Lectures:

All the lectures from a previous edition of the course are available here.

• 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
• Lectures 8: The Wiener filter: Karhunen-Loeve decomposition of stochastic processes, stationary processes, estimation of Gaussian processes
• Lecture 9: X-ray tomography, X-ray propagation and Beer's law
• Lecture 10: X-ray tomography, backprojection, Radon inversion formula, ill-posedness of the inverse problem
• Lecture 11: X-ray tomography, regularized inversion
• Lecture 12: Non-uniform fast Fourier transforms (NUFFTs)
• Lecture 13: Magnetic Resonance Imaging (MRI), nuclear magnetic resonance (NMR), Bloch phenomenological equations, relaxation times
• Lecture 14: Magnetic Resonance Imaging (MRI), simple imaging experiment, signal equation
• Lecture 15: Magnetic Resonance Imaging (MRI), roles of relaxation times, selective excitation, a taste of compressed sensing
• Lecture 16: Wave optics, the phenomenon of diffraction, history, Fresnel-Kirchhoff integral formula
• Lecture 17: Rayleigh-Sommerfeld diffraction theory, Fresnel diffraction, Fraunhoffer diffraction, examples
• Lecture 18: Lenses, thin lenses, Fourier transform properties of thin lenses
• Lecture 19: Image formation, relation between object and image, effects of diffraction