1. Introduction

The determination of the mechanical properties of materials and structures is of great importance in both scientific and engineering fields. A range of methods have been developed to measure mechanical properties (e.g. uniaxial tensile tests [1], bending tests [2], bulge tests [3] and high-speed dynamic tests [4]). Among these methods, the most commonly used method is uniaxial tensile or compressive testing by the use of a Universal Testing Machine (UTM) to determine the properties such as Young’s modulus, Poisson’s ratio, elongation percentage, yield strength and failure strength. To accurately determine these mechanical properties, the surface strains of the tested specimen must be precisely measured.

Both electrical-resistance strain gauges [5, 6] and mechanical extensometers [7, 8] can be used in material testing to measure surface strains of a test specimen. While providing high-precision strain measurement in many applications, these contact techniques may incur some practical problems that adversely affect the convenience of the tensile tests. When strain gauges are being used, they need to be physically attached to the specimen and the strain measurement range is generally less than 2%. For mechanical extensometers, their weight and method of attachment may influence the mechanical response of the test specimen. Furthermore, both strain gauges and mechanical extensometers are difficult to be used in some extreme environmental conditions (e.g. high or low temperature).

To overcome these disadvantages of the contact techniques, the non-contact video extensometers [9–17] using a digital video camera and various digital image processing algorithms have been proposed. The current video extensometers can be divided into feature-based [9, 10] and intensity-based [11–17] video extensometers. For intensity-based video extensometers, the 2D-DIC technique is generally used to track motions of gauge points. Zhang et al developed a bi-axial video extensometer based on DIC and employed it to study the mechanical behavior of a ceramic composite grid panel [11]. Huang et al proposed an automatic DIC method for real-time monitoring of clamping force of a bolted joint [12]. Wu et al. proposed a real-time video extensometer for displacement or strain measurement for dynamic tests [13]. Pan et al. developed an advanced video extensometer for non-contact, real-time, high-accuracy strain measurement in material testing [14]. However, the accuracy of strain measurement using a common optical extensometer with 2D-DIC is not sufficient for experimental applications due to the effect of out-of-plane motion. Although telecentric lens has been proven to be non-sensitive to the out-of-plane displacement [18], disadvantages of telecentric lens, e.g. high-cost and limited field of view (FOV), may limit its application [18, 19]. More importantly, there still exist non-negligible strain errors if large out-of-plane displacement occurred when using the telecentric lens [18, 19].

To eliminate the effect of out-of-plane motion on 2D-DIC measurements, Pan et al. adopted a non-deformable reference specimen to enable the application of correction to the displacement of a deformed specimen, such that highly accurate strain measurement could be achieved using an ordinary 2D-DIC system [15]. The results of our previous work indicated that an optical extensometer, consisting of a single large-format lens and two separate image sensors, could achieve a strain resolution of 2-3$\mu \epsilon$ using the correction sheet [16]. To minimize the effect of out-of-plane motion on 2D-DIC, we also proposed a dual-reflector image method [17]. By averaging the strain in two optical extensometers formed on the front and rear surfaces of a specimen, the effect of any slight out-of-plane motion can be eliminated [17]. However, the use of a compensation specimen or correction sheet requires the attachment of an object [15, 16], which makes the method not a practical choice. The dual-reflector image method seems to be a practical choice, but the use of dual mirrors in the vicinity of the specimen makes the measurement system more complex.

As we all know that three-dimensional (3D) DIC offers an alternative that simultaneously measures all three components of displacement without introducing in-plane displacement errors [18, 20]. However, 3D-DIC requires the stringent synchronization between two digital cameras and requires complicated system calibration of binocular stereovision [20], which makes the measurement rather inconvenient. Recently, the single-lens 3D-DIC technique has attracted a lot of attention [21–25]. Genovese et al. presented a single camera pseudo-stereo system using a biprism in front of the camera objective to split the scene into two equivalent lateral stereo views in the two halves of the sensor [21]. Pankow et al. developed a single-lens 3D-DIC system using a single camera and a series of mirrors and they applied the system for high-speed out-of-plane displacements measurement [22]. Xia et al. developed a diffraction assisted image correlation for 3D displacement measurement using a single camera and 2D-DIC algorithm [23]. For the single-lens 3D-DIC systems, the stringent synchronization of cameras is not needed and the setup is more compact and simple. However, in these existing single-lens 3D-DIC systems, metric calibration or distortion calibration still needs to be carried out [22, 24, 25] which still make the measurement rather inconvenient.

