# Performing Stereo-Measurements with Optimal Patterns Processed by Localised Spectrum Analysis

Thomas Jailin, B. Blaysat, A. Vinel, R. Langlois, F. Sur, M. Grédiac

► To cite this version:

Thomas Jailin, B. Blaysat, A. Vinel, R. Langlois, F. Sur, et al. Performing Stereo-Measurements with Optimal Patterns Processed by Localised Spectrum Analysis. Experimental Mechanics, 2025, 65 (9), pp.1461-1472. 10.1007/s11340-025-01217-7. hal-05308609

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Submitted on 15 Oct 2025

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Distributed under a Creative Commons CC BY 4.0 - Attribution - International License
Performing stereo-measurements with optimal
patterns processed by Localised Spectrum Analysis

Thomas Jailin¹*, Benoît Blaysat¹,²†, Adrien Vinel¹†,
Raphaël Langlois¹,³†, Frédéric Sur⁴†, Michel Grédiac¹†

¹*Université Clermont Auvergne, Clermont Auvergne INP, CNRS,
Institut Pascal, Clermont-Ferrand, France.

²Institut Universitaire de France, Paris, France.

³Present address: Université Paris-Saclay, CentraleSupélec, ENS Paris-Saclay, CNRS, LMPS - Laboratoire de Mécanique Paris-Saclay, Gif-sur-Yvette, France.

⁴LORIA, Université de Lorraine, CNRS, INRIA, Nancy, France.

*Corresponding author(s). E-mail(s): thomas.jailin@uca.fr;

Contributing authors: benoit.blaysat@uca.fr;

adrien.vinel@sigma-clermont.fr; raphael.langlois@ens-paris-saclay.fr;

frederic.sur@loria.fr; michel.grediac@uca.fr;

†These authors contributed equally to this work.

**Abstract**

**Background:** Localised Spectrum Analysis is a full-field measurement method which enables processing optimal patterns (i.e. checkerboard patterns) and obtaining high metrological performance. However, LSA was still limited to 2D cases.

**Objective:** This paper presents stereo-LSA, an extension of the method for stereo image pairs, enabling the measurement of the 3D displacement field on sample surfaces engraved with optimal checkerboard patterns.

**Methods:** A standard stereo procedure is followed. The LSA formulation is updated to take into account global rotations induced by the stereo angle.

**Results:** The technique is applied to three different cases. A synthetic case is used to validate the stereo implementation, and a comparison is proposed with stereo digital image correlation, routinely used in the community.

**Conclusions:** Stereo-LSA enables measuring 3D-displacement field on a sample surface prepared with checkerboard patterns, leading to high metrological performance.
# 1 Introduction

Introduced to the community a few decades ago, full-field measurement techniques are now widely used in experimental solid mechanics [1]. They enable obtaining large amounts of data per test and make it possible to analyse heterogeneous phenomena, which was not formerly possible with local means of measurement. The most common technique to measure the kinematic field on a sample surface is undoubtedly the Digital Image Correlation (DIC) method [2–4]. This can be explained by its good metrological performance and ease of use. A simple can of spray paint is indeed often sufficient to apply the random speckle pattern needed for DIC onto the sample surface.

In DIC, the quality of the measurement is closely tied to the texture present on the sample surface [5–9]. Many studies have focused on the impact of the pattern on the measurements [10–15], and have proposed good practices to obtain “good” speckles [16]. Bomarito et al. showed in [17] that the theoretically optimal pattern is the checkerboard pattern because it maximises the grey level gradient in the images and, as such, optimises the metrological performance of the measurement process. However, DIC is not well-suited to such periodic patterns because its solution is often trapped in local minima of the optical residual. DIC practitioners usually make use of random patterns [9], which facilitates the algorithm convergence and the deposition step but decreases the metrological performance of the technique. To tackle this issue and work on optimal patterns, the localised spectrum analysis (LSA) method was developed [18–20]. Similarly to DIC, LSA minimises the optical residual but with the advantage of working in the frequency domain by taking benefit from the periodicity of the pattern. This leads to higher metrological performance. Yet, it may be worth noting that DIC achieves similar metrological performance to LSA when applied to a checkerboard pattern and initialised with a close initial guess. LSA also enables the use of well-controlled and reproducible patterns on the tested specimens, which can be of great interest for standardisation. Following that latter idea, a recent study proposed to print the checkerboard pattern on polyimide films and glue them to the specimen surface [21]. The pattern properties can thus be characterised before the mechanical test, and the method can be used at a low cost (i.e., without the need for a laser engraver). So far, LSA has been applied to many materials [18, 22, 23], but only in 2D applications. No extension has been proposed in stereo to measure out-of-plane motion during a test. The aim of this paper is to fill that gap by proposing stereo-LSA, an extension of LSA to 3D.

