## 2 Basic Principles  
### Conventional Methods and Their Deficiencies in Boundary Deformation Measurement  
The basic idea of the traditional subset-based DIC method is to match (or track) the same physical points located in the reference (or source) image and deformed (or target) image. Relevant work in this area has been discussed in detail in articles [3; 13]. Althoughit is simple in principle and implementation, the existing DIC technique exhibits certain deficiencies. For example, to measure the displacement of a point located near the edge of a specimen, the subset may contain unwanted or foreign pixels outside of the specimen, as illustrated in Figure 1. In such cases, errors will be induced in the correlation and interpolation calculation. That is, the boundary deformation cannot be calculated, and the displacements in a strip along the specimen edge will be missing. Therefore, a smaller subset size is usually selected to obtain more deformation information near the specimen edges. However, a smaller subset will lead to a larger standard deviation (SD) in the displacements. A conflict always exists between the standard deviation and measuring area loss near the edges. A numerical experiment can be used to understand and explain these two conflicting aspects more intuitively. In this experiment, a speckle image with a size of 500 \(\times\) 500 pixels and 256 gray levels was used as the reference image (Figure 1). The speckle pattern was obtained by randomly spraying black and white paints onto the surface of a flat specimen. To form the deformed image, random Gaussian noise with a gray-scale intensity variance of 4 was added to the reference image. Based on these two images, the displacement field was calculated with subset sizes ranging from 11 \(\times\) 11 to 77 \(\times\) 77 pixels, with an interval of 6 pixels. As no actual deformation occurred, the SD error of the measured displacement field can be expressed as  
\[\sigma=\sqrt{\frac{1}{N-1}\sum_{i=1}^{N}d_{i}^{2}} \tag{1}\]  
where \(N\) is the total number of measuring points and \(d_{i}\) represents the displacement component (\(u\) or \(v\)) of point \(i\). The SD error reflects the deviations of the measured displacements from their mean values. Figure 2 illustrates the SD error of the u-displacement and measuring area loss versus the subset size. It is clear that, with an increase in the subset size, the SD error decreased, but the area loss increased. To solve this contradiction, i.e., to obtain the inner displacement field precisely with a relatively larger subset, as well as the boundary deformation, conventional techniques such as the extrapolation and half-subset methods can be used. The basic principles of these two methods in boundary deformation measurement are briefly introduced next. Extrapolation is an interpolation method whereby the displacements near the specimen boundary are obtained using those calculated by the conventional subset-based DIC method inside the specimen (Figure 3a). However, the extrapolated boundary deformation may not be accurate if the deformation pattern near the specimen edge differs from that inside the specimen. For example, owing to the high-stress concentration, the material near  
Figure 1: Speckle image with subset located near the specimen edge. the edge may already yield, while it remains elastic in other regions. In this case, a reliable result cannot be always assured. The half-subset method proposed by [PERSON] et al. [9] is another approach for tackling the boundary deformation problem. In this method, for a subset centered at a point close to or on the AOI boundary (Figure 3b), a binary mask is established to identify the valid points within the AOI and the invalid points outside the AOI but within the subset. During the correlation analysis, only the labeled valid points are used, while the invalid points are excluded. Further details regarding the half-subset method can be found in reference [9].