**1. Introduction**

The mechanical properties of materials are significant in the utilization of materials. Aerospace, engine, petrochemical, and other areas are developing rapidly. In these fields, materials must work stably at high temperatures [1–3]. Therefore, it is vital to study the mechanical properties of materials at high temperatures. At present, a variety of measurement methods for material deformation in a high-temperature environment have been developed, among which the digital image correlation (DIC) method [4–6] has attracted more and more research interest. The digital image correlation method is a non-contact optical measurement technique which measures the displacement and strain caused by deformation of the material. Due to its advantages of simple operation, wide application range, and full-field measurement [7–10], digital image correlation has become one of the most active optical measurement methods and is widely used in scientific research and engineering practice.

However, the digital image correlation measurement is based on the images of the specimen. The quality of the images dramatically affects the measurement accuracy of the DIC measurement. In high-temperature environments, heat waves caused by heat sources can distort the images. How to eliminate the influence of thermal disturbance on measurement accuracy is a problem that needs to be solved when using the DIC method to measure objects at high temperatures. Many experts and scholars have studied this problem and made many efforts to eliminate the influence of heat waves on DIC measurement results by improving experimental devices and algorithms.

One type of effort mainly focuses on hardware improvement. Novak et al. [11] added an "air knife" in their experimental devices. The "air knife" is positioned to blow across the sample surface, and its role is to minimize the thermal turbulence and thoroughly mix air in the lens of sight of the imaging systems, thereby reducing apparent distortions caused by heat waves. Jenner et al. [12] designed a unique system for high-temperature measurement, including a camera, loading mechanism, heating furnace, etc. to measure the strain of high-strength steel at high temperatures. L. Chen et al. [13] used an air controller to mix the air between the furnace window and lens. Bao et al. [14] used a color speckle and color camera to separate the displacement due to heat flow disturbance, which improved the measurement accuracy of DIC. This method can realize real-time correction but requires making a specific spot pattern on the specimen surface. Pan et al. [15] conducted experiments in a vacuum wind tunnel and adopted violet illumination and filters to eliminate thermal radiation and thermal disturbances. No significant disturbance of the heat waves was observed in the vacuum. The influence of heat waves can be reduced by improving the experimental system, but the corresponding system will become more complex, and the costs will increase. Besides, it is difficult to eliminate the influence of heat waves by only improving the system hardware [16]. Other solutions have to be found to solve the problem.

Another type of effort aiming at the improvement of algorithms has been carried out. There have been different ideas on algorithms to remove the effect of heat waves. The first idea is to simulate the heat wave environment through numerical simulation. Zhang et al. [17] used numerical simulation models to correct the effects of heat waves. They proposed using numerical simulation to obtain a heat flow field model, combining the ray-tracing principle to analyze the image distortion caused by the heat flow field and correcting the high-temperature deformation measurement results. The results proved that this method is feasible if the experimental parameters are accurately controlled so that the simulation model is consistent with the actual situation. However, actual environments iare complex and variable, and it is difficult to predict the distortion caused by heat waves accurately. Another algorithm idea is to regard the effect of heat waves on the image as a kind of noise and process the image using a denoise algorithm to achieve better DIC measurement results. Song et al. [16] proposed a high-temperature strain measurement method by combining the digital image correlation method and the Improved Random Sample Consensus (IRANSAC) smoothing algorithm. The IRANSAC algorithm reduces the noise from the airflow disturbance. This method smooths the noisy displacement field and reduces the noise level. Due to the heat waves on images being close to Gaussian noise, Y. Hu et al. [18] used a classical algorithm, the inverse filtering method, to remove the Gaussian noise and to process the images disturbed by the heat waves. The results proved that the method is useful to some degree. In some cases, the denoise algorithm can remove the noise caused by heat waves. However, with DIC as a measurement method, using the denoise algorithm may cause new errors, which affect the measurement results of DIC. A third idea is to research the characteristics of the disturbance caused by heat waves on imaging and according to the characteristics, correct the images. Hao et al. [19] extended the principal component analysis (PCA) method to extract the disturbance characteristics and then corrected the calculation results of DIC. Since the distortion caused by heat waves is random, Su et al. [20] proposed the grayscale-average technique, which corrects the distortion by using multiple measurements to average. The results proved that the signal to noise ratio of the processed images was significantly improved. However, in this method, it was assumed that a point on the image, under the influence of the heat waves, is oscillating around the real value. However, after research, it was found that a point on the image, through the heat waves, oscillates around an offset value. After the average processing, most of the distortion was corrected, but there was still a little distortion. Moreover, this small distortion was not negligible in DIC, either.