Stereo-LSA follows the same steps as stereo-DIC, namely the calibration of the optical system and the matching between the images obtained from each camera [4]. Three different experiments were set up to validate and present the method. The first one is a fully synthetic test, used to validate the method. The second application
consists of a real buckling test on a metal sheet specimen. A comparison between stereo-DIC and stereo-LSA is proposed. The third example is presented to observe the austenite-to-martensite transformation phase transformation of a shape memory alloy during cooling. This transformation induces heterogeneous strain maps, higher deformation gradients and smaller out-of-plane displacements than the first test case. A comparison of the topography obtained with LSA and that obtained with an optical profilometer is presented to validate the method against a well-established technique.

The present paper is organised as follows: the LSA method and the updates developed for stereo measurements are detailed in the second section. The three application tests are then presented in the third section, where a comparison with classic stereo-DIC is made. A discussion is finally proposed to conclude the paper.

# 2 Methods

## 2.1 Principle of LSA

LSA is one of the spectral techniques developed to retrieve the displacement field from images of a specimen marked with a periodic pattern, taken before and after deformation [24]. Some brief reminders about LSA are given in this section. The interested reader can refer to [18] for more details, and to [20] to have a comparison with DIC.

First, we consider an image of a checkerboard obtained on a specimen, as depicted in Figure 1. The grey level (GL) distribution within the image at each pixel $\underline{x}$ reads such as:

$$l(\underline{x}) = \text{frng}[2\pi k_{\alpha} \cdot \underline{x} + \varphi_{\alpha}(\underline{x})] + \text{frng}[2\pi k_{\beta} \cdot \underline{x} + \varphi_{\beta}(\underline{x})], \quad (1)$$

where frng is a $2\pi$-periodic function, $k_{\alpha}$ and $k_{\beta}$ are the wave vectors of each carrier wave perpendicular to each other. The wave vectors are convenient supports to define the properties of a periodic signal. Their norm corresponds to the inverse of the signal period and is assumed equal along each direction, i.e. $p = 1/\|k_{\alpha}\| = 1/\|k_{\beta}\|$, and their direction corresponds to the signal repetition direction. Functions $(\varphi_{\alpha}, \varphi_{\beta})$ are the phase modulations of the two perpendicular signals forming the checkerboard pattern. They model the difference between the true signal and a perfect checkerboard. Finally, “.” is the 2D Euclidean dot product.

In practice, the phase modulation is computed pixelwise at first order, by calculating the arguments of the windowed discrete Fourier transform (WDFT) at the frequencies and directions of each wave carrier [25]. The WDFT, denoted by $F_{w_{\sigma}}(l)$, of an image $l$ reads:

$$F_{w_{\sigma}}(l)(\underline{x}, \underline{k}) = \sum_{\underline{x}} w_{\sigma}(\underline{x} - \underline{x})l(\underline{x})e^{-2j\pi\underline{k}\cdot\underline{x}}, \quad (2)$$

where $j$ is the imaginary unit defined by $j^2 = -1$ and $w_{\sigma}$ is a Gaussian window of standard deviation $\sigma$ computed such as:
<figure><img src="image_2.png" /><figcaption>Fig. 1: Example of a checkerboard obtained onto a sample with a square size of about 30 µm. The checkerboard was deposited with a MDU1000C laser marker from Keyence. The image obtained from a Dino-Lite Edge 3.0 microscope. Here, (α, β) are the periodicity directions and (1, 2) are the image system directions.</figcaption></figure>

$$w_{\sigma}(\mathbf{x}) = \frac{1}{2\pi\sigma^2} \exp\left(-\frac{\|\mathbf{x}\|^2}{2\sigma^2}\right). \qquad (3)$$

The pros and cons of different windows are discussed in [26] and the conclusion is that the Gaussian window leads to the best compromise between competing constraints.

If the kernel is aligned with the periodicity directions, the phase modulation can finally be written such as:

$$\varphi_{\ell}^{i}(\mathbf{x}) \approx \text{angle}(F_{w_{\sigma}}(I)(\mathbf{x}, \underline{k}_{\ell})) + 2\pi c_{\ell}^{i}(\mathbf{x}); \quad \forall \ell \in \{\alpha, \beta\}, \forall i \in \{0, t\}, \qquad (4)$$

where functions $c_{\ell}^{i}$ are introduced to correct the 2π-jumps of the “angle” function [27], i is the time step of interest between a reference step 0 and a deformed step t. It is important to notice here that if the phases are not extracted along the directions of the periodicity an error is introduced and this equality no longer holds.

