- 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).
Module MAT-65-12-M-7
Fourier Analysis with Applications in Image Processing (M, 9.0 LP)
Module Identification
Module Number | Module Name | CP (Effort) |
---|---|---|
MAT-65-12-M-7 | Fourier Analysis with Applications in Image Processing | 9.0 CP (270 h) |
Basedata
CP, Effort | 9.0 CP = 270 h |
---|---|
Position of the semester | 1 Sem. irreg. |
Level | [7] Master (Advanced) |
Language | [EN] English |
Module Manager | |
Lecturers |
+ 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 |
Livecycle-State | [NORM] Active |
Courses
Type/SWS | Course Number | Title | Choice in Module-Part | Presence-Time / Self-Study | SL | SL is required for exa. | PL | CP | Sem. | |
---|---|---|---|---|---|---|---|---|---|---|
4V+2U | MAT-65-12-K-7 | 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.
Contents
Competencies / intended learning achievements
Upon successful completion of this module, the students have studied and understand the basic problems and concepts of classical Fourier analysis, a branch of analysis with many practical applications. They have mastered the important and current techniques and are able to apply them to selected tasks in image processing. They understand the proofs presented in the lecture and are able to comprehend and explain them. In particular, they are able to outline the conditions and assumptions that are necessary for the validity of the statements.
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.
Literature
- 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.
Requirements for attendance (informal)
Knowledge from the courses [MAT-12-28-K-3] and [MAT-14-11-K-3] as well as basic knowledge in the field of image processing are helpful, but not necessarily required.
Modules:
Courses
Requirements for attendance (formal)
None
References to Module / Module Number [MAT-65-12-M-7]
Module-Pool | Name | |
---|---|---|
[MAT-65-MPOOL-7] | Specialisation Image Processing and Data Analysis (M.Sc.) | |
[MAT-AM-MPOOL-7] | Applied Mathematics (Advanced Modules M.Sc.) | |
[MAT-RM-MPOOL-7] | Pure Mathematics (Advanced Modules M.Sc.) |
Notice