introductionintroduction per henrik hogstad associate professor agder university college faculty of...

IntroductionIntroductionIntroductionIntroduction

Per Henrik Hogstad

Associate Professor

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

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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

taatata

)()(0,

Time

Frequency

ta

aaa

1

)()(0,

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

EndEnd

introductionintroduction per henrik hogstad associate professor agder university college faculty of...

Documents

fourier wavelet

continuous wavelet

wavelet x

cwtthe continuoustime

frequency spectrum

better frequency resolution

cwt resolve events

analysis of signals