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Projection-based coupling of infrared thermography and stereocorrelation-based digital image correlation

This paper proposes a projection-based coupling method that utilizes a pinhole camera model and radial basis functions to accurately map infrared thermography temperature data onto 3D surface coordinates from stereocorrelation-based digital image correlation, enabling full-field analysis of deformation and thermal gradients on curved surfaces.

Original authors: Jendrik-Alexander Tröger, Lutz Müller-Lohse, Stefan Hartmann

Published 2026-06-30

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CC BY 4.0

Original authors: Jendrik-Alexander Tröger, Lutz Müller-Lohse, Stefan Hartmann

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to understand how a piece of metal bends and heats up when you twist it. To do this, you need two different kinds of "eyes":

The Shape Eye (Digital Image Correlation): This uses regular cameras to take 3D pictures of the object's surface. It tells you exactly where every tiny point on the object is moving in space. Think of it as a 3D GPS for the surface.
The Heat Eye (Infrared Thermography): This uses a special heat-sensing camera. It sees the temperature of the surface but only in a flat, 2D picture. It's like a thermal map, but it doesn't know if the surface is curved or flat; it just sees pixels of heat.

The Problem:
Usually, these two "eyes" speak different languages. The Shape Eye says, "Point A is here in 3D space." The Heat Eye says, "Pixel B is hot." If the object is flat, it's easy to match them up. But if the object is curved (like a tube or a half-shell), matching the 3D point to the 2D heat pixel becomes a nightmare. You might think a point is hot, but you're actually looking at the wrong spot because the surface is bent.

The Solution (The Paper's Big Idea):
The authors created a "universal translator" using a mathematical trick called the Pinhole Camera Model.

Think of this like a shadow puppet show.

Imagine you have a 3D puppet (the object) and a light source (the camera).
The authors figured out exactly how to project the 3D coordinates from the Shape Eye onto the 2D screen of the Heat Eye.
They did this by taking a single photo of a special "reference object" (a brass shape with dots on it) that both cameras could see. This allowed them to calculate a Projection Matrix—essentially a set of instructions that says, "If a point is at 3D location X, it will appear at 2D pixel Y on the heat camera."

Once they have this map, they can take the 3D coordinates of the object and instantly know exactly what the temperature is at that specific 3D point, even if the surface is curved.

The "Superpower" (Interpolation):
The paper doesn't just stop at matching points. They also used a mathematical tool called Radial Basis Functions (RBFs).

Imagine you have a few scattered dots of temperature data on a curved surface. RBFs are like a magical smoothing blanket that stretches over those dots to create a continuous, smooth temperature map.

Why is this cool? Because this smooth map allows the computer to calculate things that are usually impossible to measure directly:
- Temperature Gradients: How fast the heat is changing as you move across the curve (like feeling the slope of a hill).
- Temperature Rates: How fast the heat is changing over time (like feeling how quickly a cup of coffee is cooling down).

The Experiments:
The team tested this on two things:

A Plastic Half-Shell: They heated it up. Because plastic is a poor conductor of heat, the heat stayed near the source. Their method successfully mapped the heat to the curved shape and calculated how fast the temperature was changing, though the plastic's slow reaction made the "rate" calculation a bit noisy.
A Metal Tube: They twisted and pulled a metal tube. Metal conducts heat well. Their method successfully mapped the heat and stress on the curved tube, showing exactly where the metal was heating up due to the twisting and how fast that heat was rising.

The Bottom Line:
This paper presents a method to take two separate, independently calibrated camera systems (one for shape, one for heat) and glue their data together perfectly using a simple camera model. This allows scientists to see the full story of how curved objects deform and heat up simultaneously, providing a rich dataset for understanding materials without needing to build a custom, expensive camera rig.

Technical Summary: Projection-Based Coupling of Infrared Thermography and Stereocorrelation-Based Digital Image Correlation

Problem Statement
In experimental solid mechanics, full-field measurement techniques like Digital Image Correlation (DIC) and Infrared Thermography (IRT) are standard for analyzing surface deformation and temperature fields, respectively. However, a significant challenge arises when coupling these datasets, particularly for non-flat (curved) surfaces. DIC, especially stereocorrelation-based 3D-DIC, provides three-dimensional material point coordinates in a Lagrangian frame. In contrast, IRT measures pixel temperatures in an Eulerian (two-dimensional) frame. Expressing both displacement and temperature fields within the same Lagrangian frame (i.e., at specific material points) is essential for constitutive model calibration, model discovery, and numerical validation. Existing approaches often rely on complex hybrid multiview frameworks that require integrated calibration or specific hardware setups (e.g., dichroic mirrors), or they are limited to planar specimens where through-thickness gradients are neglected. Furthermore, many commercial systems utilize subset-based (local) DIC, making it difficult for researchers without access to core code to integrate these systems for thermo-mechanical analysis.

