# Video tracking of targets for vibration measurement of large-scale structures under seismic excitation

Stefania Lo Feudo*<fnref n="1" />, Yunhyeok Han<fnref n="1" />, Gwendal Cumunel<fnref n="2" />, Ryan Hoult<fnref n="3" />,
Alex Bertholet<fnref n="3" />, Denis Garnier<fnref n="2" />, Paulo Candeias<fnref n="4" />, António A. Correia<fnref n="4" />,
João Pacheco de Almeida<fnref n="3" />

## Abstract

Development of video-based vibration techniques and reliability of displacement measurements have important implications in several areas, including structural testing and monitoring of large structures. Through the use of numerous targets placed on the structure, 3-dimensional (3D) analysis and modal identification can be performed with minimal installation costs, no mass increase, and contactlessly. This study applies and validates video tracking of targets for vibration analysis of a large-scale structure, namely a reinforced concrete U-shaped wall, tested experimentally under seismic excitation on a shake table. 2D grids of targets are placed on two orthogonal surfaces and filmed by several cameras. Values obtained from a motion capture system, digital image correlation, and traditional displacement sensors are compared. Results confirm that the 3D motion of the U-shaped wall surfaces and foundation can be accurately extracted from video analysis.

**Keywords:** Experimental vibration; DIC; motion capture system; seismic response; U-shaped wall; large-scale experiments.

## 1 Introduction

Structural testing is crucial for simulating earthquake loading on reinforced concrete (RC) structures, as it provides essential insights into their behavior and vulnerability under moderate and strong ground motions, thereby enhancing the safety and reliability of these constructions in earthquake-prone regions. While conventional instrumentation is commonly used to measure displacement and acceleration histories at specific locations, state-of-the-art optical instrumentation now enables the measurement of a wide range of engineering parameters over larger areas. Notably, video analysis represents an interesting approach for measuring vibration of structures<fnref n="3" />. Digital image correlation (DIC) is one of the main tools for video-based vibration analysis⁵⁵;4;39;5⁹. In DIC, local displacement is estimated by comparing small regions into two

<footnote n="1">¹ QUARTZ Lab (EA7393), Vibrations, Acoustics, Structures and Mechanical Shapes, ISAE-Supméca, Saint-Ouen-sur-Seine, France</footnote>

<footnote n="2">² Lab Navier, ENPC, CNRS, F-77455, Gustave Eiffel University, Marne-la-Vallée, France</footnote>

<footnote n="3">³ Institute of Mechanics, Materials and Civil Engineering, UCLouvain, Louvain-la-Neuve, Belgium</footnote>

<footnote n="4">⁴ National Laboratory for Civil Engineering (LNEC), Lisbon, Portugal</footnote>

<footnote>*Corresponding author: stefania.lofeudo@isae-supmeca.fr</footnote>
subsequent images taken at successive instants. In the optical flow method, local speed is evaluated from pixel intensity gradient$^{26;27;16}$. Phase-based motion magnification method amplifies motion by looking at the phase of the image Fourier's transform$^{10;54;57;52}$.

By using two or more cameras, stereo vision techniques enable accurate 3D displacement measurement of a structure$^{55;4;46}$. Ngeljaratan and Moustafa$^{35}$ proposed 3D target-tracking for monitoring a bridge under earthquake shaking and pedestrian loading. The DIC was validated by comparison with conventional instrumentation. As a result, Root Mean Square Error (RMSE) in the displacement histories and peak values between the DIC and string potentiometers was less than 9 %. Calibration and correlation-based target tracking was performed with two cameras on a 3-story RC building specimen and a brick-walled RC frame under uni-axial near-fault ground motion by Yang$^{58}$.

To extract vibration displacement field or motion at discrete locations, a surface treatment is often necessary. To this end, speckle patterns or targets can be employed. The former are preferred for high-resolution images and suitable for DIC applications. A viable alternative in the case of large-scale structures and in order to reduce computational costs is to glue or paint on the structure's surface several discrete targets. Yang$^{58}$ tracked 8 black-white markers fixed at the nodes of the above-mentioned 3-story RC structure. The difference between LVDT and image analysis was less than 2 mm for a maximum displacement of 74.6 mm. Lo Feudo et al.$^{32}$ measured vibrations of a clamped-clamped beam with local non-linearity by detecting and tracking virtual sensors. Notably, targets consisted of two concentric circles with noise on the white circular crown. Maximally Stable Extremal Regions algorithm$^{34}$ was employed to extract the local features and to define regions of interest (ROI) around each target. Then, feature points were detected and tracked within each ROI$^{53}$. Several applications of target-tracking were addressed on past studies, such as measuring horizontal displacements and rotations of flexible steel column structures in two load cases$^{40}$; motion analysis of flexible beams and cables with commercial software EVaRT4.6$^{36}$; identification of mode shapes of a cantilever beam$^{49}$; bridge deflection under static$^{7}$ and dynamic load$^{15}$; 3D reconstruction of a wing dynamics with a single camera$^{45}$.

