arxiv: 2605.03747 · v1 · submitted 2026-05-04 · ⚛️ physics.app-ph

Recognition: unknown

Generalized Virtual-Wave Theory for Photothermal Coherence Tomography under Arbitrary Excitation Toward Non-Contact Industrial Inspection of Composite Materials

Pengfei Zhu , Julien Lecompagnon , Philipp Daniel Hirsch , Mathias Ziegler

Authors on Pith no claims yet

Pith reviewed 2026-05-08 02:22 UTC · model grok-4.3

classification ⚛️ physics.app-ph

keywords virtual wave theoryphotothermal tomographycomposite inspectionheat diffusionFredholm integralnon-destructive testingdepth localization

0 comments

The pith

A Fredholm integral maps arbitrary thermal excitation responses to virtual wave fields obeying a wave equation.

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

The paper derives a way to convert measured heat diffusion patterns, produced by any form of heating such as pulses or chirps, into fields that behave like propagating waves. This conversion starts from the heat equation with a general source term and produces an integral relation that respects causality and the fact that heat flow cannot be reversed. A sympathetic reader would care because ordinary thermal images blur rapidly and make it hard to tell how deep a defect lies inside a composite, while the mapped wave fields show distinct wavefronts and reflections that support standard imaging methods. The inverse step is solved by ADMM or truncated SVD to recover stable reconstructions from noisy data. Experiments on carbon-fiber samples with embedded defects then demonstrate sharper boundaries and more reliable depth estimates than conventional thermography.

Core claim

Starting from the heat equation with a general source term, we derive a Fredholm integral mapping between the measured diffusion field and a virtual wave field governed by a wave equation, explicitly enforcing causality and thermodynamic irreversibility. The resulting ill-posed inverse problem is solved using ADMM or truncated SVD, depending on the excitation characteristics. Numerical and experimental results demonstrate that the proposed method converts blurred thermal responses into wave-like fields with clear wavefronts and reflections, enabling improved depth localization and tomographic reconstruction.

What carries the argument

Fredholm integral mapping from the measured diffusion field to a virtual wave field governed by a wave equation

If this is right

Converts blurred thermal responses into wave-like fields with clear wavefronts and reflections.
Enables improved depth localization and tomographic reconstruction of subsurface features.
Yields enhanced contrast and sharper boundaries compared with conventional thermographic techniques.
Provides more reliable depth interpretation in carbon fiber reinforced polymer samples.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same mapping could be inserted into existing wave-based tomography codes already used in ultrasonics.
Adaptive choice of excitation waveform might be used to emphasize defects at particular depths.
The approach may apply to other diffusive transport problems where a virtual wave field could be defined.

Load-bearing premise

The Fredholm integral mapping derived from the heat equation remains accurate and yields stable reconstructions under real experimental noise for arbitrary excitations without further approximations.

What would settle it

A controlled experiment in which the reconstructed virtual wave fields fail to produce the expected reflections or arrival times from known embedded defects whose positions are verified by independent methods.

Figures

Figures reproduced from arXiv: 2605.03747 by Julien Lecompagnon, Mathias Ziegler, Pengfei Zhu, Philipp Daniel Hirsch.

**Figure 1.** Figure 1: The transformation from diffusion-wave field to virtual-wave field for Dirac pulse excitation. a Schematic image of virtual wave method. b Temperature variation between defect area and sound area. c Virtual-wave variation between defect area and sound area. d Virtual-wave tomography results. e Time-domain photothermal tomography results. f Amplitudebased tomography results. g Phase-based tomography result… view at source ↗

**Figure 2.** Figure 2: The transformation from diffusion-wave field to virtual-wave field for modulated excitation. a Temperature signals under lock-in excitations in the time and frequency domain. b Reconstructed virtual wave signals under lock-in excitations in the time and frequency domain. c Reconstructed virtual wave field under lock-in excitations in the time and frequency domain. d Reconstructed virtual wave field under c… view at source ↗

**Figure 3.** Figure 3: The photothermal experimental setup and results. a Experimental setup. b The excitation signal waveform. c The photograph of tested sample. d Original results from pulse excitation. e Original results from lock-in excitation. f Original results from chirp-pulsed excitation. 4.2. Experimental results To validate the proposed generalized virtual wave theory, photothermal experiments were conducted. The laser… view at source ↗

**Figure 4.** Figure 4: The signal analysis for photothermal experiments. a Temporal and frequency-domain signal under pulse excitation. b Temporal and frequency-domain signal under lock-in excitation. c Temporal and frequency-domain signal under chirppulsed excitation. d Frequency-domain results from pulse excitation. e Frequency-domain results from lock-in excitation. f Frequency-domain results from chirp-pulsed excitation. s … view at source ↗

**Figure 5.** Figure 5: The virtual wave reconstruction of photothermal experiments. a Temporal and frequency-domain signal under pulse excitation. b Time-domain virtual wave images under pulse excitation. c Frequency-domain virtual wave images under pulse excitation. d Temporal and frequency-domain signal under lock-in excitation. e Time-domain virtual wave images under lock-in excitation. f Frequency-domain virtual wave images … view at source ↗

