## Module Handbook

• Dynamischer Default-Fachbereich geändert auf MAT

# Module MAT-65-12-M-7

## 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 1 Sem. irreg. [7] Master (Advanced) [EN] English Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) Steidl, Gabriele, Prof. Dr. (PROF | DEPT: MAT) [MAT-SPAS] Analysis and Stochastics [MAT-88.105-SG] M.Sc. Mathematics [NORM] Active

## Notice

This module cannot be included in the Master's examination together with the module [MAT-65-10-M-4] or the module [MAT-65-14-M-7] due to large content overlaps. Students who have already used the course [MAT-65-10-K-4] for one of the specialization modules of the Bachelor's programme in Mathematics can use the second part of the course as module [MAT-65-14-M-7] "Distributions and Wavelets" for their master's studies.

## 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")

## Contents

• 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).

## 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 of the module (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.

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.)