Optimally Matched Wavelets

This thesis addresses the problem of constructing a discrete wavelet approximating the shape of a given pattern. For the design of a biorthogonal wavelet basis we present an approach, which is based on the lifting scheme. The lifting scheme is a parametrisation of all biorthogonal wavelets. It reduces our problem to a linear least squares problem. The special structure of the problem allows for an efficient optimisation algorithm. Every refinable function can be used as a dual generator, if it respects the perfect reconstruction constraints. The smoothness of the generator also implies the smoothness of the dual wavelet. Strategies for obtaining also a smooth primal wavelet function are discussed. The most promising way includes a slight modification of the discrete wavelet transform, leading to a special case of the so called double density transform. With this modification we can achieve both good matching and high smoothness of the wavelets. It doubles the amount of data and is thus redundant. The method is applied to the analysis of MEG data and to the condition monitoring on linear guideways. Furthermore the transfer matrix of refinable functions is explored in various ways. Interesting properties and efficient computations of the spectral radius, the sum of eigenvalue powers, and the determinant are investigated. An explicit lifting decomposition of CDF filter banks is shown. The mathematical details are presented in a notation inspired by functional programming. Since the FOURIER transform is avoided where this is sensible, the results are easily accessible for implementation in computer programs. Symbols for function scaling and translation that were only used for illustrative purposes in former wavelet related papers are now integrated into a strict formalism.