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arxiv: 2605.19963 · v1 · pith:OEMBT3PHnew · submitted 2026-05-19 · 📡 eess.SP

ADOPT: Analytical Demodulation of Periodic Textures for In-Plane Wave Tracking

Jalal Jouidi , Florent Chatelain , Lucie Bailly , Pierre Granjon , Nicolas Le Bihan , Stefan Catheline This is my paper

Pith reviewed 2026-05-20 03:57 UTC · model grok-4.3

classification 📡 eess.SP
keywords in-plane wave trackinganalytic signalphase demodulationperiodic texturedigital image correlationdisplacement estimationdispersion curve
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The pith

ADOPT demodulates periodic textures with an oriented analytic signal to track in-plane wave displacements.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper models wave-induced surface deformation as a spatial phase shift applied to an underlying periodic pattern. It introduces ADOPT to build an oriented two-dimensional analytic signal, apply orientation-selective filtering to isolate the relevant frequency content, and recover both the local displacement amplitude and direction from the extracted phase. Simulations establish that this approach yields lower estimation error than digital image correlation when noise levels are low and displacements are small, while also requiring less computation. Experiments on silicone membranes driven by impulsive excitation recover both the instantaneous wave field and the resulting dispersion relation. The method therefore supplies a direct, physics-informed route from image sequences to quantitative wave kinematics.

Core claim

Wave-induced in-plane deformation is modeled as a spatial phase modulation of a periodic carrier; ADOPT recovers the displacement field by constructing an oriented two-dimensional analytic signal, performing orientation-selective filtering, and extracting the local phase that directly encodes the displacement vector.

What carries the argument

oriented two-dimensional analytic signal that isolates spectral components via orientation-selective filtering before phase extraction

Load-bearing premise

Wave-induced deformation can be accurately modeled as a spatial phase modulation of a periodic carrier, with orientation-selective filtering successfully isolating the relevant spectral components without significant interference from other effects.

What would settle it

Independent laser-vibrometer measurements of surface displacement on the same membrane specimen under identical impulsive excitation, compared directly against ADOPT phase estimates.

Figures

Figures reproduced from arXiv: 2605.19963 by Florent Chatelain, Jalal Jouidi, Lucie Bailly, Nicolas Le Bihan, Pierre Granjon, Stefan Catheline.

Figure 1
Figure 1. Figure 1: Examples of periodic surface texture patterns extracted from the standard Brodatz [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: In-plane wave propagation (zero out-of-plane displacement) in a square membrane of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Pattern on the surface and corresponding 2D Fourier transform. [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Orientation detection and filtering principle, where the regions highlighted in cyan [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Displacement estimation of the U component (in pixels). Left: filtered modulated pattern along θu. Center: phase of the oriented analytic signal. Right: estimated displacement Uˆ. trum [θu−π/2, θu+π/2] is retained, and the rest set to zero. The inverse Fourier transform yields the complex analytic signal ˜I + d,u, whose phase gives an estimate of the wrapped displacement with expression: Uˆ = 1 2πξp arg D … view at source ↗
Figure 6
Figure 6. Figure 6: Wavefront reconstruction (U and V in pixels for SNR= 20 dB). Left column: ground truth, Middle column: DIC (MSE = -11 dB), Right column: proposed method (MSE = -21 dB), DIC images are smaller due to edge interpolation limits. estimation is given in [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: The averaged MSE of Uˆ and Vˆ versus SNR. Comparison between DIC, the proposed method, MLE, and the CRB. DIC interpolation and the limitations of its model introduce distortions and edge artifacts, whereas our method preserves the linear wavefront across the field. 0 0.25 0.5 0.75 1 −40 −32.5 −25 −17.5 −10 Normalised frequency ξn MSE (dB) CRB MLE ADOPT (a = 0.01 ℓ) DIC (a = 0.01 ℓ) ADOPT (a = 0.005 ℓ) DIC … view at source ↗
Figure 8
Figure 8. Figure 8: The averaged MSE of Uˆ and Vˆ versus normalized frequency. Comparison between DIC, the proposed method, MLE, and the CRB for SNR = 20 dB. 12 [PITH_FULL_IMAGE:figures/full_fig_p012_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Left: silicone membrane with a speckle pattern. Right: silicone membrane with a periodic pattern. Overall setup and testing protocol – As shown in [PITH_FULL_IMAGE:figures/full_fig_p014_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Experimental setup used to apply a vertical impulse (along the [PITH_FULL_IMAGE:figures/full_fig_p015_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: Estimation of the displacement along the [PITH_FULL_IMAGE:figures/full_fig_p016_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Resulting spatio-temporal displacement maps for DIC (left) and ADOPT (right) [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Time-domain and frequency-domain representations of the signal extracted along the [PITH_FULL_IMAGE:figures/full_fig_p017_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Estimated dispersion curves. Top: displacement amplitudes below 22 µm. Bottom: displacement amplitudes below 280 µm. The solid line shows the mean curve, and the shaded region corresponds to ± one standard deviation. These experimental results confirm the observations made in simulation. While DIC ex￾hibits large variance and inaccurate dispersion estimates for small displacements, converging only when de… view at source ↗
read the original abstract

