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

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.

• The paper develops a causality-preserving Fredholm reconstruction that transforms pulsed, harmonic, and chirp-pulsed photothermal data into wave-like signals with linear depth mapping.
• The method pairs ADMM sparse inversion with pulsed signals and truncated SVD with harmonic data, resolving simulated and experimental CFRP defects that conventional thermography blurs.
• Experiments show major time-domain CNR gains, including 34.87 for pulses and 10651.57 for chirp excitation, while homogeneous-material assumptions and virtual-wave-speed calibration remain open challenges.

Motivation and problem statement

Photothermal imaging is attractive for non-destructive testing of composite materials because it is contactless and non-invasive, but its quantitative capabilities are constrained by the parabolic character of the heat equation. Diffusion transport has no wavefront, no finite propagation velocity, and a mathematically smoothing effect on the measured signal. As a consequence, conventional thermographic depth interpretation relies on artificially defined quantities such as the diffusion length $L_{\mathrm{diff}} = \sqrt{\alpha t}$ or the frequency-domain diffusion length $\mu = \sqrt{\alpha/(\pi f)}$, both of which yield blurred boundaries and ambiguous depth assignment when material properties or instrumentation change.

The virtual-wave concept introduced by Burgholzer et al. links the diffusion equation to a wave equation via analytic continuation in the complex frequency plane, converting diffusive thermal responses into propagating signals with interpretable reflections. However, existing formulations are restricted to idealized Dirac pulse excitation modeled as an initial condition. The paper under discussion removes this restriction: it derives a generalized virtual-wave reconstruction (GVWR) framework valid for arbitrary boundary-injected excitations—pulsed, harmonic (lock-in), and chirp-pulsed—and validates it numerically and experimentally on carbon fiber-reinforced polymer (CFRP) laminates with embedded Teflon inserts (2605.03747).

Generalized theory

The starting point is the heat equation with a general source term $Q(\mathbf{r},t)$, written in Helmholtz-like operator form, paired with a scalar wave equation driven by the same temporal waveform. In the frequency domain, the diffusion field satisfies a Helmholtz equation with complex wavenumber $\sigma(\omega) = \sqrt{i\omega/\alpha}$ (with $\mathrm{Re}[\sigma] > 0$), while the virtual wave field satisfies a Helmholtz equation with real wavenumber $k(\omega) = \omega/c$. Substituting $\omega \to -ic\,\sigma(\omega)$ into the wave-domain solution yields the frequency-domain correspondence between the two fields, which the authors explicitly characterize as a mathematically constructed correspondence rather than a physical equivalence.

Transforming back to time produces a Fredholm integral equation of the first kind,

$$T(\mathbf{r},t) = \int_{-\infty}^{\infty} T_{\mathrm{virt}}(\mathbf{r},t')\, K(t,t')\, dt',$$

with an analytically evaluable kernel

$$K(t,t') = \frac{c^2}{\kappa}\frac{t'}{2\sqrt{\pi}(\alpha t)^{3/2}} \exp\!\left(-\frac{c^2 t'^2}{4\alpha t}\right) H(t)\,H(t'),$$

derived in closed form via a completing-the-square argument on the Fourier integral (Appendix A). The Heaviside factors enforce causality and encode thermodynamic irreversibility: the temperature at time $t$ aggregates contributions from all prior virtual times $\mu = \sqrt{\alpha/(\pi f)}$0.

The discretized system $\mu = \sqrt{\alpha/(\pi f)}$1 is severely ill-conditioned for three reasons stated by the authors: temporal spreading renders columns of $\mu = \sqrt{\alpha/(\pi f)}$2 nearly linearly dependent; short-time virtual-wave features are exponentially attenuated $\mu = \sqrt{\alpha/(\pi f)}$3, analogous to ultrasound attenuation; and direct inversion amplifies noise into unphysical oscillations. Two regularization strategies are adopted depending on excitation type:

• ADMM with $\mu = \sqrt{\alpha/(\pi f)}$4 sparsity for pulsed excitations, where the virtual-wave signal consists of a small number of temporally localized reflection events ($\mu = \sqrt{\alpha/(\pi f)}$5). The regularization parameter is selected via the L-curve criterion.
• Truncated SVD for harmonic excitation, where the response is spectrally compact and its energy concentrates in the leading singular subspace, satisfying the Picard condition. Since singular values decay approximately as $\mu = \sqrt{\alpha/(\pi f)}$6 with $\mu = \sqrt{\alpha/(\pi f)}$7, truncation before the noise floor intersects the singular spectrum enforces the thermodynamic irreversibility constraint rather than merely regularizing numerically. The analysis yields a diffusion-limited bandwidth $\mu = \sqrt{\alpha/(\pi f)}$8 and a minimum resolvable virtual time $\mu = \sqrt{\alpha/(\pi f)}$9, explicitly analogous to a diffraction limit.

A notable conceptual contribution is the explicit mapping between excitation waveform class and inversion strategy: ADMM and T-SVD are presented not as interchangeable numerical tools but as corresponding to distinct assumptions—temporal sparsity versus spectral compactness—of the virtual field.

Numerical validation

Simulations use stainless steel plates ($Q(\mathbf{r},t)$0, defects 1 mm deep and 10 mm wide), with excitation modeled as a realistic 2 ms square pulse rather than an ideal Dirac function. Under conventional processing, the defect appears already in the first frame and smears progressively with time—a direct manifestation of the Markovian nature of diffusion—so that neither amplitude nor phase tomograms exhibit sharp boundaries, and reconstructed depths deviate from the true value under calibrated colormaps. After GVWR with ADMM, the defect region shows a clearly resolved reflected echo, and the sound area exhibits a back-surface echo; the authors emphasize that recovering the sample thickness is impossible from any raw photothermal signal because diffusion transport is one-directional.