The intensity conservation assumption is then used to link the reference image $I^0$ to a deformed image $I^t$. Considering a displacement field $\underline{u}$ between the two images, this assumption reads:

$$I^0(\mathbf{x}) = I^t(\mathbf{x} + \underline{u}(\mathbf{x})). \qquad (5)$$

Plugging Equation (1) into Equation (5) gives the following system:

$$\left\{ \begin{array}{l} 2\pi \underline{k}_{\alpha} \cdot \mathbf{x} + \varphi_{\alpha}^{0}(\mathbf{x}) = 2\pi \underline{k}_{\alpha} \cdot (\mathbf{x} + \underline{u}(\mathbf{x})) + \varphi_{\alpha}^{t}(\mathbf{x} + \underline{u}(\mathbf{x})) \\ 2\pi \underline{k}_{\beta} \cdot \mathbf{x} + \varphi_{\beta}^{0}(\mathbf{x}) = 2\pi \underline{k}_{\beta} \cdot (\mathbf{x} + \underline{u}(\mathbf{x})) + \varphi_{\beta}^{t}(\mathbf{x} + \underline{u}(\mathbf{x})). \end{array} \right. \qquad (6)$$

Then, using the perpendicularity between $\underline{k}_{\alpha}$ and $\underline{k}_{\beta}$, Equation (6) can be simplified, which leads to:
$$ \underline{u}(\underline{x}) = \sum_{\ell=\{\alpha, \beta\}} \frac{\varphi_{\ell}^{0}(\underline{x}) - \varphi_{\ell}^{t}(\underline{x} + \underline{u}(\underline{x}))}{2\pi \|\underline{k}_{\ell}\|} \underline{e}_{\ell}, \qquad (7) $$

where $(\underline{e}_{\alpha}, \underline{e}_{\beta})$ are the normalised vectors of the periodicity directions used to make the projection in the image system, defined such as:

$$ \underline{e}_{i} = \frac{\underline{k}_{i}}{\|\underline{k}_{i}\|}, \qquad (8) $$

with $||\underline{k}_{\alpha}|| = ||\underline{k}_{\beta}||$ by assumption.

It is worth saying that even if the optical flow conservation is used through Equation (5), this is a weak hypothesis. Indeed, brightness and contrast changes mainly impact the amplitude of the signal and have negligible impacts on the phase modulations estimated by Equation (4), which are then used by LSA to retrieve the displacement.

## 2.2 Camera calibration and 3D reconstruction

While the above equations can be directly applied to obtain the 2D displacement fields between two images, additional steps are needed when stereo measurements are sought. In that case, at least two cameras looking at the same region are required. These cameras must first be calibrated to know their position compared to each other and the specimen. Assuming that each camera $i$ can be defined as a pinhole model, a link is made between a point in the world coordinate system $M_w = (X_W, Y_W, Z_W)$ and its projection in the image coordinate system $\underline{x}^i = (\underline{x}_1^i, \underline{x}_2^i)$. Indeed, according to [4], we have:

$$ \underline{x}_1 = c_1 + f_1 \frac{R_{11}X_W + R_{12}Y_W + R_{13}Z_W + t_X}{R_{31}X_W + R_{32}Y_W + R_{33}Z_W + t_Z}, \qquad (9) $$

$$ \underline{x}_2 = c_2 + f_2 \frac{R_{21}X_W + R_{22}Y_W + R_{23}Z_W + t_Y}{R_{31}X_W + R_{32}Y_W + R_{33}Z_W + t_Z}, $$

where $(c_1, c_2)$ and $(f_1, f_2)$ define the principal point in the image system (which corresponds to the intersection of the optical axis and the image plane) and the focal length of the camera, respectively. These are the four intrinsic parameters. $R_{ij}$ and $t_i$ are the rotation and translation components defining the extrinsic parameters. For the sake of clarity, the upper-scripts on the image coordinates and the camera parameters were omitted in Equation (9). The different transformations from the world system to the image system can be seen in Figure 2.