This paper addresses the problem of tracking in-plane waves from image sequences using periodic surface patterns. Wave-induced deformation is modeled as a spatial phase modulation of a periodic carrier. We propose ADOPT (Analytical Demodulation of Periodic Texture), a method based on an oriented two-dimensional analytic signal to estimate displacement phase and orientation. The approach relies on a physical model describing longitudinal and transverse in-plane waves. Orientation-selective filtering isolates relevant spectral components, and phase extraction provides a stable reconstruction of the displacement field. A theoretical analysis using the Cramer--Rao bound evaluates performance limits of ADOPT. Simulations show that the proposed method outperforms state-of-the-art Digital Image Correlation (DIC) at high signal-to-noise ratios, especially for small displacements where DIC becomes limited. Moreover, ADOPT is more computationally efficient. Experiments on silicone membranes with periodic patterns confirm accurate estimation of wave fields and dispersion curves under impulsive excitation. Overall, the proposed framework provides a robust and efficient solution for wave-induced displacement estimation.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper proposes ADOPT, a method for estimating in-plane wave displacements from image sequences of periodic textures. Wave deformation is modeled as spatial phase modulation of a known carrier; an oriented 2-D analytic signal (after orientation-selective filtering) extracts the displacement via the argument of the demodulated signal. A Cramér-Rao bound analysis is presented, simulations claim outperformance over DIC especially at high SNR and small displacements, and silicone-membrane experiments under impulsive excitation are used to recover wave fields and dispersion curves.

Significance. If the central modeling assumptions hold, ADOPT offers a computationally lighter, phase-based alternative to DIC for high-precision tracking of small-amplitude waves on periodic surfaces. The combination of a physical model, analytic-signal demodulation, and CRB-derived performance limits is a clear strength; the experimental validation on real membranes further supports practical utility in wave-propagation studies.

major comments (2)
  1. [§2] §2 (physical model) and abstract: the derivation of the oriented analytic signal (Eqs. 4–6) and the subsequent CRB analysis rest on the assumption that displacement appears strictly as a spatial phase shift of a constant-amplitude periodic carrier. No analysis or simulation is provided for cases with spatially varying amplitude (e.g., local thickness or illumination changes) or overlapping sidebands from longitudinal/transverse components; under those conditions the quadrature component is no longer pure phase and the extracted displacement contains systematic bias not bounded by the reported CRB.
  2. [Simulations and experiments] Simulations and experiments section: all reported simulations use ideal sinusoidal carriers with a single dominant orientation; the silicone-membrane experiments likewise employ a single dominant pattern orientation. Consequently the interference/cross-talk case raised by the phase-modulation model is not stress-tested, leaving the claim of robust outperformance over DIC at high SNR dependent on unverified assumptions.
minor comments (2)
  1. Notation for the 2-D analytic signal and orientation filter could be clarified with an explicit block diagram or pseudocode to aid reproducibility.
  2. Figure captions should explicitly state the SNR values, displacement amplitudes, and number of Monte-Carlo trials used in the DIC comparison plots.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive and detailed comments on the modeling assumptions and validation aspects of our work. We address each major comment point by point below, proposing revisions to strengthen the manuscript where the concerns are valid.

