| A novel approach to wavelet filter design is introduced. Motivated by the fact that different transforms lead to different time-frequency representations of the same signal, we present a general framework for the design of a mother wavelet best adapted to a specific signal or to a class of signals for a specific task. The filter is obtained through an unconstrained optimization over a compact manifold of an objective function. We explicitly reparametrize the filter's coefficients using a lattice decomposition so that the orthogonality constrains are automatically satisfied, and derive an expression for the gradient of the objective function with respect to the lattice angles. Any gradient-based optimization can then be used to find the optimal solution. We show that the framework we have developed is not constrained to wavelet transforms, but can be extended directly to a generic wavelet packet basis. It is then possible to optimize the specific basis and the filters jointly. |
