Fiducial marker and blob detection-based motion compensation algorithm for Thermoelastic Stress Analysis measurements

Non-contact techniques for measuring stress and strain on structures offer advantages including preservation of structural integrity, remote and continuous monitoring, full-field measurement capability, non-destructive testing, and adaptability to various materials. Among them, thermoelastic stress analysis is one of the most established non-contact techniques to measures stress distributions in structures by analysing temperature-induced deformations. This research proposes a methodology for the compensation of the edge effect, a undesirable phenomenon resulting from rigid displacements in the analysed structure when using thermoelasticity approach. These displacements give rise to virtual stress distributions along the sample’s edges. For this purpose, a fiducial marker was used to track rigid displacements in thermal images, that were then processed using a blob detection algorithm, for compensating for edge effects under different loading conditions. The algorithm was capable of measuring the displacement of a sample at different frequencies (5-25Hz) and distances from the target (300-600 mm), with the lowest uncertainty of 0.2 mm. The impact of compensation was found The impact of the compensation was found to be significant, especially at frequencies lower than 10 Hz, where the displacement was the highest.


Introduction
With the development of digital image analysis along the last decades, many non-contact measurement techniques gained enormous popularity [1].In particular, techniques developed to measure stress without contact gave the possibility to provide comprehensive full-field measurements of stress distributions, facilitating non-destructive testing for ongoing operations and displaying remarkable adaptability across a wide spectrum of materials, from metals and composites, to biological tissues [2].In addition, without the need to physically install probes or sensors on the surface of the sample, the integrity of the measurand is preserved.
Thermoelastic Stress Analysis (TSA) is one of such non-contact techniques, used to measure full-field stress distributions over the surface of structures that undergo external loading [3].Contrary to other contactless stress measurement techniques, TSA has the great advantage of not necessitating a surface treatment of the analysed sample [4, 5,6], unlike for example Digital Image Correlation (DIC) [7].The technique is based on the thermoelastic effect, which states that, in a homogenous elastic material under adiabatic conditions, the rate of change of AIVELA-2023 Journal of Physics: Conference Series 2698 (2024) 012001 IOP Publishing doi:10.1088/1742-6596/2698/1/012001 2 temperature (∆T ) and the rate of change of stress (∆σ) are linearly related [8].Being necessary to measure a temperature variation in order to obtain the stress distribution using TSA, a cyclic load is needed to achieve a condition where conductive and convective heat exchanges can be neglected.The TSA provides for the analysed thermal video to be processed via Lock-In [9], which allows to remove all the ∆T variations that have frequency components different from the cyclic load.
The application of this kind of loads generates a rigid displacement of the body, that occurs at the same frequency as the load itself.Contributions given by the material of the sample or by the way the test bench is anchored to the ground, could also amplify the rigid motion.This means that when the Lock-In is applied, a ∆T is found that is not related to the presence of any kind of stress, but caused by the motion itself.This issue, defined as edge effect, affects all the moving boundaries framed in the scene.Considering a single pixel near the edge of the sample, if there is a rigid motion, then the pixel will alternately see the specimen and the background; clearly, due to the fact that this phenomenon will occur at the frequency analysed via Lock-In, the pixel will present a ∆T equal to the temperature difference between the background and the sample.A Motion Compensation method was proposed by Wang et al. [10], where, while the IR camera acquired the thermal video for the TSA purpose, another camera, working in the visible spectrum, filmed the displacement of a visible speckle realised on the surface of the sample.Through DIC applied to the visible light video, the displacement field was measured and used as compensation for the thermal video.As this approach needed the two videos to be synchronised both spatially and temporally, Sakagami et al. [11] proposed a method based on DIC applied directly to the thermal images.This technique required the test to be performed twice, once with the surface of the sample treated with an IR speckle, the second time with the surface painted in matte black for the TSA purpose.A similar approach was proposed by Silva et al. [12].To avoid both issues of repeating the test twice or synchronising visible and IR video, Tocci et al. [13] proposed a motion compensation based on Optical Flow.Although it solved many criticalities of the previous methods, this approach heavily relied on the presence of sample features in the scene, as well as not always being computationally easy during the post-processing of the video.
In this work an approach is proposed, where the thermal video analysed via TSA gets directly pre-processed and compensated, using a blob detection algorithm to track the position of a fiducial marker, placed on the surface of the sample itself.In this way, no surface treatment is needed and the test has to be performed only once, without the need for a second camera.The proposed method is capable of performing the compensation with an extremely low impact in terms of computational time needed, and can be easily applied by setting just a few parameters.
The manuscript is organized as follows.Sec.2 is dedicated to the description of the compensation method and to give a theoretical background on TSA; here, the experimental set-ups used to test the algorithm are illustrated.In Sec.3 the results of the tests are shown.Finally, in Sec. 4, the conclusions are drawn.

