A Novel Algorithm for the Decomposition of Non-Stationary Multidimensional and Multivariate Signals

Abstract

The decomposition of a signal is a fundamental tool in many fields of research, including signal processing, geophysics, astrophysics, engineering, medicine, and many more. By breaking down complex signals into simpler oscillatory components, we can enhance the understanding and processing of the data, unveiling hidden information contained in them. Traditional methods, such as Fourier analysis and wavelet transforms, which are effective in handling mono-dimensional stationary signals, struggle with non-stationary datasets and they require the selection of predefined basis functions. In contrast, the empirical mode decomposition (EMD) method and its variants, such as Iterative Filtering (IF), have emerged as effective non-linear approaches, adapting to signals without any need for a priori assumptions. To accelerate these methods, the Fast Iterative Filtering (FIF) algorithm was developed, and further extensions, such as Multivariate FIF (MvFIF) and Multidimensional FIF (FIF2), have been proposed to handle higher-dimensional data. In this work, we introduce the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) technique, an innovative method that extends FIF to handle data that varies simultaneously in space and time, like the ones sampled using sensor arrays. This new algorithm is capable of extracting Intrinsic Mode Functions (IMFs) from complex signals that vary in both space and time, overcoming limitations found in prior methods. The potentiality of the proposed method is demonstrated through applications to artificial and real-life signals, highlighting its versatility and effectiveness in decomposing multidimensional and multivariate non-stationary signals. The MdMvFIF method offers a powerful tool for advanced signal analysis across many scientific and engineering disciplines.

1. Introduction

Signal decomposition of 1D and higher-dimensional datasets plays an important role in applied mathematics and signal processing, and it has many applications in data analysis [1,2,3,4], image processing [5,6,7], machine learning [8,9,10], geophysics [11,12,13], communications [14,15,16], and many other fields. Decomposing complicated signals into simpler oscillatory components can help in understanding the underlying structures and enable efficient data representation [17,18], noise reduction [17,19], and feature extraction [18,20,21]. Customary decomposition techniques like Fourier analysis and wavelet transforms effectively process one-dimensional signals and methods that can handle multidimensional and multivariate signals are increasingly important as real-world data grows in complication. However, these methods have limitations. Fourier-based techniques [22] assume the stationarity of the signal, whereas wavelet-based methods [23] require the a priori selection of a proper basis to be used. Furthermore, both approaches are linear in nature.

To address these limitations, in 1998, the pioneering empirical mode decomposition method was published [23]. EMD separates signals into their oscillatory components, known as Intrinsic Mode Functions (IMFs), by iteratively subtracting local averages defined by the envelopes connecting the maxima and minima of the signal. While effective in many applications, EMD and derived algorithms suffer from several limitations, including their reliance on envelope interpolation and the lack of a rigorous mathematical framework. For these reasons, in the last two decades, many alternative non-linear methods, which require a priori selection of the base to be used as common in every technique based on optimization, have been developed, like, for instance, the sparse Time Frequency representation [24], the geometric mode decomposition [25], the symplectic geometry mode decomposition [26,27], the empirical wavelet transform [28], and the variational mode decomposition [29,30]. Among them, the only one based on iteration, and therefore, not requiring any a priori assumption on the signal under investigation, like the number of IMFs contained in the signal or the basis to be used in the decomposition, is the Iterative Filtering (IF) method.

The Iterative Filtering (IF) algorithm’s structure is based on EMD’s ones, with the key difference in the way the signal’s moving average is computed. IF computes the moving average $\mathcal{M}\left(f\right)\left(x\right)$ as the convolution of the signal f with a compactly supported filter or window w, instead of relying on the average of two envelopes, like in the EMD

(1)$\mathcal{M}\left(f\right)\left(x\right)={\int }_{-L}^{L}f(x+t)w\left(t\right)\mathrm{d}t,$

where L represents the filter length, or half-support of the filter function w. Unlike traditional approaches, IF determines L adaptively based on the signal itself. For instance, as suggested in [17,31], L can be computed using the relative distances between subsequent extrema of the signal. This self-adaptive approach makes IF inherently non-linear, enabling it to adjust to the signal’s characteristics. Building on IF, the Fast Iterative Filtering (FIF) algorithm was developed to accelerate the decomposition process using the Fast Fourier Transform (FFT). FIF preserves the adaptive nature of IF while significantly reducing computational costs, making it suitable for high-dimensional data. In addition to its speed, FIF introduces the ability to select filter properties, such as the Fokker–Planck filter, which ensures smoothness and compact support.

In recent years, FIF has been extended to handle increasingly complex datasets, leading to the development of the Multivariate Fast Iterative Filtering (MvFIF) [32], Multidimensional Iterative Filtering (MIF) [33], and Multidimensional Fast Iterative Filtering (FIF2) [34] algorithms. These extensions enable the decomposition of multichannel signals and multidimensional data, respectively, adapting the principles of FIF to higher dimensional and temporal domains.

