ABSTRACTWavelet transformation, analogously to fast Fourier transformation, turns a discrete vector of length N into a vector of N coefficients, which scale the transformation's base functions. It is a linear operation which can be efficiently implemented as a recursion. Requiring the base function of the transformation to be orthogonal and approximative yields a set of constraining equations for discrete wavelets. Daubechies wavelet filter coefficients are the minimum phase solutions of this nonlinear equation set. I implemented a pyramidal algorithm which performs the transformation based on Daubechies wavelets of lengths from 4 to 20 samples. In the electronic version of this report, interactive figures allow readers to explore the transformation by submitting their own input and choosing the wavelet's length. By compressing and recovering a photograph of a face, I demonstrate the conditions for a successful wavelet compression; the matrix representation of the picture in the wavelet space is sparse and samples with small values can be neglected without affecting essential features of the uncompressed picture. Seismic data render rather densely in the wavelet domain. Finally, lacking clear criteria to define seismic data quality, a compression scheme based on data reduction does not seem to be practical. A more promising seismic application of wavelet transformation is interpolation. As with fast Fourier transformation, I utilize discrete wavelet transformation for interpolation of regularly sampled seismic traces by padding zeroes. |