Course Wavelets

Course

Meets requirements of following modules(MID)

• in active programs
  • Ma TIN2012 WL

Course Organization

Instruction language

• english (german only if whole auditory is german speaking)

Study Level

• Graduate

Prerequisites

• first year undergraduate mathematics course
• basic knowledge of digital signal processing, in particular concerning the z-transform
• basic experience in programming

Textbooks, Recommended Reading

• Strang, Gilbert; Nguyen Truong: Wavelets and Filter Banks
• C. Sidney Burrus; Ramesh A. Gopinath; Haitao Guo: Introduction to Wavelets and Wavelet Transforms. A Primer
• Bäni, Werner: Wavelets. Eine Einführung für Ingenieure

Instructors

• Prof.Dr. Alexander Stoffel

Supporting Scientific Staff

Transcipt Entry

Assessment

Course components

Lecture/Exercise

Objectives

Contents

• wavelets and filter banks: analogy to Fourier series, the Haar wavelet and its filter bank, description of filters by impulse response, z-transform and matrices, subsampling and upsampling
• two-channel filter bank, condition of perfect reconstruction, quality considerations
• construction of appropriate filters, construction of the corresponding scaling function and wavelet
• the lifting scheme, Deslauriers-Dubuc-filters
• denoising and image compression

Acquired Skills

• The students know what wavelet and Fourier transform have in common and what the differences are. They are able to describe simple filters by the impulse response and the z-transform and to calculate the corresponding matrix.
• They are able to explain filter banks using the example of the Haar filter bank. They know the quality criteria for filter banks and are able to explain the disadvantages of the Haar wavelet.
• They know the method to construct filters for biorthogonal wavelets and are able to give the most important examples.
• The students are able to explain the lifting scheme using the example of the lazy wavelet and the hat wavelet. They are able to explain the corresponding generalisation to Deslauriers-Dubuc-Filters.
• The students understand simple algorithms for denoising and data compression using wavelets. They are able to explain advantages and disadvantages of different wavelets for those applications.

Operational Competences

• The students are able to implement simple programs in Scilab to visualize advantages and disadvantages of different wavelet transforms and to illustrate calculation schemes.

Additional Component Assessment

• no

Lab

Objectives

Contents

• influence of quantization errors at the reconstruction of image data from their coefficients, importance of vanishing moments for data compression
• denoising algorithms for audio data
• comparison of different thresholding algorithms for denoising image data
• coding experiments for wavelet coefficients to achieve data compression

Acquired Skills

• The students are able to implement a Scilab program which visualizes the influence of quantization errors at the reconstruction of image data. They are able to draw conclusions on the properties of scaling functions and wavelets. Furthermore they are able to implement a program for determining the number of vanishing moments and they explain the importance of the number of vanishing moments for data compression using example signals.
• The students are able to implement a Scilab program which tests denoising algorithms for one dimensional signals and visualizes the results. They extend this program to denoise an audio signal and optimize the result by varying the value of the threshold.
• The students are able to implement a Scilab program which tests different thresholding algorithms for denoising of image data. They analyze the results concerning the quality of the denoised images.
• The students implement a Scilab program for experiments with coding algorithms for wavelet coefficients and subsequent data compression. They compare and interprete the practical results using different wavelet transforms for image compression.

Operational Competences

• The students are able to implement Scilab programs which test and compare the quality of different wavelet transforms in view of practical applications. They document the results of such tests and explain them to other students.

Additional Component Assessment

• no