wavelets medical image processing projects. per henrik hogstad -mathematics -statistics...
TRANSCRIPT
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WaveletsWavelets Medical Image ProcessingMedical Image ProcessingProjectsProjects
WaveletsWavelets Medical Image ProcessingMedical Image ProcessingProjectsProjects
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Per HenrikPer HenrikHogstadHogstadPer HenrikPer HenrikHogstadHogstad
- 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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ResearchResearchResearchResearch
SINTEF Unimed Ultrasound in Trondheim
The Norwegian Radiumhospital in Oslo
Sørlandet Hospital in Kristiansand / Arendal
Mathematics - Computer Science - Medicine
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MathematicsMathematicsComputer ScienceComputer ScienceMedicineMedicine
MathematicsMathematicsComputer ScienceComputer ScienceMedicineMedicine
SINTEF Unimed Ultrasound in Trondheim- Detection of blood vessel in ultra sound image
The Norwegian Radiumhospital in Oslo- Linear accelerator- Computing patient position- Databases- Image processing ( Wavelets)
Sørlandet Hospital in Kristiansand- Bone thickness- Blood vessel thickness in liver
Sørlandet Hospital in Arendal- IR diagnostic
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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
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DNRDNR
Linear Accelerator
DatabasesPatient Position
Image processing
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RadiatRadiationion Theraphy - Patient Position Theraphy - Patient PositionRadiatRadiationion Theraphy - Patient Position Theraphy - Patient Position
Referance picture Control picture
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Digital Image ProcessingDigital Image ProcessingDigital Image ProcessingDigital Image Processing
Digital Image Processing
Manipulation of images by computer
Input Image Computer Output Image
Digital Image
Processing
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Image TranformationImage Tranformation
Original Image
Tranformed Image
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HistogramHistogram
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Histogram EqualizationHistogram Equalization
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ConvolutionConvolution
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Fourier-transformationFourier-transformation of a square wave of a square waveFourier-transformationFourier-transformation of a square wave 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
dxexfuf uxj 2)()(ˆ
dueufxf uxj 2)(ˆ)(
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Image TransformationFourier Transformation (2-dim)Image TransformationFourier Transformation (2-dim)
dxxfexfFxfeuFuf uxjuxj )()()()()(ˆ 22
dydxeyxfvuf vyuxj )(2),(),(ˆ
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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…
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•• Wavelet transform has been perhaps the most exciting development in the last decade to bring together researchers in several different fields:
Seismic GeologySignal processing (frequency study, compression, …)Image processing (image compression, video compression, ...)Denoising dataCommunicationsComputer scienceComputer scienceMathematicsMathematicsElectrical EngineeringQuantum PhysicsMagnetic resonanceMusical tonesDiagnostic of cancerEconomics…
Interesting applicationsInteresting applicationsThe subject of Wavelets is expanding at a tremendous rateThe subject of Wavelets is expanding at a tremendous rateInteresting applicationsInteresting applicationsThe subject of Wavelets is expanding at a tremendous rateThe subject of Wavelets is expanding at a tremendous rate
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Introduction to WaveletsIntroduction to WaveletsIntroduction to WaveletsIntroduction to Wavelets
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Wavelets are building blocksthat can quickly decorrelate data.
At the present day it is almost impossible to give a precise definition of wavelets. The research field is growing so fast and novel contributionsare made at such a rate that even if one manages to give a definition today,it might be obsolute tomorrow.
One, very vague, way of thinking about wavelets could be:
Wavelets = Building blocksWavelets = Building blocksWavelets = Building blocksWavelets = Building blocks
•• Wavelets are Wavelets are building blocksbuilding blocks for general functions. for general functions.• • Wavelets have Wavelets have space-frequency localizationspace-frequency localization..• • Wavelets have Wavelets have fast transform algorithmsfast transform algorithms..
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•• Wavelets are mathematical functionsthat can cut up data into different frequency components, and then study each componentwith a resolution matched to its scale.
• • Wavelets have advantages over traditionalFourier methods in analyzing physical situation where the signal is transientor contains discontinuities and sharp spikes.
