# Generalized Scale Factor Calibration Method for an Off-Axis Digital Image Correlation-Based Video Deflectometer

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## Abstract

**:**

## 1. Introduction

## 2. Off-Axis Digital Image Correlation-Based Video Deflectometer

## 3. Scale Factor Calibration Based on a Generalized Off-Axis Imaging Model

#### 3.1. Accurate SF Calibration Method

_{c}xy in the unit of pixels is established with the camera sensor plane center O

_{c}as the origin point, x-axis to the right, and y-axis downwards. With the optical center O as the origin point, the camera coordinate system O-XYZ in the unit of mm is also established with the X and Y parallel to x and y axes of the image coordinate system, and the Z axis points to the test structure along the optical axis of the camera. For clarity, Figure 2b plots the imaging relation of point P

_{1}in Figure 2a according to ideal pinhole imaging model. In Figure 2b, O

_{c}(x

_{c}, y

_{c}) denotes the image center, f is the focal length of the camera lens, l

_{ps}is the physical size of the pixel, and L is the distance from the optical center of the camera to the measurement point, which can be measured using a laser rangefinder; β is the pitch angle that can be measured by a laser rangefinder or a theodolite.

_{1}on the structure plane moves to P

_{2}with a vertical displacement V. Correspondingly, point p

_{1}(x, y) in the camera sensor target moves to point p

_{2}(x’, y’), v is the image displacement in pixels. Since vertical displacement is the primary displacement component in structure deflection measurement, the horizontal image displacement is approximately zero. Based on the principle of trigonometric geometry, we have $\Delta O{p}_{1}{p}_{2}\sim \Delta O{P}_{1}{P}_{2}\prime $, therefore

_{SF}as

_{1}is related to the distance L from the optical center of the camera to the measurement point, the pitch angle of the camera β, the focal length f, the pixel size of the camera sensor l

_{ps}, the center of image coordination (x

_{c}, y

_{c}), and image coordinates of the measurement point before and after deformation.

_{ps}are known parameters of the fixed focal length lens and the camera, L can be measured by a rangefinder, and β can be measured by the theodolite. Additionally, the image coordinate (x, y) of each measurement point is directly obtained after specifying measurement points in the reference image. Image coordinates (x, y’) of each measurement point in the deformed image can be computed using a subset-based DIC algorithm. Based on these parameters, the SF can be efficiently calculated using Equation (a) without affecting the real-time calculation performance of the video deflectometer.

#### 3.2. Comparison with Existing Off-Axis SF Calibration Method

_{SF}. To indicate the performance of the proposed SF calibration method, it is necessary to compare the proposed Equation (a) against the improved Feng’s Equation (b) and Pan’s Equation (c), as will be shown in the section below.

## 4. Experiments

#### 4.1. Experiment Configuration

#### 4.2. Laboratory Translation Experiments

#### 4.2.1. Experiment Setup

#### 4.2.2. Experimental Results

#### 4.2.3. Discussion

_{SF}and distance L. For quantitative analysis, the relationship between K

_{SF}and other parameters such as focal length f, pixel size l

_{ps}, pitch angle β, and the image coordinate of the measurement point (including the image coordinate before and after deformation), the common camera and lens parameters are selected to simulate the proposed calibration model shown in Equation (a) K

_{SF}and compare the model as shown in Equation (b) K

_{SF1}and Equation (c) K

_{SF2}.

_{SF1}calculated by Equation (b) and the approximate K

_{SF2}calculated by Equation (c) are almost the same. The difference between the proposed accurate calibration method and the approximate calibration method is almost negligible when the camera’s pitch angle is less than 20°, the focal length of the camera lens is greater than 25 mm, or the pixel size is smaller than 5 µm.