Projection of laser beams as in$^{28;47;18}$ or emitting diodes,$^{12}$ can be used instead of glued stickers. Optical motion tracking (OMT), for example OptiTrack system, is also used to measure 3D displacements from infrared reflecting markers located on the object to be tracked. The use of this type of system is common in biodynamics and is beginning to be used for structural dynamics$^{8;31}$.

Optical methods, in fact, offer a high density of measurement points, or even capture the vibration field across a large area, with minimal (or no) impact in terms of added mass. Additionally, video tracking of discrete targets serves as an effective tool for vibration analysis, enabling modal identification and 3D analysis. In this framework, Lo Feudo et al.$^{32}$ identified natural frequencies and modal amplitudes of a beam with local non-linearity by means of a data assimilation method based on extended Kalman and modal filters. The uncertainty of each mode was estimated as equal to 0.01 pixel through the observation noise. Yang et al.$^{56}$ employed the autoregressive exogenous (ARX) model to perform modal analysis of an arch dam in Southwestern China from its earthquake response. The refined Frequency Domain Decomposition (rFDD) algorithm was developed to apply operational modal analysis (OMA) on seismic structural response by Pioldi et al.$^{44}$. The rFDD and an improved Data-Driven Stochastic Subspace Identification (SSI-DATA) were compared for modal identification of a 10-story frame structure from simulated signals$^{43}$. Deviations with respect to real values were evaluated and discussed. Astroza et al.$^{2}$ compared several input-output and output-only methods for estimating modal parameters on a full-scale 5-story base-isolated RC building tested on the NEES@UCSD shake table. Data obtained from ambient vibration and low-amplitude white noise base excitation were used for modal identification. Results showed a good agreement in terms of mode shapes and modal frequencies. On the other hand, damping ratios resulted to be more method-dependents.
In Ngeljaratan and Moustafa's study³⁵ the system identification was performed before and after the seismic tests on a bridge structure by using results of target tracking. A set of discrete targets was bonded on columns and girders along a line. Different methods were compared for modal identification, like the Fast Fourier Transform (FFT) and the Autoregressive Covariance (AR), the Peak-Picking (PP), the Least Square Complex Exponential (LSCE), and SSI methods.

This paper presents some results obtained for a large-scale seismic test performed on a RC U-shaped core wall in the framework of the ERIES-ALL4wALL project. The structure was instrumented with traditional sensors, like string potentiometers, accelerometers, and Linear Variable Differential Transformers (LVDTs). Also, two different acquisition systems for video analysis were employed with two sets of targets. One is the motion capture system OptiTrack, the other is composed by three middle speed/resolution cameras filming circular and cross targets tracked by DIC. Thanks to the response obtained at a large number of locations, it is possible to reconstruct the 3D vibration field and to extract modal parameters. The Enhanced Frequency Domain Decomposition (EFDD) method will be herein employed. The first element of novelty is the video-based dataset itself, given the lack of large shake-table experiments on large-scale RC members wherein the above-mentioned methods have been applied simultaneously. In particular, this paper aims to validate the measurements obtained from video analysis. The dataset, to be made freely available for public download, will contain the results obtained from post-processing and not the raw videos (since each sequence requires more than 5.5 GB of memory for each camera).

The second innovative aspect is related to the challenge of employing multiple stereo camera pairs to perform 3D vibration analysis and 3D mode shape identification. To the authors' knowledge, existing works applied video analysis to large structures only tracking 2D grids of targets³⁵;⁵⁸. On the contrary, in the present case two orthogonal target grids were attached to the structure (for OMT) or painted (for DIC). This enables the identification of 3D mode shapes.

The use of multiple simultaneous views adds challenge to the experimental setup. Each group of targets must be filmed by a pair of cameras, with adequate lighting for each. In addition, since dynamic testing is involved, the movement of the structure (unknown a priori) must be considered to prevent the targets from falling out of the camera's field of view. For DIC, target sizing is also a key issue. In fact, painted targets must be small enough to assume rigid behavior within the zone, but large enough to result in changes in pixel intensity after small movements. The distance and focal length of the camera must also be selected. Therefore, the third novelty of this paper is the presentation of a simple procedure for choosing camera placement, setting up, and for sizing targets by building a virtual environment in Blender 4.0 software.

The paper is organized as follows. Section 2 presents the RC U-shaped core wall properties and the ground motions used for the seismic tests. Section 3 presents two video acquisition systems. Methods used for camera settings, calibration, target tracking and 3D reconstruction are given. In Section 4, displacement at the targets location is compared with that provided by traditional physical sensors. Following this, measurements obtained from video analysis are used to identify modal parameters in operational conditions. Finally, the conclusions are drawn in Section 5.

