[Paper Review] Multidimensional Iterative Filtering method for the decomposition of high-dimensional non-stationary signals

Antonio Cicone, Haomin Zhou|arXiv (Cornell University)|Jul 26, 2015

Machine Fault Diagnosis Techniques14 references3 citations

TL;DR

This paper introduces the Multidimensional Iterative Filtering (MIF) method for decomposing high-dimensional, non-stationary signals into intrinsic mode functions (IMFs) using extended Fokker–Planck (FP) filters. The method leverages data-adaptive, smooth, compactly supported filters to achieve stable and convergent decomposition, demonstrating effective separation of mixed components in 2D and 3D signals, including real hyperspectral image data.

ABSTRACT

Iterative Filtering (IF) is an alternative technique to the Empirical Mode Decomposition (EMD) algorithm for the decomposition of non-stationary and non-linear signals. Recently in [1] IF has been proved to be convergent for any $L^2$ signal and its stability has been also showed through examples. Furthermore in [1] the so called Fokker-Planck (FP) filters have been introduced. They are smooth at every point and have compact supports. Based on those results, in this paper we introduce the Multidimensional Iterative Filtering (MIF) technique for the decomposition and time-frequency analysis of non-stationary high-dimensional signals. And we present the extension of FP filters to higher dimensions. We illustrate the promising performance of MIF algorithm, equipped with high-dimensional FP filters, when applied to the decomposition of 2D signals. [1] A. Cicone, J. Liu, and H. Zhou, Adaptive local iterative filtering for signal decomposition and instantaneous frequency analysis, arXiv:1411.6051, 2014.

Motivation & Objective

• Address the challenge of time-frequency analysis for non-stationary, high-dimensional signals in applications like structural health monitoring, art analysis, and hyperspectral imaging.
• Overcome limitations of traditional linear methods (e.g., Fourier, wavelets) and EMD/EEMD by developing a data-adaptive, non-optimization-based decomposition technique.
• Extend the convergence and stability properties of 1D Iterative Filtering (IF) to multidimensional signals using generalized Fokker–Planck filters.
• Enable decomposition of signals with mixed characteristics (e.g., smooth vs. non-smooth components) in 2D and 3D domains.
• Improve performance of existing classifiers in hyperspectral imaging by pre-processing with MIF to reduce false alarms.

Proposed method

• Propose the Multidimensional Iterative Filtering (MIF) algorithm as an extension of 1D Iterative Filtering for non-stationary, high-dimensional signals.
• Introduce higher-dimensional Fokker–Planck (FP) filters with compact support and smoothness, derived from the 1D FP filter framework.
• Use convolution-based moving average computation via FP filters instead of spline interpolation, enabling rigorous convergence analysis.
• Apply iterative sifting process: compute IMF candidates by subtracting filter-convolved signal components from residuals until convergence.
• Define IMFs as the limit of iterative filtering steps, ensuring zero moving average and satisfying intrinsic mode function properties.
• Apply MIF to 2D and 3D signals, including synthetic mixtures and real hyperspectral images, with post-processing via ACE classifier.

Experimental results

Research questions

• RQ1 Can the Iterative Filtering method be generalized to handle non-stationary, high-dimensional signals while preserving convergence and stability?
• RQ2 How can Fokker–Planck filters be extended to higher dimensions to maintain smoothness and compact support for effective signal decomposition?
• RQ3 Can MIF effectively separate components of different natures (e.g., smooth and non-smooth) in multidimensional signals?
• RQ4 Does MIF improve the performance of downstream signal classification tasks, such as in hyperspectral image analysis?
• RQ5 What are the theoretical conditions ensuring convergence of the MIF algorithm in higher dimensions, and how do they relate to filter design?

Key findings

• The MIF algorithm successfully decomposes 2D synthetic signals into distinct IMFs, separating smooth and non-smooth components using a single fixed FP filter.
• In a 2D example, the first IMF captured non-smooth, non-stationary features while the second IMF represented a smooth, stationary signal, demonstrating effective component separation.
• For hyperspectral imaging, MIF pre-processing reduced false alarms in ACE-based classification, improving detection accuracy by removing spurious noise and artifacts.
• The method achieved stable decomposition without requiring prior basis selection or optimization, relying solely on data-driven filtering.
• The use of multidimensional Fokker–Planck filters enabled smooth, compactly supported filtering that preserved signal features while ensuring convergence.
• The results suggest that MIF is a promising alternative to EMD and EEMD in multidimensional settings, particularly where data adaptivity and stability are critical.