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Mathematics - ICT - MedicineMathematics - ICT - Medicine
PresentationPresentation
Per Henrik Hogstad
University of Agder
- Mathematics- Statistics- Physics (Main subject: Theoretical Nuclear Physics)- Computer Science
- Programming / Objectorienting- Algorithms and Datastructures- Databases- Digital Image Processing- Supervisor Master Thesis
-Research- Mathematics of Wavelets + Computer Application Wavelets / Medicine- SimReal+ (SimReal + SimVideo)
MathematicsBachelor / MasterMathematicsBachelor / Master
Bachelor - Master
Fourier / WaveletsTransformation TheoryVector Field - Vector CalculusLinear Algebra - Vector SpaceMatrix
Derivation / IntegrationTrigonometryVectorAlgebra
Basic Mathematics
University
ProjectsProjects
External ProjectsMathematics / Physics / Statistics / ICT
Internal ProjectsSimReal+ (SimReal + SimVideo)
SimReal+Interactice Learning Tool in Mathematics / Physics / Statistics / Computer Science Video Lesson / Live Streaming / Video Simulation / Interactive Simulation / Exercise / Applicaction
Video LessonStreamingVideo SimulationInteractive SimulationExerciseApplication
Comments / ResultsComments / Results
- A great supplement, and very good academic contents
- A great way to repeat past lectures
- I wish this could be used in all my courses
- Enables me to get a better understanding of the subject
- Extremely impressive that you have spent all this time to create such a fantastic tool.
Survey Exam - Physics
Comments from the students
SimReal+ - ResearchSimReal+ - Research
Brekke : LICE conferenceLondon International Conferenceon Education
Brekke/Hogstad : JournalNew teaching methodsUsing computer technologyin physics and mathematics
Elira Curri : Master Thesis (UiA, Albania)Ninni Marie Hogstad : Master Thesis (UiA)
MathematicsParabel
SimReal+Calculator - Graph - Function Analysis - Built in Function
SimReal+Calculator - Regression Analysis
Projects - SimReal+Projects - SimReal+
External ProjectsMathematics / Physics / Statistics / ICT
Internal ProjectsSimReal+ (SimReal + SimVideo)
Mathematical ApplicationsMathematical Applications
External ProjectsExternal Projects
SINTEF Unimed Ultrasound in Trondheim
The Norwegian Radiumhospital in Oslo
Sørlandet hospital in Kristiansand / Arendal
Mathematics - Computer Science - Medicine
The 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 Radiumhospital in OsloLinear AcceleratorThe Norwegian Radiumhospital in OsloLinear Accelerator
- Control of the Linear Accelerator- Databases (patient/employee/activity)- Computations of patient positions
PatientDatabase
Computer
Linear Accelerator
Mathematical Transformation of Medical Images Mathematical Transformation of Medical Images
Detector Detector
Metal Detector
Mathematics
Image Transformation Image Transformation
Transformation Home Car RepairHospital
Transformation Computing - Addition
2
d xxfexfFuF u xj )()()( 2
4
+
16
=
20
2
+
8
=
10
2
2
Room 1 Room 2
TransformationTransformation
Transformation Computing - Logarithm
2
d xxfexfFuF u xj )()()( 2
8
*
32
=
256
3
+
5
=
8
xy 2
yx 2lo g
Rom 1 y
Rom 2 x
TransformasjonTransformasjon
yxay ax log
Music - Digital
Ren tone Reell toneDigitalisering TabellFourierTransform Sammensetn av rene toner
Integrasjon Derivasjon
Analog Digital
Transformation TheoryTransformation Theory
TransformasjonTransformasjonf(x)f(x) F(u)F(u)Room 1 Room 2
f(x) = T-1(F(u))f(x) = T-1(F(u))
F(u) = T[f(x)] F(u) = T[f(x)]
Transformation TheoryIntegral TransformationTransformation TheoryIntegral Transformation
f(…)f(…) F(…)F(…)Room 1 Room 2
f(…) = T-1(F(…))f(…) = T-1(F(…))
F(…) = T[f(…)] F(…) = T[f(…)]
...(...)...)]([ dfxfT
Transformation TheoryIntegral TransformationWavelet - Laplace - Fourier
Transformation TheoryIntegral TransformationWavelet - Laplace - Fourier
f(…)f(…) F(…)F(…)
Fourier
jsdttfestfL s t
)())](([
Wavelet
Laplace
d xxfexfFuF u xj )()()( 2
d xxfxfbafWbaW baba )()(),]([),( ,,
N=2
N=10
Fourier TransformationFourier SerieFourier TransformationFourier Serie
1
1
0
])12sin[(12
14
2sin
2cos
2)(
n
nnn
xnn
T
xnb
T
xna
axf
N
n
xnn
xf1
])12sin[(12
14)(
