bruk av wavelets (en relativt ny matematisk metode) innen medisinsk bildebehandling
DESCRIPTION
13 2 4TRANSCRIPT
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Bruk av Bruk av WaveletsWavelets (en relativt ny matematisk metode)(en relativt ny matematisk metode)innen innen medisinsk bildebehandlingmedisinsk bildebehandling
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IntroductionIntroduction
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]
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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
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ResearchResearch
SINTEF Unimed Ultrasound in Trondheim
The Norwegian Radiumhospital in Oslo
Sørlandet hospital in Kristiansand / Arendal
Mathematics - Computer Science - Medicine
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Mathematical Image OperationMathematical Image Operation - - ApplicationApplication
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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…
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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
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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
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Fourier transformation
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Fourier transformation
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Fourier transformation
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Fourier transformation
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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…).
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Fourier - Wavelet Fourier - Wavelet
t
a=1/2
a=1
a=2
t
Signal
Time Inf
Fourier
Freq Inf
Wavelet
Time InfFreq Inf
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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
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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
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Wavelet TransformWavelet TransformMorlet WaveletMorlet WaveletFourier/WaveletFourier/Wavelet
Fourier
Wavelet
xex x
2ln2cos)(
2
f
F[f]
[f]Wψ [f]W
a1
ψ2
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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
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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
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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
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Matcad ProgramMatcad ProgramWavelet TransformWavelet Transform
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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
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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
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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
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Discrete Wavelet-transformation
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Compress 1:50
JPEG Wavelet
Original
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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
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The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography
DiameterRelative contrastNumber of microcalcifications
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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
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The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography
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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
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Ultrasound Image - Edge detectionUltrasound Image - Edge detectionSINTEF – Unimed – Ultrasound - TrondheimSINTEF – Unimed – Ultrasound - Trondheim
- Ultrasound Images- Egde Detection
- Noise Removal- Egde Sharpening- Edge Detection
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Edge DetectionEdge DetectionConvolutionConvolution
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Edge detectionEdge detectionWaveletWavelet
1σ
2
2
2x2
2π1 e
σx2Ψ(x)
Mexican Hat
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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.
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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, || )(
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Edge detectionEdge detectionWavelet - Different EdgesWavelet - Different Edges
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Noise RemovalThresholding
Hard Soft Semi-Soft
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Noise RemovalSyntetic Image 45 Wavelets - 500.000 test
Original
Original + point spread function + white gaussian noise
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Noise RemovalSyntetic Image
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Noise Removal Ultrasound Image
Original
Semi-soft
Soft
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Edge sharpening
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Edge detectionEdge detection
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Edge detectionEdge detection
Scalogram
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Edge detectionEdge detection
Scalogram
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Edge detectionEdge detection
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End