1. Introduction

Three-dimensional (3D) deformation measurement is a fundamental problem in photo-mechanics and related fields. In the past decade, it was significantly altered by the establishment of stereo digital image correlation (stereo-DIC) [1,2]. As an optical measurement technique, stereo-DIC measures the full-field 3D deformation data with a simple optical arrangement, and thus goes beyond the traditional techniques (e.g., strain gauge and mechanical extensometer) in many land-based tasks. Huge parts of daily experiments and engineering tests, such as characterization of mechanical and geometrical properties for materials [3,4], biomechanical analysis [5,6], deformation control in industrial fabrication [7,8], and structural health monitoring [9,10], can be powered by stereo-DIC. When it comes to fluid-immersed applications (such as submerged biological tissues and materials characterization, marine structures measurement, etc.), regular stereo-DIC faces challenges posed by light refraction.

In fluid-immersed measurements, the cameras are often placed outside a fluid-filled tank or in a sealed housing and then, observe the object to be measured through a transparent window. For such scenarios, light rays reflected from the surface of the object will pass through the media with a graded refractive index, such as water, glass, and air. Because of light refraction, rays gathered by the cameras are not transmitted straightly but bent at interfaces between different media. However, cameras used in the regular stereo-DIC system are represented as the perspective model (also known as the pinhole camera model) to describe the projection transformation from 3D points in the world to 2D points in images [1,11]. The vision geometry in both forward and backward projection is assumed to correspond to the single viewpoint assumption where light travels in straight lines. If the light rays are bent due to the refraction, this assumption does not hold [12], leading to erroneous 3D reconstruction. This will further lead to errors in deformation estimates since stereo-DIC is highly relying on the recovered object shape in retrieving 3D displacement. According to the law of refraction, i.e. Snell’s law [13], the refractive distortion depends on the refractive index of medium and the incident angle of the entering ray. Adopting dome windows is accordingly a natural yet efficient way to compensate for the refractive error. However, they need to be carefully fabricated and assembled to fit the optics lens and exhibits undesirable field curvature which leads a plane to be projected as a paraboloid [14,15], thereby limiting its use in daily stereo-DIC measurement tasks.

In contrast, flat windows are more flexible and practical and then, the foci of this study. Although the literature has shown that the refractive distortion can be partially corrected by adjusting the focal length and lens distortion parameters [16,17], these approximate methods are insufficient because the flat refractive effect is highly nonlinear and depends on the location of an object [18,19]. A more attractive approach to eliminate errors caused by the refraction of light is to use specific image formations, such as refractive camera models [12,20–22], generalized camera models [18,23], and the combined pinax camera model [15], to explicitly describe the effect of light refraction. These methods have facilitated great progress in refractive imaging and measurement, however, the complexity of the theoretical models leads to they are rarely concerned in practical stereo-DIC applications. Particularly, related work can be found in a range of literature for stereo-DIC. For example, authors in [24] developed a non-linear stereo calibration method by explicitly modeling the refraction at the refracting interface(s). The framework is theoretically comprehensive to represent the refractive effects, which however is complicated in finding the optimal calibration due to the high nonlinearity of light refraction. In contrast to developing a specific calibration model, the refraction error can be eliminated partially by carefully arranging the cameras with their optical axes perpendicular to the refracting interface and then, performing submerged stereo calibration with the conventional calibration framework [25,26]. In addition, the refraction error can also be suppressed to some degree by using a specific image registration method that takes into account the refraction-induced image distortion [27]. Though both methods are simple and practical, they, similar to the aforementioned approximate methods, cannot compensate for the refractive distortion completely. More recently, a single-camera stereo-DIC was specially proposed for underwater 3D deformation measurement by combining a bilateral telecentric lens and a bi-prism [28]. Because of such specific optical configuration, the device is allowed to perform measurement without stereo camera calibration but is also limited to small-scale object measurement accordingly. A notable current limitation of most of the existing methods for measuring submerged objects is that they are rarely compatible with the popular stereo-DIC systems which rely on the perspective cameras, resulting in a major practical limitation given that building an easy-to-use stereo-DIC system for fluid-immersed measurement in daily experiments is not easy. This implies that flexible, accurate, and compatible methods for measuring submerged objects with stereo-DIC are an important area for further development in fluid-immersed full-field optical deformation characterization.