## 2 Experimental tests

Most experimental testing of RC structures and their assemblages employ quasi-static approaches, typically applied by hydraulic actuators⁴¹. Instead of attempting to replicate a specific strain rate or displacement history from a given earthquake, quasi-static methods subject the structure to predetermined static loading cycles based on engineering parameters such as ductility factors. Such tests typically follow protocols that incrementally increase drifts until failure is reached, commonly defined as a 20 % drop with respect to the maximum observed strength.
RC U-shaped core walls are popular in construction because they offer lateral stiffness in all directions while enclosing elevators or stair shafts²². Over the past two decades, numerous quasi-static tests have been conducted on RC U-shaped walls³⁰;¹¹;¹⁹;²⁴. Recent quasi-static tests at UCLouvain tested two RC U-shaped core walls for axial-flexure and axial-torsion, respectively²¹. DIC measurements were validated with conventional instrumentation (e.g., micrometers and LVDTs) and state-of-the-art Distributed Fiber Optic Sensors (or, DFOS)²¹;²⁰. Quasi-static tests inevitably provide inaccurate estimates of a structure's strength and ductility when subjected to real earthquake dynamic loads⁴¹. Cyclic quasi-static loadings also struggle to reproduce other potentially important features of the seismic response, including higher-order mode effects or energy dissipation by rocking impact.

A continuation of the aforementioned quasi-static tests involved dynamically testing two large-scale 40-ton RC U-shaped core walls on a shake table. Shake table tests, which reproduce the dynamic motions of recorded earthquakes, offer a more realistic method for assessing structural performance⁴¹. Despite their importance, such dynamic tests are less frequent due to their technical complexity, limited access to specialized equipment, and higher costs. To the authors' knowledge, the only shake table tests on U-shaped walls to date were those performed on the Azalée shake table²³, where low ductility steel was used. This can hinder our understanding of structural behavior under earthquake-induced actions. Dynamic shake table tests are therefore crucial in structural engineering because they accurately replicate real-world conditions, providing invaluable data for improving earthquake resilience.

The Engineering Research Infrastructures for European Synergies (ERIES) Transnational Access program provided an ideal opportunity to conduct these dynamic tests, enhancing our understanding of the seismic performance of RC core walls and allowing for a comparison with previous quasi-static test results. The current project is called ALL4wALL - Smart ALLoys for WALLs: towards durable structures with long service lives and minimal seismic residual displacements. The user group of the ERIES ALL4wALL project is composed of seven institutions: Université catholique de Louvain (Belgium), Ecole des Ponts ParisTech (France), ISAE-Supméca (France), University of Liège (Belgium), University of Ljubljana (Slovenia), University of Pavia (Italy), and York University (Canada).

This paper specifically focuses on the first specimen, referred to as unit UWS1. Figure 1(a) illustrates UWS1, comprising a foundation, a wall, two intermediate slabs with a central opening corresponding to the interior perimeter of the wall cross-section, and a top collar with increased thickness at the head of the wall. This half-scaled core wall unit, as depicted in Figure 1(b), shares the same cross-section and reinforcement detailing as quasi-static units previously tested at UCLouvain²¹. The specimen features a wall thickness ($t_w$) of 100 mm, a web length ($L_w$) of 1300 mm, and a flange length ($L_f$) of 1050 mm. Figure 1(b) also illustrates the segmentation of the wall into "Flange One," "Web," and "Flange Two" segments used in this research. The foundation measures 2100 mm × 2100 mm × 500 mm and is heavily reinforced with 20 mm diameter bars. The top collar, characterized by increased wall thickness at the wall head to support significant masses at the top, features an open design rather than a continuous thick top slab spanning over the flanges, due to considerations of warping restraint³⁰.

The wall unit was tested on the large triaxial shake table at the National Laboratory for Civil Engineering (LNEC) in Lisbon, Portugal. The orientation of the wall with respect to the cardinal directions is shown in Figure 1(b). The positive conventions are taken from north to south to and from west to east. Large masses, representative of a real (scaled) building, are required to excite sufficiently large inertial forces. Therefore, several 1.13 ton and 0.59 ton mass blocks are connected to the collar (head) of the wall. A total of 20 (i.e., 4 × 5) 1.13 ton mass blocks are placed atop of the collar of the wall unit, together with four 0.59 ton mass blocks. The above means that the applied total mass above the collar is 24.96 ton, or roughly 28 ton if the collar is included. M30 threaded bars were used to fasten the foundation of the unit to the shake table and also the applied mass on the top collar, with the aim of avoiding (minimizing)
<figure><img src="image_1.png" /><figcaption>Figure 1: Unit UWS1 (a) three-dimensional elevation view mass blocks and (b) cross-section and reinforcement detailing.</figcaption></figure>

relative displacements between the foundation and the shake table, and between the added mass and the top collar.

The dynamic testing of unit UWS1 involved nine distinct ground motion levels (GM0 to GM8). Each level corresponds to a specific combination of ground motion components' scale factors for the west-east (WE) and north-south (NS) directions. The ground motions were recorded during a moment magnitude $M_w$ 6.5 event$^{9,51}$ recorded on October 30, 2016, in Central Italy, by station MZ04$^{33}$. MZ04 recorded the 5% damped pseudo spectral acceleration response in the HNE and HNN directions, as depicted in Figure 2(a). For the dynamic testing, the HNN ground motions were applied in the WE shake table direction, while the HNE ground motions were applied in the NS direction. The peak ground acceleration (PGA) in these directions is approximately 0.808 g and 0.645 g, respectively, which corresponds to a scale factor of 1 (100%). This paper focuses its investigations and analyses on some specific ground motion levels: unidirectional GM0, which represents 10% of the WE ground motion, see Figure 2(b); bidirectional GM8, representing 100% of both WE and NS ground motions; and Noise 5, representing white Gaussian noise applied in the WE direction for system identification, such as OMA, after the application of GM4 wherein ground motions were scaled by 50% in both directions.

