# Reconstruction of the Image Metric of Periodic Structures in an Opto-Digital Angle Measurement System

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

- -
- the mutual inclination of the mark, the lens and the photodetector matrix,
- -
- lens distortion,
- -
- the manufacturing error of the mark.

## 2. 2D-Optical Pattern

^{×}(1

^{×}, 40 mm WD CompactTL™ Telecentric Lens, Edmund Optics product). The position of the 2D pattern perpendicular to the optical axis was adjusted using an autocollimator and structural elements.

## 3. Theory

_{x}= H cos φ, and a

_{y}= H sin φ. Then:

_{x}, b

_{y}—the shift of the rotation axis from the spot (0,0), ${a}_{x}=H\mathrm{cos}\phi ,{a}_{y}=H\mathrm{sin}\phi $.

^{×}, so there is no difference between the step of the pattern and the step of its image. Equations (2) contain 4 unknown coefficients ${a}_{x},{a}_{y},{b}_{x},{b}_{y}$, therefore two spots are enough for coefficients determination. On the other hand, each pair of spots provides its own coefficient values due to errors in coefficient determination. A suitable approach in this case is to consider an overdetermined system of 2N equations of the form (2), where N is the number of spots, and its solution by the least squares method.

- (a)
- errors in determination of the coordinates of the spot on the camera sensor;
- (b)
- the presence of lens aberrations;
- (c)
- deviations from the parallelism of the planes of the pattern and the camera sensor;
- (d)
- differences between the actual spots positions and the ideal grid.

## 4. Experiment

## 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.**2D-optical pattern. The position of three emphasized elements provides the non-equivocal determination of the pattern orientation.

**Figure 3.**The structure of vectors that display the displacement of the elements of the pattern image relative to the ideal grid (enlarged 200 times compared to the scale of the pattern).

**Figure 4.**Modulus of shift vectors’ length for various points of the sensor. Different colors correspond to different ranges of length, measured in 1 tenth of the pixel size.

**Figure 5.**Plot of the central section of the two-dimensional distortion function of the 5th order, calculated from a series of 12 frames.

**Figure 7.**Image of the vector field after performing the correction using distortion and pattern displacements correction data (enlarged 200 times compared to the scale of the pattern).

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**MDPI and ACS Style**

Korolev, A.N.; Lukin, A.Y.; Filatov, Y.V.; Venediktov, V.Y.
Reconstruction of the Image Metric of Periodic Structures in an Opto-Digital Angle Measurement System. *Sensors* **2021**, *21*, 4411.
https://doi.org/10.3390/s21134411

**AMA Style**

Korolev AN, Lukin AY, Filatov YV, Venediktov VY.
Reconstruction of the Image Metric of Periodic Structures in an Opto-Digital Angle Measurement System. *Sensors*. 2021; 21(13):4411.
https://doi.org/10.3390/s21134411

**Chicago/Turabian Style**

Korolev, Alexander N., Alexander Ya. Lukin, Yurii V. Filatov, and Vladimir Yu. Venediktov.
2021. "Reconstruction of the Image Metric of Periodic Structures in an Opto-Digital Angle Measurement System" *Sensors* 21, no. 13: 4411.
https://doi.org/10.3390/s21134411