In this paper, a method based on the background-oriented schlieren (BOS) technique is proposed to correct the heat wave distortion based on the study of characteristics of distortions due to heat waves. The background-oriented schlieren technique [21] can realize the measurement of refractive index field information that causes the light refraction. By recording the image of the background spot pattern in the presence or absence of heat wave distortion, the distortion displacement field caused by the heat waves can be obtained. According to the obtained distortion displacement field, the DIC measurement results are the corrected. The remainder of this paper is organized as follows: In Section 2, the characteristics of distortions due to heat waves are analyzed, and the flow of the proposed method is introduced in detail. The experiment system is shown in Section 3. Section 4 is the experiment results and discussion. The error level of DIC measurement with or without the influence of the heat waves is analyzed, and experiments show the characteristics of heat waves on imaging distortions. Also, experiments confirmed the effectiveness of the proposed method. The conclusion is in Section 5.

#### **2. Theoretical Background**

#### *2.1. The Principle of the Influence of Heat Waves on DIC Measurement*

The measurement of the DIC method is based on the images of the measured object taken by the camera. If heat waves exist between the camera and the measured object, the heat waves may distort the images taken by the camera, affecting the measurement accuracy of DIC. The way the heat waves cause image distortion can be divided into two aspects. One aspect is the temperature difference between the area affected by the heat waves and the other areas. This temperature difference causes a heterogeneous refractive index field, which causes the light to refract. The other aspect is the hot air flow caused by heat waves.

First of all, the refraction of light caused by the temperature difference is analyzed, regardless of the fluidity of the hot air.

Figure 1 is a schematic diagram of a simplified test system with a region of heterogeneous refractive index between the object to be measured and the imaging system. The heterogeneous refractive index region is caused by heat waves.

**Figure 1.** Schematic diagram of simplified experimental setup.

The path of light through a heterogeneous refractive index field is governed by Fermat's principle. According to Fermat's principle, light travels along the shortest path of the optical distance. In general, the path a ray of light follows is governed by the set of differential equations [22,23]:

$$\frac{d^2\mathbf{x}}{dz^2} = \left[1 + \left(\frac{d\mathbf{x}}{dz}\right)^2 + \left(\frac{d\mathbf{y}}{dz}\right)^2\right] \left[\frac{1}{n}\frac{\partial n}{\partial \mathbf{x}} - \frac{d\mathbf{x}}{dz}\frac{1}{n}\frac{\partial n}{\partial z}\right] \tag{1a}$$

$$\frac{d^2y}{dz^2} = \left[1 + \left(\frac{dx}{dz}\right)^2 + \left(\frac{dy}{dz}\right)^2\right] \left[\frac{1}{n}\frac{\partial n}{\partial y} - \frac{dy}{dz}\frac{1}{n}\frac{\partial n}{\partial z}\right] \tag{1b}$$

where *n* is the refractive index of air and the *z*-axis is aligned with the optical axis of the imaging system. In order to simplify the analysis process, only the paraxial ray is discussed, and the angle between the ray and the *z*-axis is a small angle. Thus, *dx dz* 1 and *dy dz* 1. Furthermore, it is assumed that the refractive index changes in the same magnitude in all three spatial directions. With these simplifying assumptions, the light ray enters the heterogeneous refractive index field at the same location it would have passed through if the refractive index field had been homogeneous, that is, θ2*<sup>a</sup>* = θ1. However, when the light ray leaves the heterogeneous refractive index field, it will propagate at a new angle θ2*b*.

The refractive index of air can be expressed as *n*(*x*, *y*, *z*) = *n*<sup>0</sup> + *n*(*x*, *y*, *z*) [24], *n*<sup>0</sup> is the refractive index at homogenous room temperature. *n*(*x*, *y*, *z*) represents the variation of the refractive index from the base value. Because of *<sup>n</sup> <sup>n</sup>*0, there is <sup>1</sup> *<sup>n</sup>* <sup>≈</sup> <sup>1</sup> *n*0 . Under the above assumptions, the propagation equation of light can be simplified as

$$\frac{d^2\mathbf{x}}{dz^2} = \frac{1}{n\_0} \frac{\partial n\nu}{\partial \mathbf{x}}\tag{2a}$$

$$\frac{d^2y}{dz^2} = \frac{1}{n\_0} \frac{\partial n\nu}{\partial y} \tag{2b}$$

The analysis is now focused on a light ray in the *x–z* plane as shown in Figure 1. Because of *dx dz* = tan(θ), according to the Equation (2a), there is

$$\tan(\theta\_{2b}) = \tan(\theta\_1) - \frac{1}{n\_0} \int\_{-\frac{M}{2}}^{\frac{|V|}{2}} \left(\frac{\partial n\nu}{\partial \mathbf{x}}\right) d\mathbf{z} \tag{3}$$

