1. Introduction

In recent years, accurate measurements of surface full-field 3D shape, motion and deformation of immersed objects have been attracting growing attentions in underwater archeology, in vitro biological tissues measurements and submerged engineering structures inspections [1–3]. One of the most attractive techniques to fulfill these missions is (pseudo-) stereo digital image correlation (stereo-DIC) — a well-established non-contact optical method for in-air 3D full-field profile, displacement and strain measurements [4–8]. However, for submerged objects, due to the refraction of lights at the interfaces of two dissimilar transparent media, the images captured and later processed by a regular stereo-DIC system are severely distorted and cannot be explicitly corrected by the distortion models used in regular camera calibration.

To compensate the distortion associated with refraction, several methods have been proposed [9–13]. These methods can be classified into two categories: the refraction approximating methods and the refraction accurate modeling methods. For the refraction approximating methods, various simplified approximations have been proposed to model and/or correct the refraction [9–11]. To represent the images distortion of in-air objects captured by submerged cameras, a caustic curve was adopted by Barta and Horvath [9], which is a function of several factors including index of water, height of spatial point above the water, and the distance between the cameras and the spatial points in horizontal direction. For the condition of measuring submerged object with in-air cameras, Haile proposed an elastic image registration technique [10] to determine the transformation functions from distorted to undistorted images, and then undistort the images for DIC usage. In addition, Gupta et al. [11] found that the error from refraction is small by placing the cameras with their optical axes perpendicular to the viewing windows and calibrating the cameras with a submerged calibration target. All of the aforementioned refraction approximating methods can be simply implemented, but they can only suppress the refraction distortion to some extent instead of completely.

Some accurate methods have also been proposed to completely eliminate the refraction distortion [12,13]. These methods were usually implemented in two ways: rigorously modeling the refraction and capturing distortion-free images. In the first way [12], analytical formulations were developed to model the optical refraction at all interfaces between optically dissimilar media. These formulations are theoretically sufficient to describe all refraction along rays traveling, which however are complicate in finding the solutions due to the nonlinearity of these formulations. The method to obtain distortion-free images directly is more attractive for much easier algorithms. It can be realized in a theoretically simple way [13] by mounting a hemispherical dome port in front of lens. However, the entrance pupil of the lens must overlap the center of sphere of the dome port, which is complicate in manufacturing perfect spherical shell and locating entrance pupil of the lens (if the lens needs to be adjusted during experiments, the situations may be more troublesome).

In this work, a calibration-free method using the bilateral telecentric lens-based stereo-DIC (BTL-stereo-DIC) is proposed for high-accuracy underwater full-field 3D profile and deformation measurements. The BTL-stereo-DIC based on single bilateral telecentric lens imaging and bi-prism-assisted pseudo stereovision has been validated effective and accurate for in-air 3D measurement [14,15]. To generalize it to underwater applications, we re-derived its imaging model by further taking consideration of water refraction. To be specific, a bilateral telecentric lens is fixed above the water to vertically view the upper surface of a (semi-) submerged bi-prism. The system is then adjusted to make the field of view (FOV) of the lens roughly bisected by the lower edge of the bi-prism. Through the refraction of the left and right wedges of the bi-prism, the underwater samples can be separately captured by left and right parts of the camera sensor from two different perspectives at initial and deformed states. The captured images are then segmented into two equal parts (i.e., left and right virtual images), which are matched using the well-established subset-based DIC algorithm. Using the registered image points, 3D surfaces can be reconstructed at each state using a set of derived linear formulations. Further comparison of the reconstructed 3D points at different states, 3D surface full-field displacements and strains can be retrieved. For validation, 3D underwater shape reconstruction and translation tests were carried out. Underwater membrane inflation experiment was also performed to verify the practicability of the proposed technique.

2. Methodology

2.1 Optical arrangement

Figure 1 schematically shows the optical arrangement of the calibration-free single camera stereo-DIC system. The system mainly comprises a (semi-) submerged bi-prism, a bilateral telecentric lens vertically placed above the water and a digital camera connected to the lens. By carefully adjusting the vertical position of the bilateral telecentric lens, two views of the sample can project through the left and right parts of the bi-prism and then be imaged by the camera, leading to two virtual images on the camera sensor. To sharply capture both virtual images, the high-quality bilateral telecentric lens should be arranged with its optical axis perpendicular to the upper surface of the bi-prism. It should be noted that the FOV on the test sample should be halved by the lower edge of the bi-prism. In addition, the active imaging [16] based on a combination of monochromatic light illumination and coupled bandpass filter imaging is adopted. It offers two eminent benefits: the elimination of chromatic aberration caused by the refraction of the rays at air-water interface and suppression of ambient light variations.

