Fig. 1. Section of a speckle pattern before (square) and after (parallelogram) the deformation $\mathbf{y}\left(\mathbf{X}\right)$. The red squares represent exemplary subsets with the grayscale values $f\left(\mathbf{X}\right)$ and $g\left(\mathbf{y}\right(\mathbf{X}\left)\right)$.

Fig. 2. Speckle simulation setup.

Fig. 3. (a) True value of the displacement gradient in x-direction (red) and random error of $d_x$ (blue), i. e., the standard deviation of $d_x$ in y-direction. Dotted vertical lines indicate the four numbered sections of the displacement field. Sections I and IV are characterized by a constant displacement value, while sections II and III have a constant second-order displacement gradient. Subset size S = 20 × 20 pixel. The dashed lines show an alternative displacement field with a constant first-order gradient of 10 × 10⁴ με in sections III and IV. (b) DSP result d_x in x-direction. The displacement field is evaluated from simulated speckle patterns.

Fig. 4. Random error of displacement field d_x for three different subset sizes S. The random error is evaluated from laser speckle patterns (DSP) and evaluated directly from the deformed surface topography (DIC).

Fig. 5. (a) Mean in y-direction of the displacement field $d_x(x,y)$ yields $\overline{d_x}\left(x\right)$. The DSP result (S = 20 pixels) is compared to its true value and a moving mean of the true value over 20 pixels in x-direction. (b) Laplacian of the displacement d_x and systematic error over $x$. The systematic error is the deviation of $\overline{d_x}$ from the true value.

Fig. 6. Systematic error over the Laplacian for different subset sizes $S$. The solid lines are linear fits through the data points and the error bars represent the respective random error. The error values are evaluated from laser speckle patterns (DSP) and evaluated directly from the deformed surface topography (DIC).

Fig. 7. (a) Displacement field $d_y$ measured with DSP during the laser hardening process. The vertical, black line indicates the position of the laser, which is moving in the negative $x$-direction. (b) First-order gradient ${\nabla }_x{d}_y$ of the displacement field. (c) Second-order gradient ${\Delta }_x{d}_y$ of the displacement field.

Fig. 8. (a) Displacement field $d_y$ measured with DSP during grinding. (b) Local displacements and Laplacian in $y$-direction. The position of the observed $y$-interval is indicated by a dotted, white line in (a).

Fig. 9. (a) Displacement field $d_y$ measured with DSP during single-tooth milling. (b) Measured local displacements in $y$-direction and approximate correction of systematic errors. The error bars indicate the random error that is caused by the first-order gradient. The position of the observed $y$-interval is indicated by a dotted, white line in (a).