In this paper, we propose a flexible and accurate stereo-DIC based on the flat refractive geometry to measure the 3D shape and deformation of fluid-immersed objects. We refer to it as a refractive stereo-DIC to tell it apart from the regular one. In the proposed refractive stereo-DIC, we consider the refractive distortion by reformulating the 3D reconstruction process in stereo-DIC. Based on Snell’s law, we establish a complete refractive geometry for parallel flat refracting interfaces using backward ray tracing and then, propose a refractive 3D reconstruction accordingly to trace the true object shape from that resulted by the regular triangulation. Given media, the introduced refractive geometry depends only on the refracting interface(s) and the 3D shape recovered by the triangulation, allowing the refractive 3D reconstruction to be performed flexibly without the need of modifying the camera model and image formation. By building the refractive 3D deformation measurement procedures, we show it can be integrated into the existing frameworks conveniently to establish a refractive stereo-DIC system. To demonstrate and highlight the contributions of our method, we evaluate the feasibility and correctness with experiments on measuring 3D shape and displacement of underwater specimens, respectively.

The rest of this paper is organized as follows. Section 2 presents the principles of refractive stereo-DIC, including geometry of flat refraction in Section 2.1, refractive 3D shape reconstruction in Section 2.2, and deformation measurement procedures in Section 2.3. Experimental verifications and results are shown in Section 3. We conclude in Section 4.

2. Refractive stereo-DIC measurement

2.1 Geometry of flat refraction

The geometry of refractive stereo-DIC is established with backward ray tracing of geometric optics based on the refraction law. Without loss of generality we consider two-layer refracting interfaces between three transmission media with different optical densities. We suppose for simplicity of exposition both interfaces are a practical flat surface and parallel to each other. Given a stereo-DIC system composed of left and right cameras, the refractive ray geometry is built with the complete Snell’s law in Fig. 1. Details are presented as follows.

 

Fig. 1. Refractive geometry for two flat interfaces. $O-XYZ$ and $O'-X'Y'Z'$ are the reference 3D coordinate frame and the right camera frame, respectively. The end node of each $Y$-axis is visualized with a green circle containing a cross. See text for more details.

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In Fig. 1, both cameras are placed in air, indicated by refractive index $n_1 = 1$, to image a randomly speckled object surface through double medium with refractive indices $n_2$ and $n_3$ respectively (the most common types are glass and water). The two interfaces are denoted by $\Pi _1$ and $\Pi _2$, respectively, and are assumed to be parallel at an interval of $d$. In order to observe a 3D point $\mathbf {Q}$ on the surface, we suppose the left and right cameras cast two light rays $\mathbf {L}_1$ and $\mathbf {L}_1^\prime$, respectively. If there is no refraction (i.e. $n_1=n_2=n_3$), they travel in straight and converge at the point $\mathbf {P}$, corresponding to the regular stereo-DIC measurement. For the refractive scenario, $\mathbf {L}_1$ and $\mathbf {L}_1^\prime$ are bent at points $\mathbf {P}_1$ and $\mathbf {P}_1^\prime$ respectively when they pass the interface $\Pi _1$. According to Snell’s law [13], their directions in propagation are changed and thus turn to travel along the lines $\mathbf {L}_2$ and $\mathbf {L}_2^\prime$ correspondingly. Subsequently, $\mathbf {L}_1$ bends again at point $\mathbf {P}_2$ and turns to travel along the line $\mathbf {L}_3$ after it runs through the interface $\Pi _2$; so does the ray $\mathbf {L}_1^\prime$ at $\mathbf {P}_2^\prime$ and the refracted ray path is denoted by line $\mathbf {L}_3^\prime$. Finally, $\mathbf {Q}$ is observed and imaged by both cameras when the rays hit the object along $\mathbf {L}_3$ and $\mathbf {L}_3^\prime$, respectively. The corresponding angles of refraction for both light rays are denoted by $\theta _i$ and $\theta _i^\prime ~(i=1,2,3)$, respectively.