In this work, we introduce the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) technique which extends FIF to handle datasets that vary, at the same time, across space and time, and we present its application to handle artificial and real-life signals. To the best of our knowledge, this algorithm is the first of its kind. There have been few algorithms proposed so far in the literature for the decomposition of non-stationary signals defined in higher dimensions, possibly including also time, like the fast and adaptive empirical mode decomposition for multidimensional, multivariate signals (FA-MVEMD) [35] and the fast empirical mode decomposition method for multidimensional signals based on serialization (Serial-EMD) [36]. However, these methods handle time as if it were an extra spatial dimension and not as a dimension per se. This is fine as far as the time and space variations are separated. However, in real-life signals, this is usually not the case, and space and time variations are tangled, for instance in plasma density simulations [37]. Furthermore, if we want to remove different ranges of frequencies from time and space it will be impossible to do it with these algorithms. This is a common problem in applications. Think, for instance, of what is shown in [38]. In that work, the datasets need to be filtered in high frequency in time and low frequency in space or vice versa. This task cannot be accomplished with the algorithms proposed so far in the literature.

The rest of this work is organized as follows. We first briefly recall the multidimensional and the multivariate extensions of the Fast Iterative Filtering. Then, in Section 2, we present the proposed approach to handle data that are, at the same time, multidimensional and multivariate. Next, in Section 3, we present a few examples of applications to both artificial and real-life signals. Finally, we derive the conclusions in Section 4.

1.1. Multidimensional Fast Iterative Filtering

The Multidimensional Fast Iterative Filtering (FIF2) method allows us to decompose an n-dimensional signal $f\in {\mathbb{R}}^n$ into simple oscillatory components, called Intrinsic Mode Functions (IMFs), by approximating the higher dimensional moving average of f and iteratively subtracting it. Unlike the empirical mode decomposition (EMD), which relies on maxima and minima envelopes, FIF computes the moving average using a convolution of f with a multidimensional filter w, defined as a non-negative, even, and continuous function compactly supported in $\Omega \subset {\mathbb{R}}^n$, with ${\int }_{\Omega }w\left(z\right)d^{n}z=1$.

The algorithm assumes the periodicity of f at its boundaries, a limitation that can be addressed by pre-extending the signal to enforce periodicity [39]. The filter support size $\Omega$ is calculated based on the signal’s characteristics, such as its extrema count or Fourier spectrum. This signal-dependent determination of $\Omega$ makes FIF2 non-linear and ensures adaptive decomposition.

The FIF2 method involves two nested loops:

Inner Loop: Extracts an IMF by iteratively applying the filtering process until a stopping criterion is met.

Outer Loop: Applies the same process to the residual signal, producing successive IMFs until the residual has at most one local extremum, classifying it as a trend signal.

The computational complexity of FIF2 is $\mathcal{O}(mlogm)$, where m is the maximum among all dimensions of the signal under investigation. The method’s robustness and efficiency are enhanced by choosing a Generalized Fokker–Planck filter [33]. These filters belong to the space $C^{\infty }\left({\mathbb{R}}^n\right)$ (i.e., the set of infinitely differentiable functions on ${\mathbb{R}}^n$) and are compactly supported. The $C^{\infty }$ of the filter ensures that, when we apply it iteratively, we do not reduce the smoothness of the signal. Moreover, the compact support guarantees that the computed average is truly local. To the best of our knowledge, generalized Fokker–Planck (F-P) filters are the only filters that possess both smoothness and a compact support. These filters are widely employed in various practical applications, such as [40,41,42,43]. This method has been proven to be a priori convergent and stable [33,34,44].

1.2. Multivariate Fast Iterative Filtering

The Multivariate Fast Iterative Filtering (MvFIF) method extends FIF to handle n-dimensional signals [32]. For a signal $f\in {\mathbb{R}}^n$ which is varying over time $t\in \mathbb{R}$, the algorithm computes a single filter length L and applies the 1D FIF to each channel.

To determine L, the signal is treated as a sequence of column vectors $\mathbf{v}\left(t\right)$ rotating in ${\mathbb{R}}^n$. The filter length is derived from the angle of rotation $\theta \left(t\right)$ between consecutive vectors:

(2)$\theta \left(t\right)=arccos\left({\displaystyle \frac{\mathbf{v}\left(t\right)}{∥\mathbf{v}\left(t\right)∥}}·{\displaystyle \frac{\mathbf{v}(t-1)}{∥\mathbf{v}(t-1)∥}}\right),$

where the symbol · denotes dot product between two vectors, and $\left|\right|.\left|\right|$ represents a norm of the ${\mathbb{R}}^n$ space. The filter length is set as twice the average distance between extrema in $\theta \left(t\right)$. This approach ensures that the decomposition adapts to the highest frequency rotations embedded in the signal. The computational complexity of MvFIF is $\mathcal{O}(nmlogm)$, where n is the number of channels and m is the signal length. Also, this method has been proven a priori convergent and stable [32].

