Fast unified computation of the multi-dimensional discrete sinusoidal transforms

摘要

Based on a one-to-one index mapping approach, conversions of two types of m-dimensional discrete W transforms into the multiple sums involving a number of one-dimensional discrete W transforms (1-D DWTs) are first presented. A unified fast (m−1)-dimensional polynomial transform [(m−1)-D PT] and 1-D DWTs algorithm for computing these two types of m-D DWTs are then proposed. By revealing the relationships among the m-dimensional discrete W transforms (m-D DWTs), the m-dimensional generalized discrete Fourier transforms (m-D GDFTs), the m-dimensional discrete cosine transforms (m-D DCTs), and the m-dimensional discrete sine transforms (m-D DSTs), the generalized fast algorithms for 14 types of the m-D discrete sinusoidal transforms including four types of m-D DWTs and the same number of types of m-D GDFTs, and three types of m-D DCTs and the same number of types of m-D DSTs are presented. The number of multiplications for all 14 types of the m-D discrete sinusoidal transforms needed by the proposed algorithm is only 1/m times that of the widely used corresponding row–column methods. The number of additions required by the proposed algorithm is also reduced considerably. Finally, with the help of the software platform Visual C++ 6.0 for Windows 98, the computation time comparisons between the proposed and the reported algorithms are given. The numerical experiments show that the proposed algorithms are not only highly efficient but are also very simple in computational structure.