 # Introduction Per Henrik Hogstad Associate Professor Agder University

• Slides: 47 Introduction Per Henrik Hogstad Associate Professor Agder University College Faculty of Engeneering and Science Dept of Computer Science Grooseveien 36, N-4876 Grimstad, Norway Telephone: +47 37253285 Email: Per. [email protected] no 1 3 2 4 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 Fourier transformation Fourier transformation 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 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]   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  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. Edge detection Wavelet - Max Energy Scale Edge detection Wavelet - Different Edges Noise Removal Thresholding Hard Soft Semi-Soft Noise Removal Syntetic Image 45 Wavelets - 500. 000 test Original + point spread function + white gaussian noise Original Noise Removal Syntetic Image Semi-soft Noise Removal Ultrasound Image Original Soft Edge sharpening Edge detection Edge detection Scalogram Edge detection Scalogram Edge detection End