_{ps}is 4.8 um. The simulated result of full-field SFs before and after deformation (vertical translation 100 pixels) for the proposed K

_{SF}are shown in Figure 10, and the K

_{SF2}before deformation is shown. The K

_{SF}

_{2}after deformation is not shown in Figure 10 because it equals to the SF before deformation. The K

_{SF}

_{1}in Equation (b) is not calculated because it does not depend on the image coordinate and it always equals to the approximate SF in Equation (c) when (𝑥, 𝑦) is at the center of the image.

_{SF}and Pan’s K

_{SF2}, and K

_{SF2}is relatively small. For accurate calibration, SF is greatly affected by the image y-coordinate of the measured point, but less affected by the image x-coordinate. Additionally, it is affected by the position of the measured point after deformation. The SF is the smallest when the y-coordinate is at the center of the image after deformation. For approximate calibration, the influence of the x-coordinate and y-coordinate on K

_{SF2}is almost the same. The K

_{SF2}of the measurement point at the center of the image is always the largest, no matter before or after deformation. It can be seen from Figure 10d that the difference between K

_{SF}and K

_{SF2}is almost zero when the y-coordinate of the measured point moves to the image center after deformation. It indicates that the K

_{SF}almost equals the K

_{SF2}when 𝑦’ is in the middle position.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic illustration of off-axis DIC-based video deflectometer for deflection monitoring.

**Figure 2.**Imaging model of off-axis DIC: (

**a**) geometric model with off-axis imaging of camera; (

**b**) the off-axis imaging relation diagram of the measurement point.

**Figure 4.**Schematic diagram of camera roll angle correction: (

**a**) original model; (

**b**) corrected model.

**Figure 7.**Image displacement of two camera lenses with different focal lengths: (

**a**) f = 8 mm, (

**b**) f = 50 mm.

**Figure 9.**SF variation with the camera and lens parameters: (

**a**) SF-pitch angle curve for different focal lengths of the lens, (

**b**) SF-pitch angle curve for different pixel sizes.

**Figure 10.**Simulated results of full-field SFs before and after deformation for two calibration methods: (

**a**) the proposed calibration method before deformation, (

**b**) Pan’s calibration method before deformation, (

**c**) the proposed calibration method after vertical translation 100 pixels, (

**d**) difference between the proposed and Pan’s calibration after vertical translation 100 pixels.

Focal Length (mm) | Pitch Angle | Distance L (m) | ||||||||
---|---|---|---|---|---|---|---|---|---|---|

P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P9 | ||

8 | 23.6° | 2.284 | 2.280 | 2.276 | 2.257 | 2.251 | 2.246 | 2.226 | 2.221 | 2.216 |

50 | 22.1° | 2.262 | 2.258 | 2.254 | 2.235 | 2.230 | 2.226 | 2.204 | 2.198 | 2.193 |

Focal Length (mm) | RMSE (mm) | ||||||||
---|---|---|---|---|---|---|---|---|---|

P1 | P2 | P3 | P4 | P5 | P6 | P7 | P8 | P9 | |

8 | 0.044 | 0.017 | 0.012 | 0.059 | 0.015 | 0.026 | 0.052 | 0.051 | 0.014 |

50 | 0.007 | 0.024 | 0.025 | 0.021 | 0.009 | 0.002 | 0.010 | 0.006 | 0.014 |

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**MDPI and ACS Style**

Tian, L.; Ding, T.; Pan, B.
Generalized Scale Factor Calibration Method for an Off-Axis Digital Image Correlation-Based Video Deflectometer. *Sensors* **2022**, *22*, 10010.
https://doi.org/10.3390/s222410010

**AMA Style**

Tian L, Ding T, Pan B.
Generalized Scale Factor Calibration Method for an Off-Axis Digital Image Correlation-Based Video Deflectometer. *Sensors*. 2022; 22(24):10010.
https://doi.org/10.3390/s222410010

**Chicago/Turabian Style**

Tian, Long, Tong Ding, and Bing Pan.
2022. "Generalized Scale Factor Calibration Method for an Off-Axis Digital Image Correlation-Based Video Deflectometer" *Sensors* 22, no. 24: 10010.
https://doi.org/10.3390/s222410010