The generalized unified computation of multidimensional discrete orthogonal transforms

Abstract

By introducing a form of reorder for multidimensional data, we propose a unified fast algorithm that jointly employs one-dimensional W transform and multidimensional discrete polynomial transform to compute eleven types of multidimensional discrete orthogonal transforms, which contain three types ofm-dimensional discrete cosine transforms (m-D DCTs), four types ofm-dimensional discrete W transforms (m-D DWTs) (m-dimensional Hartley transform as a special case), and four types of generalized discrete Fourier transforms (m-D GDFTs). For real input, the number of multiplications for all eleven types of them-D discrete orthogonal transforms needed by the proposed algorithm are only 1/m times that of the commonly used corresponding row-column methods, and for complex input, it is further reduced to 1/(2m) times. The number of additions required is also reduced considerably. Furthermore, the proposed algorithm has a simple computational structure and is also easy to be implemented on computer, and the numerical experiments show that the computational efficiency is consistent with the theoretic analysis.