read point-by-point responses

  1. Referee: [§2] §2 (physical model) and abstract: the derivation of the oriented analytic signal (Eqs. 4–6) and the subsequent CRB analysis rest on the assumption that displacement appears strictly as a spatial phase shift of a constant-amplitude periodic carrier. No analysis or simulation is provided for cases with spatially varying amplitude (e.g., local thickness or illumination changes) or overlapping sidebands from longitudinal/transverse components; under those conditions the quadrature component is no longer pure phase and the extracted displacement contains systematic bias not bounded by the reported CRB.

    Authors: We agree that the core derivation and CRB analysis in §2 are developed under the phase-modulation model with constant-amplitude carrier, which enables the oriented analytic signal to extract displacement via the argument. This is a deliberate modeling choice aligned with the high-contrast periodic textures used in our target applications. We acknowledge that the manuscript does not explicitly quantify bias under spatially varying amplitude or overlapping sidebands. In the revised version we will add an explicit statement of these modeling assumptions in §2 and the abstract, together with a short robustness analysis (including a targeted simulation with mild amplitude modulation) to bound the resulting phase error relative to the ideal CRB. revision: yes

  2. Referee: [Simulations and experiments] Simulations and experiments section: all reported simulations use ideal sinusoidal carriers with a single dominant orientation; the silicone-membrane experiments likewise employ a single dominant pattern orientation. Consequently the interference/cross-talk case raised by the phase-modulation model is not stress-tested, leaving the claim of robust outperformance over DIC at high SNR dependent on unverified assumptions.

    Authors: The simulations were intentionally restricted to ideal single-orientation carriers to isolate the performance gain of ADOPT over DIC under the exact conditions for which the CRB was derived, thereby providing a controlled benchmark. The silicone-membrane experiments similarly reflect typical laboratory patterns with one dominant orientation. We recognize that the current results do not directly stress-test cross-talk from multiple orientations or longitudinal/transverse sideband overlap. We will therefore augment the revised manuscript with additional simulation cases that superimpose weak secondary orientations and wave components, reporting the resulting displacement error and comparison with DIC to substantiate the robustness claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper states a physical model (wave displacement as spatial phase modulation of a known periodic carrier) as an input assumption in the abstract and §2, then applies standard oriented 2-D analytic signal processing (Eq. 4–6) to extract phase. The Cramér–Rao bound is derived under those explicit model assumptions rather than fitted to data. Simulations and experiments serve as external validation rather than re-deriving the inputs. No self-citation chains, fitted parameters renamed as predictions, or ansatz smuggled via prior work appear in the provided sections; the derivation remains self-contained against the stated model and established signal-processing results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests primarily on the domain assumption that wave deformations act as phase modulations on periodic carriers and that orientation-selective filtering isolates clean spectral components; no free parameters or invented entities are explicitly introduced in the abstract.

axioms (2)
  • domain assumption Wave-induced deformation is modeled as a spatial phase modulation of a periodic carrier
    Explicitly stated in the abstract as the basis for the demodulation approach.
  • domain assumption Orientation-selective filtering isolates relevant spectral components for phase extraction
    Described as part of the method to enable stable displacement reconstruction.

pith-pipeline@v0.9.0 · 5719 in / 1377 out tokens · 56401 ms · 2026-05-20T03:57:14.585782+00:00 · methodology

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