# Digital Shearography for NDT: Phase Measurement Technique and Recent Developments
## 3 Phase Measurement Technique
### Phase Measurement Technique: Temporal Phase-Shift Shearography  
Temporal phase-shift digital shearography refers to a technique of introducing different phase-shift amounts in a time series to obtain phase information [23]. Several methods exist for  
Figure 4: Comparison between (**a**) real-time subtraction version and (**b**) phase-shift shearography. Figure 3: (**a**) Holographic NDT result, (**b**) shearographic NDT result. introducing the phase shift, including the piezoelectric transducer method (PZT), polarization phase-shift method, and grating phase-shift method. Among them, the piezoelectric transducer is the most commonly used in practical applications due to its simple use. It produces a precisely designed movement through a piezoelectric crystal driven mirror, introducing a known phase shift between the two images in the time series. The previous section introduced the basic principles of shearography, where the intensity formula for a single speckle pattern image is as follows:  
\[I=I_{0}(1+\gamma\cos\phi)=A+B\cos\phi \tag{6}\]  
There are three unknowns in the formula: background intensity \(I_{0}\), contrast \(\gamma\), and phase difference \(\phi\). Therefore, at least three equations are needed. In theory, three equations are sufficient to obtain the relative phase difference. However, as can be seen in Table 1, the calculation relationship of the \(4+4\) algorithm is the simplest and has a higher extraction accuracy. As a result, it has become the most widely used algorithm in temporal phase-shift shearography. Called the \(4+4\) algorithm, four images are captured before loading to solve for \(\phi\), then four images are captured after loading to solve for \(\phi^{\prime}\)[24]. The four intensity equations before loading with a known phase can be expressed as follows:  
\[ \begin{array}{l}I_{1}=A+B\cos\phi\\ I_{2}=A+B\cos(\phi+\frac{\pi}{2})\\ I_{3}=A+B\cos(\phi+\pi)\\ I_{4}=A+B\cos(\phi+\frac{3\pi}{2})\\ \end{array}  \tag{7}\]  
Then, the phase difference before loading can be calculated by the following formula:  
\[\phi=\arctan\frac{(I_{4}-I_{2})}{(I_{1}-I_{3})} \tag{8}\]  
Similarly, the phase difference after loading can be calculated using the same formulas, and the relative phase change can be obtained by subtracting the phase difference before and after loading. Compared with the direct subtraction method, the measurement sensitivity is much improved, but it requires four images to be captured under each loading condition, making it unsuitable for dynamic measurements. The phase-shift algorithm has been improved to adapt to continuous or dynamic measurements, resulting in the development of various fast temporal phase-shift algorithms. The phase-shift steps before loading have no effect on the acquisition rate of the system. The fast temporal phase-shift algorithms reduce the number of phase-shift steps after or during loading and can realize dynamic measurement when no phase shift is required [25; 26; 27]. The \(4+1\) temporal phase-shift algorithm takes four images before loading, using the same procedure as described in the previous section. A single image is recorded on each loading, and no phase-shifting step is required. For each load, a phase map is obtained from the four phase images before loading and one real-time image during loading. The \(4+1\) temporal phase-shift algorithm provides the ability to achieve dynamic measurements, but its poor phase map quality is not suitable for high-precision quantitative measurements, limiting its use for qualitative measurements such as NDT. A \(4+2\) algorithm has also been developed, requiring only one phase-shift step during loading. The phase map quality is markedly better than the \(4+1\) algorithm, meaning that the fringes in the phase map are clear and complete. Compared with the \(4+4\) algorithm, the phase map quality is reduced within a reasonable range to achieve an increase in the acquisition rate, thereby realizing dynamic measurement in a low dynamic range.