Some studies on multidimensional Fourier theory for Hilbert transform, analytic signal and space-time series analysis

TL;DR

This paper introduces the Fourier frequency vector (FFV) for multidimensional Fourier transform, enabling a physical interpretation of negative frequencies and proposing a multidimensional Hilbert transform and analytic signal with improved properties for space-time analysis.

Contribution

It presents the Fourier frequency vector (FFV) concept and develops the multidimensional Hilbert transform and analytic signal, extending classical 1D methods to multidimensional space-time analysis.

Findings

Defined Fourier frequency vector (FFV) for MDFT

Proposed multidimensional Hilbert transform (MDHT) and analytic signal (MDAS)

Demonstrated image decomposition into AM-FM components using Fourier methods

Abstract

In this paper, we propose the Fourier frequency vector (FFV), inherently, associated with multidimensional Fourier transform. With the help of FFV, we are able to provide physical meaning of so called negative frequencies in multidimensional Fourier transform (MDFT), which in turn provide multidimensional spatial and space-time series analysis. The complex exponential representation of sinusoidal function always yields two frequencies, negative frequency corresponding to positive frequency and vice versa, in the multidimensional Fourier spectrum. Thus, using the MDFT, we propose multidimensional Hilbert transform (MDHT) and associated multidimensional analytic signal (MDAS) with following properties: (a) the extra and redundant positive, negative, or both frequencies, introduced due to complex exponential representation of multidimensional Fourier spectrum, are suppressed, (b) real part of MDAS is original signal, (c) real and imaginary part of MDAS are orthogonal, and (d) the magnitude envelope of a original signal is obtained as the magnitude of its associated MDAS, which is the instantaneous amplitude of the MDAS. The proposed MDHT and associated DMAS are generalization of the 1D HT and AS, respectively. We also provide the decomposition of an image into the AM-FM image model by the Fourier method and obtain explicit expression for the analytic image computation by 2DDFT.