Completed

  • Objective:

    Beside the Fast Multipole Method wavelet based approaches are of increasing interest for the fast calculation of large matrices.

    Method:

    The first part of the project is the implementation of the wavelet method for the compression of the data. Next step is the investigation whether wavelet and FMM approaches can be used together and whether an additional speed up is possible.

    Application:

    The aim of the project is the development of an algorithm that allows for a fast calculation of large matrices. The final aim is the possibilities to handle large acoustic problems numerically.

  • Objective:

    Gabor multipliers are an efficient time-variant filtering tool used implicitly in many engineering applications in signal processing. For these operators, the result of a Gabor transform (the sampled version of the Short Time Fourier Transform) is multiplied by a fixed function (the time-frequency mask or symbol). The result is then synthesized.

    Transforms other than the Gabor transform, particularly the wavelet transform, are more suitable for certain applications. The concept of multipliers can easily be extended in this case. This results in the introduction of operators called wavelet multipliers, which will be investigated in detail in this project. The project aims to precisely define wavelet multipliers' mathematical properties and optimize their use in applications.

    Method:

    The problem will be approached using modern wavelet theory, harmonic analysis tools, and numeric tools. Formulation and demonstration of analytic statements will be conducted jointly with systematic numeric experiments in order to study the properties of wavelet multipliers.

    The following topics will be investigated in the project:

    • Eigenvalues and eigenvectors of wavelet multipliers
    • Invertibility and injectivity of wavelet multipliers
    • Reproducing kernel invariance
    • Discretization and implementation of wavelet multiplier
    • Best approximation of operators by wavelet multipliers and identification of wavelet multipliers

    Application:

    The applications of wavelet multipliers in signal processing are numerous and include any application requiring time-variant filtering. Some applications of wavelet multipliers will be investigated further in the parallel projects:

    • Mathematical Modeling of Auditory Time-Frequency Masking Functions
    • Improvement of Head-Related Transfer Function Measurements
    • Advanced Method of Sound Absorption Measurements

    Publications:

    • Anaïk Olivero: "Expérimentation des multiplicateurs temps-échelle" (On the time-scale multipliers) Master thesis under the supervision of R. Kronland-Martinet and B. Torrésani, June 2008
  • Objective:

    Time-variant filters are gaining more importance in today's signal processing applications. Also, there wavelet analysis has numerous applications. The goal of this project is to investigate time-variant systems based on wavelet analysis.

    Method:

    The concept of multipliers can be easily extended to wavelet frames. This means the coefficients of a wavelet analysis are multiplied by a fixed symbol and then resynthesized. The influence of the special structures of these sequences on the resulting operators will be investigated.

    The theory of Pseudo-Differential Operators (PDO) can be translated to the wavelet case. How operators of interest in the investigation of multipliers, like the Kohn-Nirenberg correspondence, are translated to this case is of particular interest. Natural starting points for the research are:

    Use dilations in the definition of the spreading function instead of modulation.

    Define a special wavelet kernel function by using a weak formulation:

    < K f , g > = < k , Wg f >

    Application:

    A very useful application for this project is an analysis-modification-synthesis system based on the wavelet analysis. With some language manipulation, this could be called a "Wavelet Phase Vocoder".

    The application investigated in this project is the measurement of reflection coefficients. The wavelet analysis is preferable for signals containing transient parts. It is essential to separate the impulse responses of different reflections in order to calculate the absorption coefficient of a sound-proof wall. The impulse responses can be easily separated in a scalogram, and they can be extracted by using a wavelet multiplier.

    Project-completion:

    This project ended on 28.02.2008 and is incorporated into the 'High Potential'-Project of the WWTF, MULAC.

  • Objective:

    Weighted and controlled frames were introduced to speed up the inversion algorithm for the frame matrix of a wavelet frame. In this project, these kinds of frames are investigated further.

    Method:

    The frame multiplier concept is closely linked to the weighted frames concept. The frame operator of the weighted frame is simply a frame multiplier of the original frame. This project aims to explore this synergy. Finding an efficient way to invert the frame operator by applying weights to a given frame would be especially interesting. Weights are searched, for which the frame bounds are as "tight" as possible, meaning the spectrum is more concentrated. The first numerical experiments to find optimal weights have been conducted.

    Application:

    Weighted frames have already been applied to wavelets on the sphere. Also, the original work by Duffin and Schaefer dealt with the problem of finding such weights for a sequence of exponentials.

    Partners:

    • J. P. Antoine, Unité de physique théorique et de physique mathématique – FYMA