2. Proposed Approach

Given a discrete n-dimensional signal f that varies over time $t\in I\subset \mathbb{R}$, we want to decompose it into IMFs. We propose in this work what we call the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) algorithm. The idea is to extract an IMF in space and one in time iteratively until the signal does not contain any oscillations anymore, both in space and time. In particular, for $f\left(t\right)\in {\mathbb{R}}^n$, $t\in I$ fixed, we start estimating the filter support ${\Omega }_t\in {\mathbb{R}}^n$ to be applied to $f\left(t\right)$, as performed in the FIF2 algorithm. We estimate ${\Omega }_t$ for every time instant t and then compute

(3)$\widehat{\Omega }=min\left\{{\Omega }_t\right\},$

which will be the support of the filter we apply to all space tensors $f\left(t\right)$ for every fixed $t\in I$. It is important to underline that the optimal computation of the filter support in 2D and higher dimensional spaces is currently an open problem [33,34]. Solving this problem is out of the scope of the present work. In the numerical section, to compute the support $\widehat{\Omega }$, we rely on what was performed for the FIF2 algorithm [34]. Furthermore, we select a Generalized Fokker–Planck filter $w_s$, as in [33], as a filter function and we impose its support to be $\widehat{\Omega }$. Then, we convolve it with the signal to produce its moving average ${\mathcal{M}}_s\left(f\right)=f*w_s$. We then subtract it from the signal and iterate the procedure producing $f_{k+1}=f_k-{\mathcal{M}}_s\left(f_k\right)$, for $k\in \mathbb{N}$, until a stopping criterion is satisfied.

Regarding the stopping criterion, there are many possible options. One way of doing it, as mentioned in [17,31] for the monodimensional case, is to consider the relative change in $f_k$ and discontinue the inner loop at a certain $k=N_s\in \mathbb{N}$ as soon as a prefixed threshold of $\delta \ge 0$ is reached. This can be expressed as

(4)$∥f_{k+1}-f_k∥/∥f_k∥\le \delta .$

The previous steps will produce the first estimate of the first IMF in space, ${\mathrm{IMF}}_s^{\left(1\right)}=f_{N_s}$. To speed up calculations, we can compute convolution in the frequency domain, where it becomes a simple product. This is made possible, with low computational complexity, thanks to the FFT method. To give the measure of the acceleration induced by this alternative approach, we can consider that if the signal under investigation is $f\in {\mathbb{R}}^2\times \mathbb{R}$ and after discretization is a cube of dimension $n\times m\times p$, then the computational complexity of the convolution passes from $\mathcal{O}(p\times {(n\times m)}^2)$ in time domain to $\mathcal{O}(p\times (n\times m)log(n\times m\left)\right)$ if performed in the frequency domain via FFT.

After computing the first IMF in space, we can subtract it from the signal to obtain a new signal $f=f-{\mathrm{IMF}}_s^{\left(1\right)}$. We can then compute the signal moving average over time. To do so, we apply a multivariate approach, similar to the Multivariate Fast Iterative Filtering (MvFIF) method [32]. We consider now the vector $\tilde{\mathbf{v}}\left(t\right)=(f,t)\in {\mathbb{R}}^n\times \mathbb{R}$ obtained by fixing $t\in I$ and combining the signal f and the time stamp t, and we measure the $\tilde{\mathbf{v}}$ vector rotations $\tilde{\theta }\left(t\right)$ as the time passes

(5)$\tilde{\theta }\left(t\right)=arccos\left({\displaystyle \frac{\tilde{\mathbf{v}}(t+1)}{∥\tilde{\mathbf{v}}(t+1)∥}}·{\displaystyle \frac{\tilde{\mathbf{v}}\left(t\right)}{∥\tilde{\mathbf{v}}\left(t\right)∥}}\right)$

We point out that in this formula, differently from previous approaches, the time variability is now taken into account as an $n+1$ variable [32]. We can now define the filter length L of a Fokker–Planck filter $w_t$ that we want to convolve with the signal to obtain its moving average. L, which is the half support size of the filter, can be computed as the double average distance between subsequent extrema in the signal $\tilde{\theta }\left(t\right)$. We point out that this approach is very natural if we consider a multivariate IMF as a vector in ${\mathbb{R}}^n$ rotating around the time axis. Computing the double average distance between subsequent extrema in $\tilde{\theta }\left(t\right)$ allows us to estimate the average scale of the highest frequency rotations embedded in this given signal.

We can now convolve the computed filter $w_t$ with the signal to produce its moving average ${\mathcal{M}}_t\left(f\right)=f*w_t$. We then subtract it from the signal and iterate the procedure, producing $f_{k+1}=f_k-{\mathcal{M}}_t\left(f_k\right)$, for $k\in \mathbb{N}$, until the stopping criterion (4) is satisfied. Also, in this case, we can consider the relative change in $f_k$ and discontinue the inner loop at a certain $k=N_t\in \mathbb{N}$ as soon as a prefixed threshold is reached. This is how we produce the first estimate of the first IMF in time, ${\mathrm{IMF}}_t^{\left(1\right)}=f_{N_t}$. If we subtract this other first IMF from the signal, we obtain a new signal $f=f-{\mathrm{IMF}}_t^{\left(1\right)}$, and we can repeat the entire previous procedure to extract the subsequent IMFs in space and time. We iterate this procedure until there are no more oscillations left both in space and time.

