Parallel implementation of 2D Daubechies - D4 transform in a cluster

Abstract

Wavelet transform is an important mathematical tool with strong application in signal processing. From the mathematical point of view many different wavelet transforms are developed, where orthogonal wavelet transforms are more important. Also, various algorithms for their computations are developed. In this paper we present the parallel implementation of the orthogonal wavelet transform named Daubechies D4 transform, which consists of four taps filters. At the beginning, a short mathematical introduction of the Daubechies D4 transform is presented. After that the sequential and parallel algorithm implementations are analyzed. A parallel implementation for the PC cluster environment is done. We have analyzed two ways of parallel implementation, one with row mapping and the other with block mapping (mesh architecture of the processing elements). For practical implementation the C++ programming language and MPI paradigm (for parallel environment) are used. Finally, the mathematical complexity of these implementations is presented. In particular, we have analyzed separately the complexity of computations and communications. It is shown, also, that the complexity is between O(log2 n) and O(n2 / p). At the end, a comparison of sequential and parallel implementations is presented according to the results obtained by testing the practical implementations.