WaveletsWavelets Medical Image ProcessingMedical Image ProcessingProjectsProjects

WaveletsWavelets Medical Image ProcessingMedical Image ProcessingProjectsProjects

Per HenrikPer HenrikHogstadHogstadPer HenrikPer HenrikHogstadHogstad

- 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

MathematicsMathematicsComputer ScienceComputer ScienceMedicineMedicine

MathematicsMathematicsComputer ScienceComputer ScienceMedicineMedicine

SINTEF Unimed Ultrasound in Trondheim- Detection of blood vessel in ultra sound image

The Norwegian Radiumhospital in Oslo- Linear accelerator- Computing patient position- Databases- Image processing ( Wavelets)

Sørlandet Hospital in Kristiansand- Bone thickness- Blood vessel thickness in liver

Sørlandet Hospital in Arendal- IR diagnostic

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

DNRDNR

Linear Accelerator

DatabasesPatient Position

Image processing

RadiatRadiationion Theraphy - Patient Position Theraphy - Patient PositionRadiatRadiationion Theraphy - Patient Position Theraphy - Patient Position

Referance picture Control picture

Digital Image ProcessingDigital Image ProcessingDigital Image ProcessingDigital Image Processing

Digital Image Processing

Manipulation of images by computer

Input Image Computer Output Image

Digital Image

Processing

Image TranformationImage Tranformation

Original Image

Tranformed Image

HistogramHistogram

Histogram EqualizationHistogram Equalization

ConvolutionConvolution

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

dxexfuf uxj 2)()(ˆ

dueufxf uxj 2)(ˆ)(

Image TransformationFourier Transformation (2-dim)Image TransformationFourier Transformation (2-dim)

dxxfexfFxfeuFuf uxjuxj )()()()()(ˆ 22

dydxeyxfvuf vyuxj )(2),(),(ˆ

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…

•• 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

Introduction to WaveletsIntroduction to WaveletsIntroduction to WaveletsIntroduction to Wavelets

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..

•• 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

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

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…).

Image TranformationDetailsImage TranformationDetails

Original Image

Details Image

The restof the Image

Discrete Wavelet-transformation

Compress 1:50

JPEG Wavelet

Original

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

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

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

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

Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet

Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet

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

WaveletsBasic KnowledgeWaveletsBasic Knowledge

- Informatics- Programming / Object oriented (Java / C++)

- Mathematics- Lineær algebra (Vektor Space / Basis Functions / Matrices / … )- Fourier Analysis

- Statistics

- Physics

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

Wavelet Transform

Wavelet TransformTumour

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

Krsand

ThicknessThicknessof blood vesselof blood vesselin liverin liver

ThicknessThicknessof blood vesselof blood vesselin liverin liver

a

bxaxba 2/1

, || )(

xex x

2ln

2cos)(

2

Morlet

Krsand

ThicknessThicknessThicknessThickness

Mexican HatKrsand

ArendalArendalDiagnostics - IR radiationDiagnostics - IR radiationArendalArendalDiagnostics - IR radiationDiagnostics - IR radiation

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 Computation

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

Aorta with Prothesis

Ultra Sound Image

Edge DetectionEdge DetectionConvolutionConvolutionEdge DetectionEdge DetectionConvolutionConvolution

Edge detectionEdge detectionWaveletWaveletEdge detectionEdge detectionWaveletWavelet

1σ

2

2

2

x2

2π1 e

σ

x2Ψ(x)

Mexican Hat

Edge detectionEdge detectionEdge detectionEdge detection

Scalogram

Edge detectionEdge detectionOne beamOne beamEdge detectionEdge detectionOne beamOne beam

Edge detectionEdge detectionEdge detectionEdge detection

Scalogram

Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy

Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy

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

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

Methods for prepreparing of Imagein front of Wavelet transformMethods for prepreparing of Imagein front of Wavelet transform

1 Noise removalHardSoftSemi-soft

2 Egde sharpening

3 Different Wavelets

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

Different WavletsDifferent WavletsDifferent WavletsDifferent Wavlets

Edge ComputingEdge ComputingEdge ComputingEdge Computing

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

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

EndEnd