## 2 Theory and Algorithm
### Low-Level Variation Magnification  
In practical measurement, lower frequency structural boundary variations commonly have a larger vibration amplitude and will occupy more energy of the global edge translation signals. By contrast, low-level high-frequency variations of the detected edge profile may be covered by the noise, which makes these signals difficult to extract. To enhance these low-level components, we use phase-based Eulerian video magnification (PEVM) to preprocess subtle changes in video before the deviation extraction. In contrast to the intensity-based linear video magnification method [27], which handles the pixel values of the image sequence independently in the time line, PEVM visualizes subtle motions in video by band-pass filtering of temporal image phases. Variations in specific frequency bands can be magnified and visualized in the reconstructed video. Considering that linear temporal phase variations are sensitive to large motions, such as camera or object motion, we follow the acceleration magnification method of [PERSON] et al. For a 1D signal \(I(x,t)\) at positon \(x\) and time \(t\), linear methods [27; 28; 29] assume a displacement function \(\sigma(t)\) such that \(I(x,t)=g(x+\sigma(t))\). The magnified counterpart of \(I(x,t)\) is defined as:  
\[\tilde{I}(x,t)=g\left(x+\left(1+\alpha\right)\sigma\left(t\right)\right) \tag{10}\]  
The second-order Taylor expansion yields:  
\[\tilde{I}(x,t)\approx g(x)+(1+\alpha)\sigma(t)\frac{\partial g(x)}{\partial x }+\frac{1}{2}(1+\alpha)^{2}\sigma(t)^{2}\frac{\partial^{2}g(x)}{\partial x^{ 2}} \tag{11}\]  
By assuming that the large object motion is approximately linear, the band-pass filtering and magnification processes will be conducted on the second-order non-linear item:  
\[C(x,t)=\sigma(t)^{2}\frac{\partial^{2}g(x)}{\partial x^{2}} \tag{12}\]  
The reconstructed magnified signal can be calculated as:  
\[\tilde{I}(x,t)=I(x,t)+\alpha C(x,t) \tag{13}\]  
where \(\alpha\) is the amplification factor. The difference between linear and acceleration magnification in the 1D signal is illustrated in Figure 2. The red lines show the signal magnification results for factor \(\alpha=2\). The purple curves show the magnified linear and acceleration variations of the 1D signal. Comparedwith the linear method, acceleration magnification handles only small motion and is more robust to large translational motions. Motion can be represented by a phase shift; as such, phase-based acceleration magnification operates on the second-order temporal phase variations of the image sequence. For a given image \(I(x,y,t)\) in timeline \(t\), the complex steerable pyramid [31] decomposes the image signal into multiple frequency bands at scale \(\omega\) and orientation \(\theta\):  
\[(I(x,y,t)\otimes\Phi_{\omega,\theta})=A_{\omega,\theta}(x,y,t))e^{i\phi_{ \omega,\theta}(x,y,t)} \tag{14}\]  
where \(\otimes\) is convolution, \(A_{\omega,\theta}(x,y,t)\) is the temporal amplitude, and \(\phi_{\omega,\theta}(x,y,t)\) is the temporal phase. The second-order derivative of the Gaussian filter \(G_{\theta}(x,y,t)\), namely the Laplacian filter, is applied to extract subtle phase variations of interest [32; 33]:  
\[C_{\sigma}\left(\phi_{\omega,\theta}(x,y,t)\right)=\frac{\partial^{2}\phi_{ \omega,\theta}(x,y,t)}{\partial t^{2}}\otimes G_{\theta}(x,y,t)=\phi_{\omega, \theta}(x,y,t)\otimes\frac{\partial^{2}G_{\theta}(x,y,t)}{\partial t^{2}} \tag{15}\]  
where \(C(\cdot)\) is the temporal Laplacian filter with the kernel as \(\sigma=\frac{r}{4\sqrt{2}\omega}\). The parameter \(r\) is the frame rate, and \(\omega\) is the target frequency. Therefore, the magnified phase with amplification factor \(\alpha\) is given by:  
\[\hat{\phi}_{\omega,\theta}=\phi_{\omega,\theta}(x,y,t)+\alpha C_{\sigma}\left( \phi_{\omega,\theta}(x,y,t)\right) \tag{16}\]  
Eventually, the reconstructed magnified image can be calculated by using the image amplitude and the magnified phase. A simulation test is provided to illustrate the acceleration magnification result.