Distortion models can also be used to take into account flaws within the optical system. In our case, radial, tangential, and thin prism distortions are considered, leading to twelve additional parameters. Since the cameras are fixed during the experiment,
<figure><img src="image_3.png" /><figcaption>**Fig. 2:** Stereo projection of a point $M_W$ expressed in the world coordinates to a point $\underline{x}^i$ in the image system of camera $i$ ($j$, respectively).</figcaption></figure>

intrinsic, extrinsic, and distortion parameters are identified following a standard calibration procedure relying on calibration grids and the OpenCV Python library [28]. Once all the parameters are identified, Equation (9) provides, for each image point $\underline{x}$, two equations per camera containing three unknowns, namely the world coordinates of the point of interest. Using two (or more) cameras thus makes it possible to solve the problem using a standard least square algorithm. Two cameras will be used in this study.

It is then possible to reconstruct the topography of the observed surface at a certain time step $t$, given that the correspondence between the points $\underline{x}^i$ in the image from the camera $i$ are known in the image from another camera $j$. This matching step is done by performing a 2D LSA analysis between the images obtained from the $i^{th}$ and the $j^{th}$ camera. If the surface of the specimen of interest is a plane, it can be shown that the transformation between the images of the surface in both cameras is a planar homography. Since we are interested in small out-of-plane displacements, it is possible to assume that the specimen is planar, the images being mapped through a homography. As the camera are equipped with lenses with long-focus length, and are positioned in such a way that their optical axis makes a moderate angle (less than 30 degrees) with a normal direction to the specimen, we further simplify by assuming that images of the specimen are mapped through a similitude, that is a geometrical transform made of a rotation, translation, and scale change [29]. This will be discussed in Section 2.4.

If one is interested in the 3D displacement of the sample surface between two different time steps, a matching step is also necessary between the images of different time
steps. The problem description can follow either a Lagrangian or an Eulerian descrip-
tion. In the former, the reference image remains the first one, and each deformed step
is processed independently of the other. In the latter, the reference image is taken as
the previous one of the analysed image, and the displacement increment is added at
each step. The Eulerian description may help convergence of DIC in case of a large
transformations of the specimen during the acquisition process (since Eulerian point-
of-view only consider deformations between short time intervals), but displacement
errors are summed all over the process. In our case, the Lagrangian description was
used. It follows that the reference image of the left camera can be seen as the principal
image. Indeed, all other images will be compared to it, either to compute the dis-
placement obtained from different steps or to obtain the matching between the images
obtained from different cameras.

It is also important to say that the object has to remain within the in-focus area during testing to maintain image sharpness. This in-focus area is directly linked to the depth of field, which depends on the optical setup used and the working distance of the camera. Leaving this area has an even greater impact in LSA since the pattern is small.

## 2.3 Reducing sensor noise effect considering image stacks

It is often recommended in full-field measurement methods to record several images of the unloaded reference state to create a reference image by averaging them [30]. This allows for minimising the impact of sensor noise on the displacement estimations. When working with several cameras, this reference image has to be created for each. Nevertheless, directly averaging the images may result in a biased estimation of the noiseless reference image. Indeed, micro-vibrations may occur while capturing the stack of reference images, which can induce a bias in the resulting averaged reference image [31]. One could correct each image of the stack by the displacement field obtained w.r.t. the first image, but this would imply the need of interpolations of the images, where the grey level gradients are sought to be maximal, thus leading to a potentially non negligible bias.

Since LSA processes phase modulation maps, it is possible to address this problem by working on the phase maps rather than on the raw images. The advantage is that the phase maps exhibit much lower gradients, thus leading to limited interpolation errors. The following steps are used to obtain a reference phase modulation from the stack of images captured in the reference state:

(i) the rigid body motion (RBM) of each image of the stack is estimated relative to the first image;

(ii) a phase modulation corrected for the RBM is generated for each image;

(iii) a reference phase modulation is computed by averaging all the corrected phase modulations for each camera. This reference phase modulation is then used in the LSA process.

This method has been implemented in the OpenLSA open-source code [27], and is
used in the following.
## 2.4 Taking into account the global rotation

An important comment has to be made at this stage. In the standard version of LSA, the phases of all the images are extracted along the periodicity directions of the reference image. This enables obtaining easily Equation 7, since the phases and wave carriers of the two analysed images are expressed along the same directions. However, this cannot be done when using several cameras, which cannot be aligned along the same direction. Indeed, the different observation angles of each camera induce an apparent transformation in the images. It follows that the wave vectors ($k_{\alpha}^i, k_{\beta}^i$) and ($k_{\alpha}^j, k_{\beta}^j$) obtained in the images from camera i and j, respectively, cannot be considered identical any longer. Figure 3 shows the difference observed in the spectrograms between two images when a global rotation is present between two images. We can see that rotated images give rotated spectrograms. The extraction of the phases of image j along the periodicity directions of image i, as previously done in Equation 4, would introduce a bias in the extracted phases.

