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

2. IntroductionIntroduction 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 37253285Email: Per.Hogstad@hia.no

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4. IntroductionIntroduction 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 + TestWavelets/Medicine

5. ResearchResearch SINTEF Unimed Ultrasound in Trondheim The Norwegian Radiumhospital in Oslo Sørlandet hospitalin Kristiansand / Arendal Mathematics - Computer Science - Medicine

6. Mathematical Image Operation - Application

7. 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 …

8. 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)

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

10. Fourier transformation

14. 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…).

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

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

17. Wavelet Transform Morlet Wavelet Fourier/Wavelet Fourier Wavelet

18. Wavelet Transform Morlet Wavelet Fourier/Wavelet Fourier Wavelet

19. Wavelet Transform Morlet Wavelet - Visible Oscillation

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

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

22. Matcad Program Wavelet Transform

23. CWT - DWT CWT DWT Binary dilation Dyadic translation Dyadic Wavelets

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

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

26. Discrete Wavelet-transformation

27. Compress 1:50 JPEGWavelet Original

28. 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

29. The Norwegian Radiumhospital Mammography Diameter Relative contrast Number of microcalcifications

30. The Norwegian Radiumhospital Mammography - Mexican Hat - 2 Dim

31. The Norwegian Radiumhospital Mammography

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

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

34. Edge Detection Convolution

35. Edge detection Wavelet Mexican Hat

36. 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.

37. Edge detection Wavelet - Max Energy Scale

38. Edge detection Wavelet - Different Edges

39. Noise Removal Thresholding HardSoftSemi-Soft

40. Noise Removal Syntetic Image 45 Wavelets - 500.000 test Original Original+ point spread function + white gaussian noise

41. Noise Removal Syntetic Image

42. Noise Removal Ultrasound Image Original Semi-soft Soft

43. Edge sharpening

44. Edge detection

45. Scalogram

46. Edge detection Scalogram

47. Edge detection

48. End