Blob detection
The methodology proposed in this research relies on the tracking of a fiducial marker displacement in thermal images for compensating rigid motions.For this purpose, while the structure object of TSA measurement is generally conditioned with a high-emissivity paint, a low-emissivity material is selected to realise a marker to obtain a high-contrast region in the thermal video.In this way, the marker position can be tracked with a standard blob detection algorithm, using area, circularity, inertia, and convexity of the blob as tracking parameters [14].Fig. 1 shows the structure of the blob detection algorithm used in this research.Once the blob detection parameter are set-up, and the region of interest (ROI) where the TSA is applied, the blob and its centroid are detected in the first frame.This serves as a reference to determine the motion of the marker during the thermal video.Completed the steps for the first frame, for each of the other frames the algorithm proceeds as follows: (i) Threshold: a threshold is applied, so only the marker appears in the frame as a black blob (circled in red in Fig. 2); (ii) Blob centroid detection: the blob detection algorithm detects the position of the blob and calculates its centre of mass; (iii) Blob displacement: the initial position of the centroid of the blob is subtracted from its current position in the analysed frame; in this way it is possible to measure how much the marker moved; (iv) ROI compensation: the ROI, chosen at the beginning of the process, is moved from its original position by a number of pixels equal to the blob displacement obtained during the previous step.
Fig. 2 shows the original frame with a zoomed-in on the marker in the infrared range, and the threshold and blob detection applied as described above.The result is a sequence of thermal images where the ROI containing the stressed part of the sample undergoes no rigid motion, while the background is in motion.In Fig. 3 the resulting measured displacement is shown.The blob detection based algorithm performances have been tested with the experimental setup (see Fig. 4), where: a LDS -V650 electromagnetic shaker was used to generate the vertical motion of the specimen; a laser triangulation sensor (OMRON ZX1LD100), with a resolution 0.007 mm, was used as reference; the tested sample was a 90 × 45 × 20 mm wood block, whose surface was treated with a matte black spray paint; a highly reflective low-emissivity (∼ 0.09) circular aluminium patch marker was adhered on the front of the specimen.To investigate the influence of the distance between the thermal camera and the target and the displacement frequency of the sample on the algorithm results, 20 tests were conducted.The camera-target distances are 300, 400, 500, 600 mm, and the frequencies are 5, 10, 15, 20, 25 Hz.

Thermoelastic Stress Analysis
The fundamental principle behind TSA lies in the concept that volume variations, and consequently deformations, induce temperature changes in the specimen itself [15].Lord Kelvin established the relationship between the temperature change (∆T ) and the stress change (∆σ) through the following expression: Here, α represents the thermal expansion coefficient, ρ corresponds to the sample material density, C p is the specific heat capacity at constant pressure, T is the absolute temperature of the body and I is the stress variation.Eq. ( 1) elucidates the direct proportionality between temperature variation and the first invariant of the Cauchy's Stress Tensor [16].This implies that to measure the stress distribution accurately, it is imperative to measure ∆T .However, due to conductive and convective heat exchanges, such variations tend to be highly transient.The application of a cyclic load at a sufficiently high frequency allows achieving a pseudo-adiabatic condition, where these heat exchanges can be neglected [8].Given that the ∆T generated by the stress is typically low in amplitude, the thermal video is processed using a digital lock-in amplifier (LIA) [9].This amplifier isolates the component at the same frequency as the cyclic load from measurement noise.The loading frequency is contingent on both the material of the sample and its shape, typically falling within the range of 2 Hz to 25 Hz, except for some polymers that could require excitations up to 100 Hz [3].Using a LIA, a target frequency is designated for analysis and used to generate two square waves: one in phase and one in antiphase with the reference signal.Sequentially, the reference signal is multiplied by each square wave and filtered using a low-pass filter, acting as an integrator [17].This process yields two values, I x and I y , as: where θ represents the phase shift between the applied cyclic load and the temperature variation.I x and I y respectively denote the real and imaginary components of a complex number, with the magnitude representing the amplitude of ∆T at the analysed frequency.By calculating the phase of this imaginary number, it becomes possible to ascertain whether ∆T is associated with tensile or compressive stress.
A second test bench was designed to estimate the influence of motion compensation for TSA measurements.A Nylon-6 200 × 30 × 4 mm rectangular shape specimen was realised, with a hole of 40 mm in diameter at the centre of the sample.The sample was spray-painted using a matte black paint and the high-emissivity marker was used to obtain a high-contrast blob in the scene.An electrodynamical shaker (Sentek L1024M-PA115) was used to subject the sample to a tensile-compressive load.Finally, a black cloth was used to remove electromagnetic sources from the surrounding.A photonic cooled infrared camera (Flir A6751sc) was used, with a spatial resolution of 640 × 512 pixels, placed at 500 mm from the sample, acquiring at a sampling rate of 5 Hz, 10 Hz, 15 Hz, and 20 Hz.