<figure><img src="image_4.png" /><figcaption>Fig. 3: Spectrogram of a synthetic checkerboard in (a) the reference state, and (b) rotated by 20°.</figcaption></figure>

Equation (6) was updated to take these changes into account by considering that two images of the specimen taken with different cameras are related by a similarity, decomposed into a translation, a rotation, and a scale change, as mentioned in Section 2.2. If we consider two images captured from two cameras, 0 and 1, we can write a more general equation such as:

$$ \begin{cases} 2\pi \underline{k}_{\alpha}^{0} \cdot \underline{x} + \varphi_{\alpha}^{0}(\underline{x}) = 2\pi \underline{k}_{\alpha}^{1} \cdot (\underline{x} + \underline{u}(\underline{x})) + \varphi_{\alpha}^{1}(\underline{x} + \underline{u}(\underline{x})) \\ 2\pi \underline{k}_{\beta}^{0} \cdot \underline{x} + \varphi_{\beta}^{0}(\underline{x}) = 2\pi \underline{k}_{\beta}^{1} \cdot (\underline{x} + \underline{u}(\underline{x})) + \varphi_{\beta}^{1}(\underline{x} + \underline{u}(\underline{x})) \end{cases} \quad (10) $$

with
$$ \begin{pmatrix} \underline{k}_{\alpha}^{0} \\ \underline{k}_{\beta}^{0} \end{pmatrix} \frac{1}{\|\underline{k}_{\alpha}^{0}\|} = \underline{R}(\theta^{1} - \theta^{0}) \begin{pmatrix} \underline{k}_{\alpha}^{1} \\ \underline{k}_{\beta}^{1} \end{pmatrix} \frac{1}{\|\underline{k}_{\alpha}^{1}\|} \qquad (11) $$

and

$$ \begin{pmatrix} \underline{e}_{\alpha}^{0} \\ \underline{e}_{\beta}^{0} \end{pmatrix} = \underline{R}(\theta^{1} - \theta^{0}) \begin{pmatrix} \underline{e}_{\alpha}^{1} \\ \underline{e}_{\beta}^{1} \end{pmatrix}. \qquad (12) $$

In these equations $\underline{R}(\theta^1 - \theta^0)$ corresponds to the rotation of the wave vectors between the two cameras. This formulation also enables having wave vectors with different norms between the two images, which means that the frequency of the carrier changes between the images.

Combining these last three equations, we obtain:

$$ \underline{u}(\underline{x}) = \left( \frac{\|\underline{k}_{\alpha}^{0}\|}{\|\underline{k}_{\alpha}^{1}\|} \underline{R}(\theta^{0} - \theta^{1}) - \underline{I}_{d} \right) \underline{x} + \sum_{\ell = \{\alpha, \beta\}} \frac{\underline{R}(\theta^{0} - \theta^{1}) \varphi_{\ell}^{0}(\underline{x}) - \varphi_{\ell}^{1}(\underline{x} + \underline{u}(\underline{x}))}{2\pi \|\underline{k}_{\ell}\|} \underline{e}_{\ell}. \quad (13) $$

This allows accounting for global rotations and scale change between two images. It enables performing LSA between images obtained from different observation angles. Yet, it does not deal with high localised rotations since the phase modulation would not be extracted along the periodicity directions in the locally rotated areas.

This method is now available in the OpenLSA code [27]. As an example, an LSA was performed between two synthetic images of a checkerboard, where a global rotation was imposed between the reference image and the rotated image. Different angles were studied from 1 to 30 °. Noise was added to the intensity of the images to be representative of real conditions. Figure 4 shows the standard deviation of the error of the norm of the displacement between the results obtained from the LSA and the ground truth. This error remains below 0.01 pixel when the new formulation is used and keeps increasing with the angle when the old formulation is used.

# 3 Application and results

Three test cases are now presented. The first one is a synthetic case used to validate the method. A comparison is made between stereo-LSA and stereo-DIC. In the second case, stereo-LSA and stereo-DIC are used to analyse a real buckling test on a metal sheet. In the third application, stereo-LSA is applied to study the phase transformation of a shape-memory alloy (SMA) under cooling.