This paper proposes a self-calibration single-lens 3D video extensometer for non-contact, non-destructive and high-accuracy strain measurement. Firstly, a single-lens 3D imaging system with a triangular prism mirror and two mirrors is constructed to acquire stereo images of the test sample surface, so the problems of synchronization and out-of-plane displacement can be solved easily. Moreover, a speckle-based self-calibration method which calibrates the single-lens stereo system using the reference speckle image of the specimen instead of the calibration targets is proposed, which will make the system more convenient to be used without complicated calibration. In the speckle-based calibration method, points in left and right half images are matched using 2D-DIC to calibrate the stereo system automatically. Furthermore, an efficient and robust inverse compositional Gauss-Newton (IC-GN) algorithm [26] combined with a robust stereo matching stage is employed to achieve high-accuracy and real-time subset-based stereo matching. The advantages of IC-GN algorithm has been fully disclosed in our previous work [26], compared with the traditional Newton-Raphson algorithm. To our knowledge, it’s the first time to propose the concept of self-calibration 3D video extensometer. The remainder of the paper is organized as follows. The methods for the proposed self-calibration single-lens 3D video extensometer are presented in Section 2. In Section 3, the details of the experimental procedures are introduced and the results demonstrating the new self-calibration single-lens 3D video extensometer are given. Advantages, limitations and further applicability of the proposed single-lens 3D video extensometer are discussed in Section 4. The conclusions for the study are drawn in Section 5.

2. Methods

2.1 Optical setup of the single-lens 3D video extensometer

A schematic illustration of the established single-lens 3D video extensometer for non-contact, non-destructive and high-accuracy strain measurement is shown in Fig. 1(a). The system consists of a CMOS camera, a short-focus lens, a reflection stereo imaging device and a blue ring LED light source. The blue LED light source is used to provide quasi-monochromatic illumination. The reflection stereo imaging device is made up of a triangular prism mirror and two mirrors. The schematic of the optical arrangement of the reflection stereo imaging device is shown in Fig. 1(b). Compared with the system developed in [22], the use of the triangular prism mirror can make the system more compact. The angle between the mirrors with the vertical direction is 30 degrees. The angle between the reflective surfaces on the prism with the vertical direction is 45 degrees. According to the arrangement of the reflection stereo imaging device, the stereo angle between left and right half images should be 60 degrees.

 

Fig. 1 (a) Schematic diagram of the single-lens 3D video extensometer and (b) optical arrangement of the reflection stereo imaging device.

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The actual device diagram of the single-lens 3D video extensometer is shown in Fig. 2, with a working distance of 55mm. A short-focus lens (12mm, Edmund Optics, Barrington, NJ) and a COMS camera with a resolution of 1280 by 1024 pixels (5.3 pixel size) and imaging speed of 25 frames per second (UI-1242LE, IDS, Germany) were used to record images of the specimen. In the refraction stereo image device, two mirrors and a triangular prism mirror with a reflective surface size of 20 by 20 mm were installed to produce stereo images, as shown in Fig. 2(c). The width of the whole setup is 70 mm, the height of the camera is 54 mm and the diameter of the reflection stereo imaging device is 75 mm. The overall size of the designed single-lens 3D video extensometer is smaller than an ordinary SLR camera, as shown in Fig. 2, which means that the proposed single-lens 3D video extensometer is convenient to be used.

 

Fig. 2 The actual setup of the single-lens 3D video extensometer.