The convergence and stability of this newly proposed method follow directly from the convergence and stability of the FIF2 and MvFIF algorithms.

The pseudo-code of MdMvFIF is given in Algorithm 1 and a MATLAB implementation is available online (http://www.cicone.com).

When the space and time variations are separated, i.e., in the vector $\tilde{\mathbf{v}}=(f,t)$ the variation over time t is the same for every point $f\in {\mathbb{R}}^n$, we could further accelerate calculations via the convolution of the data tensor with an $n+1$ dimensional filter. We call this algorithm the Space-Time Fast Iterative Filtering (ST_FIF), the pseudo-code of which is given in Algorithm 2.

In this pseudo-code, DFT and iDFT stand for Discrete Fourier Transform and inverse DFT, respectively, whereas ${\mathrm{IMF}}_i$ represents the i-th IMF.

ST_FIF produces a decomposition that is somehow equivalent to what is also performed in the fast and adaptive empirical mode decomposition for multidimensional, multivariate signals (FA-MVEMD) [35] and the fast empirical mode decomposition method for multidimensional signals based on serialization (Serial-EMD) [36]. As we mentioned in the introduction, these methods can be applied under the assumption that the space and time variations in the data are not tangled; otherwise, the decompositions produced will contain artifacts. Furthermore, these methods have a second limitation—as also explained in the introduction, the IMFs produced contain time and space frequencies combined together. If we want to filter different frequency ranges for time and space it will not be possible with this algorithm. For these reasons, in this work, we focus only on implementing the first algorithm and show in the next section a few applications for synthetic and real-life signals.

Algorithm 1 Multidimensional and Multivariate Fast Iterative Filtering IMFs = MdMvFIF $\left(f\right)$

IMFs = {} i=0 while there are oscillations left in space or time do    i=i+1    compute the spatial filter support $\widehat{\Omega }$ of the space filter function $w_s$    compute the 2D filter $w_s$    compute DFT of signal f and of the filter $w_s$    set $k=0$    while the stopping criterion (4) is not satisfied do      ${\mathrm{IMF}}_s^{\left(i\right)}=\mathrm{iDFT}{\left(I-\mathrm{diag}\left(\mathrm{DFT}\left(w_s\right)\right)\right)}^k\mathrm{DFT}\left(\mathbf{f}\right)$      $k=k+1$    end while    ${\mathrm{IMFs}}_s={\mathrm{IMFs}}_s\cup \left\{{\mathrm{IMF}}_s^{\left(i\right)}\right\}$    $f=f-{\mathrm{IMF}}_s^{\left(i\right)}$    compute the time filter length L based on $\tilde{\theta }\left(t\right)$, computed as in (5)    compute the filter $w_t$    compute DFT of signal f and of the filter $w_t$    set $k=0$    while the stopping criterion (4) is not satisfied do      ${\mathrm{IMF}}_t^{\left(i\right)}=\mathrm{iDFT}{\left(I-\mathrm{diag}\left(\mathrm{DFT}\left(w_t\right)\right)\right)}^k\mathrm{DFT}\left(\mathbf{f}\right)$      $k=k+1$    end while    ${\mathrm{IMFs}}_t={\mathrm{IMFs}}_t\cup \left\{{\mathrm{IMF}}_t^{\left(i\right)}\right\}$    $f=f-{\mathrm{IMF}}_t^{\left(i\right)}$ end while ${\mathrm{IMFs}}_s={\mathrm{IMFs}}_s\cup \left\{f\right\}$

Algorithm 2 Space Time Fast Iterative Filtering IMFs = ST_FIF $\left(f\right)$

IMFs ={} while the stopping criterion is not satisfied do    compute the time filter length L for the filter based on $\tilde{\theta }\left(t\right)$, computed as in (2)    compute the spatial filter support $\Omega$    compute the $n+1$ filter w whose support is $\widehat{\Omega }\times [-L,L]$    compute DFT of signal f and of the filter w    set $k=0$    while the stopping criterion (4) is not satisfied do      ${\mathrm{IMF}}_i=\mathrm{iDFT}{\left(I-\mathrm{diag}\left(\mathrm{DFT}\left(w\right)\right)\right)}^k\mathrm{DFT}\left(\mathbf{f}\right)$      $k=k+1$    end while    IMFs = IMFs $\cup \left\{{\mathrm{IMF}}_i\right\}$    $f=f-{\mathrm{IMF}}_i$ end while IMFs = IMFs $\cup \left\{f\right\}$

3. Numerical Examples

To show the abilities of the MdMvFIF algorithm in handling data, we apply it to two artificial signals, a simulation of plasma density and two real-life datasets. For the sake of simplicity, especially in plotting results, in the following, we focus on signals that are 2D in space; x and y will define the two-dimensional spatial domain, which evolves over time, denoted by t, producing 3D data cubes. It is important to point out that a data cube with 2D spatial and 1D temporal dimensions may contain $p+q$ active frequencies—$p\in \mathbb{N}$ in space and $q\in \mathbb{N}$ in time. As a result, the MdMvFIF technique will produce $p+q$ IMFs plus a trend: p IMFs capturing the spatial frequencies, and the remaining q capturing the temporal frequencies. Each spatial IMF corresponds to a single active spatial frequency, while each temporal IMF corresponds to a single active temporal frequency. The proposed approach can handle signals of any dimension in space plus time.