 

Fig. 1 Optical arrangement of the proposed technique using a bilateral telecentric imaging system and a semi-submerged bi-prism.

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2.2 Measuring principles

The imaging model of the underwater BTL-stereo-DIC is detailed using geometrical ray tracing. As is shown in Fig. 2(a), a bi-prism, whose index of refraction, prism angle and thickness are respectively notated as n₁, α and t₀, is fixed beneath the bilateral telecentric lens. For easy explanation, a world coordinate frame OXYZ is defined on the bi-prism. Specifically, it has OXY plane attaching to upper surface of the bi-prism, origin O locating at the center of upper surface of the bi-prism, Y axis parallel to bi-prism ridges and Z axis perpendicular to upper surface of the bi-prism. It should be noted that, in practice, offsets (ΔX, ΔY) more or less exist between the optical axis and Z axis. Therefore, a coordinate frame O_LX_LY_L whose origin O_L locates on the optical axis of the bilateral telecentric lens is also specified on a plane parallel to OXY, and the offsets between O_LX_LY_L and OXY are of course (ΔX, ΔY).

 

Fig. 2 Imaging model of the BTL-stereo-DIC with (a) semi-submerged and (b) submerged bi-prisms.

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Consider a spatial point P(X, Y, Z), which emits two light rays passing through left and right wedges of the bi-prism and impinging the O_LX_LY_L plane on (X_L1, Y_L1) and (X_L2, Y_L2), respectively. Suppose the rays respectively project to the sensor with image coordinate of (x₁, y₁) and (x₂, y₂). Due to the orthographic projection of the bilateral telecentric lens, the following equations hold:

Later, using the projection points on sensor plane and several inherent parameters, the coordinates of X and Z can be derived. It begins by finding the projection points A and B on the OXY plane, which are expressed as

When the light rays transmit from air to glass through A and B, no bending will be encountered as the light rays are perpendicular to the interfaces of two different media. The light rays will travel along straight lines through points A and B to points C and D, respectively. Based on the geometrical relationship, the coordinates of C and D are

At the glass-water interface, refraction of the lights is inevitable. Obviously, the angles of incidence are α at the glass-water interface. Therefore, the angles of refraction β of both rays are obtained using the well-known Snell’s law as

Thus, for any point P₁ on line CP and P₂ on line DP, their coordinate can respectively be given as

As point P is the intersection point of line CP and DP, the following equation holds:

From Eq. (8) and Eq. (9), the scalar λ₁ and λ₂ are respectively solved as

Substituting λ₁ into Eq. (8a) or λ₂ into Eq. (8b) leads to the coordinate of point P:

In this equation, the coordinates of P consist of two parts: the first part changes linear with (x₁, y₁) and (x₂, y₂); the second part is invariant to spatial point P. For different points, the second part keeps constant, and it is equivalent to a rigid-body translation imposed on final results. As such, the second part has no effect on 3D shape, displacement and deformation measurements, which of course can be neglected from Eq. (11). Then it yields the coordinate of P:

After an inspection on Eq. (12), one can find that the coordinates are independent to offsets (ΔX, ΔY) and bi-prism thickness t₀. Therefore, (ΔX, ΔY) and t₀ do not need to be measured, which can greatly simplify the operation. Note in this equation, all of these three coordinate components of P have a same coefficient, the scalar s. If only strain is required, s needs not to be determined, or can even be removed from Eq. (12).

We may start a practical consideration of asking why the bi-prism is partly semi-submerged – after all, if the bi-prism can be fully immerged the method can possess better flexibility. It is lucky to find that if the upper surface of the bi-prism is covered by water (or more complicate, a glass window is added), as is shown in Fig. 2(b), no more refraction will be introduced to the optical paths if the water-air and/or glass-air interface is perpendicular to the optical axis. Thus, in these conditions Eq. (12) is also applicable.