## 3.1 Test case 1: synthetic case

This fully synthetic case is used to validate the method. In this test, a virtual rect- angular sheet measuring 25 mm in width and 20 mm in height was simulated. A
<figure><img src="image_5.png" /><figcaption>**Fig. 4:** Standard deviation of the error of the norm of the displacement between the results obtained from the LSA and the ground truth. In the new formulation (orange curve), the global rotation is taken into account. There is one data point per degree.</figcaption></figure>

checkerboard pattern with a pitch of 70 µm was applied to its surface. Two cameras are used to obtain the images of this virtual sheet. The calibration parameters of the two cameras are assumed to be known, including radial, tangential, and thin prism distortions. These sets of parameters are close to the ones used in the subsequent experiments to ensure they are representative of real conditions. A pair of images was obtained from the virtual sheet in its reference configuration using the specified camera parameters. Particular attention was paid to the pixel integration procedure to accurately generate images from the 3D description of the checkerboard. The sheet was then deformed. The in-plane displacement along the x-axis was set to 1 mm, and the displacement along the y-axis was set to -2 mm. The out-of-plane displacement along the z-axis followed a linear evolution, with 0 mm on the left side and 1 mm on the right side. Another pair of images was generated in the deformed configuration. A white noise with a standard deviation of 2.31 GL was then added in the four images. An example of an image obtained with this procedure is given in Figure 5.a. The image scale is about 1 pixel ≈ 15.7 µm. The out-of-plane displacement field can be seen in Figure 5.b.

Stereo-LSA and stereo-DIC were then used to retrieve the 3D displacement fields. The DIC algorithm was initialised with the LSA results to avoid being trapped in local minima. Linear subsets with a step size of 1 are used to allow consistent comparison with the pixelwise LSA results. The subset size is set to 21 in order to obtain the same spatial resolution as LSA, as detailed in [20]. The comparison with the ground truth (GT) is given in Table 1. As expected, it can be seen that both techniques lead to similar results, in very good agreement with the ground truth. The RMSE on the displacement is 0.080 µm for LSA and 0.003 µm for DIC. While LSA exhibits a
<figure><img src="image_6.png" /><figcaption>Fig. 5: (a) Synthetic image obtained from the virtual sheet; (b) Out-of-plane displacement fields imposed to the sheet.</figcaption></figure>

higher error, it is still negligible relative to the applied transformation on the specimen, which is on the order of millimetres. It may be worth reminding that the pattern is optimal, which explains the quality of the results obtained in both LSA and DIC. This validates the implementation of stereo-LSA, which will be applied on real experiments in the next two cases. This is also consistent with the results obtained in the 2D case presented in [20].

<table><caption>Table 1: Comparison of the results of stereo-LSA and stereo-DIC with the ground truth of the synthetic test. “std” stands for “standard deviation”.</caption><tr><td>std(U<sub>x</sub><sup>GT</sup> - U<sub>x</sub><sup>LSA</sup>)</td><td>0.05 µm</td></tr><tr><td>std(U<sub>y</sub><sup>GT</sup> - U<sub>y</sub><sup>LSA</sup>)</td><td>0.05 µm</td></tr><tr><td>std(U<sub>y</sub><sup>GT</sup> - U<sub>y</sub><sup>DIC</sup>)</td><td>0.05 µm</td></tr><tr><td>std(U<sub>z</sub><sup>GT</sup> - U<sub>z</sub><sup>LSA</sup>)</td><td>0.17 µm</td></tr><tr><td>std(U<sub>z</sub><sup>GT</sup> - U<sub>z</sub><sup>DIC</sup>)</td><td>0.18 µm</td></tr></table>

## 3.2 Test case 2: buckling of a metal sheet

This test case aims to validate the method on a real experiment and compare the results obtained from stereo-DIC and stereo-LSA.

### *Material, setup and experimental procedure*

The specimen is an aluminium sheet with dimensions 250 × 50 × 2 mm³. A speckle pattern was deposited on one half of the sample to perform DIC measurements. A
periodic pattern was engraved on the second half of the sample to perform LSA (see Figure 6). It was first painted with white undercoating, followed by the engraving of the checkerboard with a square size of 70 µm [22]. It can be seen that the checkerboard is slightly tilted with respect to the image system in order to avoid aliasing in the images [32]. The sample was then placed in a Zwick tensile machine equipped with a 20 kN load cell. Two cameras were used to record images of the sample. The calibration of the setup was carried out using a standard calibration grid. More information about the optical system can be found in Table 2. As mentioned in the previous section, one hundred images were acquired with each camera before the loading to compute the reference phases associated with each pattern seen by the cameras. A compression load was then applied to obtain the bending of the specimen and a pair of images was acquired in the deformed state.

