# Calibration and Image Reconstruction in a Spot Scanning Detection System for Surface Defects

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## Abstract

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## 1. Introduction

## 2. Spot Scanning Surface Defect Evaluation System

#### 2.1. System Layout

#### 2.2. Ideal Scanning Trace and Image Reconstruction

#### 2.3. Deviations in the System and their Consequences

## 3. System Calibration and Image Reconstruction Methods

#### 3.1. Modeling of the Practical Scanning Trace

- Step 1:
- Calculate the corresponding scanning time of the sequence number, as $t=n/f$;
- Step 2:
- Define the line equation of the rotation axis using Equation (3) with parameters ${\theta}_{r}$ and ${\phi}_{r}$;
- Step 3:
- Calculate the plane equation of the sample surface after rotation about the rotation axis using Equation (4) with the parameter $\omega $;
- Step 4:
- Define the line equation of the illumination line using Equation (5) with parameters ${x}_{i},{y}_{i},{\theta}_{i},{\phi}_{i}$ and $v$;
- Step 5:
- Calculate the intersection point of the sample plane and the illumination line;
- Step 6:
- Transform the coordinate of the intersection point from the system coordinate system to the surface coordinate system using Equation (6) with parameters ${\theta}_{r},{\phi}_{r}$, and $\omega $;
- Step 7:
- Transform the surface coordinate of the intersection point to the image coordinate using Equation (2) with parameters $v$ and $\omega $.

#### 3.2. System Calibration Method

#### 3.2.1. Extraction of Feature Sequence Number Sets

#### 3.2.2. Establishment of Constraint Function

#### 3.2.3. Nonlinear Optimization

#### 3.3. System Adjustment and Image Reconstruction Method

- Initialize an image matrix $Ig(u)$ and a weight matrix $Wt(u)$ with the same size determined by the actual scanning range and ${k}_{0}$; initialize the sequence number $n=0$;
- Calculate the image coordinate corresponding to the sequence number by $u={g}_{{\widehat{\mathsf{\psi}}}^{\prime}}\left(n\right)$, and round it to the integer coordinate ${u}_{int}$; define a weight value according to the distance between them as $w=1-\Vert u-{u}_{\mathrm{int}}\Vert $;
- Update the image matrix by $Ig({u}_{\mathrm{int}})=Ig({u}_{\mathrm{int}})+w{I}_{n}$ with ${I}_{n}$, the corresponding intensity in the acquired discrete intensity sequence, and the weight matrix by $Wt({u}_{\mathrm{int}})=Wt({u}_{\mathrm{int}})+w$;
- Update the sequence number by $n=n+1$; if the traversal has not been completed (i.e., $n<N$), go back to step 2;
- Compute the element-wise quotient of the two matrices, i.e., $Ig(u)=Ig(u)/Wt(u)$;
- Find the positions at which the elements in $Wt(u)$ have the value of zero and fill the elements at these positions in $Ig(u)$ (no-value pixels) using the average of 8-neighborhood.

## 4. Experiment Results and Discussions

#### 4.1. Test Environment

#### 4.2. Experiments of System Calibration

#### 4.3. Experiments of Image Reconstruction

## 5. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 2.**Schematic of the ideal scanning. (

**a**) System state at scanning time $t$; (

**b**) Scanning trace and the distribution of sampling points (for intuitive understanding the spacing is enlarged).

**Figure 3.**Simulated practical scanning traces (upper row) and the corresponding images (lower row) reconstructed using the ideal scanning trace with different system deviations. (

**a**) Ideal case; (

**b**) Angle deviation of the rotation axis; (

**c**) Position deviation of the initial illuminated spot; (

**d**) Rotation velocity deviation.

**Figure 4.**Schematic of system state at different time during practical scanning. (

**a**) Initial time ($t=0$); (

**b**) Scanning time $t$.

**Figure 6.**Flow chart of the extraction of feature points and corresponding images. (

**a**) Original image; (

**b**) Selected defect regions; (

**c**) Feature point of a dig; (

**d**) Feature points of a scratch.

**Figure 7.**Reconstructed images after system calibration. (

**a**) Calibrated image; (

**b**) Adjusted image; (

**c**) Enlarged image of the scratches with widths of 0.5–6 µm.

**Figure 8.**(

**a**) The maximum straightness errors of scratches in the adjusted image with comparison to those in the original image; (

**b**) The distance errors of dig pairs in the adjusted image.

**Figure 9.**Gray value distributions in a background area near the center (upper row) and a background area far from the center (lower row) of the reconstructed images with different methods: (

**a**,

**c**) The proposed image reconstruction method; (

**b**,

**d**) The nearest-neighbor interpolation method.

Parameter | ${\mathit{\theta}}_{\mathit{r}}$$(\xb0)$ | ${\mathit{\phi}}_{\mathit{r}}$$(\xb0)$ | ${\mathit{\theta}}_{\mathit{i}}$$(\xb0)$ | ${\mathit{\phi}}_{\mathit{i}}$$(\xb0)$ | ${\mathit{x}}_{\mathit{i}}\text{}\left(\text{\xb5}\mathbf{m}\right)$ | ${\mathit{y}}_{\mathit{i}}\text{}\left(\text{\xb5}\mathbf{m}\right)$ | ${\mathit{v}}_{}\text{}(\text{\xb5}\mathbf{m}/\mathbf{s})$ | ${\mathit{\omega}}_{}\text{}\left(\mathbf{RPS}\right)$ | Constraint Function |
---|---|---|---|---|---|---|---|---|---|

setting value | 0 | 0 | 135 | 90 | 0 | 0 | 20 | 4 | 1.7 $\times $ 10^{6} |

calibrated value | 0.038 | 102.08 | 134.99 | 88.83 | 37.45 | 960.52 | 20.034 | 4.000060 | 54.60 |

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## Share and Cite

**MDPI and ACS Style**

Wu, F.; Cao, P.; Du, Y.; Hu, H.; Yang, Y.
Calibration and Image Reconstruction in a Spot Scanning Detection System for Surface Defects. *Appl. Sci.* **2020**, *10*, 2503.
https://doi.org/10.3390/app10072503

**AMA Style**

Wu F, Cao P, Du Y, Hu H, Yang Y.
Calibration and Image Reconstruction in a Spot Scanning Detection System for Surface Defects. *Applied Sciences*. 2020; 10(7):2503.
https://doi.org/10.3390/app10072503

**Chicago/Turabian Style**

Wu, Fan, Pin Cao, Yubin Du, Haotian Hu, and Yongying Yang.
2020. "Calibration and Image Reconstruction in a Spot Scanning Detection System for Surface Defects" *Applied Sciences* 10, no. 7: 2503.
https://doi.org/10.3390/app10072503