Discrete wavelet transform implementation in Fourier domain for multidimensional signal

Wavelet transforms are often calculated by using the Mallat algorithm. In this algorithm, a signal is decomposed by a cascade of filtering and downsampling operations. Computing time can be important but the filtering operations can be speeded up by using fast Fourier transform (FFT)-based convolutions. Since it is necessaty to work in the Fourier domain when large filters are used, we present some results of Fourier-based optimization of the sampling operations. Acceleration can be obtained by expressing the samplings in the Fourier domain. The general equations of the down- and upsampling of digital multidimensional signals are given. It is shown that for special cases such as the separable scheme and Feauveau's quincunx scheme, the samplings can be implemented in the Fourier domain. The performance of the implementations is determined by the number of multiplications involved in both FFT-convolution-based and Fourier-based algorithms. This comparison shows that the computational costs are reduced when the proposed implementation is used. The complexity of the algorithm is O(N log N). By using this Fourier-based method, the use of large filters or infinite impulse response filters in multiresolution analysis becomes manageable in terms of computation costs. Mesh simplification based on multiresolution "detail relevance" images illustrates an application of the implemenentation. (C) 2002 SPIE and IST.