In Eq. (12), β is derived from Eq. (5); α is the inherent parameter of the bi-prism which can be measured in advance; (x₁, y₁) and (x₂, y₂) are the projection points (in unit of pixel) of 3D point P on captured images, which respectively locates at left and right parts of the images. To retrieve the 3D coordinate components of P, (x₁, y₁) and (x₂, y₂) should be accurately found. To this end, the classical subset-based DIC method [17] is adopted and outlined at follows.

2.3 Measurement procedures

The procedures of underwater 3D profiles and deformation measurements are shown in Fig. 3, which include four consecutive steps: image capture and division, image matching, profile reconstruction and deformation measurements.

 

Fig. 3 Procedures of underwater 3D profile and deformation measurements.

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3. Experiments and results

3.1 Experimental procedures

Two validation experiments, including 3D shape reconstruction of a ball and in-plane and out-of-plane translation of a plate, were carried out to examine the effectiveness and accuracy of the proposed method. In addition, 3D surface deformation, Young’s modulus and further stress fields of an expanded circular membrane were retrieved to validate the practicability of the method.

To be specific, a semi-submerged bi-prism (prism angle: 30.58°, index of refraction: 1.52) was mounted with its upper surface parallel to the water surface. To clearly capture images of two diverse views through the refraction of the bi-prism, a bilateral telecentric lens (Xenoplan 1:5, Schneider Optics, Inc., Germany) connected with a digital camera (TXG50, Baumer Electric AG, Switzerland, 2448 × 2050 pixels resolution) were mounted in air (index of refraction: 1.00), which has a fixed FOV of 42.2 × 35.3 mm. To guarantee the perpendicular between the optical axis of the bilateral telecentric lens and the upper surface of the bi-prism, the bilateral telecentric lens and the bi-prism were carefully adjusted to make a bubble gradienter placed on them parallel to the horizon. As shown in Fig. 4(a), in the shape reconstruction test, the ball used has a nominal diameter of 59.62 mm, while in the translation tests the plate was fixed on a 2-axis translation stage (positioning accuracy: 0.01 mm). In the membrane inflation test shown in Figs. 4(b) and (c), a taut balloon (0.20 mm in thickness and 15.00 mm in diameter) was fixed on the left side of a circular U-pipe (110 mm in left side length, 1390 mm in right side length, 13.52 mm in inside diameter and 15.00 mm in outside diameter) and loaded by injecting water into the U-pipe with a 50 mL syringe. All of these immersed specimens were decorated with random speckle patterns and placed directly underneath the bi-prism in a tank filled with water (index of refraction: 1.33). To effectively suppress the inhomogeneous and fluctuation of ambient light and the chromatic aberration, active imaging [16] was employed with combined use of a monochromatic blue LED light source emitting at 450-455 nm and a band-pass filter with a center wavelength of 450 ± 2 nm. Additionally, to eliminate the thermal errors from cameras self-heating, 2-hour pre-heating of the camera was performed before image capture [21].

 

Fig. 4 Pictures of (a) in-plane and out-of-plane translation and (b) membrane inflation tests; (c) schematic experimental details of the membrane inflation experiment.

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During the experiments, several sets of images should be captured. It began by capturing a picture of a still ball. Then, the plate was translated using 2-axis translation stage first in-plane (X-direction) and then out-of-plane (Z-direction) from −3 mm to 3 mm with a step of 0.5 mm, and an image was recorded at each state. Finally, the membrane was loaded by injecting 20 mL water at each state, corresponding to 1.37 kPa pressure increment onto the membrane. The inflation progress of the membrane is also captured at each state. Due to the existence and evolution of the height difference between the water surface in two sides of the U-pipe, the loads exerted onto the membrane will increase linearly. After the experiments, the captured images were processed to retrieve 3D profiles using the pre-described measurement procedures. Specifically, the parameters involved in DIC calculation were 61 × 61 pixels in subset size and 5 pixels in grid step; the scaling factor s in Eq. (12) was calculated as 0.0172mm/pixel. After 3D profile reconstruction at each state for each sample, full-field displacements (i.e. U, V, W: the displacements along X, Y, and Z direction, respectively) and strain (ε_x, ε_y, ε_xy: the strain in X and Y direction, and shear strain, respectively) can be retrieved. Pushing a bit further, by using the measured displacement and strain fields of the membrane, the Young’s modulus and stress fields of the membrane can also be obtained.

