The generalized unified computation of multidimensional discrete orthogonal transforms

By introducing a form of reorder for multidimensional data, we propose a unified fast algo-rithm that jointly employs one-dimensional W transform and multidimensional discrete polynomial trans-form to compute eleven types of multidimensional discrete orthogonal transforms, which contain three types of m-dimensional discrete cosine transforms ( m-D DCTs) ,four types of m-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 the m-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 im-plemented on computer, and the numerical experiments show that the computational efficiency is con-sistent with the theoretic analysis.