**Figure 6.** Figure 6: The comparison between generalized virtual wave reconstruction (GVWR) and frequency multiplexed photothermal coherence tomography (FM-PCT) technique. a Photothermal tomography results from FM-PCT along various depth layers for pulse and chirp-pulsed excitations. b Cross-sectional results from FM-PCT for pulse and chirppulsed excitations. c Photothermal tomography results from GVWR for pulse excitation. d … view at source ↗

**Figure 7.** Figure 7: The special virtual wave transform results for simulation and photothermal experiments. a Virtual wave transform results of pulse excitation in simulation. b Virtual wave transform results of lock-in excitation in simulation. c Virtual wave transform results of chirp-pulsed excitation in simulation. d Virtual wave profiles vary with time of defect and sound areas in simulation under pulse excitation. e Vir… view at source ↗

read the original abstract

Photothermal imaging is a powerful noncontact and nondestructive technique for subsurface inspection of composite materials, yet its performance is fundamentally limited by the diffusive and irreversible nature of heat transport, leading to severe image blurring and ambiguous depth interpretation. The concept of virtual waves provides a route to overcome this limitation by linking diffusion fields to propagating wave fields, but existing approaches are largely restricted to idealized impulsive excitation. Here, we propose a generalized virtual-wave photothermal tomography framework that extends the diffusion-to-wave transformation to arbitrary boundary excitations, including pulsed, harmonic, and chirped waveforms. Starting from the heat equation with a general source term, we derive a Fredholm integral mapping between the measured diffusion field and a virtual wave field governed by a wave equation, explicitly enforcing causality and thermodynamic irreversibility. The resulting ill-posed inverse problem is solved using ADMM or truncated SVD, depending on the excitation characteristics. Numerical and experimental results demonstrate that the proposed method converts blurred thermal responses into wave-like fields with clear wavefronts and reflections, enabling improved depth localization and tomographic reconstruction. Experiments on carbon fiber reinforced polymer samples with embedded defects show enhanced contrast, sharper boundaries, and more reliable depth interpretation compared with conventional thermographic techniques. This work establishes a unified and physically grounded framework for wave-based photothermal tomography under realistic excitation conditions.

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.

Desk Editor's Note private letter to a colleague

The paper's real contribution is extending virtual-wave photothermal tomography to arbitrary excitations through a Fredholm integral from the heat equation, with experiments showing clearer defect imaging in composites, though the inversion's noise stability is the main unaddressed weakness.

read the letter

The one or two things your colleague should know are that this generalizes the virtual-wave idea from pulses only to any waveform via a direct Fredholm mapping, and that the carbon-fiber experiments indicate better contrast and depth localization than plain thermography. The mapping itself looks like a clean step forward without obvious circularity.

Referee Report

3 major / 2 minor

Summary. The manuscript proposes a generalized virtual-wave theory for photothermal coherence tomography applicable to arbitrary excitations. Starting from the heat equation with a general source term, it derives a Fredholm integral equation mapping the measured temperature field to a virtual wave field satisfying a wave equation with built-in causality and irreversibility. The inverse problem is solved using either ADMM or truncated SVD depending on the excitation waveform. Numerical simulations and experiments on carbon fiber reinforced polymer samples with embedded defects demonstrate improved wavefront clarity, depth localization, and defect contrast compared to standard thermographic methods.

Significance. If the derived Fredholm mapping remains accurate and the regularized inversions produce stable, physically meaningful virtual wave fields under experimental noise levels for arbitrary excitations, the work would offer a unified, physically grounded approach to overcome the diffusive limitations in photothermal imaging. This could significantly advance non-contact, nondestructive inspection of composite materials by providing wave-like features for better tomographic reconstruction. The experimental validation on real samples and use of conventional solvers are positive aspects, though the lack of detailed error analysis limits the current strength of the claims.

major comments (3)

[§2] §2 (derivation of the Fredholm mapping): The central claim that the integral mapping from the heat equation with general source term explicitly enforces causality and thermodynamic irreversibility is load-bearing, yet the manuscript provides neither the explicit kernel expression for arbitrary excitations (pulsed, harmonic, chirped) nor the intermediate steps showing how the source term is incorporated. This omission prevents verification that the operator remains well-defined and distinct across waveform types.
[§4] §4 (inverse solvers): The choice of ADMM versus truncated SVD is stated to depend on excitation characteristics, but no condition-number bounds, singular-value spectrum analysis, or noise-propagation estimates (for the 1–5 % noise typical in thermography) are given. Without these, the assertion that the reconstructions remain stable and free of non-causal artifacts under realistic conditions cannot be assessed and directly impacts the reliability of the depth-localization results.
[§5] §5 (experimental validation): The reported improvements in contrast, boundary sharpness, and depth interpretation on CFRP samples are presented qualitatively. No quantitative metrics—such as defect-depth error, SNR ratios, or statistical comparison against conventional pulsed thermography—are supplied, leaving the practical advantage of the generalized framework unsupported by measurable evidence.

