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Graph Iterative Filtering methods for the analysis of nonstationary signals on graphs

Giuseppe Scarlato, Antonio Cicone, Marco Donatelli

Abstract

In the analysis of real-world data, extracting meaningful features from signals is a crucial task. This is particularly challenging when signals contain non-stationary frequency components. The Iterative Filtering (IF) method has proven to be an effective tool for decomposing such signals. However, such a technique cannot handle directly data that have been sampled non-uniformly. On the other hand, graph signal processing has gained increasing attention due to its versatility and wide range of applications, and it can handle data sampled both uniformly and non-uniformly. In this work, we propose two algorithms that extend the IF method to signals defined on graphs. In addition, we provide a unified convergence analysis for the different IF variants. Finally, numerical experiments on a variety of graphs, including real-world data, confirm the effectiveness of the proposed methods. In particular, we test our algorithms on seismic data and the total electron content of the ionosphere. Those data are by their nature non-uniformly sampled, and, therefore, they cannot be directly analyzed by the standard IF method.

Graph Iterative Filtering methods for the analysis of nonstationary signals on graphs

Abstract

In the analysis of real-world data, extracting meaningful features from signals is a crucial task. This is particularly challenging when signals contain non-stationary frequency components. The Iterative Filtering (IF) method has proven to be an effective tool for decomposing such signals. However, such a technique cannot handle directly data that have been sampled non-uniformly. On the other hand, graph signal processing has gained increasing attention due to its versatility and wide range of applications, and it can handle data sampled both uniformly and non-uniformly. In this work, we propose two algorithms that extend the IF method to signals defined on graphs. In addition, we provide a unified convergence analysis for the different IF variants. Finally, numerical experiments on a variety of graphs, including real-world data, confirm the effectiveness of the proposed methods. In particular, we test our algorithms on seismic data and the total electron content of the ionosphere. Those data are by their nature non-uniformly sampled, and, therefore, they cannot be directly analyzed by the standard IF method.

Paper Structure

This paper contains 24 sections, 3 theorems, 41 equations, 18 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Iterative Filtering
  3. IF in the continuous setting
  4. IF in the discrete setting
  5. General discrete IF algorithm and its convergence
  6. Graph Fourier Transform Iterative Filtering (GFT-IF)
  7. Graph Fourier Transform and Graph Convolution
  8. GFT-IF Algorithm
  9. Inner loop convergence of the GFT-IF algorithm
  10. Distance Based Iterative Filtering (DB-IF)
  11. Inner loop convergence of the DB-IF algorithm
  12. Computational cost analysis
  13. Precomputation
  14. Window computation
  15. Sifting process step
  16. ...and 9 more sections

Key Result

Theorem 1

Let $W\in\mathbb{R}^{n\times n}$ such that Then the inner loop of the general discrete IF algorithm (algo:general_IF) converges to where $Q$ is a unitary matrix such that $W = Q T Q^H$ with $T$ a diagonal matrix, and $D$ is a diagonal matrix defined as

Figures (18)

  • Figure 1: In the top row, the signals sampled on a graph representing a grid of random points on the interval $[0, 2\pi)$ used for the GFT-IF and DB-IF algorithms. In particular, the signal $s$ (\ref{['fig:ex_1signal']}) is obtained as the sum of a high frequency component (\ref{['fig:ex_1base_signal_0']}) and a low frequency one (\ref{['fig:ex_1base_signal_1']}). In the bottom row, the same signals are sampled on an equispaced grid used for the classical IF algorithm.
  • Figure 2:
  • Figure 3: Examples of the window functions used in the GFT-IF (\ref{['fig:ex_1GFT_windows']}), DB-IF (\ref{['fig:ex_1DB_windows']}) and FIF (\ref{['fig:ex_1FIF_windows']}) algorithms. In the case of the GFT-IF and FIF algorithms, those windows are obtained by convolving the kernel $w$ with delta signals centered at different vertices/points of the graph/domain. In the case of the DB-IF algorithm, those windows correspond to rows of the window matrix $W$.
  • Figure 4: Results of the GFT-IF algorithm applied to the signal $s$ (shown in \ref{['fig:ex_1signal']}). The figure (\ref{['fig:ex_1IMF_0']}) shows the first IMF obtained with the GFT-IF algorithm alongside the expected result (\ref{['fig:ex_1comp_signal_0']}) and their difference (\ref{['fig:ex_1error_0']}). Similarly, the figure (\ref{['fig:ex_1IMF_1']}) shows the residual signal compared to the expected result (\ref{['fig:ex_1comp_signal_1']}) and their difference (\ref{['fig:ex_1error_1']}).
  • Figure 5: Results of the DB-IF algorithm applied to the signal $s$ (shown in \ref{['fig:ex_1signal']}). The figure (\ref{['fig:ex_1DB_IMF_0']}) shows the first IMF obtained with the DB-IF algorithm alongside the expected result (\ref{['fig:ex_1comp_DB_signal_0']}) and their difference (\ref{['fig:ex_1DB_error_0']}). Similarly, the figure (\ref{['fig:ex_1DB_IMF_1']}) shows the residual signal compared to the expected result (\ref{['fig:ex_1comp_DB_signal_1']}) and their difference (\ref{['fig:ex_1DB_error_1']}).
  • ...and 13 more figures

Theorems & Definitions (12)

  • Theorem 1: General discrete IF convergence
  • proof
  • Definition 1: Undirected weighted graph
  • Definition 2: Adjacency Matrix
  • Definition 3: Degree Matrix
  • Definition 4: Graph Laplacian
  • Definition 5: Graph Fourier Transform
  • Definition 6: Graph Convolution
  • Theorem 2: GFT-IF convergence
  • proof
  • ...and 2 more