Ingrid Daubechies

Professor of Applied and Computational Mathmatics
Associated Faculty in the Department of Electrical Engineering
Ph.D. 1980, Free University, Brussels, Belgium

My research focuses on time-frequency methods, understood in a very broad sense, with the goal of developing and refining mathematical tools for signal analysis and computer graphics. These include wavelets as well as other mathematical transforms. The eighties saw the development of a synthesis between mathematical ideas from the fields of functional and harmonic analysis on the one hand, and filtering approaches (in particular, subband filtering) from digital signal processing on the other hand. This culminated in the construction of several types of wavelet bases that have turned out to be useful in a wide variety of applications. I have been fortunate to have been involved in some of those constructions: the compactly supported orthonormal wavelet bases that I published in 1988 are widely used; the biorthogonal wavelet filters that are being considered for the JPEG-2000 image compression standard were constructed by Cohen, Feauveau, and me in 1992.

At this point, the urgency is no longer in the construction of yet other wavelet bases. My research group is more interested in developing a deeper mathematical understanding of, for example, quantization issues, or rate-distortion theory for redundant representations using wavelets or other methods. In particular, we are studying frames of wavelets, or other time-frequency localization functions. Particular types of frames have been shown to be superior to non-redundant descriptions, or bases, for pattern recognition and for denoising. For compression, no good results have been obtained with frames so far, probably because the problem of quantizing redundant data is not well understood. We are working on this, and have obtained promising results.

Another project is to use a generalization of wavelets to settings in which grids can be irregular, with the goal of transferring many standard signal processing techniques (for example, filtering) to other fields, such as computer graphics, in which surfaces are given by finely meshed but typically irregular triangulations. Multiresolution techniques and wavelet-like constructions in this setting allow then to "filter" and edit such surfaces.

We are also working on applications of wavelets in the multiple-resolution structures of data network traffic (in collaboration with AT&T), or in homogenization of transport equations for complex media (where one is interested in a correct assessment of the influence of the fine scales on macroscopic quantities, without having to resolve the fine scales).

All my research projects are mathematical in nature, but they are inspired by my frequent contacts with many engineers and motivated by a genuine concern for their usefulness in applications as well as their mathematical interest. Every student and postdoctoral fellow is encouraged to participate to a high degree on these interdisciplinary contacts.