Frequency / Transient signals / DiscontinuityFrequency / Transient signals / DiscontinuityFrequency / Transient signals / DiscontinuityFrequency / Transient signals / Discontinuity
Adopting a whole Adopting a whole new mindsetnew mindset or perspective in prosessing data or perspective in prosessing data
DataData
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Wavelets - Different scalesWavelets - Different scalesWavelets - Different scalesWavelets - Different scales
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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…).
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Image TranformationDetailsImage TranformationDetails
Original Image
Details Image
The restof the Image
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Discrete Wavelet-transformation
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Compress 1:50
JPEG Wavelet
Original
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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
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tdtc
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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
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WW
WW
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WW
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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
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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
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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
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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
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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
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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
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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]Wavelet TransformWavelet TransformMorlet Wavelet - Non-visible OscillationMorlet Wavelet - Non-visible Oscillation [2/2] [2/2]
xex x
2ln
2cos)(
2
][fW 1ψ
Scalogram
][fWa
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(x)f2
][fW 2ψ
Scalogram
][fWa
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(x)f1
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WaveletsBasic KnowledgeWaveletsBasic Knowledge
- Informatics- Programming / Object oriented (Java / C++)
- Mathematics- Lineær algebra (Vektor Space / Basis Functions / Matrices / … )- Fourier Analysis
- Statistics
- Physics
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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
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Wavelet Transform
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Wavelet TransformTumour
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The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammographyThe Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography
DiameterRelative contrastNumber of microcalcifications
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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
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1σ
cosθsinθ
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y
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y
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y
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a
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2a 1a yx
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The Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammographyThe Norwegian RadiumhospitalThe Norwegian RadiumhospitalMammographyMammography
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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
Krsand
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ThicknessThicknessof blood vesselof blood vesselin liverin liver
ThicknessThicknessof blood vesselof blood vesselin liverin liver
a
bxaxba 2/1
, || )(
xex x
2ln
2cos)(
2
Morlet
Krsand
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ThicknessThicknessThicknessThickness
Mexican HatKrsand
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ArendalArendalDiagnostics - IR radiationDiagnostics - IR radiationArendalArendalDiagnostics - IR radiationDiagnostics - IR radiation
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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 Computation
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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
Aorta with Prothesis
Ultra Sound Image
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Edge DetectionEdge DetectionConvolutionConvolutionEdge DetectionEdge DetectionConvolutionConvolution
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Edge detectionEdge detectionWaveletWaveletEdge detectionEdge detectionWaveletWavelet
1σ
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Edge detectionEdge detectionEdge detectionEdge detection
Scalogram
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Edge detectionEdge detectionOne beamOne beamEdge detectionEdge detectionOne beamOne beam
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Edge detectionEdge detectionEdge detectionEdge detection
Scalogram
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Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy
Edge DetectionEdge DetectionWavelet -Wavelet - Scale EnergyScale Energy
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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 ScaleEdge detectionEdge detectionWavelet - Max Energy ScaleWavelet - Max Energy Scale
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Edge detectionEdge detectionWavelet - Different EdgesWavelet - Different EdgesEdge detectionEdge detectionWavelet - Different EdgesWavelet - Different Edges
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Methods for prepreparing of Imagein front of Wavelet transformMethods for prepreparing of Imagein front of Wavelet transform
1 Noise removalHardSoftSemi-soft
2 Egde sharpening
3 Different Wavelets
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Noise RemovalThresholdingNoise RemovalThresholding
Hard Soft Semi-Soft
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Noise RemovalSyntetic Image 45 Wavelets - 500.000 test
Noise RemovalSyntetic Image 45 Wavelets - 500.000 test
Original
Original + point spread function + white gaussian noise
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Noise RemovalSyntetic ImageNoise RemovalSyntetic Image
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Noise Removal Ultrasound ImageNoise Removal Ultrasound Image
Original
Semi-soft
Soft
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Edge sharpeningEdge sharpening
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Different WavletsDifferent WavletsDifferent WavletsDifferent Wavlets
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Edge ComputingEdge ComputingEdge ComputingEdge Computing
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Mathematical Image OperationMathematical Image Operation - - ApplicationApplicationMathematical Image OperationMathematical Image Operation - - ApplicationApplication
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Mathematical Image OperationMathematical Image Operation - - ApplicationApplicationMathematical Image OperationMathematical Image Operation - - ApplicationApplication
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EndEnd