M0120 Wavelet Analysis

Faculty of Science
Spring 2005

The course is not taught in Spring 2005

Extent and Intensity
2/0. 2 credit(s) (fasci plus compl plus > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Vítězslav Veselý, CSc. (lecturer)
Guaranteed by
doc. RNDr. Vítězslav Veselý, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Vítězslav Veselý, CSc.
Prerequisites
Calculus of complex numbers, Vector and matrix calculus, Linear functional analysis, Basics of Fourier analysis of periodic and nonperiodic functions including convolution operators.
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
  • Mathematics (programme PřF, M-MA, specialization Applied Mathematics)
Course objectives
Basic course focusing both on theory and applications of wavelets which are a relatively new field of modern mathematics with a wide scope of potential usage. Following the introductory remarks to general bases systems in a abstract Hilbert space, the exposition concentrates mainly on discrete orthonormal wavelet bases (wavelet series) in the L2 space, existence conditions, construction and other related topics. Special attention is paid to compactly supported wavelets. Many advantages of wavelet expansions are emphasized, in paricular in contrast with those of Fourier type which have been used so far. On the other hand also some examples illustrate possible risks in inappropriate situations. The exposition concludes with a survey of application areas, special attention being paid to nonlinear denoising techniques.
Syllabus
  • Basis systems in Hilbert spaces: overcomplete systems (frames), Riesz (biorthogonal) and orthonormal bases as their special cases, representation (expansion) of elements in terms of these bases using pseudoinverse operators and other related techniques.
  • Operators and bases of wavelet type in L2: integral (continuous) wavelet transform and its inverse, discrete wavelet transform using frame bases, in particular the orthonormal and biorthogonal ones, multiresolution analysis for orthonormal wavelet bases, quadrature mirror filters.
  • Compactly supported Daubechies wavelets: construction and properties.
  • Time-frequency localization: wavelet transforms versus Fourier transforms.
  • Applications: denoising, compression, digital communication, etc.
Literature
  • WALTER, Gilbert G. Wavelets and other orthogonal systems. 2-nd edition. Boca Raton: CRC Press, 2001, 392 s. ISBN 1-58488-227-1. info
  • WALTER, Gilbert G. Wavelets and other orthogonal systems with applications. Boca Raton: CRC Press, 1994, 248 s. ISBN 0-8493-7878-8. info
  • VESELÝ, Vítězslav. Wavelety a jejich použití při filtraci dat (Wavelets and their application to data filtering). Eds. J. Antoch and G. Dohnal. In Proceedings ROBUST'96. Prague: JCMF Prague, 1997, p. 241-272. ISBN 80-7015-540-X. info
  • VESELÝ, Vítězslav. Kernel frame smoothing operators. Eds. J. Antoch and G. Dohnal. In Proceedings ROBUST'2000. Prague: JČMF Praha, 2001, p. 308-323. ISBN 80-7015-792-5. info
  • DAUBECHIES, Ingrid. Ten lectures on wavelets. Philadelphia, Pa.: Society for Industrial and Applied Mathematics, 1992, xix, 357 s. ISBN 0-89871-274-2. info
Assessment methods (in Czech)
Výuka: přednáška, Zkouška: ústní s písemnou přípravou
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~vesely/educ_cz.html#wavelets
  • Permalink: https://is.muni.cz/course/sci/spring2005/M0120