# Defect Isolation from Whole to Local Field Separation in Complex Interferometry Fringe Patterns through Development of Weighted Least-Squares Algorithm

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Experimental Methodology

#### 2.1. Phase Separation Method of Whole Field vs. Local Field

_{1},a

_{2}) show the fringe patterns and the corresponding unwrapped phase, respectively. Folds appear at the densest fringes in Figure 1(a

_{1}), and, as mentioned earlier, too-dense fringes interfere with the quantification of defects. The unwrapped phase presents the spherical wave characteristics of the quadratic term. Based on the weighted least-squares algorithm, the 10 × 10 grid phase of Figure 1(b

_{1}) and the weighted matrix $\omega (x,y)$ of Figure 1(b

_{2}) are obtained sequentially. Using Equation (5), the phases of the whole body and the local body are separated, as shown in Figure 1(c

_{1},d

_{1}), respectively. Figure 1(c

_{1},c

_{2}) exhibit the same phase characteristics as that of Figure 1(a

_{1},a

_{2}), which demonstrates the effectiveness of our method in separating the whole field phase. From Equation (6), Figure 1(d

_{1},d

_{2}) show the phase and strain distribution of the local body, respectively. The local-body phase is on a relatively flat plane, and the abnormal strain can be clearly displayed in the above phase distribution. The strain distribution of the defect in Figure 1(d

_{2}) is in the micron level, which is beyond the range of $\lambda $/2, indicating that interfering fringes can occur in the fringe pattern. The proposed method achieves the “fringe-pattern-free” of the local-body phase, which shows that our method can also eliminate the above-mentioned interfering fringes.

#### 2.2. System View

## 3. Experimental Results

#### 3.1. Comparison of Quantitative Methods

#### 3.2. Contrast Treatment of Simple and Complex Fringe Patterns

_{1}–a

_{4},b

_{1}–b

_{4}) show the fringe patterns, unwrapped phase, separated whole-body phase and the corresponding local-body strain distributions of single and complex fringes, respectively. Figure 4(a

_{1},b

_{1}) show fringe patterns acquired in the same sampling area, showing low-density and high-density fringe patterns due to different excitation times. Obviously, the fringe identification table in Section 3.1 obtains less defect qualitative information for low-density fringe patterns. The unwrapped phase maps of Figure 4(a

_{2},b

_{2}) are skewed, which can be misleading to actual defect quantification. Figure 4(a

_{3},b

_{3}) separate the whole-body phase maps and verify the tilt and spherical aberration described above. For the effectiveness of phase separation, we have a visual demonstration of the separation effect. According to our hypothesis, the whole-body phase and the local-body phase have different phase characteristics. As shown in Figure 4(a

_{3},b

_{3}), the whole-body phase is similar to the spherical wave, has the characteristics of quadratic term aberration, and carries the polynomial coefficients of the radius of curvature and the angle of inclination, as noted in Equation (2). As shown in Figure 4(a

_{4},b

_{4}), the local-body phase is characterized by an up-and-down phase or strain change in the defect region on a relatively flat plane. The black dashed line indicates a relatively flat area, while the red dashed line indicates the strain distribution in the defect area. The strain distribution of the local body of Figure 4(a

_{4},b

_{4}) has great similarity in the transverse direction, indicating the effectiveness of the proposed method. At the same time, due to the difference in excitation time, the local-body phase of simple fringes and complex fringes has numerical differences in the axial direction, indicating that the defect parts are susceptible to excitation and have a cumulative effect.

#### 3.3. Mural Cooling Process

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The process of phase separation method with the experimental data: (

**a**,

_{1}**a**) are wrapped phase and the corresponding unwrapped phase, respectively; (

_{2}**b**,

_{1}**b**) are the grid phase and weighted matrix $\omega (x,y)$, respectively; (

_{2}**c**,

_{1}**c**) are re-wrapped phase map and the 3D rendering of the separated whole-body phase distribution, respectively; (

_{2}**d**,

_{1}**d**) are the isolated local-body phase distribution and the corresponding 3D rendering, respectively.

_{2}**Figure 3.**A comparison of the method with the traditional identification method: (

**a**) original photo of the sampling fresco area; (

**b**) obtained fringe patterns map; (

**c**) unwrapped phase map; (

**d**) separated whole-body phase map; (

**e**) re-wrapped phase map of (

**d**); (

**f**) isolated local-body phase map.

**Figure 4.**Simple fringe vs. complex fringe: (

**a**–

_{1}**a**) are the fringe patterns, unwrapped phase, separated whole-body phase and the corresponding local-body strain distribution of simple fringes, respectively; (

_{4}**b**–

_{1}**b**) are the fringe patterns, unwrapped phase, separated whole-body phase and the corresponding local-body strain distribution of complex fringes, respectively.

_{4}**Figure 5.**Cooling process of the mural: (

**a**) obtained fringe patterns, unwrapped phase, separated whole-body phase and local-body strain distribution of 6 s; (

**b**) obtained fringe patterns, unwrapped phase, separated whole-body phase and local-body strain distribution of 102 s; (

**c**) obtained fringe patterns, unwrapped phase, separated whole-body phase and local-body strain distribution of 228 s; (

**d**) obtained fringe patterns, unwrapped phase, separated whole-body phase and local-body strain distribution of 324 s; (

**e**) obtained fringe patterns, unwrapped phase, separated whole-body phase and local-body strain distribution of 432 s; (

**f**) obtained fringe patterns, unwrapped phase, separated whole-body phase and local-body strain distribution of 540 s.

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**MDPI and ACS Style**

Chen, Z.; Zhou, W.; Yu, Y.; Tornari, V.; Artioli, G.
Defect Isolation from Whole to Local Field Separation in Complex Interferometry Fringe Patterns through Development of Weighted Least-Squares Algorithm. *Digital* **2024**, *4*, 104-113.
https://doi.org/10.3390/digital4010004

**AMA Style**

Chen Z, Zhou W, Yu Y, Tornari V, Artioli G.
Defect Isolation from Whole to Local Field Separation in Complex Interferometry Fringe Patterns through Development of Weighted Least-Squares Algorithm. *Digital*. 2024; 4(1):104-113.
https://doi.org/10.3390/digital4010004

**Chicago/Turabian Style**

Chen, Zhenkai, Wenjing Zhou, Yingjie Yu, Vivi Tornari, and Gilberto Artioli.
2024. "Defect Isolation from Whole to Local Field Separation in Complex Interferometry Fringe Patterns through Development of Weighted Least-Squares Algorithm" *Digital* 4, no. 1: 104-113.
https://doi.org/10.3390/digital4010004