Conclusions

In this paper we have discussed some properties of wavelet networks that we think are important when WNs are to be used for object representation. Wavelet networks are a combination of RBF networks and wavelet decomposition, where radial basis functions were replaced by wavelets. We have shown, that

• the optimized parameters as well as the wavelet coefficients are directly related to the underlying image structure,
• the coefficients can be computed from the projections of the wavelets onto the considered function and
• the optimized wavelets are linearly independent.

The first point is in our opinion very important: While, e.g., the radial basis functions in RBF networks do not reflect any properties of the represented functions, the wavelets of an optimized WN on the other hand do closely resemble properties of the represented function. Of course, as we have shown, it depends on the used mother-wavelet, what properties are reflected. In addition to the above properties, we have shown, that, apart from certain constrains taken from [Daugman, 1988], it is fairly straight forward, to optimize WNs (definition of a region of interest is needed, of course) and that the optimization time with under a minute for a mid-sized network is acceptable.

The above properties have been used in three small experiments, in which we have made extensive use of the wavelet subspace and the fact, that the wavelets form a basis. The wavelet subspace is isomorphic to the subspace of the ${\Bbb{L}}^2({\Bbb{R}}^2)$, spanned by the optimized wavelets. It is low dimensional and invariant to affine deformations of a template WN which makes computations in our tracking experiments more efficient. The pose estimation experiment showed that by carefully selecting the filters (e.g. by using a WN) both the error and the filtering effort can be minimized. All experiments would not have been successful, if the mapping from the ${\Bbb{L}}^2({\Bbb{R}}^2)$ into the subspace hadn't been unique. Our experiments have mainly dealt with faces, but we think that the properties of WNs are general enough be applied to general objects.

Lately, in [Reyneri, 1999], the relations ANNs, WNs and fuzzy systems have been discussed, but WNs were considered only in a very simplified fashion: Only radial wavelets were considered, which limits the potential of wavelet networks considerably. We would like to argue that, because of the close relation between the data and the basis functions, WNs offer new potential that goes, beyond the potential of RBF Networks. At least for 2-D functions and the shapes of human faces, this has been partially shown here. We think that this can be generalized to other $N$-D functions.