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.
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]
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
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
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
Matcad ProgramMatcad ProgramWavelet TransformWavelet TransformMatcad ProgramMatcad ProgramWavelet TransformWavelet Transform
CWTCWT - DWT - DWTCWTCWT - DWT - DWT
dxxfxfbafWbaW baba )()(),]([),( ,,
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
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
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
Noise RemovalSyntetic Image 45 Wavelets - 500.000 test
Noise RemovalSyntetic Image 45 Wavelets - 500.000 test
Original
Original + point spread function + white gaussian noise