<figure><img src="image_7.png" /><figcaption>Fig. 6: Images obtained from the two cameras at the beginning of the test.</figcaption></figure>

## Results and discussion

Stereo-DIC and stereo-LSA analyses were carried out on the left and right regions of the sample, respectively. DIC was performed using an in-house code, linear subsets with a size of 100 pixels, and a step size of 1. This large subset size enables fitting well the smooth curvature of the sample and allows for reducing the noise impact. LSA was performed using a Gaussian window with a standard deviation of about 6.2 pixels (the pitch of the pattern in the image). The results obtained for the out-of-plane
<table><caption>**Table 2:** Main optical system and image specifications.</caption><tr><td>Cameras</td><td>Prosilica GT 6600</td></tr><tr><td>Image resolution</td><td>6576 × 4384 pixels</td></tr><tr><td>Lenses</td><td>Samyang 100mm T3.1 ED UMC Macro</td></tr><tr><td>Aperture</td><td>f/16</td></tr><tr><td>Exposure time</td><td>25 ms</td></tr><tr><td>Stereo angle</td><td>≈ 24 °</td></tr><tr><td>Acquisition rate</td><td>1 Hz</td></tr><tr><td>Image scale</td><td>1 pixel ≈ 15.8 µm</td></tr><tr><td>Calibration grid</td><td>34 × 12 dots - step: 2 mm</td></tr></table>

displacement are shown in Figure 7.a. The results along three lines for each method were extracted and plotted in Figure 7.b. A slight twist of the sample is observed after deformation, with a more significant displacement toward the right side of the specimen. This is due to the imperfect symmetry of the test. Nevertheless, the two techniques led to the same trend, with out-of-plane displacement greater than 4.5 mm.

<figure><img src="image_8.png" /><figcaption>**Fig. 7:** (a) Out-of-plane displacement fields obtained by DIC (left) and LSA (right), displayed with transparency on the real images; (b) Out-of-plane displacement along the red and blue curves depicted on Fig.a; (c) Random error, defined as the difference between the estimation and a polynomial interpolation, obtained on the out-of-plane displacement from both techniques.</figcaption></figure>

In order to compare stereo-DIC and stereo-LSA at the same location, a DIC analysis was also performed on the checkerboard area. The DIC algorithm was initialised with the LSA displacement to ensure the good convergence of the algorithm on a periodic pattern. It was possible to reduce the subset size to 21 pixels since the pattern exhibited much higher gradients. It also enables having the same spatial resolution as
LSA. The difference obtained between the two methods is shown in Figure 8.a. The standard deviation of the difference between the two techniques is around 0.6 µm, which is considered as low given the displacements obtained during the test.

<figure><img src="image_9.png" /><figcaption>**Fig. 8:** Comparison of the out-of-plane displacement map obtained from DIC and LSA.</figcaption></figure>

### 3.3 Test case 3: Phase transformation of a SMA

In this test, the phase transformation in a shape memory alloy during cooling is studied. This test is expected to be challenging due to the high strain gradients and low out-of-plane motion.

#### *Material, setup and experimental procedure*

Stereo-LSA is now applied to an SMA to study its phase transformation under cooling. More details about the material and the experimental protocol can be found in [23]. The specimen is a cylindrical ingot, 33 mm in diameter and 3 mm thick. It is made of a single-crystal of Cu-Al<sub>13.9</sub>-Ni<sub>4.6</sub> (wt.%) This material presents an austenite structure at high temperature and a martensite structure at low temperature. Between these two temperatures, the two phases coexist within the material. The objective is here to study the austenite-to-martensite phase transformation of the material during cooling. The specimen was first painted with a white undercoating, followed by the engraving of the checkerboard with a square size of 70 µm [22]. The sample was then placed onto a Peltier cell that can be used either for heating or cooling in the [0, 55] °C temperature range. A schematic of the experimental device is given in Figure 9, as well as examples of images obtained from the two cameras.

The sample was first heated around 50 °C in order to obtain a full austenite structure in the material. The Peltier cell was then used to cool the specimen until
<figure><img src="image_10.png" /><figcaption>**Fig. 9:** Schematic of the experimental setup and examples of images obtained from the two camera.</figcaption></figure>

around 0 °C. This induced an austenite-to-martensite phase transformation in the material. During the whole test, the same cameras as for test #2 –two CCD cameras– were used to record images at a frequency of 1 Hz. The calibration of the setup was carried out prior to the test using standard calibration grids. More information about the optical system can be found in Table 3.

