FDTRImageEnhancer: Microstructure Recovery

• FDTRImageEnhancer is an open-source framework that enhances thermal conductivity mapping by combining Gaussian convolution with microstructure-aware deep learning.
• It employs a surrogate model for FDTR blurring, region-based clustering from high-resolution structural images, and a physics-informed neural network to solve the inverse problem.
• The framework recovers bulk thermal properties with errors under 0.5% while partially revealing grain-boundary contrasts in synthetic finite-element tests.

FDTRImageEnhancer is an open-source computational framework for improving thermal conductivity mapping from Frequency Domain ThermoReflectance (FDTR) phase data by combining a physics-based Gaussian convolution abstraction with microstructure-aware deep learning (Odufisan, 8 Aug 2025). It is formulated to bridge a multiscale mismatch: structural images can resolve grain and grain-boundary features at far higher spatial resolution than FDTR thermal maps, whose effective conductivity images are blurred by pump/probe spatial averaging. The framework therefore couples a surrogate model of FDTR blurring, a microstructure-constrained parameterization based on clustered structural images, and a physics-informed neural network that jointly matches phase data and analytically inverted conductivity maps. In the reported synthetic finite-element study, it recovers bulk conductivity values within less than $0.5\%$ error and qualitatively resolves grain-boundary effects that are visually obscured in conventional FDTR inversions (Odufisan, 8 Aug 2025).

1. Problem setting and conceptual scope

FDTRImageEnhancer addresses a specific inverse problem in thermal metrology rather than a generic image enhancement problem. In standard FDTR, thermal conductivity $\kappa$ and interface conductance $G$ are inferred by fitting a forward thermal model to measured phase shifts across modulation frequencies. The resulting conductivity image is an “effective” map that is spatially blurred by the finite pump and probe spot profiles, so sharp features such as grain-boundary thermal conductivity drops can become visually invisible in the inverted conductivity map (Odufisan, 8 Aug 2025).

A recurrent misunderstanding is to treat the framework as a denoiser or post-processing visual filter. The method is instead described as a microstructure-aware inverse solver. Its purpose is to reconstruct conductivity variations that are consistent with both the measurement physics and the structural prior. In that sense, the “enhancement” in FDTRImageEnhancer refers to physically informed recovery of hidden thermal contrast, not merely cosmetic sharpening.

The motivating multiscale gap is explicit. Structural images such as EBSD maps can resolve grain and grain-boundary features at nanometer scales, while FDTR maps are much blurrier because the pump and probe beams spatially average the thermal response over a finite spot size. FDTRImageEnhancer is designed to use the structural image as a constraint so that low-resolution thermal information can be interpreted in a microstructure-aware way.

2. FDTR forward abstraction and Gaussian spatial averaging

The paper frames FDTR as a measurement in which a sinusoidally modulated pump laser heats the sample and a probe laser detects the phase lag in the surface temperature response. The analytical phase is written as

$$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$

Within the framework, the central approximation is that FDTR spatial averaging can be represented as a convolution of the high-resolution structural conductivity field with a Gaussian kernel (Odufisan, 8 Aug 2025). The conductivity mapping is expressed as

$\kappa^{FDTR} = \kappa^{struct} * \Psi$

with

$$\Psi = \exp\left(\frac{-r^2}{\sigma_1}\right)\exp\left(\frac{-r^2}{\sigma_2}\right).$$

Because the product of two Gaussians is still a Gaussian, the abstraction effectively introduces two smoothing parameters, $\sigma_1$ and $\sigma_2$, that emulate the finite pump and probe spot sizes. The paper notes that these parameters depend on image resolution in pixels. The appendix justifies the convolutional surrogate through the Hankel-transform structure of radially symmetric FDTR response functions, including the identity

$$f * y = \mathcal{H}_0^{-1}\!\left(\mathcal{H}_0(f)\cdot \mathcal{H}_0(y)\right),$$

and, at the beam center $r=0$,

$\kappa$0

This abstraction is deliberately simplified. The paper explicitly presents it as a proof-of-concept surrogate for FDTR blurring rather than as a complete replacement for the underlying thermal transport physics. Its role is to make the inverse problem differentiable and computationally tractable while retaining the dominant effect of pump/probe spatial averaging.

3. Microstructure-aware reduction and learning formulation

A major obstacle in recovering a high-resolution conductivity field is dimensionality. If $\kappa$1 is allowed to vary independently at every pixel of a $\kappa$2 structural image, the inverse problem becomes highly underdetermined. FDTRImageEnhancer addresses this by segmenting the structural image with k-means clustering into a small number of regions (Odufisan, 8 Aug 2025).

In the demonstrated test case, that reduction produced only three distinct conductivity regions: a left bulk region, a right bulk region, and a grain-boundary region. This microstructure-aware reduction preserves morphology while collapsing the search space from pixelwise values to region-level parameters. The framework therefore does not learn an unconstrained thermal image; it learns conductivity values attached to structurally defined regions.

The learning model is described as a fully connected feedforward neural network with tanh activations, modest depth and width, and learnable region-wise conductivity parameters. For FDTR, the usual PDE residual used in many PINNs is replaced by a surrogate physics loss based on the Gaussian convolution abstraction. The main objective is

$\kappa$3

The weighted form is

$\kappa$4

The paper notes that the physics term can dominate numerically because thermal conductivity values are much larger than phase values. Conceptually, the network is trained to infer a conductivity field that is simultaneously structurally plausible, compatible with the surrogate FDTR spatial averaging, and consistent with the measured or analytically recovered thermal response.

