In digital image correlation methods, the sub-pixel matching error is related to the speckle image and shape function. For the gradient algorithm and curve fitting algorithm without iterative operation, the shape function is difficult to adapt to the sub-pixel distribution law of different speckle images. Although the calculation time is short, the sub-pixel matching error is large. For the Newton-Raphson algorithm with an iterative operation, the sub-pixel matching error can be greatly reduced through the infinite approximation of the target image to the original image or the original image to the target image, but the calculation is time-consuming. Generally, in the displacement range of 0.1-0.9 pixel, the sub-pixel error is mainly S-shaped, and the S-shaped error is related to the image speckle diameter. The S-shaped error is much more different for different speckle diameters. In the field of displacement measurement, the speckle image is fixed and determined, and how to select an appropriate shape function matching the known speckle image to shorten the sub-pixel matching time and improve the sub-pixel matching accuracy has become the focus of this paper.

In order to reduce sub-pixel matching error, this paper proposes a sub-pixel matching method based on parameterized shape function with an exponential variable and background variable, and the core of the method is to decompose the 3×3 correlation coefficient C(i, j) into the 3×1 correlation coefficient A(i) in the x direction and the 1×3 correlation coefficient B(j) in the y direction. Specifically, i and j vary in the set of -1, 0, and 1, and then the 3×1 correlation coefficient A(i) and the 1×3 correlation coefficient B(j) with the exponential variable and background variable are weighted. In the paper, the difference between the absolute displacement and the sum of relative displacement is defined as a sub-pixel matching error, which includes linear and S-shaped errors. The shape function parameters are calibrated by using different regulation rules of the shape function parameters of the exponential variable and background variable on the sub-pixel matching error so that the proposed method features short sub-pixel matching time and small matching error.

The experiment takes the Gaussian speckle pattern with a speckle diameter of 2 pixel, 10 pixel, and 20 pixel as an example, and 10 images with the displacement of 0.1-0.9 pixel in the x direction are generated respectively. The regulation rule of the shape function parameters on the sub-pixel matching error is experimentally studied. The research results show that the exponential variable of the shape function parameter has the primary regulation function to change the slope of the linear error and the secondary regulation function to change the direction of the S-shaped error (Fig. 4). The background variable of the shape function parameter has the primary regulation function to change the direction of the S-shaped error and the secondary regulation function to change the slope of the linear error (Fig. 5). Under the coordinated regulation of shape function parameters, the sub-pixel matching error of the above speckle image is less than 0.001 pixel (Table 1, Table 2, and Table 3). At the same time, the influence of the speckle diameter of the speckle image from 2 pixel to 20 pixel on the shape function parameters is studied (Table 4, Fig. 7, and Fig. 8). It is found that when the speckle diameter is greater than 6 pixel, the influence of the speckle diameter on the shape function parameters becomes weak, and the influence curve becomes flat, which also lays a good foundation for applying the method in the field of deformation measurement. Finally, according to the actual displacement speckle image provided by DIC Challenge, the proposed method is verified (Table 5).

The sub-pixel matching method described in this paper makes the shape function approximate to the sub-pixel distribution law of the speckle image by calibrating the shape function parameters to obtain the optimal shape function and greatly reduces the sub-pixel matching error. Since the shape function calculation only involves the addition, subtraction, multiplication, division, and exponential operation of nine correlation coefficients, as well as no more than six cycles of calculation, the sub-pixel matching time can be ignored in the digital image correlation calculation. Compared with the gradient algorithm and curve fitting algorithm without iterative operation, the sub-pixel matching method of parameterized shape function can obtain ideal sub-pixel matching accuracy for speckle images with different speckle diameters, which cannot be achieved by gradient algorithm and curve fitting algorithm. Compared with the Newton-Raphson algorithm with the iterative operation, the proposed method has comparable sub-pixel matching accuracy in the field of displacement measurement and is more suitable for online displacement measurement due to the short matching time. However, in the field of deformation measurement, there are still some problems to be solved and further studied, such as the influence of the equivalent speckle diameter change of speckle image under deformation conditions on the shape function, as well as the dynamic calibration for shape function parameters.