Methodology
The authors propose an external, projection-based coupling method that utilizes the pinhole camera model to map two-dimensional temperature data from IRT to three-dimensional point coordinates from stereocorrelation-based DIC. The methodology consists of three primary stages:

Projection Matrix Calibration:
- Two independently calibrated industrial-grade systems (one for IRT, one for 3D-DIC) are used.
- A single image of a three-dimensional reference object (an "open-book" target with point markers) is captured simultaneously by both systems.
- The reference object is made of brass, allowing the markers to be visible in both visible light (for DIC) and infrared (for IRT) due to emissivity differences, eliminating the need for prior heating of the target.
- Using the Direct Linear Transformation (DLT), a homogeneous linear system ($Am=0$) is assembled and solved via singular value decomposition to determine the projection matrix $M$ of the pinhole camera model. This matrix relates 3D object coordinates to 2D IR pixel coordinates.
Global Interpolation (Spatial and Temporal):
- To compute spatial gradients and temporal rates, a continuous surface description is required. The authors employ Radial Basis Functions (RBFs) as a global interpolation ansatz.
- Spatial Interpolation: Discrete DIC point coordinates are interpolated over surface parameters ( $\Psi$ ) to reconstruct the curved surface. Simultaneously, discrete pixel temperatures from the IR image are interpolated over pixel coordinates ( $x^\Theta$ ).
- Temporal Interpolation: The interpolation is extended to the time domain. The coefficients of the spatial RBFs are themselves interpolated using RBFs over time. This allows for the reconstruction of continuous fields in both space and time.
Computation of Derivatives:
- Temperature Gradient: The in-plane temperature gradient is computed by applying the chain rule to the interpolated temperature field, utilizing the derivative of the projection equation to map pixel coordinate derivatives to real-world coordinates.
- Temperature Rate: The material time derivative of temperature (temperature rate) is calculated by differentiating the interpolated field with respect to time, accounting for both local changes and convective contributions due to surface motion.

Key Results
The methodology was validated through two experiments:

Cylindrical Polymer Half-Shell (Purely Thermal): A polylactic acid half-shell was heated on a copper plate. The projection successfully mapped temperatures to the curved surface. While temperature gradients were successfully visualized (showing higher gradients near heat contacts), the temperature rate was found to be dominated by noise due to the low thermal conductivity of the polymer and the limitations of the IR imaging process. The authors note that regression (smoothing) rather than interpolation might be necessary for rate calculations in such low-conductivity scenarios.
Tube under Tension-Torsion (Thermo-Mechanical): A zinc die-casting alloy tube was subjected to proportional tension-torsion loading. The system successfully captured full-field strain, temperature, temperature gradients, and temperature rates. Due to the higher thermal conductivity of the metal, the temperature rate calculations yielded reasonable results, showing a homogeneous rate of approximately $1^\circ\text{C s}^{-1}$ during loading and a rapid decrease upon unloading. The global RBF approach also successfully computed principal strain directions, which are typically difficult to obtain with local interpolation methods.

Significance and Claims
The paper claims to provide a modular, straightforward approach to coupling independently calibrated industrial systems for thermo-mechanical full-field analysis. Key contributions include:

External Coupling: The method does not require modifying the internal calibration or correlation algorithms of the commercial DIC or IRT systems, making it accessible to researchers using standard equipment.
Curved Surface Applicability: Unlike many previous 2D-DIC/IRT studies, this method is explicitly designed for and validated on curved and complex geometries.
Extended Data Output: By extending the global interpolation to the temporal domain, the method enables the computation of temperature rates on curved surfaces, a capability the authors state has not yet been addressed in the literature using full-field data.
Post-Processing Nature: The approach functions as an additional post-processing step that can be embedded into existing experimental protocols without altering image acquisition or registration procedures.

The authors maintain a modest tone regarding the limitations, acknowledging that the projection accuracy is dependent on the calibration quality and that noise in the IR data can affect the calculation of temperature rates, particularly in materials with low thermal conductivity. They suggest future work should focus on uncertainty quantification and the optimization of experimental designs for heterogeneous fields.

Technical Summary: Projection-Based Coupling of Infrared Thermography and Stereocorrelation-Based Digital Image Correlation

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