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Lecturer(s)
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Černá Dana, doc. RNDr. Ph.D.
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Course content
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Lectures: 1. Ordinary k-th order spline, interpolation spline, existence and uniqueness, method of indefinite coefficients. 2. Construction and approximation properties of a linear spline, conditions for a cubic spline, construction of a cubic spline. 3. Definition of B-splines, alternative formulas, examples, positivity, compact support. 4. B-spline properties - symmetry, derivative, recurrent formula, explicit formula. 5. Marsden's identity, polynomial exactness, linear independence, cardinal splines. 6. Fourier transform, existence, inverse, transformof translation and convolution. 7. Fourier transform of B-splines, scaling equation. 8. Riesz basis, Riesz constants for B-splines, dual basis. 9. Spline representation, function approximation using B-splines, dual scaling coefficients. 10. Discrete Haar transform, Haar wavelet. 11. Definition of wavelet, multiresolution analysis, scaling equation, wavelet spaces, wavelet construction. 12. Spline wavelets, biorthogonality, wavelet approximation of functions, DWT, and IDWT. 13. Vanishing moments, sparse representation of functions. 14. Applications of wavelets and DWT. Exercises: The exercise is focused on the implementation of the discussed numerical methods in Matlab.
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Learning activities and teaching methods
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Monological explanation (lecture, presentation,briefing)
- Class attendance
- 56 hours per semester
- Preparation for credit
- 28 hours per semester
- Preparation for exam
- 28 hours per semester
- Home preparation for classes
- 38 hours per semester
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Learning outcomes
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The aim of this course is to acquaint students with the basics of the theory of splines and wavelets. The following topics are discussed: ordinary k-th order splines, B-splines and their properties, spline approximation and interpolation, spline applications, orthogonal and biorthogonal wavelets, discrete wavelet transform, wavelet approximation of functions, and wavelet applications.
Knowledge of the theory of splines and wavelets and the ability to implement the methods discussed in Matlab.
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Prerequisites
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Basic knowledge of linear algebra and calculus.
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Assessment methods and criteria
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Combined examination
Credit: active participation in seminars and tests Exam: combined
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Recommended literature
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A.H. Najmi. Wavelets: A Coince Guide. 2012. ISBN 978-14-2140-496-7.
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C.K. Chui. An Introduction to Wavelets. London, 2004. ISBN 978-01-2174-.
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COHEN, A.:. Numerical Analysis of Wavelet Methods.. Amsterdam: Elsevier, 2003. ISBN 978-0-444-51124-9.
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de BOOR, C.:. Practical Guide to Splines.. New York: Springer-Verlag, 2001. ISBN 978-0-387-95366-3.
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L. Schumaker. Spline Functions: Basic Theory. Cambridge, 2007. ISBN 978-05-2170-512-7.
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Najzar, K.:. Základy teorie splinů [skripta], Praha, Karolinum 2006.. 2006. ISBN 8024612879.
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Najzar, K.:. Základy teorie waveletů [skripta], Praha, Karolinum 2004..
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