## 3 Video analysis

This section describes properties of two acquisition systems, namely, the motion capture system and the middle speed/resolution cameras. Specific targets for each system, and methods
<figure><img src="image_2.png" /><figcaption>Figure 2: Moment magnitude *M*<sub>w</sub> 6.5 event from the 2016 Central Italy earthquake sequence on October 30: (a) 5% damped pseudo spectral acceleration (corrected using a scaled time-step) (b) acceleration-time history in the west-east (i.e., HNN) and in the north-south (i.e., HNE), corresponding to a scale factor of 100%.</figcaption></figure>

necessary for cameras' setting, target tracking and 3D reconstruction will be presented.

**3.1 Motion capture system**

OMT systems are widely used in biomechanics research to measure and analyze human move-
ment. For our study, a commercial system from OptiTrack is used. The system detects infrared
light reflected from markers using multiple cameras, allowing it to measure the position and
orientation of the markers in three-dimensional space with sub-millimeter accuracy.

To detect the markers, the cameras are equipped with light diode systems that illuminate
the camera's field of view. These diodes flash at the same frequency as the cameras' acquisition
rate. This flashing enables the cameras to detect the markers illuminated by the diode system's
light beam associated with each camera, creating 2D images with white pixels where the markers
appear against a dark background. The markers are effectively separated from the background
using a thresholding on the pixel values. Additionally, only the centroid of each marker is of
interest.

By using many cameras positioned around the structure (at least two), full 3D tracking can
be obtained. The OMT system is calibrated in order to determine the position and orientation
of each camera using a calibration wand. The position and orientation of each camera are
obtained using the so-called direct linear transformation<sup>48</sup>. Then the spatial correspondence of
the markers is performed. If a marker is detected on an image, the 2D location of its centroid is
determined in the image plane. When multiple sets of 2D points (one set for each camera) are
matched, they are reconstructed into the 3D space according to a multicollinearity constraint.
Given the coordinates of a markers centroid in at least two different camera image planes, stereo
triangulation on rays (forward intersection) is used to compute the 3D position of the markers
centroid in the scene<sup>14</sup>. Based on the projection matrix of each camera, obtained through
the calibration process, a detected marker corresponds to a ray coming from the center of the
camera and passing through the 2D location of the markers centroid. The intersection point
of rays corresponding to the same marker from multiple cameras is the position of the marker
centroid in 3D space.

The OMT system used for the tests is composed of 10 cameras: four OptiTrack PrimeX 22 cameras (2.2 MP resolution, 2048 × 1088) and six OptiTrack Prime 17W cameras (1.7 MP resolution, 1664 × 1088). The OptiTrack PrimeX 22 cameras are the cameras 3, 4, 6, and 7.
The cameras are positioned all around the structure to have different points of view (height and orientation) of the structure. The frame rate is set to 100 Hz and the exposure time to 8 ms for all cameras.

The software Motive (version 3.0.3) is used to obtain the 3D displacements of the reflective markers (19 mm in diameter) located on the structure. Before being able to take the measurements, a calibration phase is necessary. It consists of moving a calibration wand (model CW-500), having three markers whose distance is perfectly known, in the field of view of the cameras. Then, the system computes the position and orientation of each camera and number of distortions in captured images to construct a 3D capture volume in Motive. The quality of the calibration is estimated by two errors: the mean ray error and the mean wand error. The mean ray error represents how closely the tracked rays from each camera converged onto a 3D point, that is the preciseness of the calculated 3D points during wanding. The mean wand error is the mean error value of the detected wand length compared to the expected wand length. In our tests, the mean ray error is equal to 0.8 mm and the mean wand error to 0.13 mm, which is very small. Finally, a calibration square (CS-400) allows to set the ground plane at the level of the shake table.

Figure 3 shows the markers detected by each camera and Figure 4 a global view with the 3D position of the cameras and detected markers. Figure 5(i) depicts a motion capture marker.

## 3.2 Digital cameras system

Video sequences were recorded by three digital cameras Basler, two acA1300-200um and one acA1440-220um. The software Pylon Viewer was used for the image acquisition and the trigger ED developed by the R&D Vision allowed the camera synchronization. Frames were transmitted in real-time to three PC via Micro USB 3.0 links. Further details about the equipment and settings are given in Table 1 according to the good-practice guidelines for DIC²⁹. Cameras were positioned in such a way as to view a facade of the UWS1 in pairs in order to reconstruct the 3D vibration field. According to the description given in Figure 1(b), camera B1 filmed the outside surface face of the Web, camera B3 filmed the outside surface of Flange Two, and camera B2 filmed both of these normal surfaces.