MdMvFIF, like all signal analysis techniques, requires careful handling of boundaries [39]. This work focuses on the development of the method itself; therefore, in the following examples, we assume ideal boundary management and defer a detailed analysis of boundary effects to future work. For this same reason, in all subsequent examples, when we plot temporal decomposition, we select coordinate points which are away from the domain boundaries to eliminate any possible edge effects and minimize boundary-related artifacts.

In the artificial examples presented, spatial and temporal variations are separable. This means that the temporal evolution is the same at each spatial location. This simplification is intended to ease the interpretation of results, considering that, like EMD and related approaches, the MdMvFIF method does not produce a unique decomposition [45]. In contrast, for real data, the potential presence of mixed space-time modes cannot be ruled out in general, and the absence of ground truths makes it impossible to assess if the IMFs are accurately separated. Hence, from now on, we make the assumption that no significant space-time mixing is present in the data under investigation and interpret the results accordingly. All results presented in the following examples were generated using MATLAB R2023b 64 bit, and the source code is available online (http://www.cicone.com).

3.1. First Artificial Example

In this first example, we have a signal containing two non-stationary frequencies active in space and two stationary frequencies active in time. The goal of the decomposition is to separate these different types of oscillatory content into distinct IMFs. The data are shown in Figure 1 and Figure 2. In Figure 1, the left panel shows the signal at time $t=0$ on a two-dimensional spatial grid, where the x and y axes represent spatial coordinates, and the color gradient (ranging from blue to yellow) indicates signal intensity at each location. The right panel displays the time evolution of the signal at the fixed position $(x,y)=(10,10)$; the plot represents the variation in signal intensity over time, with the color scale again indicating amplitude.

In Figure 2, the left panel presents a three-dimensional visualization of the signal, with the x and y axes corresponding to spatial coordinates and the z axis representing signal intensity. The color gradient encodes variations in amplitude across space and time. The right panel illustrates the temporal evolution of the signal at multiple spatial positions, capturing how the signal propagates and changes over time across the spatial domain.

We apply MdMvFIF to decompose such signals into space and time IMFs. We plot in Figure 3 the time decomposition of the signal at the spatial position $(x,y)=(10,10)$. The three plots represent different intrinsic mode functions (IMFs), with the red curves indicating the extracted signal components and the black curves representing the ground truth. The horizontal axis corresponds to time, while the vertical axis shows the signal amplitude.

Figure 4 displays the spatial distribution of the first and second IMFs, along with the trend component of the signal produced with the proposed method. The x and x axes represent spatial coordinates, while the z axis reflects the signal amplitude. The color gradient indicates the intensity of each component across the spatial domain. Figure 5 illustrates the differences between the ground truth and the first and second IMF components produced by the MdMvFIF algorithm. The color map reflects the intensity of the differences, with x and y representing spatial positions and the color gradient showing the magnitude of variation. Since these differences are an order of magnitude of around ${10}^{-2}$, whereas the signal intensity is around 1, we can conclude that MdMvFIF can decompose this artificial signal with good accuracy.

To empirically verify the MdMvFIF algorithm’s complexity, we ran a series of tests on this example considering grids of increasing size. The CPU times and the values of the constant C in the expression time $t\approx C\times p\times m\times nlog(m\times n)$ are reported in Table 1. The results confirm the expected trend, with $C\approx 1.2\times {10}^{-6}$. All tests were performed using MATLAB R2023b 64 bit on a standard laptop equipped with an 11th Gen Intel(R) Core(TM) i7-1185G7 processor running at 3.00 GHz using 16.0 GB of RAM.

3.2. Second Artificial Example

In this second example, we have a signal containing two non-stationary frequencies active in space and two non-stationary frequencies active in time. The data are shown in Figure 6 and Figure 7. Figure 6 illustrates the signal at time $t=0$ on a 2D spatial grid, where the x and y axes represent spatial positions. The color gradient, ranging from blue to yellow, encodes signal intensity at each point. The right panel shows the time evolution of the signal at the spatial location $(x,y)=(7,10)$, with the horizontal axis representing time and the vertical axis indicating signal intensity. The color scale reflects variations in amplitude over time. Figure 7 presents a 3D visualization of the signal’s evolution over space and time, with the x and y axes representing spatial positions and the z axis corresponding to signal intensity. The color gradient reflects variations in intensity. The right panel shows a 3D representation of the signal evolution at different time points, with the same spatial axes and color gradient indicating changes in intensity.

