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

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