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2.2 Working principle of the single-lens 3D video extensometer

Assuming that the gauge points in the single-lens 3D video extensometer are named as$P_1$ and$P_2$, as shown in Fig. 3, the strain of the proposed video extensometer can be determined by calculating the relative elongation of gauge length, which can be expressed as

 

Fig. 3 Working principle of the single-lens 3D video extensometer.

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where$P_1(X_1,Y_1,Z_1)$,$P_2(X_2,Y_2,Z_2)$denote the 3D space coordinates of gauge points in the world coordinate system before deformation, $P_1^{\text{'}}(X_1^{\text{'}},Y_1^{\text{'}},Z_1^{\text{'}})$,$P_2^{\text{'}}(X_2^{\text{'}},Y_2^{\text{'}},Z_2^{\text{'}})$denote the 3D space coordinates of the same gauge points in the world coordinate system after deformation, $\overline{P_1{P}_2}$and$\overline{P_1^{\text{'}}{P}_2^{\text{'}}}$indicates the gauge length before and after deformation and can be obtained by calculating the space distance of two gauge points before and after deformation, respectively. In the proposed single-lens 3D video extensometer, the 3D space coordinates of gauge points can be reconstructed from left and right half images based on triangulation [20].

2.3 Speckle-based self-calibration method for strain measurement

To find the position of a point in 3D space based on triangulation, metric calibration of the system is a must-do step [27]. To calibrate the single-lens stereo system, a speckle-based calibration method for strain measurement is proposed. In the proposed calibration method, we calibrate the stereo system using the specimen itself instead of calibration targets. The reference speckle image of the specimen is used for system calibration directly, so we call it the self-calibration method.

After capturing the reference image, the calibration can be done automatically based on speckle analysis of the left and right reference half images. As shown in Fig. 4, the region of calibration (ROC) should be selected in the left half image and matched with the right half image. The proposed speckle-based self-calibration method can be summarized as the following steps:

 

Fig. 4 Example of speckle-based calibration of single-lens stereo system.

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Step 1: Reference image capture and subset stereo matching. After capturing the reference image, the points in ROC are matched using subset-based matching algorithm. To achieve reliable stereo matching, the automatic search scheme [28] combined with the quality-guided matching stage [29, 30] is highly recommended. The IC-GN algorithm combined with the second-order shape function is recommended to be used for stereo-matching. The IC-GN has been proven to have better noise robustness [26] and the second-order shape function can match the projection transformation better [31].

Step 2: Relative extrinsic parameters estimation and 3D reconstruction of the matched points. According to the matched points and roughly deduced intrinsic parameters of the camera, the relative extrinsic parameters between the left and right half images can be computed based on the method proposed in [32]. The intrinsic parameters of the camera can be roughly deduced from nominal values supplied by the hardware manufacturer. Using the computed relative extrinsic parameters and the roughly deduced intrinsic parameters, the 3D reconstruction of matched points can be accomplished. The intrinsic parameters include the principle point coordinates ($C_x$and$C_y$), the focal lengths ($f_x$and$f_y$) and the distortion of the single camera. The extrinsic parameters include the rotation (${[{\alpha }_x,{\alpha }_y,{\alpha }_z]}^T$) and translation (${[T_x,T_y,T_z]}^T$) vectors between the left and half image planes. For the distortion of the single camera, only the first-order coefficient ($k_1$) of the radial distortion is considered in this paper.

Step 3: Non-linear least-squares optimization. To obtain accurate intrinsic parameters and extrinsic parameters of the camera, a constrained non-linear least-squares optimization method is used, and the cost function is built according to discrepancies between actually collected control points and re-projection points from the corresponding reconstructed 3D points. The corresponding cost function is as follows

where superscript$i$denotes the sequence number of the matched points, N is the total number of matched points, $m_1^i$and $m_2^i$are matched image points,${\widehat{m}}_1^i$and${\widehat{m}}_2^i$are the re-projection of the reconstructed points$M^i$in the two half image planes,$A$and$k$are the intrinsic parameters and distortion parameters of the single camera and$R$and$T$are the relative extrinsic parameters between the left and half image planes.