### 3.2.1 Camera setting

A grid of 172 circles and crosses was painted on the outside wall surfaces. The use of a circular target allows to determine at each instant the position in space of a point located at the center of the area. The use of a cruciform target could give access, in addition to translation, to local rotation at the sensor location. This aspect will be explored in future studies and is beyond the

<table><caption>Table 1: Image acquisition equipment, settings and software information.</caption><thead><tr><th>Parameter</th><th colspan="3">Cameras</th></tr></thead><tbody><tr><td>Name</td><td>B1</td><td>B2</td><td>B3</td></tr><tr><td>Brand</td><td></td><td>Basler</td><td></td></tr><tr><td>Model</td><td>acA1300-200um</td><td>acA1440-220um</td><td>acA1300-200um</td></tr><tr><td>Image resolution (px)</td><td>1024 × 1280</td><td>1080 × 1440</td><td>1024 × 1280</td></tr><tr><td>Focal length (mm)</td><td>8</td><td>6</td><td>8</td></tr><tr><td>Diaphragm opening</td><td>3</td><td>1.9</td><td>3</td></tr><tr><td>Exposure time (μs)</td><td>9500</td><td>9500</td><td>9500</td></tr><tr><td>Lighting</td><td></td><td>Natural</td><td></td></tr><tr><td>Image acquisition rate (fps)</td><td>100</td><td>100</td><td>100</td></tr><tr><td>Image acquisition platform</td><td></td><td>Pylon Viewer</td><td></td></tr><tr><td>Syncronization</td><td></td><td>EG trigger R&D vision</td><td></td></tr></tbody></table>
<figure><img src="image_3.png" /><figcaption>(a) Camera 1 (1664 × 1088 px)</figcaption></figure>

<figure><img src="image_4.png" /><figcaption>(b) Camera 2 (1664 × 1088 px)</figcaption></figure>

<figure><img src="image_5.png" /><figcaption>(c) Camera 5 (1664 × 1088 px)</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>(d) Camera 8 (1664 × 1088 px)</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>(e) Camera 9 (1664 × 1088 px)</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>(f) Camera 10 (1664 × 1088 px)</figcaption></figure>

<figure><img src="image_9.png" /><figcaption>(g) Camera 3 (2048 × 1088 px)</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>(h) Camera 4 (2048 × 1088 px)</figcaption></figure>

<figure><img src="image_11.png" /><figcaption>(i) Camera 6 (2048 × 1088 px)</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>(j) Camera 7 (2048 × 1088 px)</figcaption></figure>

<caption>Figure 3: View of the cameras of the OMT for the ground motion test.</caption>

scope of this article. The targets were made with the help of a plexiglass stencil and a spray can. A speckle pattern was painted on the bottom surfaces of the wall. This part of the structure was filmed by other cameras with the aim of measuring the evolution of damage. This aspect is out of the scope of this paper and it will be addressed in future research studies.

In order to size the targets and define the cameras position, the experimental set-up was virtually reconstructed with the help of software Blender 4.0. At first, the LNEC laboratory floor plan was imported and the UWS1 volumetry reconstructed in 3D. Then, the three cameras were added into the Blender environment by specifying the sensor and lens properties. The cameras location was defined taking into account the available space in the laboratory and the size of the UWS1. The requirement being to keep the overall wall in the-field-of-view throughout the dynamic tests. In order to ensure the latter, the displacement of the wall, considered as a rigid body, was simulated in Blender when subjected to one of the multi-directional ground motion selected for the experimental tests. Once the position of the cameras was fixed, a grid of circular
<figure><img src="image_13.png" /><figcaption>Figure 4: Global view of the OMT system for the ground motion test (with colors reversal).</figcaption></figure>

targets was drawn in Blender on the outer surface of the wall. Camera views were simulated in Blender and the targets' size chosen. Indeed, empirical considerations from previous experience in image processing suggest sizing the targets such that the diameter in the image is greater than 10 pixels. This led to the choice to draw circular targets with a diameter equal to 5 cm for this experiment. Cruciform targets were sized following the same logic. The virtual and final real camera views are depicted in Figure 5(a)-(f). Figure 5(g)-(h) shows a view of a circular and a cross target.

### 3.2.2 Digital Image Correlation

DIC can be employed to establish a one-to-one correspondence between points in the reference and subsequent images. Consequently, a reference region of interest (ROI), defined as a rectangle subset, can be tracked, and its position in image coordinates can be obtained for all frames. In this research, we applied an algorithm previously pinpointed as a gold standard³⁷. The shape function, which represents the deformation of the rectangular subset in the deformed image, is defined as a first-order transformation function, as expressed in Eq. 1.

$$ 
\begin{aligned}
x'_{i} &= x_{i} + u + \frac{\partial u}{\partial x} \Delta x + \frac{\partial u}{\partial y} \Delta y \\
y'_{j} &= y_{j} + v + \frac{\partial v}{\partial x} \Delta x + \frac{\partial v}{\partial y} \Delta y \\
\mathbf{p} &= \begin{bmatrix} u & v & \frac{\partial u}{\partial x} & \frac{\partial u}{\partial y} & \frac{\partial v}{\partial x} & \frac{\partial v}{\partial y} \end{bmatrix}
\end{aligned} 
\quad (1) $$

where, $x_i$ and $y_j$ represent the coordinates in the image plane of the i-th and j-th points in the reference subset, while $x'_i$ and $y'_j$ denote the coordinates of the same points in the deformed subset. The displacement of the subset in the x and y directions is represented by $u$ and $v$, respectively. Finally, $\mathbf{p}$ is the vector of the shape function parameters. It is possible that the coordinates of the points in the deformed subset, namely $(x'_i, y'_j)$, may be located in a sub-pixel location. Therefore, a sub-pixel interpolation scheme must be defined. Here, the bi-cubic interpolation (Eq. 2) was chosen.