minor comments (2)

[Abstract] The abstract introduces “photothermal coherence tomography” without a brief definition or reference; adding one sentence would aid readers outside the immediate subfield.
[Figures] Figure captions for the wavefront visualizations would benefit from explicit labels for arrival times or reflection events to illustrate the claimed causality enforcement.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the detailed and constructive comments on our manuscript. We have addressed each of the major comments below and will incorporate the suggested revisions to improve the clarity and rigor of the paper.

read point-by-point responses

Referee: [§2] §2 (derivation of the Fredholm mapping): The central claim that the integral mapping from the heat equation with general source term explicitly enforces causality and thermodynamic irreversibility is load-bearing, yet the manuscript provides neither the explicit kernel expression for arbitrary excitations (pulsed, harmonic, chirped) nor the intermediate steps showing how the source term is incorporated. This omission prevents verification that the operator remains well-defined and distinct across waveform types.

Authors: We agree that including the explicit kernel expressions and the intermediate derivation steps is essential for allowing readers to verify the mapping. In the revised manuscript, we will expand Section 2 to provide the full derivation starting from the heat equation with a general source term, deriving the Fredholm integral, and specifying the kernel for different excitation waveforms including pulsed, harmonic, and chirped. This will demonstrate how the source term is incorporated and how causality and irreversibility are enforced. revision: yes
Referee: [§4] §4 (inverse solvers): The choice of ADMM versus truncated SVD is stated to depend on excitation characteristics, but no condition-number bounds, singular-value spectrum analysis, or noise-propagation estimates (for the 1–5 % noise typical in thermography) are given. Without these, the assertion that the reconstructions remain stable and free of non-causal artifacts under realistic conditions cannot be assessed and directly impacts the reliability of the depth-localization results.

Authors: We acknowledge that additional quantitative analysis of the inverse problem solvers would strengthen the manuscript. We will add to Section 4 the condition-number bounds for the operators, plots or descriptions of the singular-value spectra, and estimates of noise propagation for noise levels of 1–5% typical in thermographic experiments. This will provide evidence for the stability of the reconstructions and the absence of non-causal artifacts. revision: yes
Referee: [§5] §5 (experimental validation): The reported improvements in contrast, boundary sharpness, and depth interpretation on CFRP samples are presented qualitatively. No quantitative metrics—such as defect-depth error, SNR ratios, or statistical comparison against conventional pulsed thermography—are supplied, leaving the practical advantage of the generalized framework unsupported by measurable evidence.

Authors: We appreciate the referee highlighting the need for quantitative support. In the revised manuscript, we will include quantitative metrics in Section 5, such as defect-depth errors, signal-to-noise ratio (SNR) ratios, and statistical comparisons with conventional pulsed thermography methods. These will be computed from the experimental data on the CFRP samples to provide measurable evidence of the improvements in contrast, sharpness, and depth localization. revision: yes

Circularity Check

0 steps flagged

Derivation from heat equation is independent; no circularity detected.

full rationale

The central derivation starts from the standard heat equation with a general source term and produces a Fredholm integral operator that maps the diffusion field to a virtual wave field obeying the wave equation while enforcing causality and irreversibility. This is a direct mathematical construction, not a redefinition or fit. The subsequent ill-posed inversion is handled by standard, off-the-shelf regularizers (ADMM or truncated SVD) whose choice depends on excitation type but does not reduce the mapping itself to a fitted parameter or self-referential definition. No load-bearing self-citations, uniqueness theorems imported from prior author work, or ansatzes smuggled via citation appear in the abstract or described chain. The result is therefore self-contained against external benchmarks and does not collapse to its inputs by construction.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 1 invented entities

The framework starts from the standard heat equation (domain assumption) and introduces the virtual wave field as a mathematical construct. Regularization parameters in the solvers are free parameters chosen to stabilize the inversion.

free parameters (2)

regularization parameter in ADMM
Balances data fidelity and solution smoothness for the ill-posed inverse problem; value not specified in abstract.
truncation threshold in SVD
Determines retained singular values for regularization; chosen based on excitation characteristics.

axioms (1)

domain assumption Temperature evolution obeys the heat equation with a general source term for arbitrary excitations
Explicit starting point for deriving the Fredholm integral mapping.

invented entities (1)

virtual wave field no independent evidence
purpose: Transforms diffusive thermal response into a propagating wave field with clear wavefronts and reflections
Introduced to overcome blurring and enable tomographic reconstruction while enforcing causality and irreversibility.

pith-pipeline@v0.9.0 · 5544 in / 1474 out tokens · 105021 ms · 2026-05-08T02:22:16.993116+00:00 · methodology

discussion (0)

Forward citations

Cited by 1 Pith paper

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Reference graph

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