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

IntroductionIntroduction

Per Henrik Hogstad

Associate Professor

Agder University CollegeFaculty of Engineering and ScienceDept of Computer ScienceGrooseveien 36, N-4876 Grimstad, NorwayTelephone: +47 37253285 Email: Per.Hogstad@hia.no

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IntroductionIntroduction

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

ResearchResearch

SINTEF Unimed Ultrasound in Trondheim

The Norwegian Radiumhospital in Oslo

Sørlandet hospital in Kristiansand / Arendal

Mathematics - Computer Science - Medicine

Mathematical Image OperationMathematical Image Operation - - ApplicationApplication

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 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)

abxaxba

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 wave

f(x) square wave (T=2)

N=2

N=10

1

1

0

])12sin[(12

14

2sin2cos2

)(

n

nnn

xnn

Txnb

Txnaaxf

N

n

xnn

xf1

])12sin[(12

14)(

N=1

Fourier transformation

Fourier transformation

Fourier transformation

Fourier transformation

CWT - Time and frequency localizationCWT - Time and frequency localization

aatata

)()(0,

Time

Frequency

ˆ1)()(

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

t

a=1/2

a=1

a=2

t

Signal

Time Inf

Fourier

Freq Inf

Wavelet

Time InfFreq Inf

Filtering / CompressionFiltering / Compression

)(xf ),]([ bafW

Data compression

Remove low W-values

Lowpass-filteringReplace W-values by 0for low a-values

Highpass-filteringReplace W-values by 0for high a-values

Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet

f

[f]Wψ

F[f]

[f]Wa1

ψ2

b)1,(a [f]Wψ

b)20,(a [f]Wψ

b)10,(a [f]Wψ

Fourier

Wavelet

xex x

2ln2cos)(

2

Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet

Fourier

Wavelet

xex x

2ln2cos)(

2

f

F[f]

[f]Wψ [f]W

a1

ψ2

Wavelet TransformWavelet TransformMorlet Wavelet - Visible OscillationMorlet Wavelet - Visible Oscillation

signal Original

f

[f]Wa1

ψ2

signal Modified f

[f]Wa1

ψ2

xex x

2ln2cos)(

2

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

][fWa1

1ψ2

][fWa1

2ψ2

xex x

2ln2cos)(

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]

xex x

2ln2cos)(

2

][fW 1ψ

Scalogram

][fWa1

1ψ2

(x)f2

][fW 2ψ

Scalogram

][fWa1

2ψ2

(x)f1

Matcad ProgramMatcad ProgramWavelet TransformWavelet Transform

CWTCWT - DWT - DWT

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

dadbxbaWaC

xf ba )(),(11)( ,2

CdC 0

)( 2

abxaxba 2/1

, || )(

CWT

DWT

m

m

anbb

aa

00

0

nxx mmnm 22 )( 2/

,

m

m

nb

a

2

21 2 00 ba

Binary dilationDyadic translation

Dyadic Wavelets

voicea called group, one as processed are of pieces voctaveper voicesofnumber 2

nm,

/10

va v

m

mjkmkj chc ,12, m

mjkmkj cgd ,12,

Analysis /SynthesisAnalysis /SynthesisExampleExample

m

mkmjm

mkmjkj gdhcc 2,2,,1

Mhk

k nk

Mnkkhh 12 k

kh kNk

k 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

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

WWWW

43

210

WWWWW

WWWWWVWWWWV

WWWVWWV

WVV

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 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 RadiumhospitalMammographyMammography

DiameterRelative contrastNumber of microcalcifications

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

2

2

2σx2

2π1 e

σx2Ψ(x)

1σ

cosθsinθsinθcosθ

R

2

y

2x

a10

0a1

A

ARRP T

yx

r

y

x

bb

b

brPbrT

a

T

y

brPbr

2

1

a2π1

b,a e2)r(Ψx

y

x

aa

a

2a 1a yx

The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography

ArthritisArthritisMeasure of boneMeasure of bone

abxaxba 2/1

, || )(

xex x

2ln2cos)(

2

Morlet

External part External part

[f]Wa1

ψ2

E/I bone edge E/I bone edge

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

- Ultrasound Images- Egde Detection

- Noise Removal- Egde Sharpening- Edge Detection

Edge DetectionEdge DetectionConvolutionConvolution

Edge detectionEdge detectionWaveletWavelet

1σ

2

2

2x2

2π1 e

σx2Ψ(x)

Mexican Hat

Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy

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

abxaxba

2/1, || )(

dadbxbaWaC

xf ba )(),(11)( ,2

dbbaWaS ff

2),()(

daaaS

adadbbaW

adbdabaWdxxfE

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 Scale

440,...,2,1

2)( /

Njja Nj

dbbaWaa

aSf

f 2

22 ),(1max)(

max

abxaxba

2/1, || )(

Edge detectionEdge detectionWavelet - Different EdgesWavelet - Different Edges

Noise RemovalThresholding

Hard Soft Semi-Soft

Noise RemovalSyntetic Image 45 Wavelets - 500.000 test

Original

Original + point spread function + white gaussian noise

Noise RemovalSyntetic Image

Noise Removal Ultrasound Image

Original

Semi-soft

Soft

Edge sharpening

Edge detectionEdge detection

Edge detectionEdge detection

Scalogram

Edge detectionEdge detection

Scalogram

Edge detectionEdge detection

End