The geometry in Fig. 1 shows that the refracted ray-path pair, denoted by $\mathbf {L}_1-\mathbf {L}_2-\mathbf {L}_3$ and $\mathbf {L}_1^\prime -\mathbf {L}_2^\prime -\mathbf {L}_3^\prime$, for an observed object point is clearly different from that (i.e., $\mathbf {OP}$ and $\mathbf {OP}'$) without refraction. It varies not only with the distance from the optical axis, but also the positions and orientation of the interfaces, as well as the position of the object point. Therefore, for stereo-DIC based on the projective camera model, the presence of refraction leads to the erroneous location and displacement estimation of the measured points if the regular 3D reconstruction is applied.

2.2 Refractive 3D reconstruction

With the refractive geometry in Fig. 1, our goal is to recover the true 3D coordinates of $\mathbf {Q}$ in the reference coordinate frame $O-XYZ$ from its projections for avoiding the distortion caused by light refraction. We suppose the stereo camera system is fully calibrated as usual. The reference frame $O-XYZ$ is assumed to be aligned to the coordinate system of the left camera, thereby the optical centers of the left and right cameras are $\mathbf {O} = (0, 0, 0)$ and $\mathbf {O}^\prime = \mathbf {t} = (t_x, t_y, t_z)$, respectively, where $\mathbf {t}$ is the calibrated translation vector of the right camera relative to the reference frame. According to the refractive geometry, $\mathbf {Q}$ is the intersection of lines $\mathbf {L}_3$ and $\mathbf {L}_3^\prime$, thus its coordinates can be determined by establishing the refracted ray-paths $\mathbf {L}_1-\mathbf {L}_2-\mathbf {L}_3$ and $\mathbf {L}_1^\prime -\mathbf {L}_2^\prime -\mathbf {L}_3^\prime$ in 3D space. Detailed procedure is given below.

Given a stereo-pair of the speckled object surface captured by the left and right cameras simultaneously. The projections corresponding to $\mathbf {Q}$ are found firstly by the DIC-based stereo matching, constructing a stereo correspondence with pixel coordinates. Then we can adopt the regular triangulation algorithms [2,29] to estimate the point $\mathbf {P}$ instantly. Because of the light refraction, the point $\mathbf {P}$ is the distorted position of the true object point $\mathbf {Q}$. To track the point $\mathbf {Q}$, we need to build the refracted ray paths from $\mathbf {P}$ with the backward ray tracing.

The refractive geometry in Fig. 1 shows $\mathbf {L}_1$ and $\mathbf {L}_1^\prime$ are a segment of the lines $\mathbf {OP}$ and $\mathbf {O}^\prime \mathbf {P}$, respectively. Therefore, the normalized direction vectors of $\mathbf {L}_1$ and $\mathbf {L}_1^\prime$ are respectively given by

Let $\mathbf {O}$ be the origin of $\mathbf {L}_1$ and $\mathbf {O}^\prime$ be the origin of $\mathbf {L}_1^\prime$. In the reference frame $O-XYZ$, the rays $\mathbf {L}_1$ and $\mathbf {L}_1^\prime$ can be determined by the following equations:

When the light rays pass through the interface $\Pi _1$, they are bent to form the refracted rays $\mathbf {L}_2$ and $\mathbf {L}_2^\prime$. Both can be determined by computing the incident points $\mathbf {P}_1$ and $\mathbf {P}_1^\prime$ and the directions of both refracted rays.

According to Fig. 1, $\mathbf {P}_1$ and $\mathbf {P}_1^\prime$ are the intersections of $\mathbf {L}_1$ and $\mathbf {L}_1^\prime$ and the interface $\Pi _1$, respectively. Suppose the normal vector of the interface $\Pi _1$ is $\mathbf {N} = (N_x, N_y, N_z)$ and the distance from the origin is $D$. The equation of $\Pi _1$ is given by the following vector form

Note that the interface is needed to reconstructed with the calibrated stereo-DIC system in advance, which is introduced at the end of this section. Combining Eqs. (2) and (3), we can obtain the expressions of the incidence points $\mathbf {P}_1$ and $\mathbf {P}_1^\prime$:

Subsequently, we identify the directions of the refracted rays $\mathbf {L}_2$ and $\mathbf {L}_2^\prime$. They are denoted by unit vectors $\boldsymbol{\beta }$ and $\boldsymbol{\beta }'$, respectively. Because the refracted ray is coplanar with its incident ray and the interface normal, the vectors $\boldsymbol{\beta }$ and $\boldsymbol{\alpha }$ can be projected onto the directions parallel to and perpendicular to the normal $\mathbf {N}$ respectively, so that

Denote the determined coordinates of $\mathbf {P}_1$ and $\mathbf {P}_1'$ as $(X_1, Y_1, Z_1)$ and $(X_1', Y_1', Z_1')$ respectively, the equations for $\mathbf {L}_2$ and $\mathbf {L}_2^\prime$ are established as follows:

Revisiting the process above, it is found that Eqs. (4)–(8) give the general rule to establish the refracted ray-path for a binocular stereo camera system, thereby forming the baseline formulations for reconstructing an observed 3D point in the refracting media. Therefore, the refracted rays $\mathbf {L}_3$ and $\mathbf {L}_3'$ at the interface $\Pi _2$ are determined directly by following the baseline formulations. The interface $\Pi _2$ is expressed by $\mathbf {N}^T\mathbf {X} + D + d = 0$ since it is parallel to $\Pi _1$. The two incident points $\mathbf {P}_2$ and $\mathbf {P}_2'$ on $\Pi _2$ are computed as:

The direction vectors of the rays $\mathbf {L}_3$ and $\mathbf {L}_3'$ are respectively determined by:

With the computed incident points and the direction vectors, the refracted rays $\mathbf {L}_3$ and $\mathbf {L}_3'$ are determined according to Eqs. (8). The coordinates of the observed 3D point $\mathbf {Q}$ are finally identified as the intersection of $\mathbf {L}_3$ and $\mathbf {L}_3'$ with form of $\mathbf {Q} = \boldsymbol{\gamma }s - \mathbf {P}_2$ or $\mathbf {Q} = \boldsymbol{\gamma }'s' - \mathbf {P}_2'$, where $s$ and $s'$ are determined by solving

Despite these equations above seem complicated, they are all analytic and can be used straightforwardly without any complex loops. Meanwhile, the baseline formulations of the backward ray tracing give a concise procedure for determining the ray paths in the refractive 3D reconstruction. In Section 2.3, we will show the full measurement procedure by integrating the refractive 3D reconstruction into the stereo-DIC framework.

It is noteworthy that the pose and position, i.e. the normal vector $\mathbf {N}$ and the distance $D$, of the interface $\Pi _1$ play a crucial role in the refractive 3D reconstruction process, but are often unknown in daily experiments. Therefore, we propose to determine $\mathbf {N}$ and $D$ through reconstructing the flat interface $\Pi _1$. The method is pasting a set of removable circular markers on the interface $\Pi _1$, then reconstructing the 3D positions of all markers relative to the frame $O-XYZ$. To tackle the possible defocusing effect, we recommend detecting the markers by thresholding followed by positioning with the center-of-gravity method. For blurred images due to serious defocusing or poor illumination, several improved detection algorithms [30,31] can be adopted to extract the markers. The equation of $\Pi _1$ can be finally fitted by identifying the normal vector $\mathbf {N}$ and the distance $D$. Though only three non-collinear markers are enough to determine them, more markers allow us to find robust estimates of $\mathbf {N}$ and $D$ by integrating the interface fitter into a hypothesize-and-test framework, such as RANSAC [32]. By doing this, the 3D shape of fluid-immersed objects can be reconstructed completely and flexibly to perform 3D shape and deformation calculations without reformulating the camera model and relying on specific optics.

2.3 Deformation measurement procedures

In this section, we describe the refractive stereo-DIC by integrating the refractive 3D reconstruction above into the regular 3D shape and deformation measurement procedures. Similarly, the pipeline of refractive stereo-DIC measurement consists of five major steps: (1) speckle image acquisition, (2) calibration of the stereo camera system, (3) stereo and temporal image matching, (4) 3D reconstruction with the method in the previous section, and (5) estimation of deformation fields. An overview of the measurement procedures is shown in Fig. 2. The first step is straightforward, that is to simultaneously capture a stereo-pair of a randomly speckled object at each measurement stage. Each stereo-pair comprises a left image and a right image taken by the left and right cameras, respectively, and the stereo-pair acquired at the initial state is defined as the reference. Steps 2 to 5 are described as follows.