We apply MdMvFIF to decompose such signals into space and time IMFs. We plot the results in Figure 8 and Figure 9. In Figure 10, we report the differences between the ground truth space components and the IMFs produced with the proposed method. Also in this second example, even though the signals are non-stationary both in time and space, the proposed algorithm can separate with a good level of accuracy the oscillations contained both in space and time.

Figure 8 illustrates the time decomposition of the signal at the spatial position $(x,y)=(7,10)$. The three panels correspond to different intrinsic mode functions (IMFs), where the red lines represent the decomposed signal components and the black lines show the ground truth. The horizontal axis represents time and the vertical axis indicates signal amplitude. Figure 9 shows the spatial distribution of the first and second intrinsic mode functions (IMFs) and the trend component. The x and y axes represent spatial positions, and the z axis indicates signal amplitude, with the color gradient reflecting signal intensity.

Figure 10 displays the differences between the ground truth and the first two IMF components obtained with the MdMvFIF algorithm. The color map shows the intensity of these differences across the spatial domain, with the x and y axes representing spatial positions. Since these differences, like in the first artificial example, are an order of magnitude of around ${10}^{-2}$, whereas the signal intensity is around 1, we can conclude that MdMvFIF can also decompose this second artificial signal with good accuracy.

3.3. Earth’s Air Temperature

We now consider the first real-life signal: the Earth’s air temperature measured in kelvin on a global grid at an altitude of 2 m from the surface of our planet. This dataset is made available through the US National Center for Environmental Prediction (NCEP)/US National Center for Atmospheric Research (NCAR) Reanalysis project (the original NCEP Reanalysis data has been provided by the NOAA/OAR/ESRL PSD, Boulder, Colorado, USA http://www.esrl.noaa.gov/psd/ (accessed on 28 March 2025) and have been edited by the Climatic Research Unit, University of East Anglia https://crudata.uea.ac.uk/cru/data/ncep/ (accessed on 28 March 2025). This project aimed to produce a 40-year record of global analyses of atmospheric fields in support of the needs of the research and climate monitoring communities. The dataset is obtained through the NCEP Global Data Assimilation System (GDAS). In particular, they used global rawinsonde from the NCEP Global Telecommunications System (GTS) data with upper-air observations collected using pibals (ceiling balloons) and aircraft data and raobs (https://www.raob.com/) (accessed on 28 March 2025) from national archives in various countries; comprehensive Ocean-Atmosphere Data Set (COADS) (COADS is a joint project of NOAA/ERL, NCAR, NCDC, and many other organizations and countries) based on data collected using ships, fixed buoys, drifting buoys, pack-ice buoys, near-surface data from ocean station reports, and other data; aircraft data available from the NCEP GTS and other sources; surface land synoptic data made available from Air Force or NCEP sources; satellite sounder data obtained from several sources; Special Sensing Microwave/Imager (SSM/I) data; and satellite cloud drift winds based on the USA National Meteorological Center (NMC) tapes and from the USA Geostationary Meteorological Satellite (GMS) cloud drift winds. For more details on the project, we refer the interested reader to [46].

In this work, we consider the temperatures measured daily from 1 January 2022 to 31 December 2022 (Data are available at https://crudata.uea.ac.uk/cru/data/ncep/qs_eurasia/daily/sflux/air.2m/ (accessed on 28 March 2025) ). A sample of the data, the spatial distribution of air temperature on 10 April 2022, at a 2-m altitude, is displayed in Figure 11. In the left panel, the x and y axes represent spatial coordinates, and the color gradient (from blue to yellow) indicates temperature values in kelvin. The right panel displays the time evolution of air temperature at the spatial position $(x,y)=(50,20)$, with the horizontal axis representing time and the vertical axis showing temperature values. The color scale reflects variations in temperature amplitude.

The data $f(v_1,v_2,t)$ are stored as 3D tensors of dimension $48\times 97\times 365$.

We apply MdMvFIF to decompose such signals into space and time IMFs. We plot the results in Figure 12 and Figure 13. From these results, it is possible to see how well MdMvFIF separates high frequencies, low frequencies, and trends both in space and time variations.

3.4. Thermal Imagery from Composite Material

An infrared camera recorded the thermal evolution on the back surface of a carbon-fiber-reinforced plastic (CFRP) material for approximately 17 seconds at a sampling rate of 120 frames per second (data are publicly available at https://data.mendeley.com/datasets/v4knrwgj9y/2 (accessed on 28 March 2025). For this test, we used the dataset CFRP-006_facq-120Hz_s-Back_Img-2000).

