# Wavelets (Winter Semester 2008/09)

Lecturer: | Prof. Dr. Andreas Rieder |
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Classes: | Lecture (1060), Problem class (1061) |

Weekly hours: | 4+2 |

Lecture: | Wednesday 8:00-9:30 | Seminarraum 33 |
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Friday 8:00-9:30 | Seminarraum 31 | |

Problem class: | Friday 11:30-13:00 | HS 102 |

Lecturer | Prof. Dr. Andreas Rieder |
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Office hours: monday, 14:00-15:00, and on appointment | |

Room 3.040 Kollegiengebäude Mathematik (20.30) | |

Email: andreas.rieder(at)kit.edu | |

Problem classes | |

Office hours: | |

Room IWRMM (20.52) | |

Email: |

# Contents

Wavelet analysis is a rather new, but meanwhile well established, technique for signal and image processing with various applications in other fields. For instance, the famous

JPEG2000 standard for image compression is based upon wavelets.

In this course we will learn the mathematical foundations of wavelet analysis which belong to the field of harmonic analysis. We will motivate wavelet analysis from the shortcomings

of Fourier analysis with respect to time frequency representations of signals. Then we will study in detail the properties of the integral wavelet transform. The request for efficient evaluation of the wavelet transform leads to the concept of wavelet bases. Here, we will present the construction of orthogonal and bi-orthogonal wavelet systems. Finally, some applications will be discussed: de-noising, image compression, etc.

# Downloads

**Problem sets**

- Problem set 12 (for problem class on January 30)
- Problem set 13 (for problem class on February 6)

**Notes**

**Slides**

- Scale
- FT of L^2-functions
- Short time Fourier transform
- Filter properties of wavelet transform
- Multilevel representation of a function
- Approximation properties of wavelet transform 1
- Approximation properties of wavelet transform 2
- Cone of influence
- Wavelet transform of a chirp
- Phase space points related to wavelet frame
- Bi-orthogonal wavelet system

# Literature

- Louis, Maass, Rieder: Wavelets - Theory and Applications, Wiley 1997
- Daubechies: Ten Lectures on Wavelets, SIAM 1993
- Mallat: A Wavelet Tour of Signal Processing, Academic Press 1998
- Stark: Wavelets and Signal Processing, Springer 2005