- Fourier series (Fourier coefficients and Fourier series, convolution of periodic functions, pointwise and uniform convergence of Fourier series, Gibbs phenomenon),
- Fourier transform (Fourier transform in L1, Fourier transform in L2, Poisson’s summation formula and Shannon’s sampling theorem, Heisenberg’s uncertainty principle, Windowed Fourier transform),
- discrete Fourier transform (approximation of Fourier coefficients and aliasing formula, Fourier matrix and discrete Fourier transform, circulant matrices, Kronecker products and stride permutations, discrete trigonometric transforms),
- fast Fourier transform (Radix-2 algorithm, sparse Fourier transform, Fourier transform for non equispaced data),
- Prony’s method for the reconstruction of structured functions (Prony method, recovery of exponential sums),
- distributions (test functions and distributions, Schwartz spaces and tempered distributions, Fourier transform of tempered distributions),
- wavelets and wavelet frames (continuous wavelet transform, wavelets frames, Haar wavelets, multiresolution analysis).
Fourier Analysis with Applications in Image Processing (M, 9.0 LP)
|Module Number||Module Name||CP (Effort)|
|MAT-65-12-M-7||Fourier Analysis with Applications in Image Processing||9.0 CP (270 h)|
|CP, Effort||9.0 CP = 270 h|
|Position of the semester||1 Sem. irreg.|
|Level|| Master (Advanced)|
+ further Lecturers of the department Mathematics
|Area of study||[MAT-SPAS] Analysis and Stochastics|
|Reference course of study||[MAT-88.105-SG] M.Sc. Mathematics|
|Type/SWS||Course Number||Title||Choice in |
|SL||SL is |
required for exa.
Fourier Analysis with Applications in Image Processing
|P||84 h||186 h||-||-||PL1||9.0||irreg.|
- About [MAT-65-12-K-7]: Title: "Fourier Analysis with Applications in Image Processing"; Presence-Time: 84 h; Self-Study: 186 h
Examination achievement PL1
- Form of examination: oral examination (20-30 Min.)
- Examination Frequency: irregular (by arrangement)
- Examination number: 86228 ("Fourier Analysis with Applications in Image Processing")
Evaluation of grades
The grade of the module examination is also the module grade.
Competencies / intended learning achievements
By completing the given exercises, the students have developed a skilled, precise and independent handling of the terms, propositions and techniques taught in the lecture. In addition, they have learnt how to apply these techniques to new problems, analyze them and develop solution strategies individually or by team work.
- W. Walter: Einführung in die Theorie der Distributionen,
- W. Rudin: Functional Analysis,
- R. Strichartz: A Guide to Distribution Theory and Fourier Transform,
- G. B. Folland: Fourier Analysis and its Applications,
- I. Daubechies: Ten Lectures on Wavelets,
- S. Mallat: A Wavelet Tour of Signal Processing.