It’s noticed that the scale information hasn’t been calibrated in the proposed calibration. However, the proposed single-lens 3D video extensometer is used for strain measurement, so the scale information is not needed [32]. Meanwhile, the skewness between the rows and columns in the image hasn’t been considered in the proposed calibration. For present industrial sensor arrays, the skweless constraint can be met well, including the COMS camera used in this paper.

2.4 Robust stereo matching stage combined with the efficient and robust inverse compositional Gauss-Newton algorithm

Figure 5 shows the stereo matching stage used in this research. For the reference images, the points of interest (POIs) are chosen in the left reference half image and search its corresponding points in right half reference image using the second-order shape function. The second-order shape function can match the projection transformation better than the first-order function [31]. For the deformed images, we first match the POIs in left reference image with left deformed image using the first-order shape function. Then we search the corresponding points in right deformed image using the second-order shape function. For local uniform deformation, the first-order shape function can achieve better accuracy and better efficiency than the second-order shape function. The reason is that the second-order shape function has larger random errors than the first-order shape function [33]. It’s worth mentioning that there is no incremental correlation using this strategy, so error accumulation can be avoided and better robustness can be achieved.

 

Fig. 5 Robust stereo matching stage without accumulated errors.

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Moreover, to achieve fast and robust sub-pixel matching, the efficient and robust IC-GN algorithm is adopted. The IC-GN algorithm is first introduced into DIC community by Sutton and his collaborators in their book [20]. The efficiency of the IC-GN algorithm has been studied by the proposer of this algorithm [34] and the noise robustness of the IC-GN algorithm has been analyzed in our previous work [26]. More details regarding the IC-GN algorithm with first-order shape function and second-order shape function in DIC are not given here. Interested readers are referred to [35] and [31]. It’s worth mentioning that Su et al. proven that the sum of squared difference (SSD) criterion and the cross-correlation (CC) criterion are not equivalent for DIC analysis [36]. The interpolation bias of CC criterion is larger than that of SSD criterion. Hence, the IC-GN algorithm combined with the SSD criterion, e.g. zero-mean normalized sum of squared difference (ZNSSD) criterion or parametric sum of squared difference (PSSDab) criterion, is highly recommended.

2.5 Real-time strain measurement using the proposed 3D video extensometer

Figure 6 shows the flowchart of real-time strain measurement using the proposed single-lens 3D video extensometer. Prior to the experiment, the whole experiment equipment should be installed and the working distance should be adjusted to an appreciate value to see the speckle pattern on the specimen clearly. Then, capturing the reference image, and selecting the ROC for system calibration and POIs for strain measurement. The system calibration can be done automatically using the speckle-based method described in Section 2.3. For POIs, reasonable memory should be allocated to store pre-computed invariants [37] in the IC-GN algorithm and stereo matching of the POIs should be accomplished. Using the calibrated parameters from ROC, 3D reconstruction of the POIs can also be accomplished in the reference stages.

 

Fig. 6 Flowchart of real-time strain measurement using the proposed 3D video extensometer.

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During the experiment, the real-time correlation computation can be achieved using parallel computation. For each deformed stage, the POIs need to be correlated from left deformed image and right deformed image respectively. Two child threads should be created for each point, so the left and right subset matching can be realized simultaneously. Meanwhile, CPU with high clock speed (high main frequency) is recommended to be used to achieve the highest computation efficiency. For efficient integer-pixel searching of POIs, algorithms in [13] can be used.

3. Experiments and Results

To quantitatively validate the feasibility and practicability of the proposed single-lens 3D video extensometer, two experiments were conducted. In this first experiment, the static errors of the proposed 3D video extensometer were analysis using the non-loaded Al-alloy specimen. In the second experiment, the specimen was loaded using a Tensile testing machine (Istron 3380, US) and the measured strain was compared with the strain gauges. All the algorithms were implemented using the C + + language and tested on a desktop computer [Inter (R) Quad-Core (TM) i7-4700 CPU with a main frequency of 3.4 GHZ, 16.0 GB RAM].

