Department of Mathematics

Ford Motor Research Laboratory

Wavelet Applications *

Transient signals abound in the automotive industry --- squeaks and rattles for example --- and wavelet transform techniques are tailor made for their analysis. In much the same way as Fourier transforms can be used to gain more insight into particular signal types, wavelet transforms are particularly effective for interrogating signals that have sudden, short lived events embedded in them. The results are often displayed in what are called ``scalograms.''

While inverse wavelet transforms are well documented and can be used to recover the original signal, filtering techniques are less well understood. When images of scalograms are calculated, one can often see patterns emerge. These patterns are associated with different types and levels of sound, which in turn are likely to have been generated by specific dynamic events. Thus, there is a need to be able to automatically find such patterns and to categorize them. This is NOT a pattern recognition problem in the usual sense, as the patterns that are being searched for are not known a priori. This is a very open-ended problem and is likely to benefit from out-of-the-box thinking and techniques developed for cryptology and data mining may be useful here.

Once identified, some applications require the ability to remove or modify specific patterns in the scalogram and then transform the modified scalogram back into the time domain. A mathematically consistent way of achieving this inversion needs to be developed, computer algorithms written, and signals tested.

Another application is the modeling of the cochlea (the fluid--filled, spiral-shaped part of the inner ear in mammals). Accurately recording sound is a relatively simple task, but it is hard to meaningfully quantify the measurement so that it relates to a humans perception of what has been heard. An innovative way to find a transform function that relates the sound, as measured in the air, to the nerve signal that is generated in the ear is needed.

There is a range of possible objectives of this project depending on the mutual interest of students and posers: scalogram modification and inversion, pattern identification, sound modeling.

The ideally completed project deliverable depends upon the specific aspect of wavelet analysis is of interest, but will certainly include mathematical analysis, algorithm development, and computer simulation and testing.

* This summary prepared by R. E. Svetic with the assistance of D. Scholl and P. M. Tuchinsky, Ford Research Laboratory, Ford Motor Company; the project team will be managed by Michigan State University Mathematics Professor M. Frazier.

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