)f1

Matcad ProgramMatcad ProgramWavelet TransformWavelet TransformMatcad ProgramMatcad ProgramWavelet TransformWavelet Transform

CWTCWT - DWT - DWTCWTCWT - DWT - DWT


dadbxbaWaC

xf ba )(),(11

)( ,2

CdC 0

)(2

a

bxaxba 2/1

, || )(

CWT

DWT

m

m

anbb

aa

00

0

nxx mmnm 22 )( 2/

,

m

m

nb

a

2

2

1 2 00 ba

Binary dilationDyadic translation

Dyadic Wavelets

voicea called group, one as processed are of pieces v

octaveper voicesofnumber 2

nm,

/10

va v

m

mjkmkj chc ,12, m

mjkmkj cgd ,12,

Analysis /SynthesisAnalysis /SynthesisExampleExample Analysis /SynthesisAnalysis /SynthesisExampleExample

m

mkmjm

mkmjkj gdhcc 2,2,,1

Mhk

k nk

Mnkkhh 12

kkh kN

kk hg 1)1(

J=5J=5Num of Samples: 2Num of Samples: 2JJ = 32 = 32

1 12

0,,

12

0,,

12

0,,

0

10

00)()(

)()()(

J

jj kkjkj

kkjkj

kkJkJJ

jj

J

tdtc

tctftf

AnalysisAnalysisSynthesisSynthesisJ=5 J=5

Sampling: 2Sampling: 255 = 32 = 32

AnalysisAnalysisSynthesisSynthesisJ=5 J=5

Sampling: 2Sampling: 255 = 32 = 32

j=4j=4j=5j=5 j=3j=3 j=2j=2 j=1j=1 j=0j=05V

4V 3V 2V 1V 0V

0W4W 3W 2W 1W

4W 43 WW 432 WWW 43

21

WW

WW

43

210

WW

WWW

WWWWWV

WWWWV

WWWV

WWV

WV

V

32100

3211

322

33

44

5

1 12

0,,

12

0,,

12

0,,

0

10

00)()(

)()()(

J

jj kkjkj

kkjkj

kkJkJJ

jj

J

tdtc

tctftf

Discrete Wavelet-transformation

Compress 1:50

JPEG Wavelet

Original

ResearchResearchThe Norwegian Radiumhospital in OsloThe Norwegian Radiumhospital in OsloResearchResearchThe Norwegian Radiumhospital in OsloThe Norwegian Radiumhospital in Oslo

- Control of the Linear Accelerator- Databases (patient/employee/activity)- Computations of patientpositions- Mathematical computations

of medical image information- Different imageformat (bmp, dicom, …)- Noise Removal - Graylevel manipulation (Histogram, …)- Convolution, Gradientcomputation- Multilayer images- Transformations (Fourier, Wavelet, …)- Mammography- ...

Wavelet

The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammographyThe Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography

DiameterRelative contrastNumber of microcalcifications

The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammograpMammographhy - Mexican Hat - 2 Dimy - Mexican Hat - 2 DimThe Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammograpMammographhy - Mexican Hat - 2 Dimy - Mexican Hat - 2 Dim

2

2

2σ

x2

2π1 e

σ

x2Ψ(x)

1σ

cosθsinθ

sinθcosθR

2

y

2x

a

10

0a

1

A

ARRP T

y

xr

y

x

b

bb

brPbrT

a

T

y

brPbr

2

1

a2π

1b,a

e2)r(Ψx

y

x

a

aa

2a 1a yx

The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammographyThe Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography

ArthritisArthritisMeasure of boneMeasure of boneArthritisArthritisMeasure of boneMeasure of bone

a

bxaxba 2/1

, || )(

xex x

2ln

2cos)(

2

Morlet

External part External part

[f]Wa

1ψ2

E/I bone edge E/I bone edge

Ultrasound Image - Edge detectionUltrasound Image - Edge detectionSINTEF – Unimed – Ultrasound - TrondheimSINTEF – Unimed – Ultrasound - TrondheimUltrasound Image - Edge detectionUltrasound Image - Edge detectionSINTEF – Unimed – Ultrasound - TrondheimSINTEF – Unimed – Ultrasound - Trondheim

- Ultrasound Images- Egde Detection

- Noise Removal- Egde Sharpening- Edge Detection

Edge DetectionEdge DetectionConvolutionConvolutionEdge DetectionEdge DetectionConvolutionConvolution

Edge detectionEdge detectionWaveletWaveletEdge detectionEdge detectionWaveletWavelet

1σ

2

2

2

x2

2π1 e

σ

x2Ψ(x)

Mexican Hat

Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy

Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy


a

bxaxba

2/1, || )(

dadbxbaWaC

xf ba )(),(11

)( ,2

dbbaWaS ff

2),()(

daa

aS

a

dadbbaW

a

dbdabaWdxxfE

f

f

ff

2

2

2

2

22

)(

),(

),()(

WaveletTransform

Inv WaveletTransform

Wavelet scaledependentspectrum

Energy of the signal

A measure of the distribution of energy of the signal f(x) as a function of scale.

Edge detectionEdge detectionWavelet - Max Energy ScaleWavelet - Max Energy ScaleEdge detectionEdge detectionWavelet - Max Energy ScaleWavelet - Max Energy Scale

4

40,...,2,1

2)( /

N

j

ja Nj

dbbaWaa

aSf

f 2

22),(

1max

)(max

a

bxaxba

2/1, || )(

Edge detectionEdge detectionWavelet - Different EdgesWavelet - Different EdgesEdge detectionEdge detectionWavelet - Different EdgesWavelet - Different Edges

Noise RemovalThresholdingNoise RemovalThresholding

Hard Soft Semi-Soft

Noise RemovalSyntetic Image 45 Wavelets - 500.000 test

Noise RemovalSyntetic Image 45 Wavelets - 500.000 test

Original

Original + point spread function + white gaussian noise

Noise RemovalSyntetic ImageNoise RemovalSyntetic Image

Noise Removal Ultrasound ImageNoise Removal Ultrasound Image

Original

Semi-soft

Soft

Edge sharpeningEdge sharpening

Edge detectionEdge detectionEdge detectionEdge detection


Scalogram


Scalogram


EndEnd

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Wavelets - Fagside fag.grm.hia.no/fagstoff/perhh/htm/fag/matem/datwwww/wavelet.htm