The thermal data acquisition spanned the phases before the application of a heat pulse, during the heat pulse, and during the cooling period. CFRP is a composite material composed of a synthetic resin matrix reinforced with embedded carbon fibers [47]. This experiment focused on the back face of the CFRP specimen to analyze deeper defect depths. The heat pulse was generated using two FX60 BALCAR photographic flashes, each delivering 6.2 kJ of energy in a 2 ms pulse. The flashes were positioned at a 45° angle to the sample surface, with their pulses aligned to the center of the sample. Thermal imaging was performed with a FLIR X6900 infrared camera equipped with an InSb CCD sensor matrix. The camera captured data at a resolution of 512 × 512 pixels with 14-bit intensity resolution and operated within a spectral range of 3.0 to 5.0 µm. The camera was positioned 1.0 m from the sample at a 90° angle to its surface. The sample’s emissivity was set to 0.98, and the flashes were placed 0.5 m from the sample to ensure efficient heat delivery. This setup ensured precise and repeatable data acquisition for thermal simulations, defect detection, and subsurface characterization of the back surface of the CFRP specimen.

A sample of the data is shown in Figure 14. The left panel presents a spatial sample of the temperature on the material’s surface, with the x and y axes representing spatial coordinates, and the color scale indicating the temperature values. The right panel illustrates the time evolution of the temperature at the spatial position $(x,y)=(230,50)$, with the horizontal axis representing time and the vertical axis showing temperature amplitude. The data $f(v_1,v_2,t)$ are stored as 3D tensors of dimension $512\times 512\times 250$. We apply MdMvFIF to decompose such signals into space and time IMFs. We plot the results in Figure 15 and Figure 16.

In this case, MdMvFIF allows us to identify in an adaptive way the trend both in the spatial and time data. In applications, the trend could be the useful information we want to preserve or the disturbance we may want to remove. For instance, in [38], the authors applied FIF2 to remove the trend in space and MvFIF to remove the high-frequency oscillations, preserving only the trend in time. With MdMvFIF, it will be possible to do it in a single shot.

3.5. Plasma Turbulence Simulations

The dataset used in this example was produced by a high-resolution numerical simulation of plasma turbulence, employing a fluid Hall-magnetohydrodynamic model (HMHD) of decaying plasma turbulence. Its evolution into fully developed turbulence, as well as its statistical properties have been thoroughly characterized in [48].

In this example, we consider the plasma density studied in [48]. For the purpose of this work, the original datacube $f(v_1,v_2,t)$ of size $1024\times 1024\times 1001$ was subsampled in space to $256\times 256$ and in time we considered the first 256 values from the original 1001, producing a 3D tensor of dimension $256\times 256\times 256$. A sample of the data is shown in Figure 17. In the left panel, the x and y axes represent spatial positions, with the color scale indicating plasma density. The right panel displays the time evolution of plasma density at the spatial position $(x,y)=(20,50)$, showing the variation of density over time. We apply MdMvFIF to decompose such signals into space and time IMFs. We plot the results in Figure 18 and Figure 19.

We observe that the oscillations extracted in time, Figure 19, differ slightly from the one present in time in the original data; refer to the right panel in Figure 17. This is because in the simulated data, space and time variations are tangled. This is an example of a real-life signal that requires the handling of time as a different variable from space and not as an extra spatial dimension as it is performed in the fast and adaptive empirical mode decomposition for multidimensional, multivariate signals (FA-MVEMD) [35], the fast empirical mode decomposition method for multidimensional signals based on serialization (Serial-EMD) [36], and the ST_FIF proposed in this paper.

4. Conclusions

In this work, we introduced the Multidimensional and Multivariate Fast Iterative Filtering (MdMvFIF) method and the Space Time Fast Iterative Filtering (ST_FIF), which are extensions of the Fast Iterative Filtering (FIF) algorithm designed to decompose complex signals that vary both in space and time. The MdMvFIF overcomes several limitations present in previous signal decomposition methods, such as the separation of time and space variations and the inability to independently filter different frequency ranges in space and time. The second one is an accelerated version of the first, but it has two important limitations: it does not allow to independently filter different frequency ranges in space and time, which is a fundamental feature in applications, and it requires the data to have separated spatial and time variations, which is usually not true in real-life signals. For these reasons, in the rest of the work, we focused on testing only the first proposed algorithm.

We demonstrated via several examples that the MdMvFIF method can be applied to high-dimensional and multivariate datasets, handling not only multidimensional signals but also signals that evolve over time. By iterating the filtering process first in space and then in time, the MdMvFIF method effectively isolates Intrinsic Mode Functions (IMFs) from complex data, making it a valid tool for the study of many real-world datasets. We presented several numerical examples, including synthetic signals and real-life data, to show the versatility and robustness of the approach.

In particular, the MdMvFIF algorithm provides a powerful tool for signal analysis in applications where the interplay between spatial and temporal variations is complex and intertwined. Future work will explore further optimization of the algorithm, including strategies for speeding up calculations preserving its ability to independently filter different frequency ranges in space and time, and its integration with machine learning techniques for feature extraction and analysis.

In conclusion, the MdMvFIF technique expands the capabilities of signal decomposition, offering a non-linear, adaptive, and efficient approach for analyzing multidimensional and multivariate signals that vary over both space and time. This makes it a promising method for a wide range of applications, like in engineering, physics, economics, medicine, and more general signal processing tasks.