$$ g(x, y) = \sum_{m=0}^{3} \sum_{n=0}^{3} \alpha_{mn} x^m y^n \quad (2) $$
<figure><img src="image_14.png" /><figcaption>Figure 5: Virtual and real camera views of (a, d) B1, (b, e) B2 and (c, f) B3. B1 and B3 have a resolution of 1024 × 1280 px, and B2 of 1080 × 1440 px. All images are resized here for consistent representation. (g) and (h) are enlarged views from B3: (g) shows a circle and (h) a cross. (i) depicts a motion capture marker.</figcaption></figure>

The interpolated value at the sub-pixel location (x, y) is denoted by $g(x, y)$, while the coefficient of the bi-cubic interpolation is $\alpha_{mn}$. In order to evaluate the similarity between a reference subset and the corresponding deformed subset in subsequent images, a correlation criterion was defined. The Zero-Normalized Cross Correlation (ZNCC) or Zero-Normalized Sum of Squared
Difference (ZNSSD) correlation criterion offers the most robust performance in the presence of noise and is insensitive to illumination variations³⁸. Therefore, the ZNSSD correlation criterion, as defined in Eq. 3, is employed to quantify the similarity between subsets.

$$C_{ZNSSD}(\mathbf{p}) = \sum_{i=-M}^{M} \sum_{j=-M}^{M} \left[ \frac{f(x_i, y_j) - f_m}{\Delta f} - \frac{g(x'_i, y'_j) - g_m}{\Delta g} \right]^2 \\ \Delta f = \sqrt{\sum_{i=-M}^{M} \sum_{j=-M}^{M} [f(x_i, y_j) - f_m]^2} \\ \Delta g = \sqrt{\sum_{i=-M}^{M} \sum_{j=-M}^{M} [g(x'_i, y'_j) - g_m]^2} \quad (3)$$

where $f(x_i, y_j)$ represents the pixel intensity in the reference gray-scale image, $f_m$ is the mean intensity of the subset in the reference image, $g(x'_i, y'_j)$ is the pixel intensity in the deformed image, $g_m$ is the mean intensity of the deformed subset, and $2M+1$ is the size of the subset. An iterative spatial domain cross-correlation algorithm was employed as a sub-pixel displacement registration algorithm. The Gauss-Newton iteration method was utilized with the gradient vector of the correlation criterion and the approximation of the Hessian matrix to obtain the transformation of the subset. Past studies evaluated the accuracy of measurements performed by DIC as less than 1/10 pixel⁵;¹¹³. As shown by the authors, accuracy depends on several factors such as noise level, blur level, target style, etc.

### 3.2.3 Target tracking and 3D reconstruction

The image post-processing operations and target tracking were performed in Python with the OpenCV library. Before starting the dynamic tests, some photos necessary for calibration were taken. In particular, a ChArUco panel (10 cm × 6 cm, 50 cm×30 cm, DICT_5×5_1000) was photographed at different angles from each camera. The ChArUco panel was also placed on the wall surfaces and photographed by the cameras, see Figure 5(d-e). To place the painted targets in the 3D coordinates and to follow their position during the ground motion, the following operations were performed in the post-processing stage:

1. **Intrinsic calibration.** True focal length and distortion coefficients were determined according to the pinhole camera model and Zhang's method⁶⁰, resulting in all images being undistorted.

2. **Extrinsic calibration.** The position of the cameras was determined by selecting on images taken before the start of testing some points visible from cameras B1-B2, and B2-B3, respectively, and whose position was known. For example, the vertices of the lower and upper base and intermediate slabs. In this way, rotation matrix and translation vector of each camera with respect to the Cartesian Galilean reference system shown in Figure 5(a-c) were estimated.

3. **Target detection.** For each camera, the position in the image plane of the circular and cruciform targets was initialized by selecting by hand the features. A bounding box defining a ROI was then defined around each target. Algorithms such as MSER detectors could be used for automatic detection of the targets. A total of 168 targets were detected.

4. **Target tracking.** The algorithm of digital image correlation described in Section 3.2.2 was employed to track the position of targets during the video sequence. The size of ROI to be tracked with DIC is defined as 21 × 21 px.
5. **3D reconstruction.** The position of each target was projected into the 3D space. The position of targets at the first frame of the video sequences is represented in Figure 6(a).

<figure><img src="image_15.png" /><figcaption>Figure 6: (a) Position of the circular and cross targets position in the 3D space. (b) Similar view for the RC U-shaped wall.</figcaption></figure>

# 4 Experimental results

The seismic response of the U-shaped specimen subjected to ground motions obtained from video analysis will be used here to make comparisons to that obtained by classical sensors. Then, a method of OMA will be applied on a stationary signal and the first two mode shapes will be identified.

