LAUSR

In this paper, we propose the notion of Fourier frequency vector (FFV) which is inherently associated with the multidimensional (MD) Fourier representation (FR) of a signal. The proposed FFV provides physical meaning to the so-called negative frequencies in the MD-FR that in turn yields MD spatial and MD space-time series analysis. The one-dimensional Hilbert transform (1D-HT) and associated 1D analytic signal (1D-AS) of an 1D signal are well established; however, their true generalization to an MD signal, which possess all the properties of 1D case, are not available in the literature. To achieve this, we observe that in MD-FR the complex exponential representation of a sinusoidal function always yields two frequencies, namely negative frequency corresponding to positive frequency and vice versa. Thus, using the MD-FR, we propose MD-HT and associated MD analytic signal (AS) as a true generalization of the 1D-HT and 1D-AS, respectively, and obtain an explicit expression for the analytic image computation by 2D discrete Fourier transform (2D-DFT). We also extend the Fourier decomposition method for 2D signals that decomposes an image into a set of amplitude-modulated and frequency-modulated (AM–FM) image components. We finally propose a single-orthant Fourier transform (FT) of real MD signals which computes FT in the first orthant, and values in rest of the orthants are obtained by simple conjugation defined in this study.