Analog - DigitalFourier TransformationAnalog - DigitalFourier Transformation
WWt
ttc
W
ntWc
W
nftf
n
2,2
)sin()(sin
22sin
2)(
)5cos()2cos(2)sin(2)( ttttf
Transformation Theory
x
T[f(x)]
T-1[F(a,b)]
f(x) F(a,b)
a
b
Wavelet TransformMorlet Wavelet - Non-visible OscillationWavelet TransformMorlet Wavelet - Non-visible Oscillation
][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
WaveletsNew mathematical method with many interesting applications
WaveletsNew mathematical method with many interesting applications
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 CWTDefinition of The Continuous Wavelet Transform CWT
0 , )(, 2 aRbaRLf
The continuous-time wavelet transform (CWT)of f(x) with respect to a wavelet (x):
dxxfxfbafWbaW baba )()(),]([),( ,,
a
bxaxba
2/1, || )(
dadbxbaWaC
xf ba )(),(11
)( ,2
][ fW ),] ([ bafW
)( xf
)( xL2(R ))(0,1 x )(0,2 x )(1,2 x
Wavelet TransformMorlet WaveletFourier / Wavelet
Wavelet TransformMorlet WaveletFourier / Wavelet
Fourier
Wavelet
xex x
2ln
2cos)(
2
F [f]
[f]W ψ
f
1σ
2a 1a yx
The Norwegian RadiumhospitalMammographyMicro-calcificationsWavelet Transformation - Mexican Hat
The Norwegian RadiumhospitalMammographyMicro-calcificationsWavelet Transformation - Mexican Hat
2
2
2σ
x2
2π1 e
σ
x2Ψ(x)
brPbrT
a
T
y
brPbr
2
1
a2π
1b,a
e2)r(Ψx
y
xr
y
x
b
bb
y
x
a
aa
cosθsin θ
sin θcosθR
2
y
2x
a
10
0a
1
A A RRP T
ECG
MammographySeismic
The Norwegian RadiumhospitalMammographyMicro-calcifications
The Norwegian RadiumhospitalMammographyMicro-calcifications
DiameterRelative contrastNumber of microcalcifications
Mathematical tranformationof the mammography imaging
Mammographyphantom Phantom information
Transformed image
Wavelet TransformMammography - VideoWavelet TransformMammography - Video
OsteroporosisOsteroporosis
External part External part
[f]Wa
1ψ2
E/I bone edge E/I bone edge
a
bxaxba 2/1
, || )(
xex x
2ln
2cos)(
2
Morlet
Ultrasound Image - Edge detectionSINTEF – Unimed – Ultrasound - TrondheimUltrasound Image - Edge detectionSINTEF – Unimed – Ultrasound - Trondheim
- Ultrasound Images- Egde Detection
- Noise Removal- Egde Sharpening- Edge Detection
Prosthesis
UltrasoundUltrasound
Doppler-effekt
Lytter L beveger seg
mot / fralyd-kilden S
samtidig somS også beveger seg.
SS
L
S
S
LLL
S
S
S
S
S
S
SSSS
fvv
vv
f
vvvvvv
f
f
vv
f
vv
f
v
f
vTvvT
Venstre
Høyre
SS
LL f
vv
vvf
Svevning
To lyd-kilder med litt avvikende frekvens gir opphav til en
resultant-bølge som inneholder varierende amplitude.
Frekvensen som resultant-amplituden
varierer med kalles svevnings-frekvensen.
212121
12
12
2121
111
)1(
ffTTTT
TT
Tf
TT
TTTTnnTT
svevning
21 fff svevning
Svevningsfrekvensener differensen mellom enkelt-frekvensene
SvevningPiano
To stk piano spiller samme a-tonemed frekvens 440 Hz.
Ved å la det ene pianoet endresin a-tone-frekvens til henholdvis
441 Hz, 442 Hz og 443 Hz,hører vi en svevningstone
med økende frekvens.
Svevningstonens frekvenser lik differensen mellom frekvensene
til de to pianofrekvensene.
MicrocosmosDifferent parts of microcosmos in our daily lifeMicrocosmosDifferent parts of microcosmos in our daily life
ElectricityMobilephone TV
Induction Semiconductor
Medicine
Magnetic resonance New energy
MicrocosmosLHC - Large Hadron ColliderMicrocosmosLHC - Large Hadron Collider
MRIMagnetic Resonance ImagingMRIMagnetic Resonance Imaging
PETEmission of PositronsPETEmission of Positrons
PET radiotracer (18F fluorodeoxyglucose (FDG))+ radioactive fluorine.Natural decay includes emmission of positronsthat reach with electrons (annihilaation) in the bodyproduces energy in the form of a pair of photons.The PET skanner, which is able to detect these photons,creates three-dimensional images that show how the FDG is distributed in the area of the body.
Areas where a large amount of FDG accumulates,called ’hot spots’ because they appear more intensethan surrounding tissue, indicate that a high level of chemical activity or metabolism is occuring there.
Infrared camera - WaveletsInfrared camera - Wavelets
Calculator in MedicineSafety CalculatorCalculator in MedicineSafety Calculator
HeartCirculationHeartCirculation
Surgery SimulatorSurgery Simulator
ENDEND