## 2 Theoretical Background  
### 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.  
The path of light through a heterogeneous refractive index field is governed by [PERSON]'s principle. According to [PERSON]'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]:  
\[\frac{d^{2}x}{dz^{2}} =\left[1+\left(\frac{dx}{dz}\right)^{2}+\left(\frac{dy}{dz} \right)^{2}\right]\left[\frac{1}{n}\frac{\partial n}{\partial x}-\frac{dx}{dz} \frac{1}{n}\frac{\partial n}{\partial z}\right] \tag{1a}\] \[\frac{d^{2}y}{dz^{2}} =\left[1+\left(\frac{dx}{dz}\right)^{2}+\left(\frac{dy}{dz} \right)^{2}\right]\left[\frac{1}{n}\frac{\partial n}{\partial y}-\frac{dy}{dz} \frac{1}{n}\frac{\partial n}{\partial z}\right] \tag{1b}\]  
Figure 1: Schematic diagram of simplified experimental setup.  
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, \(\frac{dx}{dz}\ll 1\) and \(\frac{dy}{dz}\ll 1\). 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, \(\theta_{2u}=\theta_{1}\). However, when the light ray leaves the heterogeneous refractive index field, it will propagate at a new angle \(\theta_{2b}\).  
The refractive index of air can be expressed as \(n(x,y,z)=n_{0}+n\
u(x,y,z)\)[24], \(n_{0}\) is the refractive index at homogenous room temperature. \(n\
u(x,y,z)\) represents the variation of the refractive index from the base value. Because of \(n\
u\ll n_{0}\), there is \(\frac{1}{n}\approx\frac{1}{n_{0}}\). Under the above assumptions, the propagation equation of light can be simplified as  
\[\frac{d^{2}x}  
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.