# Adaptive Subset-Based Digital Image Correlation for Fatigue Crack Evaluation
## 3 The Feasibility Tests of Adaptive Subset Sizes  
The feasibility of the proposed algorithm is experimentally validated using a speckle-patterned aluminum plate specimen, as shown in Figure 2. To quantitatively investigate the surface deformation, horizontal displacements of 200 \(\upmu\)m, 500 \(\upmu\)m and 1 mm are applied to the specimen using the scanning stage in this experiment. Furthermore, the test results are compared with the conventional SSSIG algorithm. Figure 2a shows the data acquisition system, consisting of the control computer, digital camera, speckle-patterned aluminum specimen and scanning stage. The overall working principle is as follows. First, several reference images are acquired from the surface of the scanning stage-mounted specimen. To avoid image distortion, the digital camera keeps the parallel aligned condition with respect to the target specimen surface. The adaptive subset sizes can be constructed within the entire ROI from the combinations of reference images. Then, the control computer sends out a signal to the scanning stage to numerically translate the specimen in the \(x\) or \(y\) directions, and the corresponding surface is acquired as a test image for the validation test. By repeating the specimen translation and test image acquisition over the predetermined steps, the adaptive subset sizes are validated through comparing the actual translated displacement and the displacement which calculated by the DIC. Finally, the validation test results are compared with the conventional subset size determination algorithm, i.e., SSSIG. The digital camera employed in this system is the Canon EOS 5D Mark 4 with a 100 mm F 2.8 L macro IS USM lens. The scanning stage is able to shift the specimen along the \(x\) and \(y\) directions with a spatial resolution of 0.5 \(\upmu\)m. Then, the speckle patterns are made on the aluminum specimen of \(500\times 500\times 2\) mm\({}^{3}\) using a stone spray, as shown in Figure 2b. Here, the speckle patterns are spatially biased and intentionally designed to examine the feasibility of the proposed algorithm. The speckle pattern images of the ROI, which has \(300\times 300\) pixels (\(6\times 6\) mm\({}^{2}\)) on the target specimen, are taken by the digital camera. The image resolution is \(3360\times 2240\) pixels when the working distance between the lens and the target specimen surface is 230 mm. Here, a single pixel is equivalent to 20 \(\up  
To validate the optimized adaptive subset sizes, the specimen is horizontally shifted using the scanning stage and the corresponding DIC errors are computed for the entire ROI. Figure 4 shows the DIC errors when the horizontal displacements are 200 \(\upmu\)m, 500 \(\upmu\)m and 1 mm. The resultant images show that no error appears in the test cases, meaning that the spatially optimized subsets with distinctive features track the same speckle patterns well even when the target patterns are translated within the ROI. In order to compare the validation results of the proposed algorithm with the conventional algorithm ones, a single subset size is determined by SSSIG. First, SSSIG selects a random location within the ROI and calculates speckle pattern gradients within the subset while increasing the subset size. Then, a single subset size that satisfies a certain threshold is determined. According to the typical procedure of SSSIG, the two different seed points are randomly selected, as shown in Figure 5. To equivalently compare the test results, the seed points are intentionally selected among \(O_{i}\). Then, the SD error values, i.e., \(SDerror(x)\) and \(SDerror(y)\), are calculated. It is given by [31]:  
\[SDerror(x)\approx\left(\frac{N(\eta)}{\sum\ \sum\left(f_{x}\right)^{2}}\right)^{ \frac{1}{2}},\ SDerror(y)\approx\left(\frac{N(\eta)}{\sum\ \sum\left(f_{y}\right)^{2}}\right)^{\frac{1}{2}}, \tag{3}\]  
where \(N(\eta)\) is the noise variance calculated using the two images acquired with a certain time interval.