Applications of Fourier Analysis (FSI-SF0)

Academic year 2023/2024
Supervisor: prof. RNDr. Miloslav Druckmüller, CSc.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Aims of the course unit:

Introduction to Fourier analysis and illustration of its applications in image processing and analysis.
Learning outcomes and competences:
Understanding Fourier analysis and its significance for applications in technology.
Prerequisites:

Basic courses in Mathematics – Mathematics 1, 2, 3. Basics of programming in Matlab.
Course contents:

Fourier series, Fourier transform, discrete Fourier transform - basic notions, properties, applications mostly in image processing and analysis.
Teaching methods and criteria:
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes:

Accreditation: A short semestral project (either to be done on the last seminar or individually later).
Controlled participation in lessons:

Lectures are voluntary, seminars are compulsory.
Type of course unit:
    Lecture  13 × 1 hrs. optionally                  
    Computer-assisted exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture

1. Vector space, basis, vector spaces of infinite dimension
2. Unitary space, Hilbert spae
3. Fourier series
4. One-dimensional Fourier transform and its properties, convolution
5. Two-dimensional Fourier transform and its properties
6. Discrete Fourier transform
7. Spectrum visualization, spectum modification
8. Image filtration
9. Analysis of directions in image
10. Image registration - phase correlation
11. Image compression (JPG)
12. Computer tomography (CT)
    Computer-assisted exercise Sample applications and their implementation.
Literature - fundamental:
1. FOLLAND, G. B. Fourier Analysis and Its Applications. Second Edition. Providence (Rhode Island, U.S.A.): The American Mathematical Society, 2009. 433s. The Sally series, Pure and Applied Mathematics, Undergraduate Texts. ISBN 978-0-8218-4790-9.
2. ČÍŽEK, V. Diskrétní Fourierova transformace a její použití. 1st edition. Praha: SNTL - Nakladatelství technické literatury, n.p., 1981. 160s. Matematický seminář SNTL. ISBN 04-019-81.
3. BEZVODA, V., et al. Dvojrozměrná diskrétní Fourierova transformace a její použití - I.: Teorie a obecné užití. 1. vydání. Praha: Státní pedagogické nakladatelství, n.p., 1988. 181s. ISBN 17-135-88.
5. KÖRNER, T. W., Fourier Analysis, Cambridge University Press, 1995
Literature - recommended:
4. BRACEWELL, R. N. The Fourier transform and its applications. McGraw-Hill, 1965, 2nd ed. 1978, revised 1986
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-P full-time study --- no specialisation -- Cr 2 Elective 2 1 S
N-MET-P full-time study --- no specialisation -- Cr 2 Elective 2 1 S
B-MAI-P full-time study --- no specialisation -- Cr 2 Elective 1 3 S