Construction of data-adaptive orthogonal wavelet bases with an extension of principal component analysis

We show that orthonormal bases of functions with multiscale compact supports can be obtained from a generalization of principal component analysis. These functions, called multiscale principal components (MPCs), are eigenvectors of the correlation operator expressed in different vector subspaces. MPCs are data-adaptive functions that minimize their correlation with the reference signal. Using MPCs, we construct orthogonal bases which are similar to dyadic wavelet bases. We observe that MPCs are natural wavelets, i.e. their average is zero or nearly zero if the signal has a dominantly low-pass spectrum. We show that MPCs perform well in simple data compression experiments, in the presence or absence of singularities. We also introduce concentric MPCs, which are orthogonal basis functions having multiscale concentric supports. Use as kernels in convolution products with a signal, these functions allow to define a wavelet transform that has a striking capacity to emphasize atypical patterns.