The local Karhunen-Loeve bases

We propose yet another dictionary of orthonormal bases (which has the same tree structure as the popular wavelet packet or local trigonometric dictionaries) adapted to a given ensemble of signals. These orthogonal waveforms are generated by a set of locally adapted versions of the Karhunen-Loeve (KL) transform. The basis vectors in this dictionary represent local features in the time-frequency plane compared to the standard KL basis vectors. Because of the structure of the bases, the best basis selection algorithm of Coifman-Wickerhauser is readily applicable. Moreover, no a priori choice of conjugated quadrature filters or cosine/sine polarity is necessary; it is completely data driven. The computational cost to build this dictionary is comparable to or potentially less than that of the standard KL transform. As an application, we give an example of clustering geophysical acoustic waveforms.