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: [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
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