Stereo LSA is then used to measure the displacement and strain fields on the sample during test. The reference image is chosen as the one recorded just before cooling the specimen.

## Results and validation

Figure 10 shows the strain maps as well as the out-of-plane displacement map obtained at the end of the test. The martensite needles can clearly be observed on the different maps. Strain gradients between the different needles can be greater than 10%. It is worth saying that using lenses with bigger focal lengths would enable obtaining more pixels per period, thus reducing the noise on the displacement/strain maps. The needle
<table><caption>**Table 3:** Main optical system and image specifications.</caption><tr><td>Cameras</td><td>Prosilica GT 6600</td></tr><tr><td>Image resolution</td><td>6576 × 4384 pixels</td></tr><tr><td>Lenses</td><td>Samyang 100mm T3.1 ED UMC Macro</td></tr><tr><td>Aperture</td><td>f/16</td></tr><tr><td>Exposure time</td><td>25 ms</td></tr><tr><td>Stereo angle</td><td>≈ 23 °</td></tr><tr><td>Acquisition rate</td><td>1 Hz</td></tr><tr><td>Image scale</td><td>1 pixel ≈ 15.7 µm</td></tr></table>

feature can also be observed on the out-of-plane displacement map, which shows that the phase transformation has a non-negligible impact on the roughness of the sample. It is worth noting that at the end of the test, the order of magnitude of the out-of-plane displacement obtained remains below 80 µm, which is much lower than the values obtained from the previous tests.

An “Alicona-IF Portable” optical profilometer was finally used to validate these results obtained from stereo-LSA. To do so, an extra couple of images was taken at room temperature after the test to obtain the final topography from stereo-LSA. The sample was then scanned on the profilometer immediately after to minimise microstructure changes. The 3D surfaces obtained from the profilometer and the one obtained from LSA are given in Figure 11 (a comparison is not shown because the sampling densities of the two techniques differ significantly, which would result in the need for interpolation and averaging). An excellent agreement was obtained, which validates the method.

## 4 Conclusions and future work

This work developed an extension of LSA to 3D. It follows the same strategy as the one routinely used to perform stereo-correlation: calibration grids are used to calibrate the optical system, and the 3D reconstruction of the surface is obtained from 2D-LSA analyses, for which an image of one camera is used as the reference. To make it possible to deal with the apparent rotations induced by the different observation angles of each camera, the equations of LSA were slightly modified. This new formulation accounts for global rotations between two images. It enables performing LSA between images obtained from different observation angles but does not deal with highly localised rotations. Lifting this barrier is one of the future work of this study.

The method was then validated using three application cases. In the first one, the method was validated using a fully synthetic case. The second test was used to compare classic stereo-DIC and stereo-LSA and test the method in the case of large out-of-plane displacements. In the end, both methods led to the same results, the differences remaining in the order of magnitude of the noise. The third test case focused on the phase transformation of an SMA, which exhibits higher strain gradients and lower out-of-plane displacements. The results obtained from that latter case were
<figure><img src="image_11.png" /><figcaption>**Fig. 10:** Strain maps and out-of-plane displacement map obtained on the SMA at the end of the test.</figcaption></figure>

validated using an optical profilometer.

Compared to stereo-DIC, stereo-LSA allows the use of optimal patterns, resulting in the best metrological performance achievable. The periodic pattern used with LSA also offers the advantage of being easily reproducible, which supports the standardisation of the method.

A limitation is that in stereo-LSA, maintaining accurate focus is especially critical due to the small size of the pattern. Also, even though out-of-plane displacement can now be measured, the sample still needs to be flat at the beginning to allow the checkerboard pattern to be engraved. To address this problem, the study of [21] could be used to apply a checkerboard pattern on developable surfaces by using strain gauge 2.0.
<figure><img src="image_12.png" /><figcaption>**Fig. 11:** Height of the surface reconstructed in 3D using stereo-LSA and an optical profilometer.</figcaption></figure>

**Declarations**

**Conflict of interest/Competing interests** The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

**Acknowledgements**

The authors gratefully acknowledge support from the French government research programme "Investissements d'Avenir" through the IDEX-ISITE initiative 16-IDEX-0001 (CAP 20-25), and the AURA regional council. They also acknowledge with thanks Hélène Chanal for her help on the profilometer measurements.

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