*2.3. Optimized Thickness Strain Measurement*

destructive measurement of a sample's thickness strain [19]. However, as Li wrote in destructive measurement of a sample's thickness strain [19]. However, as Li wrote in The measurement of the thickness strain is the core part of calculating the R-value. Li built a dual-camera DIC system on the front and back of the test piece to realize the non-destructive measurement of a sample's thickness strain [19]. However, as Li wrote in [19], the strain measurement results contain noise, so it is necessary to remove the noise from the strain data. Since the thickness of the test sample is usually a few millimeters, the noise is amplified during the thickness strain measurement. Figure 5 shows the DP980 thickness strain history, where the *x*-axis represents the number of photos taken and the yellow circle indicates the noise data. Two smoothing algorithms, namely the Least squares algorithm and Random sample consensus (RANSAC) algorithm, were used to fit the curve and reduce noise. A comparison of the results using these two algorithms follows.

**Figure 5.** DP980 thickness strain fitted by least squares algorithm.

## 2.3.1. Least Squares Algorithm

The least squares method is the most common method used to solve curve fitting problems. The basic idea of the method is to minimize the sum of squares of the error to find the best function match for the data. In general, we can model the expected value of *y* as an nth degree polynomial, yielding the general polynomial regression model:

$$y = f(\mathbf{x}, \boldsymbol{\mathfrak{a}}) = \mathbf{a}\_0 + \mathbf{a}\_1 \mathbf{x} + \mathbf{a}\_2 \mathbf{x}^2 + \mathbf{a}\_3 \mathbf{x}^3 + \dots \mathbf{a}\_n \mathbf{x}^n \tag{4}$$

 = [<sup>1</sup> , <sup>2</sup> , ⋯ , ] where *α* = [*α*1, *α*2, · · · , *αn*] is a parameter to be determined.

 (, (, )) (, ) ( , ) ( = 1,2, ⋯ , ) The objective function *L*(*y*, *f*(*x*, *α*)) is minimized to find the optimal estimated value of the parameter *α* in the function *f*(*x*, *α*) for *m* given sets of data (*x<sup>i</sup>* , *yi*) (*i* = 1, 2, · · · , *m*).

$$L(y, f(\mathbf{x}, \mathbf{a})) = \sum\_{i=1}^{m} \left[ y\_i - f(\mathbf{x}\_i, y\_i) \right]^2 \tag{5}$$

rithm's fitting results become inaccurate The result using least squares fitting is shown in Figure 5 as a red curve. The red curve represents the results of quadratic polynomial fitting. Observation of Figure 5 indicates that the fitted curve shifts relative to the expected data. Because of this, the algorithm's fitting results become inaccurate if the data contains a large amount of error or noise. This is the limitation of the least squares algorithm.
