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Divergence Phase Index: A Riesz-Transform Framework for Multidimensional Phase Difference Analysis

Magaly Catanzariti, Hugo Aimar, Diego M. Mateos

TL;DR

The paper addresses the limitation of traditional, one-dimensional phase-difference measures by introducing the Divergence Phase Index ($DPI$), a geometry-aware metric for multidimensional signals based on the $n$-dimensional Riesz transform. It builds a rigorous framework by tying the Hilbert transform to analytic signals through distributional Fourier theory, defines multidimensional instantaneous phase components $\Phi_f^j=\tan^{-1}\left(\frac{R_j f}{f}\right)$, and uses their differences to form the DPI with amplitude-invariance and rotation-consistency. The authors validate DPI across diverse data: 1D intracranial EEG during seizures shows hypersynchronization captured by higher DPI in ictal states, while 2D images—including synthetic examples and Van Gogh’s Self-Portrait—demonstrate DPI’s ability to detect structural differences independent of intensity, with finer parcellations improving localization. They also demonstrate rotation-detection capabilities in isotropic microscopy images, illustrating DPI’s potential for image forensics and data integrity. Overall, DPI provides a rigorous, versatile tool for multidimensional phase analysis with broad implications for neuroscience, imaging, and complex systems analysis.

Abstract

We introduce the Divergence Phase Index (DPI), a novel framework for quantifying phase differences in one and multidimensional signals, grounded in harmonic analysis via the Riesz transform. Based on classical Hilbert Transform phase measures, the DPI extends these principles to higher dimensions, offering a geometry-aware metric that is invariant to intensity scaling and sensitive to structural changes. We applied this method on both synthetic and real-world datasets, including intracranial EEG (iEEG) recordings during epileptic seizures, high-resolution microscopy images, and paintings. In the 1D case, the DPI robustly detects hypersynchronization associated with generalized epilepsy, while in 2D, it reveals subtle, imperceptible changes in images and artworks. Additionally, it can detect rotational variations in highly isotropic microscopy images. The DPI's robustness to amplitude variations and its adaptability across domains enable its use in diverse applications from nonlinear dynamics, complex systems analysis, to multidimensional signal processing.

Divergence Phase Index: A Riesz-Transform Framework for Multidimensional Phase Difference Analysis

TL;DR

The paper addresses the limitation of traditional, one-dimensional phase-difference measures by introducing the Divergence Phase Index (), a geometry-aware metric for multidimensional signals based on the -dimensional Riesz transform. It builds a rigorous framework by tying the Hilbert transform to analytic signals through distributional Fourier theory, defines multidimensional instantaneous phase components , and uses their differences to form the DPI with amplitude-invariance and rotation-consistency. The authors validate DPI across diverse data: 1D intracranial EEG during seizures shows hypersynchronization captured by higher DPI in ictal states, while 2D images—including synthetic examples and Van Gogh’s Self-Portrait—demonstrate DPI’s ability to detect structural differences independent of intensity, with finer parcellations improving localization. They also demonstrate rotation-detection capabilities in isotropic microscopy images, illustrating DPI’s potential for image forensics and data integrity. Overall, DPI provides a rigorous, versatile tool for multidimensional phase analysis with broad implications for neuroscience, imaging, and complex systems analysis.

Abstract

We introduce the Divergence Phase Index (DPI), a novel framework for quantifying phase differences in one and multidimensional signals, grounded in harmonic analysis via the Riesz transform. Based on classical Hilbert Transform phase measures, the DPI extends these principles to higher dimensions, offering a geometry-aware metric that is invariant to intensity scaling and sensitive to structural changes. We applied this method on both synthetic and real-world datasets, including intracranial EEG (iEEG) recordings during epileptic seizures, high-resolution microscopy images, and paintings. In the 1D case, the DPI robustly detects hypersynchronization associated with generalized epilepsy, while in 2D, it reveals subtle, imperceptible changes in images and artworks. Additionally, it can detect rotational variations in highly isotropic microscopy images. The DPI's robustness to amplitude variations and its adaptability across domains enable its use in diverse applications from nonlinear dynamics, complex systems analysis, to multidimensional signal processing.

Paper Structure

This paper contains 12 sections, 2 theorems, 30 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Theoretical Framework
  3. Analytic functions and the Fourier version of the Cauchy-Riemann system
  4. The instantaneous phase of a signal
  5. A higher dimensional extension
  6. The Riesz transforms and the rotation of the space
  7. Results
  8. One-dimensional examples
  9. Two-dimensional examples
  10. Application of rotation properties in image analysis
  11. Discussion
  12. Conclusion

Key Result

Lemma 2.1

Let $\varphi(t)$ be a real signal defined on $\mathbb{R}$. Set $u(t,y)$ for $t\in\mathbb{R}$ and $y>0$, to be such that Then

Figures (4)

  • Figure 1: (A) Intracranial electroencephalography (iEEG) signals recorded during baseline (0--10 s) and seizure (10--20 s) conditions, filtered in the theta band ($[1, 3]\,\text{Hz}$). (B) Divergence Phase Index ($DPI$) computed across all pairwise channel combinations for baseline (left) and seizure (right) states.
  • Figure 2: (A) Top row: from left to right, the original image ($O$), the original image with reduced intensity ($O'$), and the modified image ($M$). Bottom row: Divergence Phase Index ($DPI$) matrices between each pair of images. Red indicates no significant difference between corresponding image squares $(i,j)$, while green indicates a significant difference. (B)$DPI$($O,M$) results for different image partition sizes ($N_s^2 = 4, 5, 6, 8, 9, 12$), showing that finer partitions yield more precise localization of image differences.
  • Figure 3: (A) Original image (left) and modified image (right) of Vincent van Gogh’s "Self-Portrait" with an intentional alteration in the left eye (see zoomed inset). (B) Grayscale conversions of the original image ($O$), the low-intensity version ($O'$; 10% of original intensity), and the modified image ($M$). (C) Divergence Phase Index ($DPI$) matrices computed for each image pair using a parcellation of $N_s^2 = 17$ squares, followed by binarization based on the elbow method. Red indicates no significant difference between corresponding image squares $(i,j)$, while green indicates a significant difference.
  • Figure 4: Visualization of rotation detection using the Riesz transform. The original image (top-left) is compared with several rotated versions (remaining panels) by evaluating the similarity of their $R_1$ and $R_2$ components. The angle that maximizes the match between the vector fields indicates how much each test image is rotated relative to the original. The rotations shown include nearly imperceptible cases, common rotations, and less frequent rotations.

Theorems & Definitions (10)

  • Lemma 2.1
  • proof
  • Definition 2.1
  • Definition 3.1
  • Definition 3.2
  • Definition 4.1
  • Definition 4.2
  • Proposition 4.1
  • proof
  • proof