Math 262 / CME 372: Handouts

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 Course handouts: Course Organization .html. Course Survey: .pdf. Lectures: Lecture 1: Time-invariant (linear) operators, convolutions, continuous-time Fourier transform Lecture 2: Fourier inversion formula, convolution theorem, central limit theorem via Fourier transforms Lecture 3: Parseval-Plancherel theorem, Fourier transform of square-integrable functions, Fourier transform of distributions, Fourier transform in higher dimensions Lecture 4: The (Weyl-) Heisenberg uncertainty principle and its interpretation in quantum mechanics Lecture 5: Poisson summation formula, sampling and the aliasing formula, Shannon's sampling theorem Lecture 6: Discrete convolutions, Fourier series, another look at Shannon's sampling theorem Lecture 7: Numerical accuracy of the trapezoidal rule, Fourier transform of finite signals, FFTs Lecture 8: X-ray tomography, X-ray propagation and Beer's law Lecture 9: X-ray tomography, Radon transform, backprojection, Radon inversion formula Lecture 10: X-ray tomography, ill-posedness of the inverse problem, regularized inversion Lecture 11: Non-uniform fast Fourier transforms (NUFFTs) Lecture 12: Magnetic Resonance Imaging (MRI), nuclear magnetic resonance (NMR), Bloch phenomenological equations, relaxation times Lecture 13: Magnetic Resonance Imaging (MRI), simple imaging experiment, signal equation Lecture 14: Magnetic Resonance Imaging (MRI), roles of relaxation times, selective excitation Lecture 15: Wave optics, the phenomenon of diffraction, history, Fresnel-Kirchhoff integral formula Lecture 16: Rayleigh-Sommerfeld diffraction theory, Fresnel diffraction, Fraunhoffer diffraction, examples Lecture 17: Lenses, thin lenses, Fourier transform properties of thin lenses Lecture 18: Image formation, relation between object and image, effects of diffraction Lectures 19: The Wiener filter: Karhunen-Loeve decomposition of stochastic processes, stationary processes, estimation of Gaussian processes