## 4.1 3D vibrations of the RC U-shaped core wall

Signals measured from LVDT, string potentiometer sensors, OMT and DIC were synchronized in time by maximizing their cross-correlation. Also, the mean value was subtracted from each signal.

At first, the 3D displacement obtained by applying DIC on a circular and a cross targets (numbered 1 and 2), placed on the northern extremity of the west foundation face, is compared in Figure 7. A very good correspondence is obtained, mostly in the x- and y-direction, according to the Cartesian coordinates given in Figure 6(b). As expected, the displacement is very low in the z-direction (i.e. vertical) and the signal is affected by targets' tracking errors and by the image noise, which is due to a random variation in pixel intensity present in all digital images¹⁷. In the following analyses, DIC results will refer to both cruciform and circular without distinction.

Figure 8 shows the 3D displacement of the foundation for GM0. It should be noted that the LVDT, OMT, and painted targets are not in the exact same location. This is typical in
<figure><img src="image_16.png" /><figcaption>Figure 7: Displacement of circular and cross targets (1 and 2) obtained from DIC in the (a) x-, (b) y- and (c) z-direction under GM8.</figcaption></figure>

the case of video analysis, because it is generally impossible to place a physical sensor (e.g., an accelerometer, vibrometer, or string potentiometer) at the same location of a target: (i) in the case of OMT because the physical space required is already occupied by a OMT reflexive marker that needs to be unencumbered in its 3D surrounding; (ii) in the case of painted targets because a physical sensor would cover the target in the images and hence disturb its detection and tracking. However, the foundation block is expected to remain approximately rigid throughout testing, thus no significant error is expected to contribute to these displacement values due to locations of the sensors. According to Figure 9, the error measured as the point-to-point difference between data obtained from virtual (OMT and DIC) and physical (LVDT) sensors is less than 1 mm for DIC and 1.5 mm for OMT in the only motion direction, that is along *y* (WE direction). This value is rather low considering that the distance between cameras and the structure was of several meters. These disturbances have multiple origins, like image noise, vibrations of the cameras' support, cameras' cooling fan and tracking errors. In this experiment, the smallest projection in the image of the 5 cm circular target was measured as equal to 10 px. Previous studies$^{5;1;13}$ report accuracy on the order of less than 1/10 px, which leads to 0.5 mm in the present case. This agrees with results shown in Figure 9(c), where error with respect to the LVDT is less than 0.5 mm. For the OMT system, the noise is mainly due to the intermittent loss of marker position by some cameras during the ground motion test.

<figure><img src="image_17.png" /><figcaption>Figure 8: Displacement at the foundation in the (a) x-, (b) y- and (c) z-direction under GM0, for different sensing methods.</figcaption></figure>

Figure 10 shows the 3D displacement of the foundation under GM8. A good correspondence is found between data coming from physical sensors and targets. Again, only noise is measured in the *z*-direction (vertical direction, along which no ground motion component was applied), where displacements are less than 1 mm. The collar displacement measured from a string potentiometer, OMT and DIC for target 168 are then compared in Figure 11 in the *x*- and *y*-direction, confirming the accuracy of video analysis. Also in this case, the collar can be approximated as rigid. However, unlike for the foundation, DIC results now show in the *z*-direction an increase in the motion level. This is expected and corresponds to the vertical
<figure><img src="image_18.png" /><figcaption>Figure 9: Error of displacement obtained from targets tracking at the foundation in the (a) x-, (b) y- and (c) z-direction under GM0.</figcaption></figure>

<figure><img src="image_19.png" /><figcaption>Figure 10: Displacement at the foundation in the (a) x-, (b) y- and (c) z-direction under GM8.</figcaption></figure>

<figure><img src="image_20.png" /><figcaption>Figure 11: Displacement at the collar in the (a) x-, (b) y- and (c) z-direction under GM8.</figcaption></figure>

elongation of the unit due to its flexural flexibility, enhanced by crack opening and closing. Differences between OMT and DIC are also to be expected, as the markers are located in different locations. The displacement in the z-direction was not measured by a physical sensor, and therefore a comparison cannot be made. This fact is a reminder that specific placement of the traditional sensor must be planned if a video-based measurement is to be validated in the vertical direction. Finally, Figure 12 shows the torsional movement of the U-shaped wall at the foundation and at the collar under GM8. Rotations were obtained by converting data in polar coordinates.