As with other decomposition methods, MdMvFIF is sensitive to boundary effects. In this work, ideal boundary conditions were assumed in order to focus on the development and evaluation of the method itself. Addressing boundary treatment in a more robust and systematic way will be an important direction for future work.

Another potential limitation lies in the handling of mixed space-time modes, which may not be fully resolved due to the intrinsic structure of the algorithm. While this issue does not affect the artificial examples - where spatial and temporal variations were designed to be separable—it remains an open question when dealing with real - world data, where no ground truth is available. Investigating the behavior of the algorithm in such cases will be the focus of future research.

Finally, the computation of the filter support plays a crucial role in the performance of this class of methods. As discussed in this work, its optimal definition in higher dimensions is still an open problem, and we plan to address this issue in the future.

Author Contributions

Conceptualization, R.C., A.C., E.P. and H.Z.; methodology, R.C., A.C., E.P. and H.Z.; software, R.C., A.C., E.P. and H.Z.; validation, R.C., A.C., E.P. and H.Z.; formal analysis, R.C., A.C., E.P. and H.Z.; investigation, R.C., A.C., E.P. and H.Z.; resources, R.C., A.C., E.P. and H.Z.; data curation, R.C., A.C., E.P. and H.Z.; writing—original draft preparation, R.C., A.C., E.P. and H.Z.; writing—review and editing, R.C., A.C., E.P., and H.Z.; visualization, R.C., A.C., E.P. and H.Z.; supervision, R.C., A.C., E.P. and H.Z.; project administration, R.C., A.C., E.P. and H.Z.; funding acquisition, R.C., A.C., E.P. and H.Z. All authors have read and agreed to the published version of the manuscript.

Data Availability Statement

Codes and examples available at www.cicone.com.

Conflicts of Interest

The authors declare no conflicts of interest. The funders had no role in the design of the study; in the collection, analyses, or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.

Footnotes

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content.

Figures and Table

Figure 1 Example 1. (Left panel): Signal at time $t=0$. (Right panel): Time evolution for position $(x,y)=(10,10)$.

Figure 2 Example 1. (Left panel): 3D signal evolution in the second space variable and time. (Right panel): 3D signal evolution in space at different time stamps.

Figure 3 Example 1. Time decomposition for data at position $(x,y)=(10,10)$.

Figure 4 Example 1. From left to right, first, second IMF and trend over space.

Figure 5 Example 1. Differences between the ground truth and the first and second IMF and trend produced over space by MdMvFIF algorithm, respectively.

Figure 6 Example 2. (Left panel): Signal at time $t=0$. (Right panel): Time evolution for position $(x,y)=(7,10)$.

Figure 7 Example 2. (Left panel): 3D signal evolution in the second space variable and time. (Right panel): 3D signal evolution in space at different time stamps.

Figure 8 Example 2. Time decomposition for data at position $(x,y)=(7,10)$.

Figure 9 Example 2. From left to right, first, second IMF and trend over space.

Figure 10 Example 2. Differences between the ground truth and the first and second IMF and trend produced over space by the MdMvFIF algorithm, respectively.

Figure 11 Example 3. (Left panel): air temperature measured on 10 April 2022. (Right panel): Time evolution for position $(x,y)=(50,20)$.

Figure 12 Example 3. Decomposition over space of the data into two IMFs plus a trend.

Figure 13 Example 3. Time decomposition for data at position $(x,y)=(50,20)$.

Figure 14 Example 4. (Left panel): sample of the superficial heat of the CFRP material, frame 100. (Right panel): sample of the time evolution for position $(x,y)=(230,50)$.

Figure 15 Example 4. Decomposition over space of the data into three IMFs plus a trend.

Figure 16 Example 4. Time decomposition for data at position $(x,y)=(230,50)$. In red the original signal, in blue the trend in time computed via MdMvFIF.

Figure 17 Example 5. (Left panel): sample of the plasma density HMHD simulation at $t=0$. (Right panel): sample of the time evolution for the plasma density at pixel $(x,y)=(20,50)$.

Figure 18 Example 5. Decomposition over space of the data into two IMFs plus a trend.

Figure 19 Example 5. Time decomposition for data at position $(x,y)=(20,50)$.

Table 1

CPU times and estimated constants C for various grid configurations.

n	m	p	$\mathit{p}·\mathit{m}·\mathit{n}·log\left(\mathbf{mn}\right)$	t	C
200	200	300	55,224,719.9	79.195048	1.43405 × 10−6
300	300	300	133,764,547.8	168.615102	1.26054 × 10−6
400	400	300	249,797,759.2	300.553451	1.20319 × 10−6
500	500	300	404,845,500.7	559.767067	1.38267 × 10−6
300	300	200	89,176,365.17	100.982945	1.13240 × 10−6
300	300	300	133,764,547.8	168.615102	1.26054 × 10−6
300	300	400	178,352,730.3	210.953230	1.18279 × 10−6
300	300	500	222,940,912.9	293.803903	1.31786 × 10−6

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