Bidimensional Multivariate Empirical Mode Decomposition With Applications in Multi-Scale Image Fusion

Empirical mode decomposition (EMD) is a fully data-driven technique designed for multi-scale decomposition of signals into their natural scale components, called intrinsic mode functions (IMFs). When EMD is directly applied to perform fusion of multivariate data from multiple and heterogeneous sources, the problem of uniqueness, that is, different numbers of decomposition levels for different sources, is likely to occur, due to the empirical nature of EMD. Although the multivariate EMD (MEMD) has been proposed for temporal data, which employs real-valued projections along multiple directions on a unit hypersphere in the $n$ -dimensional space to calculate the envelope and the local mean of multivariate signals, in order to guarantee the uniqueness of the scales, its direct usefulness in 2D multi-scale image fusion is still limited, due to its inability to maintain the spatial information. To address this issue, we propose a novel bidimensional MEMD (BMEMD) which directly projects a bidimensional multivariate signal, which is composed of multiple images, on the unit hypersphere in the $n$ -dimensional space. This is achieved via real-valued surface projections and the mean surface is estimated by interpolating the multivariate scatter data so as to extract common spatio-temporal scales across multiple images. Case studies involving texture analysis and multi-focus image fusion are presented to demonstrate the effectiveness of the proposed method.