## 4.2 Modal identification

The identification of modal parameters is obtained from EFDD in operational conditions by using the motion of painted targets tracked by DIC. Only the response of the RC U-shaped wall excited by a white Gaussian noise (Noise 5) in the WE direction is considered. Calculations are performed with the open-source Python module named PyOMA$^{42}$. Relative displacements with respect to the foundation are considered. To perform the EFDD, the Singular Value Decomposition (SVD) of the Power Spectral Density (PSD) matrix is evaluated by using 3D
<figure><img src="image_21.png" /><figcaption>Figure 12: Torsional movement at the (a) foundation and (b) collar of the wall unit under GM8 ground motion.</figcaption></figure>

displacements at the targets location. Figure 13 depicts the singular value components at each frequency line. Then, the Power Spectral Density function, identified around a resonance peak, is returned in time domain using the Inverse Discrete Fourier Transform (IDFT). Modal frequencies are obtained by determining the number of zero-crossing as a function of time, and damping by the logarithmic decrement of the autocorrelation function$^{6;25;50}$. At Noise 5, two modal frequencies are identified: 2.19 Hz and 5.06 Hz, with damping ratio 6.7% and 3.7%, respectively. A mode shape is evaluated as the first singular vector at a resonant peak selected on the SVD plot. The two mode shapes are here identified and depicted in Figure 14. Black dots denotes the initial position of the targets, whereas color dots represents a mode shape. Blue line represents the sensor trajectory. Examination of the two deformations suggests a first flexural-torsional mode and a second torsional mode. It is noted that the identified modal frequencies and shapes refer to the system composed by the shake table and the RC wall. In general, they will not match the results of eigenvector/eigenvalue analyses obtained from numerical models of the wall with a fixed base. More elaborate comparisons will be performed in future works.

<figure><img src="image_22.png" /><figcaption>Figure 13: Singular value decomposition at Noise 5 extracted from response of painted targets processed by DIC.</figcaption></figure>

# 5 Conclusion

This investigation proposes and validates two video-analysis methods for 3D vibration measurement of a large-scale structure under seismic excitation. The structure under study is a RC U-shaped wall tested in two directions on the large shake table at LNEC, in Lisbon (Portugal).
<figure><img src="image_23.png" /><figcaption>Figure 14: (a) First and (b) second mode shapes identified by EFDD.</figcaption></figure>

The structure was equipped with two-dimensional grids of targets placed on two orthogonal surfaces. These dynamic tests were monitored by two systems, namely a motion capture system (OMT) and a set of three middle speed/resolution cameras. In both cases, the 3D vibration field was reconstructed and compared where possible with data obtained from traditional sensors (LVDTs and string potentiometers). Point-to-point error displacement between virtual and physical sensors was shown to be less than 19 % for a maximum displacement at the foundation of 7.7 mm for the OMT and 13 % for targets tracked by DIC. For larger displacements, the error is much lower. For the maximum displacement at the collar in the *y*-direction under GM8, the error is equal to 0.6% for the DIC and to 2% for OMT.

Video systems provided accurate displacement values in all directions and over a large number of targets, enabling three-dimensional reconstruction of the vibration field. Moreover, application of an operational modal identification, namely, EFDD method, allowed the identification not only of modal frequencies and damping, but also of mode shapes. In the present case, a flexural-torsional and a torsional mode emerge from the study of the response to a mono-directional white noise ground motion. The displacement data obtained from the validated target constitutes a valuable database that can be used for refined comparisons with nonlinear dynamic models of the RC U-shaped wall, contributing to a better understanding of these elements behavior under seismic excitation. The numerical simulation of modal frequencies and shapes requires, however, the inclusion of the shake table in the model to account for the wall-table dynamic interaction.

The technique herein applied in laboratory conditions can also be extended to the outdoor environment. Practical considerations regarding camera placement and choice of targets size selection, demonstrated through a virtual experiment in Blender 4.0, offer valuable insights for researchers and engineers setting up optical measurements to identify the dynamic properties of large structures in general. This method is particularly useful when experimental campaigns are limited due to high costs, the need for outdoor testing, or requirements for land occupation permits.
**Data availability statement**

The datasets generated during and/or analyzed during the current study are available from the
corresponding author on reasonable request.

Author contributions

**Stefania Lo Feudo**: writing original draft (lead), conceptualization, investigation and re-
sources (equal), data curation (support). **Yunhyeok Han**: data curation (lead), conceptu-
alization and investigation (equal), writing original draft (support). **Gwendal Cumunel**:
conceptualization and resources (equal), data curation, investigation and writing original draft
(support). **Ryan Hoult**: data curation, investigation, funding acquisition and writing original
draft (support). **Alex Bertholet**: resources (equal), investigation (support). **Denis Garnier**:
resources (equal), investigation (support). **Paulo Candeias**: resources (equal), data curation
and investigation (support). **António A. Correia**: resources (equal), writing review & editing,
funding acquisition and project administration (support), **João Pacheco de Almeida**: funding
acquisition, project administration and writing review & editing (lead), conceptualization and
data curation (support).

Funding

This work is part of the transnational access project "ERIES ALL4wALL", supported by the
Engineering Research Infrastructures for European Synergies (ERIES) project (www.eries.eu),
which has received funding from the European Unions Horizon Europe Framework Programme
under Grant Agreement No. 101058684. This is ERIES publication number J5.

Acknowledgments

The investment and availability of the technical staff at LNEC, during the preparation and
execution of the test, was fundamental to the successful completion of these tests. The authors
also express their gratitude to PhD student Erika Licciardello for her help in recording video
sequences.

Disclosure

The authors have nothing to report.

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