For modulated excitations, harmonic (1 Hz) cases are inverted with T-SVD and chirp-pulsed cases (0.2–0.6 Hz sweep, 12 s duration, 2 ms pulse width) with ADMM after peak-based separation exploiting linearity of heat diffusion. Chirp-pulsed excitation yields highly compressed outputs with negligible side-lobe power, consistent with thermal-diffusion-wave radar behavior, and resolves two subsurface defects in the frequency domain.

Experimental results on CFRP

Experiments employ a cooled MWIR camera (ImageIR 9300) at 100 Hz frame rate with 52 µm pixel pitch, coaxial laser illumination at 940 nm through a dichroic mirror, and 403 measurement points at 0.743 mm spacing. The specimen is a CFRP laminate stack ($Q(\mathbf{r},t)$1) with a sandwiched Teflon insert. Three waveforms are compared: 120 ms pulses, 0.2 Hz lock-in, and chirp-pulsed excitation (0.2–0.6 Hz, 16 s, 100 ms pulse width).

Raw data show the expected limitations: pulse excitation reveals only obscured delamination contrast decaying by 3.33 s without resolvable boundaries; lock-in excitation fails to detect the defect in the time domain entirely; chirp-pulsed excitation provides higher contrast but with spectral aliasing artifacts. After virtual-wave reconstruction, all three modalities produce wave-like signals with localized reflection peaks absent from the original diffusion data.

Quantitative contrast-to-noise ratios ($Q(\mathbf{r},t)$2) demonstrate the magnitude of the improvement:

Domain	Pulse (raw → VW)	Lock-in (raw → VW)	Chirp-pulsed (raw → VW)
Time	1.17 → 34.87	0.03 → 2.50	4.20 → 10651.57
Frequency	4.30 → 12.88	3.29 → 2.70	16.85 → 2.96

Two features of this table warrant emphasis. First, the time-domain CNR under chirp-pulsed excitation increases by more than three orders of magnitude to 10651.57—an exceptionally large figure that should be interpreted cautiously, since CNR values of this size depend strongly on the sound-region standard deviation approaching zero after regularization-induced smoothing. Second, frequency-domain CNR actually decreases after virtual-wave reconstruction for lock-in and chirp-pulsed cases (3.29 → 2.70 and 16.85 → 2.96); the authors attribute the moderate frequency-domain gains to representations dominated by specific effective components rather than temporally resolved events, but the regression for chirp-pulsed data is not separately explained and represents a genuine trade-off of the method.

Comparison with existing coherence tomography

Against frequency-multiplexed photothermal coherence tomography (FM-PCT) and enhanced truncated-correlation photothermal coherence tomography (eTC-PCT), GVWR offers two structural advantages. First, FM-PCT applied to chirp-pulsed data suffers spectral aliasing—the delamination appears at multiple discontinuous depth locations—whereas GVWR reconstructs a continuous, well-defined delamination profile under both pulse and chirp-pulsed excitation. Second, and more fundamentally, conventional techniques inherit a nonlinear depth mapping ($Q(\mathbf{r},t)$3 in time domain; $Q(\mathbf{r},t)$4 in frequency domain), whereas GVWR recovers a linear mapping $Q(\mathbf{r},t)$5, directly analogous to ultrasonic time-of-flight. On this basis the authors claim GVWR constitutes a "true tomographic" framework, a claim supported by the cross-sectional slices showing delamination confined to a thin depth region consistent across both excitation types.

Limitations and open questions

Several restrictions are acknowledged or implicit. The special virtual wave transform (SVWT)—the prior Dirac-pulse formulation reproduced in Appendix B—completely fails under lock-in and chirp-pulsed excitation, which motivates the generalization but also underscores how sensitive these transforms are to excitation-model mismatch. Both SVWT and the present GVWT assume homogeneous, isotropic media with constant diffusivity; layered composites, anisotropy, and spatially varying parameters break the separability required for the kernel, and extension to heterogeneous/anisotropic materials remains future work. The ill-posedness is fundamental rather than incidental: diffusion exponentially suppresses high-frequency content, so deep-feature resolution is bounded by the SNR-dependent limit $Q(\mathbf{r},t)$6 derived above, and no regularization can recover information below that floor. The anomalous frequency-domain CNR reduction for broadband excitations is reported but not fully analyzed. Finally, the linear depth relation $Q(\mathbf{r},t)$7 presumes a well-defined virtual wave speed $Q(\mathbf{r},t)$8, which is a free parameter of the construction rather than a material property; how $Q(\mathbf{r},t)$9 should be chosen or calibrated for quantitative absolute depth in new materials is left open.

Conclusion

This work extends the virtual-wave transformation of photothermal imaging from idealized Dirac excitation to arbitrary boundary-driven waveforms via a causality-preserving Fredholm integral formulation, pairing ADMM sparse inversion with pulsed data and truncated SVD with harmonic data according to the temporal structure of the excitation. Simulations and CFRP experiments show conversion of blurred diffusive responses into fields with genuine wavefronts and interface echoes, order-of-magnitude time-domain CNR gains, elimination of spectral aliasing relative to FM-PCT, and a linear depth mapping analogous to ultrasonics. The framework's validity currently rests on homogeneity and isotropy assumptions, and its quantitative depth accuracy depends on the treatment of the free virtual wave speed—both of which define the immediate open problems for industrial deployment.