3.2 Profile reconstruction of a sphere surface

Figure 5 gives the measured contour of the ball. A set of highly concentric circles are presented in the contour, qualitatively proving a good shape reconstruction. To quantitatively evaluate the accuracy, the measured surface is then fitted with a sphere formula. The fitting yields a sphere surface with diameter of 59.36 mm and a root mean square error (RMSE) less than 0.02 mm. Compared with the reference diameter (59.62 mm) measured by Vernier caliper, the fitted diameter has a relative error of about 0.44%, confirming that the proposed method is an accurate method to reconstruct submerged 3D surface.

 

Fig. 5 Contour of the reconstructed ball profile.

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3.3 In-plane and out-of-plane translation of a plate

To validate the accuracy of BTL-stereo-DIC for submerged displacements measurement, the displacement fields of the in-plane and out-of-plane translation tests are retrieved and analyzed. For clarity, in the translation experiments, we use U and W displacement to represent the U and W displacement of in-plane and out-of-plane translation tests, respectively.

In Fig. 6(a), the U and W displacements are averaged and plotted versus the imposed displacements, and both of them are almost overlapping to the applied displacements at first glance. Later, the errors of the displacement, including the mean error and standard deviation (SD) for both tests are obtained and shown in Fig. 6(b). It clearly shows that the mean errors of U displacement are significantly larger than the mean errors of W displacement (about 4 times larger). On the contrary, the SD of U displacement is obviously smaller than that of W displacement (nearly 60% smaller). Quantitatively, for in-plane and out-of-plane translation tests, the maximum absolute mean errors (mean biases) are 0.054 mm and 0.003 mm, while maximum SD are 0.011 mm and 0.007 mm. Both mean errors and SD in the translation tests have small values, which are good validations for the displacement measurement ability of the underwater BTL-stereo-DIC. However, we should note that the U displacement has better precision yet worse accuracy over the W displacement.

 

Fig. 6 (a) Mean U-displacement for in-plane and mean W-displacement for out-of-plane translation tests versus the imposed displacements; (b) mean error and standard deviation of the measured displacements for both in-plane and out-of-plane tests.

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For the SD error, just like that of regular stereo-DIC, it is larger in out-of-plane direction than that along in-plane direction in most cases. As for the mean error, it contradicts to the common sense, the accuracy should be better for in-plane translation test. The result may be attributed to several reasons. First, the imposed U displacements may have larger loading direction deviation, which results larger errors on U displacements. In addition, the upper surface of the used bi-prism may not flat enough. As the measured surface need to motion from one side to the other side of the FOV for in-plane translation test yet only small motion in the FOV for out-of-plane translation test, the uneven upper surface of the bi-prism will lead to less influence to out-of-plane translation.

3.4 Deformation measurement of a membrane

The proposed method was also used for full-field displacement, strain, stress and Young’s modulus measurements of an underwater membrane [22]. Figure 7 gives the deformation progress of the membrane under uniform pressure loading. Figures 7(a) and 7(b) are the images captured at initial and deformed states, respectively. Processing the images using the pre-described procedures, the full-field 3D profiles and displacements can be obtained. The curve shown in Fig. 7(c) is the central deflection of the membrane at each state versus the imposed fluid pressure. Inspecting the reconstructed profiles of the specimen, one can find a set of gradually expanded quadric surfaces, just as expected.

 

Fig. 7 Images captured by the camera at initial (a) and deformed (b) state; (c) Measured 3D profiles and central deflection of the membrane versus the applied fluid pressure.

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By subtracting the 3D profile at initial state from those of other states, the displacement fields can be derived. For instance, Fig. 8(a) respectively shows the measured U, V and W displacement fields at final state. The U and V displacements are believed to be antisymmetric with respect to X-axis and Y-axis, respectively. While the W displacement distribution shows a set of circles with good concentricity. However, there are obvious deviations between the axes of symmetry and their theoretical axes (X-axis for U displacement and Y-axis for V-displacement). These deviations can be attributed to three possible reasons: namely, the inhomogeneous of the membrane, the inhomogeneous pretension applied to the balloon and the slant of the measured surfaces. In addition, the antisymmetric distributed displacement fields in X and Y direction will lead to symmetric strain fields in X and Y direction, which possess the axis of both X-axis and Y-axis. Further, symmetric sheer strain field is estimated, which is symmetric to two different axes, and they are the lines passing through the origin and having tilt angles of 45° and 135°. Although there is some noise on the strain fields, these trends can be clearly observed in Fig. 8(b).