 

Fig. 2. Overview of the refractive stereo-DIC measurement pipeline.

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3. Experiments and results

The performance of the proposed refractive stereo-DIC method has been evaluated on both 3D shape reconstruction and 3D deformation estimation. Both tests consider the scenario of a pair of cameras measuring a standard specimen situated at a glass tank filled with water. The experimental setup is shown in Fig. 3(a). The stereo measurement system was composed of two cameras, both were equipped with an imaging sensor with a resolution of $2048 \times 2048$ pixels and a 25 mm prime lens. The working distance and the horizontal field of view were about 550 mm and 240 mm, respectively. The thickness of the glass is 5 mm, and the refractive indices of the glass and water are 1.52 and 1.33 respectively. The left and right cameras were calibrated by adopting Zhang’s method based on a chessboard with $12 \times 9$ corners prior to the experiments. To reconstruct the refracting interfaces, we adhered a number 44 of circular markers to the front face of the water tank. A pair of images for the markers are shown in (b). Subsequently, a standard sphere with diameter 40 mm and a disk were used to evaluate the performances in 3D shape reconstruction and deformation measurement, respectively. Random speckle patterns were fabricated on each of the specimen surfaces. Speckle image samples for both specimens are shown in (c). Details of the experiments and results are given in the following subsections.

 

Fig. 3. (a) Experimental setup. (b) Stereo-pair of 44 circular markers. (c) Speckle images of the underwater sphere and disk specimens.

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3.1 Underwater 3D shape reconstruction

We reconstructed the 3D shape of the underwater sphere to validate that the presented refractive geometry provides a good approximation to a complex refractive optical system. For that, a stereo-pair of the randomly speckled sphere was captured by the left and right cameras, and a number 5876 of stereo correspondences were established according to the foregoing stereo-DIC matching algorithm. Following the refractive 3D reconstruction method described in Section 2.2, the observed segment of the sphere surface was recovered successively.

Figure 4 shows the measured point clouds for the sphere with the proposed and regular stereo-DIC methods. Both were reconstructed in the reference coordinate frame(i.e., $O-XYZ$ in Fig. 1). For comparison and visualization, they were moved from the measured centers $(-3.50, 10.22, 654.74)$ mm and $(-4.82, 10.15, 621.00)$ mm to the origin respectively and then, were plotted in the same viewport with reversed $Z$-coordinates. Intuitively, the measured profile in (b) has a narrower depth range than that in (a), implying that the former seems larger than the latter in terms of geometrical size. To verify this point, two spheres were fitted from the measured point clouds. Results show that both point clouds were fitted well to the spheres respectively, but the resulted sphere radii were different. The radius corresponding to the profile in (a) is 20.08 mm, which is close to the ground-truth 20 mm (half of the true diameter), while that of the profile in (b), 25.98 mm, is significantly larger than the ground-truth. In addition, we generated an ideal surface of the measured sphere at the centers of the reconstructed 3D profiles, respectively, for evaluating the discrepancies of the measure profiles. The discrepancy maps are illustrated in Fig. 5(a) and (b). We find discrepancies in (a) almost range from 0 to 0.1 mm, while in (b) are obviously large, ranging from 4.35 to 7.11 mm. In (c), we compare the the spherical surface measured by our method with that measured in air with the regular stereo-DIC by computing the differences in radial. The comparison shows that the 3D profiles measured by both methods are in good agreement. The results clearly show that a good reconstruction of the observed sphere surface could be obtained by using the proposed method. From a pure geometry point of view, the results of 3D shape reconstruction show that the introduced flat refractive geometry is reasonable and correct, and demonstrate our 3D reconstruction method based on the geometry is capable of retrieving the true 3D shape of underwater objects with high accuracy.

 

Fig. 4. 3D sphere shape measured by the refractive stereo-DIC (a) and the regular version (b), respectively.