Blob detection
Both the tracking results and the signals acquired using the laser triangulation sensor were analysed and compared in the frequency domain, to avoid effects coming from the synchronisation of the signals.For each experiment, two aspects were taken into account when comparing the image analysis signal with the reference one: the shift in frequency between the main peaks of the two spectra, and their amplitude difference.The displacement obtained by using the marker tracking was measured in pixels.Hence, it was previously scaled considering the sample dimensions in the frame and its real-world dimensions.A different scaling factor was calculated for each of the distances tested.
No particular trends were observed when examining the frequency shift between the Fast Fourier Transform (FFT) peaks of the blob detection-based signal and the laser triangulation signal.Despite identifying a mean shift of 0.013 Hz, it's crucial to note that, given the 20-second duration of the infrared videos, this shift falls below the frequency resolution of the analysed signal spectrum.
Figure 6 illustrates the results of comparing amplitude differences.Despite an exhaustive analysis, no significant dependencies on distance and frequency were identified.Notably, the algorithm exhibited its poorest performance at 15 Hz, displaying an amplitude difference of 0.25 mm.An average uncertainty of 0.13 mm was computed, accompanied by a standard deviation of 0.07 mm. Figure 6: Amplitude difference between the peaks of the triangulation sensor (reference) and the blob detection

Thermoelastic Stress Analysis
When assessing the effectiveness of the motion compensation algorithm applied to TSA, it's essential to account for the fact that as the load frequency increases, the amplitude of the sine displacement decreases.Consequently, at higher frequencies, the imperative to compensate for motion becomes less significant.An initial qualitative comparison is depicted in Fig. 7. To determine the efficacy of the Marker-Based motion compensation algorithm, a quantitative metric, as depicted in Fig. 8, was formulated.This metric involves analysing the temperature profile along the vertical pixels positioned in the midpoint of each ∆T map.Specifically, we calculated the ratio between the maximum value of the uncompensated map and its counterpart in the compensated map.This derived metric, termed the Edge Correction Factor (ECF), serves as a representative measure of the compensation impact on each test.

Conclusions
An innovative approach utilising fiducial markers and blob detection was employed as a motion compensation technique for thermal images utilised in thermoelastic stress analysis measurements.The algorithm underwent initial testing to assess its performance in comparison to a laser triangulation sensor.These tests were conducted at varying distances from the target (300-600 mm) and different displacement frequencies .When comparing the signal peaks of the two sensors in the frequency domain, two critical aspects were considered: frequency shift and amplitude difference.A ∆f lower than the available frequency resolution indicated a negligible shift.The mean ∆s (where s represents the amplitude of the peak) was found to be 0.13 mm, deemed satisfactory for the study's objective, which wasn't focused on creating a high-precision displacement measurement technique.Subsequently, the method was employed to pre-process a thermal video analysed using TSA, facilitating stress distribution measurement.Evaluating the stress concentration factor as a comparison metric revealed a tenfold reduction in edge effects compared to the uncompensated video.This improvement was particularly pronounced at lower frequencies (5 Hz and 10 Hz), where displacement amplitudes were higher.
Future investigations will primarily concentrate on directly comparing the marker-based algorithm's performance with other previously proposed algorithms.Ongoing refinements will aim to extend compensation capabilities beyond rigid motions, considering the potential for macro deformations in the sample.

Figure 1 :
Figure 1: Structure of the fiducial marker-based motion compensation algorithm.

Figure 3 :
Figure 3: Displacement of the centroid of the blob from the detection algorithm.

Figure 5 :
Figure 5: Schematic of TSA measurement test setup.

Figure 7 :
Figure 7: Comparison between the non-compensated and compensated ∆T maps at different loading frequencies

Figure 8 :
Figure 8: Temperature profile (red line) in compensated and uncompensated map.The points used to calculate the ECF have been highlighted both in the plots of the profile and in the maps

Figure 9 :
Figure 9: ECF over the loading frequency of the sample