The geometric size of the Al-alloy specimen is shown in Fig. 7(a). The chemical composition (wt %) of the sample is Si 0.4-0.7, Fe 0.4, Cu 0.1, Mn 0.4-1.0, Mg 4.0-4.9, Cr 0.05-0.25, Zn 0.25, Ti 0.5, and a remaining balance of Al. The positive and negative sides of the specimen are shown in Figs. 7(b) and 7(c). The positive side of the specimen was painted with a speckle pattern using the traditional spray painting technique. Actually, to ensure the quality of the speckle pattern on the surface of specimens, the water transferring technique which was reported in our previous work [38] is highly recommended. For the water transferring technique, the speckle pattern was designed by the computer and transferred to the surface of specimen in a harmless, convenient and repeatable way. Two strain gauges with a size of 3 mm by 4mm were adhered on the negative side of the specimen, as shown in Fig. 7(c). The schematic of the experimental setup is shown in Fig. 7(d).

 

Fig. 7 The schematic of the experimental setup: (a) the geometric size of the Al-alloy specimen, (b) the positive side of the painted specimen, (c) the negative side of the specimen adhered with two stage gauges, (d) the field experimental setup.

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3.1 Static errors analysis of the proposed single-lens 3D video extensometer

To analyze the static errors of the proposed single-lens 3D video extensometer, 50 images were captured on the unloaded specimen with a frame rate of 25 Hz. The first image was used as the reference image and the rest 49 images were used as the deformed images. As the specimen is not loaded, the measured strain value can represent the static errors. The ROC and POIs were selected as shown in Fig. 4 and Fig. 3 respectively. And the subset size is selected as 31 by 31 pixels in this experiment.

Table 1 shows the initial value and calibrated parameters using the proposed speckle-based calibration method. For calibration, the subset size is selected as 31 by 31 pixels and the step size is set as 20 pixels in ROC. A total of 375 points are used for calibration. The re-projection error is only 0.07 pixels, which verifies the correctness of the calibrated results. Figure 8 shows the reconstructed 3D points in ROC after non-linear least-squares optimization. The calibrated stereo angle is 57.28 degrees, which is very close to the pre-set stereo angle by installation. As can be seen from Table 1, the translation vector has been normalized to$T_x$. The scale information is still unknown. However, the scale information is not necessary for strain measurement [32]. Based on the calibrated results, the proposed single-lens 3D video extensometer can already be used for strain measurement.

Table 1. The initial value and calibrated parameters using the proposed speckle-based calibration method

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Fig. 8 Reconstructed 3D points in ROC after non-linear least-squares optimization.

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The measured strain values of the 49 images are shown in Fig. 9. It can be seen from Fig. 9 that the static errors of the proposed extensometer in most of time is less than 10$\mu \epsilon$. The absolute maximum error and standard deviation (STD) are 20.58$\mu \epsilon$ and 6.72$\mu \epsilon$, respectively. The measured results successfully prove the feasibility and robustness of our proposed extensometer. It’s worth mentioning that the strain can be measured in real-time using the method described in Section 2.5.

 

Fig. 9 Static errors of the proposed single-lens 3D video extensometer.

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3.2 Uniaxial tensile tests with an Al-alloy specimen

To further evaluate the performance of the proposed single-lens 3D video extensometer, the Al-alloy specimen was loaded and the measured strain was compared with the strain gauge. The loading value is about 0.2 KN for each step and one image was captured after each loading for comparison. The calibration parameters and calculation parameters used in this experiment are the same with that in Section 3.1.

The measured strain value by both strain gauge and the single-lens 3D video extensometer are shown in Fig. 10. It can be seen from Fig. 10 that the strain value measured by the single-lens 3D video extensometer is in good accordance with strain gauge. The maximum difference of the measured strain between strain gauge and the single-lens 3D video extensometer is 22.01$\mu \epsilon$, with a STD of 8.82$\mu \epsilon$. The differences between the two methods in most of time are less than 10$\mu \epsilon$, which is the same with the static errors. With the increase of loading, the measurement errors do not increase, which mean that the effects of out-of-plane displacement has been eliminated. Due to the effects of out-of-plane displacement, the measured strain errors of 2D video extensometers increase with the increase of loading [16–18], as shown in Fig. 11, although the specimen was preloaded. It can be seen from Fig. 11 that the proposed single-lens 3D video extensometer is almost not affected by out-of-plane displacement within the range of focus.