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Fig. 5. Discrepancy maps between the 3D shapes measured by our method (a) and the regular stereo-DIC method (b) and the ideal spherical surface; (c) the discrepancy map between the spheres measured in water with our method and in air with the regular stereo-DIC.

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3.2 Underwater 3D deformation evaluation

The performance in underwater 3D deformation was evaluated with the disk specimen. The thickness of the disk is $\delta = 0.5$ mm and the diameter is $\phi = 80$ mm. In order to verify our method as clear as possible, in- and out-of-plane rigid body translation and out-of-plane deformation of the disk specimen were conducted, respectively.

3.2.1 In-plane and out-of-plane rigid translation

Here the specimen was moved by a two-axis translation stage with a precision of $\pm$0.05 mm, as shown in Fig. 3(a). In both tests, the same large displacement range, i.e. from -5 mm to 5 mm, was adopted. For each test, we moved the disk specimen to the zero position of the stage firstly and captured a pair of images as the reference stereo-pair. Subsequently, we adjusted the translation stage to shift the specimen to -5 mm position and then, moved it to the 5 mm position stepwise with a stride of 1 mm. At every step, a stereo-pair of the disk specimen was recorded for displacement computation. With the measurement procedures in Section 2.3, both in-plane and out-of-plane displacements were estimated. Then, the absolute errors relative to the imposed translation values were computed. The displacement results and errors were averaged finally. Results are shown in Fig. 6(a). For comparison, the displacements in both tests were estimated by the regular stereo-DIC, corresponding results are shown in Fig. 6(b).

 

Fig. 6. Measured mean displacements versus the imposed translations for the proposed refractive stereo-DIC (a) and the regular stereo-DIC (b). In (b), the SD errors are magnified ten-fold for visualization.

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On close inspection of Fig. 6, we see that, for in-plane rigid displacement, both methods produced similar results that are consistent with the imposed displacements; the absolute errors of both range about from -0.01 to 0.02 mm with the maximum standard deviation (SD) 0.01 mm. Comparison of (a) and (b) shows, however, that the measured results are significantly different for out-of-plane rigid displacement. Out-of-plane displacements measured by our method well agree with the actual translations of the disk specimen, and maintain a reasonable error fluctuation from -0.06 to 0.02 mm with the maximum SD of 0.01 mm. While the displacement errors produced by the regular stereo DIC increase linearly with increasing of the distance from the reference position of the specimen; and the errors, range from -1.15 to 1.26 mm with the maximum SD 0.03 mm, are much larger than that of our method. These results imply our method shows the expected performance in fluid-immersed 3D deformation measurement, even the object is subjected to large displacements. Beyond this, we find that the influence of light refraction is mainly manifested in the out-of-plane direction, while hardly affects the in-plane displacement measurement . The main reason is that the specimen was moved in a plane that is almost parallel to the refracting interface (i.e., the front face of the water tank) for the in-plane displacement case. This phenomenon is reasonable according to the flat refractive geometry presented in Section 2.1.

3.2.2 Out-of-plane deformation

To measure the out-of-plane deformation, we applied a concentrated displacement load perpendicular to the disk surface at the center of the disk specimen. The loading device is a 0.01 mm micrometer with precision of $\pm$0.004 mm as shown in Fig. 3(a). Before loading the specimen, a pair of speckle images of the disk were captured as the reference stereo-pair. Then loading started and 10 loaded states with a known displacement step of 0.1 mm were achieved finally. The corresponding speckled stereo-pairs were recorded in phase. A sample of the speckled disk image has been shown in Fig. 3(c). Following the measurement procedures in Section 2.3, the displacement fields in X-, Y- and Z-directions were computed.

