introductionintroduction per henrik hogstad associate professor agder university college faculty of...
TRANSCRIPT
IntroductionIntroductionIntroductionIntroduction
Per Henrik Hogstad
Associate Professor
Agder University CollegeFaculty of Engeneering and ScienceDept of Computer ScienceGrooseveien 36, N-4876 Grimstad, NorwayTelephone: +47 37253285 Email: [email protected]
1 3
2
4
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
Fourier transformationFourier transformation
Fourier transformationFourier transformation
Fourier transformationFourier transformation
Fourier transformationFourier transformation
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…).
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
Discrete Wavelet-transformation
Compress 1:50
JPEG Wavelet
Original
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 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
Edge detectionEdge detectionEdge detectionEdge detection
Edge detectionEdge detectionEdge detectionEdge detection
Scalogram
Edge detectionEdge detectionEdge detectionEdge detection
Scalogram
Edge detectionEdge detectionEdge detectionEdge detection
EndEnd