 

Fig. 10 Comparisons of measured strain between strain gauge and the proposed single-lens 3D video extensometer.

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Fig. 11 Comparisons of measured strain errors between 2D video extensometer and the proposed single-lens 3D video extensometer.

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Another four sets of experiments were also performed and the results are consistent with the results in Fig. 10. The maximum difference of the measured strain between strain gauge and the single-lens 3D video extensometer in the five sets of experiments is just 24.21$\mu \epsilon$and the maximum STD is 12.37$\mu \epsilon$. The correctness of the measured results further verifies the effectiveness of the proposed speckle-based calibration method. Based on the proposed single-lens 3D video extensometer, high-accuracy and real-time strain measurement can be achieved.

3.3 Discussions on the measured results

The maximum strain error of the proposed single-lens 3D video extensometer is within 30$\mu \epsilon$, which means that strain value of 300$\mu \epsilon$can be measured accurately if the allowable percentage error of strain is 10%. The strain accuracy of the proposed single-lens 3D video extensometer depends on two factors, namely the accuracy of 3D reconstruction of gauge points and the defined gauge lengths. The accuracy of 3D reconstruction of gauge points is mainly affected by the registration accuracy of stereo matching and stereo-vision parameters [39].

In order to maximize the accuracy, the robust IC-GN algorithm combined with a robust stereo matching stage has been employed to achieve high-accuracy subset-based stereo matching in this paper. Although not shown here, our experimental results indicate that there are almost no differences between the proposed calibration method and the classic planar calibration method [27] for strain measurement in terms of accuracy.

4. Discussions

4.1 Advantages of the proposed self-calibration single-lens 3D video extensometer

For a common 2D video extensometer, errors due to out-of-plane displacement are unavoidable and unacceptable in practical measurements. To eliminate the effects of out-of-plane displacement on strain measurement without introducing complicated system calibration, a self-calibration single-lens 3D video extensometer is proposed in this paper. The merits of the proposed video extensometer are as follows

4.2 Limitations of the proposed self-calibration single-lens 3D video extensometer

Although the proposed 3D video extensometer has the advantages as described above, there still exist limitations at the present system.

The limitations described above are unavoidable in the present system, but they will not limit the board application prospects of the proposed 3D video extensometer.

4.3 Further applicability of the proposed self-calibration single-lens 3D video extensometer

In section 3, the measurement accuracy of the proposed self-calibration single-lens 3D video extensometer has been verified using experiments. The strain measurement accuracy can be further improved by the following methods.

For cameras with higher resolution or temporal smoothing, the strain acquisition speed may be reduced. For spatial smoothing using more points, the computation efficiency should be lessened. With the development of computer technology and image sensing technology, the applicability of the proposed self-calibration single-lens 3D video extensometer should be further improved in the near future.

5. Conclusions

In this paper, a self-calibration single-lens 3D video extensometer is proposed for non-contact, high accuracy and real-time strain measurement. Compared with the commonly used 2D video extensometer, the errors due to out-of-plane displacement are eliminated. Moreover, both stringent synchronization and complicated system calibration of binocular stereovision is not required in the proposed 3D video extensometer, compared with the existing 3D-DIC system. The implementation, advantages, limitations and further applicability of the proposed 3D video extensometer have been described and discussed in detail. Two experiments were conducted to verify the performance of the proposed 3D video extensometer. The measurement results are in good accordance with the data obtained by the strain gauge technique.

With the development of the computational efficiency of DIC [37], the proposed self-calibration single-lens 3D video extensometer is expected to be used for in situ, real-time and full-field deformation measurement of materials and structures. Based on the characteristics of self-calibration, application of the proposed single-lens 3D system on microscope deformation measurement is also expected.