4. Analytical inversion, synthetic data, and reported recovery accuracy

The framework is validated on synthetic FDTR data generated from finite-element simulations. The workflow includes a training case for fitting the Gaussian smoothing model and a more difficult test case featuring a heterointerface or grain boundary (Odufisan, 8 Aug 2025). The finite-element setup provides structural ground-truth images, frequency-dependent phase maps, and analytically inverted conductivity and conductance maps. The reported phase maps are shown at $\kappa$5 and $\kappa$6 MHz, with random noise added to emulate experimental uncertainty.

For the conductivity-smoothing fit, the representative best-fit parameters are

$\kappa$7

The paper reports an RMSE of $\kappa$8 for this Gaussian smoothing fit to the synthetic conductivity map. That analytical inversion then supplies the blurred thermal supervision signal used during learning.

For the more difficult test case, the ground-truth conductivities are $\kappa$9 in the left region, $G$0 in the right region, and $G$1 at the grain boundary. The learned values are summarized below (Odufisan, 8 Aug 2025).

Region	Ground truth	Learned
Left bulk	$G$2	$G$3, $G$4 error
Right bulk	$G$5	$G$6, $G$7 error
Grain boundary	$G$8	$G$9, $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$0 error

These results establish an asymmetry in recoverability. Bulk regions are recovered extremely accurately, with errors below $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$1, while the thin grain-boundary region is strongly underestimated. The overall convergence RMSE is reported as

$$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$2

Training used $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$3 epochs, required about $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$4 hours runtime, and was executed on an A100 GPU via Google Colab Pro+.

5. Qualitative behavior and what the framework actually reveals

The main qualitative claim of FDTRImageEnhancer is not that the low-resolution FDTR conductivity map becomes visually sharp in a conventional image-processing sense, but that the learned conductivity evolution reveals microstructural thermal contrast hidden by spatial averaging (Odufisan, 8 Aug 2025). In the analytically inverted FDTR conductivity map, the transition between the two bulk regions appears smooth, and the grain boundary is visually obscured. During neural-network training, however, the conductivity evolution reportedly proceeds from smooth bulk-like behavior toward a progressively sharper deviation at the boundary as the physics loss dominates.

The paper notes that grain-boundary features became distinguishable after roughly $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$5 epochs and that the physics loss decayed approximately exponentially. Figures tracking conductivity versus epoch show the grain-boundary drop becoming clearer over training, even though the low-resolution thermal image itself does not visibly resolve it. This is the central qualitative success of the method: extracting microstructural thermal contrast that is hidden in the thermal image but encoded indirectly through the combination of phase measurements and structural prior.

A plausible implication is that the method is most useful when the structural prior is informative and the target thermal heterogeneity aligns with segmented microstructural regions. The reported results support that interpretation for bulk regions and only partially for the grain boundary, whose conductivity was not accurately recovered.

6. Assumptions, limitations, reproducibility, and broader applicability

The paper is explicit that FDTRImageEnhancer is a proof-of-concept with substantial simplifying assumptions (Odufisan, 8 Aug 2025). The FDTR physics is reduced to a two-parameter Gaussian convolution. The structural image resolution of $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$6 may still be insufficient to constrain sharp interfacial features. Interface conductance $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$7 was fixed uniformly across the structure in the test case, and density and heat capacity were held constant. The poor recovery of grain-boundary conductivity indicates that the inverse problem remains underconstrained for thin interfacial regions.

The implementation was also limited by compute resources. The paper suggests that higher-resolution EBSD images and more powerful HPC runs would improve performance. It also proposes several future directions: richer learned convolution kernels, supervised physics-constrained CNNs, per-region fitting of $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$8, more extensive convergence and sensitivity studies, and better parallelization and memory efficiency.

Reproducibility is a stated objective. The code is open-source, example datasets are included, and the full Python implementation is hosted at the project GitHub repository. The documented workflow is modular: preprocess and cluster the structural or EBSD image, estimate or fit $$\phi^{analytical} = \tan^{-1}\left(\frac{\Im[\hat{H}(\omega)]}{\Re[\hat{H}(\omega)]}\right).$$9, format phase data by frequency, run analytical inversion to obtain $\kappa^{FDTR} = \kappa^{struct} * \Psi$0 and $\kappa^{FDTR} = \kappa^{struct} * \Psi$1 maps, and then run the main FDTRImageEnhancer script while monitoring $\kappa^{FDTR} = \kappa^{struct} * \Psi$2 convergence.

Although demonstrated on FDTR, the paper presents the method as intentionally general. Its transferable principle is to represent measurement-induced spatial averaging with a physics-informed convolution and then solve the inverse problem with a microstructure-constrained learning model. The authors explicitly suggest extension to other inverse thermal transport and mechanics problems, including extracting localized strain from fracture images, quasi-brittle damage localization tasks, and inverse heat conduction problems in which measurement resolution is coarser than structural information. This suggests that FDTRImageEnhancer is best understood as a hybrid inverse-modeling framework rather than a domain-specific thermal imaging utility alone.