Due to the existence of the heat waves, the light is refracted. In Figure 1, the light emitted from the point A on the object plane appears at the point B in the perspective of the image plane. According to the geometric relationship, the distortion *x*<sup>0</sup> can be expressed as

$$\mathbf{x}\_{0} = \left[ \tan(\theta\_{1})(L+\mathcal{W}) \right] - \left[ \tan(\theta\_{2b})(L+\mathcal{W}) \right] = (L+\mathcal{W})\frac{1}{n\_{0}}\int\_{-\frac{\mathcal{W}}{2}}^{\frac{\mathcal{W}}{2}} \left(\frac{\partial n\mathcal{\prime}}{\partial \mathbf{x}}\right) d\mathbf{z} \tag{4}$$

According to Equation (4), if such an assumption is made, the air region affected by heat waves is stable and does not flow, and only the temperature is different from the other region, then *n* and *W* are constant. The distortion *x*<sup>0</sup> is also a constant.

However, hot air flows and continuously changes at random. If the heat waves caused by the heat source is stable and a point on the object plane is observed on the image plane after passing through the region affected by the heat waves, there will occur a main distortion *x*0. However, due to the fluidity of the heat waves, as shown in Figure 1, this point will randomly oscillate around *x*0, and based on the main distortion *x*0, a random distortion Δ*x* occurs. This phenomenon will be verified in the experiment. The correction method proposed in this paper is to use the BOS technique to correct the main distortion and use the time-averaging method to correct the random distortion. In this way, the DIC measurement results with the influence of heat waves removed can be obtained.

#### *2.2. Principle of Background-Oriented Schlieren*

The background-oriented schlieren (BOS) technique was proposed by Meier [25]. The BOS technique combines particle image velocimetry (PIV) technology for flow field velocity measurements with traditional schlieren technology. It can measure the refractive index field using the offset of particles in the background pattern [26]. Since its introduction, the BOS technique has attracted the attention of many scholars and is still continuously developed [21]. At present, the BOS technique is mainly used in the field of density measurements of a fluid field, fluid field visualization, temperature measurements, aero-optical wavefront measurements, and optical transfer function measurements [25,27,28]. The spot pattern in the digital image correlation method can be used as the background pattern in the BOS technique. The BOS technique can be divided into two steps. The first step is to extract the

distortion displacement field information by using the high precision PIV algorithm. The second step is to use the distortion displacement field information to construct the remapping function to complete the correction of the image. The technical principle of the BOS technique is shown in Figure 2. First, the image of the background pattern without flow field interference is taken as the reference image. Then, in the presence of flow field interference, the background pattern is imaged again as the measurement image. Next, the displacement of the corresponding particles in the two images is extracted by the PIV algorithm to obtain the refraction information of the light. Then the remapping function is constructed by using the obtained disturbance information to complete the correction of the image with distortion. The extraction of the displacement of the particles on the background pattern is the key to the background-oriented schlieren technique.

**Figure 2.** Schematic diagram of optical distortion correlation method based on the BOS technique.

#### *2.3. Correlation Algorithm Flow*

When using DIC to measure the displacement and strain of the object, it is necessary to spray spots on the surface of the object. The sprayed spots can be used as the background pattern in the BOS technique. The two methods can be combined. The flow chart of the proposed correction method is shown in Figure 3.

At first, after spraying the spots, when there is no heat source, the object to be measured is photographed to get the reference image. Next, several images of the spot pattern through the heat waves are taken as the background images with disturbance information. The PIV calculation between the background images and the reference image is performed to get disturbance displacement maps. These disturbance displacement maps are averaged over time to get the displacement distribution diagram of the main distortion.

Second, the measured object is loaded and then images of the object through heat waves are taken. These images using the displacement distribution diagram of the main distortion are remapped to eliminate the main distortion in the images.

Third, the images whose main distortion have been removed are averaged to eliminate random distortion. After that, the image with the heat wave disturbance removed is obtained. Then, performing the DIC measurement using this image will give the displacement and strain fields of the specimen, which have corrected the heat wave disturbance.

**Figure 3.** Flow chart of the correction method.