We firstly evaluated the total central displacement since its actual value at each state is known. Figure 7 shows the measured central displacements and the absolute errors versus the applied displacements. For comparison, the results computed by the regular stereo-DIC are also plotted in the figure. We find that, with increasing of the out-of-plane load, the displacements measured by our method in (a) are in good agreement with the actual ones. The corresponding absolute errors in (b) are in a small yet stable range, with the maximum of 0.01 mm and SD of 0.003 mm. The results provide a good validation for our method in both correctness and accuracy. For the results of the regular stereo-DIC, we find the measured displacements in (a) are significantly smaller in amplitude under the influence of light refraction and the errors in (b) increase linearly with the increase of load. This is similar to the results in the test of out-of-plane rigid translation in the previous section. Both translation and deformation experiments seem to show, when measuring the movement or deformation of an object through media with graded refractive index, light refraction will cause non-negligible measurement errors if the direction of movement or deformation is not parallel to the refracting interface(s). E.g., for the out-of-plane translation in the previous section, the maximum absolute error is 1.26 mm, which is about 25.20% of the corresponding translation; for the test in this section, the maximum error is 0.27 mm, which is about 27.00% of the maximum displacement.

 

Fig. 7. (a) Measured displacements versus the out-of-plane deformation and (b) the corresponding absolute errors.

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Based on the aforementioned validations, we show the U, V and W displacement fields measured in the load states of 0.1 mm and 1.0 mm in Fig. 8. Comparing the displacements in three directions with each other in both states, we find the disk mainly undergoes an out-of-plane displacement, shown as W fields in (c) and (f), when it is subjected to a concentrated load at the center, while the displacements in the other two directions are very small, see U fields in (a) and (d) and V fields in (b) and (e). The measured results are in accordance with the theory. Considering the ratio of the characteristic dimensions, $\delta$ and $\phi$, of the disk specimen is small (1/160) and the disk was was fixed with eight equally spaced bolts (see Fig. 3(c)), the case in this experiment could be approximated to a bending problem of an edge-clamped circular plate under the action of the central concentrated load. According to the theory of thin plates [38], the in-plane displacements $U$ and $V$ could be approximately zero if we ignore the thickness effect, and the out-of-plane displacement (also known as deflection) $W$ shows a centrosymmetric distribution and reaches the maximum at the center. To show the radial displacement distribution of the disk, we further estimated and inspected the total displacements along a horizontal line passing through the center of the disk for 0.1 mm and 1.0 mm load states, respectively. The total displacements, denoted by $T$, and their radial distribution curves are shown in Fig. 9. We find, although the sampled displacement curve at the load state 0.1 mm is not smooth enough, the overall distribution in the radial direction agrees well with the theory predicting and reaches the maximum at the center with an error of 0.01 mm; the radial displacement distribution sampled at the load state of 1.0 mm is in perfect agreement with the theoretical analysis, and the error of the peak displacement at the center is also 0.01 mm. As a result, these analyses prove the measured displacement fields are reasonable and correct, thereby validating the performance of our method from the perspective of full-field displacement estimation.

 

Fig. 8. 3D displacement fields measured by the proposed refractive stereo-DIC: (a)-(c) for the load of 0.1 mm and (d)-(f) for 1.0 mm.

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Fig. 9. Measured total displacement fields and the distribution curves in radial direction at the load states of 0.1 mm (a) and 1.0 mm (b).

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4. Conclusion

In this work, we propose the refractive stereo-DIC, a new extension of the widely used land-based full-field optical deformation measurement technique, to measure the 3D shape and deformation of fluid-immersed objects. We describe a refractive geometry with the backward ray tracing and provide a formulation for the refractive geometry which forms the baseline of an accurate refractive stereo reconstruction for achieving 3D measurement in refracting media. The method maintains the perspective camera model adopted in the regular stereo-DIC and thus, can be treated as a flexible post-processing approach for readily extending the existing framework to fluid-immersed optical 3D deformation characterization. We show the necessity and performance of our method in refractive 3D shape and deformation measurements. Experimental validation on underwater 3D shape reconstruction demonstrates our method provides a correct description to a complex refractive stereo imaging system without modifying the image formation. Tests on rigid translation and deformation of an underwater specimen reveal that our method is accurate and precise for 3D displacement measurements with both large and small amplitudes. Particularly, we found that light refraction has nearly no effect on the deformation in a plane parallels to the refracting interface(s). Beyond these validations, we expect our method potentially promotes further development of the 3D deformation characterization and stereo reconstruction for objects in refracting media by extending existing stereo-DIC systems and universal stereo visions.