 

Fig. 8 (a) Displacement and (b) strain fields of the membrane at the final states of the test.

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Based on the measured deformation information, the Young’s modulus E_M of the membrane can also be characterized by a formula depicted in [23,24]:

In this formula, w₀ is the central deflection shown in Fig. 7. R_M and d_M are the radius and the thickness of the circular membrane, which are measured in advance using a Vernier caliper. ν_M is the Poisson’s ratio. Δp and σ₀ are the fluid pressure increment and residual stress, respectively. During the inflation experiment carried out in this work, only fluid pressure Δp and deflection w₀ are changeable. Thus, the fluid pressure Δp can be regarded as a function of w₀:

Fitting Eq. (15) with the imposed fluid pressure and measured central deflection yields a = 1.086 × 10¹¹, b = 3.176 × 10⁶ Based on the fitted polynomial expression, one can either obtain the imposed fluid pressure from measured central deflection or predict central deflection from applied fluid pressure, which are shown in Figs. 9(a) and 9(b). In both figures, good superpositions are found between the measured (imposed) variables and the fitted values. The largest deviation in both figures exist at the second state, which may be contributed to the relatively large error when the imposed pressure is small, or the air connection between left and right tubes at begin states. In conclusion, if the results at second state is excluded, a good superposition will be found with the maximum deviation between fitted and imposed fluid pressure as 0.19 kPa and maximum deviation between fitted and measured central deflection as about 0.03 mm.

 

Fig. 9 (a) The imposed and predicted fluid pressure versus measured central deflection; (b) the measured and predict central deflection versus the applied fluid pressure.

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According to Eq. (14), we can derive the Young’s modulus of the membrane from the following equation:

The balloon made of emulsion has the property of hyperelasticity and is usually considered as incompressible material. Therefore, its Poisson ratio ν_M is designated as 0.5 and Young’s modulus is estimated as E_M = 368 kPa, which can further be used for stress field estimation based on the assumption of plane stress condition. The derived stress fields, including stress in X-direction (σ_x), stress in Y-direction (σ_y) and shear stress (τ_xy) are shown in Figs. 10(a)-10(c), respectively. Obvious symmetric distribution of these stress fields can be observed, and their axis of symmetry are the same as those of strain fields. Further, von Mises stress field, the stress irrelevant to directionality is also derived from the Cauchy stress components (σ_x, σ_y, τ_xy). In this application, the von Mises stress should be concentrically distributed, and is well presented in Fig. 10(d). Although some noise is involved, we can still conclude that the trends of the measured stress fields well satisfy what they supposed to be.

 

Fig. 10 Stress fields in (a) X-direction and (b) Y-direction; (c) shear stress (d) von Mises stress fields.

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4. Conclusions and discussions

In this paper, a simple yet effective calibration-free method based on BTL-stereo-DIC that combines bilateral telecentric imaging system and bi-prism-assisted pseudo stereovision, is proposed for underwater full-field 3D profile, motion and strain measurements. Its optical arrangements, measurement principles and implementation procedures are detailed. In addition, its accuracy is well validated by sphere surface reconstruction and in-plane and out-of-plane translation tests, while its practicability is proved by the measurements of full-field deformation and further the determination of Young’s modulus and stress fields of an inflated circular membrane. Compared with existing underwater measurement methods, the proposed method possesses prominent advantages at the following several aspects:

The proposed method, however, is not generally applicable for all underwater applications. First, the FOV is limited to a fixed small size due to the adoption of a bilateral telecentric lens, and further halved because there are two different views share a same FOV. Second, the samples can only be measured at a given submerging depth range because the prism angles of the bi-prism are non-adjustable. Fortunately, in many applications, such as in vitro biological tissues measurements, the samples usually have small scale and the submerging depth is not important. Therefore, one can change the liquid depth depending on the selected bi-prism to permit clear imaging of both views. Comprehensively considering the pros and cons, we conclude that the proposed method pretty suits for the underwater measurements of small-scale samples at both static and high-speed conditions, and has great potential in in vitro biological tissues measurements and underwater materials characterization.