Mathematics - ICT - Medicine 1

Per Henrik Hogstad Presentation 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)

Mathematics Bachelor / Master Fourier / Wavelets Transformation Theory Vector Field - Vector Calculus Linear Algebra - Vector Space Matrix Derivation / Integration Trigonometry Vector Algebra Basic Mathematics University 3

Projects External Projects Internal Projects Mathematics / Physics / Statistics / ICT Internal Projects SimReal+ (SimReal + SimVideo)

SimReal+ Interactice Learning Tool in Mathematics / Physics / Statistics / Computer Science Video Lesson / Live Streaming / Video Simulation / Interactive Simulation / Exercise / Applicaction Video Lesson Streaming Video Simulation Interactive Simulation Exercise Application Studier av svingninger (spesielt resonans) for å hindre at f.eks. bruer kollapser under påvirkning av vindkast. 5

Comments / Results Survey Exam - Physics Comments from the students - 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.

SimReal+ - Research Brekke : LICE conference London International Conference on Education Brekke/Hogstad : Journal New teaching methods Using computer technology in physics and mathematics Elira Curri : Master Thesis (UiA, Albania) Ninni Marie Hogstad : Master Thesis (UiA)

Mathematics Parabel Studier av svingninger (spesielt resonans) for å hindre at f.eks. bruer kollapser under påvirkning av vindkast. 8

SimReal+ Calculator - Graph - Function Analysis - Built in Function Studier av svingninger (spesielt resonans) for å hindre at f.eks. bruer kollapser under påvirkning av vindkast.

SimReal+ Calculator - Regression Analysis Studier av svingninger (spesielt resonans) for å hindre at f.eks. bruer kollapser under påvirkning av vindkast.

Projects - SimReal+ External Projects Internal Projects Mathematics / Physics / Statistics / ICT Internal Projects SimReal+ (SimReal + SimVideo)

Mathematical Applications 12

Mathematics - Computer Science - Medicine External Projects Mathematics - Computer Science - Medicine SINTEF Unimed Ultrasound in Trondheim The Norwegian Radiumhospital in Oslo Sørlandet hospital in Kristiansand / Arendal

The 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 Oslo Linear Accelerator - Control of the Linear Accelerator - Databases (patient/employee/activity) - Computations of patient positions Patient Database Linear Accelerator Computer

Mathematical Transformation of Medical Images

Detector Metal Detector Mathematics

Image Transformation

Transformation Car Repair Hospital Home

Transformation Computing - Addition Room 1 Room 2 4 + 16 = 20 2 + 8 = 10 Transformation

Transformation Computing - Logarithm Rom 1 y Rom 2 x 8 * 32 = 256 3 + 5 = 8 Transformasjon

Music - Digital Ren tone Reell tone Digitalisering Tabell Analog Digital Ren tone Reell tone Digitalisering Tabell FourierTransform Sammensetn av rene toner Integrasjon Derivasjon

Transformation Theory F(u) = T[f(x)] Transformasjon f(x) F(u) Room 1 Room 2 f(x) = T-1(F(u))

Transformation Theory Integral Transformation F(…) = T[f(…)] f(…) F(…) Room 1 Room 2 f(…) = T-1(F(…))

Transformation Theory Integral Transformation Wavelet - Laplace - Fourier Wavelet Laplace f(…) F(…) Fourier

Fourier Transformation Fourier Serie

Analog - Digital Fourier Transformation

Transformation Theory T[f(x)] f(x) F(a,b) T-1[F(a,b)] b x a

Wavelet Transform Morlet Wavelet - Non-visible Oscillation

Wavelets New mathematical method with many interesting applications Divide a function into parts with frequency and time/position information Signal Processing - Image Processing - Astronomy/Optics/Nuclear Physics Image/Speech recognition - Seismologi - Diff.equations/Discontinuity … Wavelets er en relativt ny matematisk metode (har skutt fart først på 80- og 90-tallet). Wavelets har etter hvert fått mange anvendelsesområder. Navnet Wavelets benyttes ofte (slik som her) som navn på en bestemt matematisk metode. Imidlertid bør her nevnes at Wavelets også (mer korrekt) benyttes som navn på en spesiell gruppe såkalte basisfunksjoner i forbindelse med transformasjon av funksjoner. På bildet vises noen slike anvendelses-områder: - Detaljskalering av en datakilde (signal, bilde, ...) - Medisinsk bildebehandling (terskling og segmentering for bl.a. å fremheve detaljer) - Wavelets-tranformering av bilde (for komprimering eller gjenfinning av detaljer) - Seismologi (forutsi jordskjelv, studere forhold under havbunnen (lete etter olje, ...), ...) - ECG-diagram (bestemme eventuelle avvik, ...) - Røntgenbilde (bestemme eventuelle svulster, ...)

Definition of The Continuous Wavelet Transform CWT The continuous-time wavelet transform (CWT) of f(x) with respect to a wavelet (x): L2(R )

Wavelet Transform Morlet Wavelet Fourier / Wavelet

Mammography The Norwegian Radiumhospital Mammography Micro-calcifications Wavelet Transformation - Mexican Hat Seismic ECG

The Norwegian Radiumhospital Mammography Micro-calcifications phantom Phantom information Diameter Relative contrast Number of microcalcifications Mathematical tranformation of the mammography imaging Transformed image

Wavelet Transform Mammography - Video

Osteroporosis External part External part E/I bone edge E/I bone edge Morlet Osteroporosis External part External part E/I bone edge E/I bone edge

Ultrasound Image - Edge detection SINTEF – Unimed – Ultrasound - Trondheim - Ultrasound Images - Egde Detection - Noise Removal - Egde Sharpening - Edge Detection Prosthesis

Ultrasound 38

Doppler-effekt Lytter L beveger seg mot / fra lyd-kilden S samtidig som S også beveger seg.

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. Svevningsfrekvensen er differensen mellom enkelt-frekvensene

Svevning Piano To stk piano spiller samme a-tone med frekvens 440 Hz. Ved å la det ene pianoet endre sin a-tone-frekvens til henholdvis 441 Hz, 442 Hz og 443 Hz, hører vi en svevningstone med økende frekvens. Svevningstonens frekvens er lik differensen mellom frekvensene til de to pianofrekvensene.

Microcosmos Different parts of microcosmos in our daily life Mobile phone Electricity TV Medicine Magnetic resonance New energy Induction Semiconductor 42

Microcosmos LHC - Large Hadron Collider 43

MRI Magnetic Resonance Imaging 44

PET Emission of Positrons PET radiotracer (18F fluorodeoxyglucose (FDG)) + radioactive fluorine. Natural decay includes emmission of positrons that reach with electrons (annihilaation) in the body produces 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 intense than surrounding tissue, indicate that a high level of chemical activity or metabolism is occuring there. 45

Infrared camera - Wavelets 46

Calculator in Medicine Safety Calculator 47

Heart Circulation 48

Surgery Simulator 49

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