Bruk av Wavelets en relativt ny matematisk metode

Bruk av Wavelets (en relativt ny matematisk metode) innen medisinsk bildebehandling

Introduction Per Henrik Hogstad Associate Professor Agder University College Faculty of Engineering and Science Dept of Computer Science Grooseveien 36, N-4876 Grimstad, Norway Telephone: +47 37253285 Email: Per. Hogstad@hia. no

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Introduction 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

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

Mathematical Image Operation - Application

Wavelets New mathematical method with many interesting applications Divide a function into parts with frequency and time/position information Signal Processing Image/Speech recognition … - Image Processing - Astronomy/Optics/Nuclear Physics - Seismologi - Diff. equations/Discontinuity

Definition of The Continuous Wavelet Transform CWT The continuous-time wavelet transform (CWT) of f(x) with respect to a wavelet (x): L 2(R)

Fourier-transformation of a square wave f(x) square wave (T=2) N=1 N=2 N=10

Fourier transformation

CWT - Time and frequency localization Time Frequency 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 rapidly varying high-frequency components are superimposed on slowly varying low-frequency components (seismic signals, music compositions, pictures…).

Fourier - Wavelet t Signal Fourier Wavelet Time Inf Freq Inf a=1/2 a=1 a=2 t

Filtering / Compression Data compression Remove low W-values Lowpass-filtering Highpass-filtering Replace W-values by 0 for low a-values Replace W-values by 0 for high a-values

Wavelet Transform Morlet Wavelet Fourier/Wavelet Fourier Wavelet

Wavelet Transform Morlet Wavelet - Visible Oscillation

Wavelet Transform Morlet Wavelet - Non-visible Oscillation [1/2]

Wavelet Transform Morlet Wavelet - Non-visible Oscillation [2/2]

Matcad Program Wavelet Transform

CWT - DWT CWT DWT Binary dilation Dyadic translation Dyadic Wavelets

Analysis /Synthesis Example J=5 Num of Samples: 2 J = 32

Analysis Synthesis J=5 Sampling: 25 = 32 j=5 j=4 j=3 j=2 j=1 j=0

Discrete Wavelet-transformation

Original Compress 1: 50 JPEG Wavelet

Research 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 Mammography Diameter Relative contrast Number of microcalcifications

The Norwegian Radiumhospital Mammography - Mexican Hat - 2 Dim

The Norwegian Radiumhospital Mammography

Morlet Arthritis Measure of bone External part E/I bone edge

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

Edge Detection Convolution

Edge detection Wavelet Mexican Hat

Edge Detection Wavelet - Scale Energy Wavelet Transform Inv Wavelet Transform Wavelet scale dependent spectrum Energy of the signal A measure of the distribution of energy of the signal f(x) as a function of scale.