Course detail

Applications of Fourier Analysis

FSI-SF0Acad. year: 2024/2025

Fourier series, Fourier transform, discrete Fourier transform - basic notions, properties, applications mostly in image processing and analysis.

Language of instruction

Czech

Number of ECTS credits

2

Mode of study

Not applicable.

Guarantor

prof. RNDr. Miloslav Druckmüller, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

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

Rules for evaluation and completion of the course

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


Lectures are voluntary, seminars are compulsory.

Aims

Introduction to Fourier analysis and illustration of its applications in image processing and analysis.


Understanding Fourier analysis and its significance for applications in technology.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

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. (CS)
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. (CS)
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. (EN)
KÖRNER, T. W., Fourier Analysis, Cambridge University Press, 1995 (EN)

Recommended reading

BRACEWELL, R. N. The Fourier transform and its applications. McGraw-Hill, 1965, 2nd ed. 1978, revised 1986 (EN)

Classification of course in study plans

  • Programme B-MAI-P Bachelor's 3 year of study, summer semester, elective
  • Programme N-MAI-P Master's 1 year of study, summer semester, elective
  • Programme N-MET-P Master's 1 year of study, summer semester, elective

Type of course unit

 

Lecture

13 hod., optionally

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

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

13 hod., compulsory

Teacher / Lecturer

prof. RNDr. Miloslav Druckmüller, CSc.

Syllabus

Sample applications and their implementation.