# Visual Calibration for Multiview Laser Doppler Speed Sensing

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## Abstract

**:**

## 1. Introduction

## 2. MLDSS Parameters

#### Basic Equations

## 3. MLDSS Calibration

#### 3.1. Geometric-Only Calibration

#### 3.2. Statistical Calibration by Minimizing Motion-Reconstruction Error

#### 3.3. System Setup and Nonideal Factors

**Temporal misalignment.**As we described above, the 3D tracking system runs slower than the MLDSS system so that they are temporally misaligned. Using f to denote the frame index of MLDSS, and ${t}_{f}$ to denote the corresponding time, we align the MLDSS measurements to the 3D tracking system. Equation (5) is thus reformulated by linearized integration and interpolation:

**Mechanical velocity offset.**A galvanometer scanner is used for controlling laser direction. While we assumed the scanner fully stopped when sampling the LDV measurement, the scanner could possibly slightly oscillate and cause error in the speed. Because the control sequence of the galvanometer is fixed and the optical path of each laser is different, this offset can be regarded as systematic to each laser. We thus refined speed reconstruction as

**Weighing error terms.**In solving the multiobject optimization problem denoted by Equation (6), it is important to balance the influence of each error term and data samples in order to not let partial data be dominant. Here, we introduce a diagonal weight matrix W. Error term ${E}_{k}$ for each sample k is thus:

**Outlier Measurements.**Speckle noise [10] is one of the main noise sources of the LDV. It causes undesired short-term peaks in velocity measurements. Such noise can largely be relieved in the low-frequency domain with the use of a tracking filter [1,9]. However, considering signal intensity is low due to potential loss in the laser focus, there occasionally is speckle noise in measurements. In order to reduce its influence on calibration accuracy, a modified Huber kernel [25] is introduced to the problem:

**Complete maximum likelihood estimation.**With the above consideration, we can naturally extend Equation (6) by estimating the complete set of parameters:

#### 3.4. Summary

- Make a calibration object with four or more asymmetric placed markers;
- register the calibration object in the 3D tracking system as a trackable rigid body;
- simultaneously capture target motion and speed with the MLDSS and the 3D tracking system, and build a dataset following the instructions in Section 4;
- estimate initial parameters using the geometric-only method introduced in Section 3.1 (optional); and
- refine all parameters by solving Equation (13).

## 4. Data Collection

## 5. Evaluation

#### 5.1. Cross-Validation

#### 5.2. Sensing Daily Object

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**System to be calibrated and its parameters. Laser beams emitted from the laser Doppler velocimeter (LDV) are reflected by the galvanometer scanner and mirror array MR1~MR4, and finally hit a moving-target object. By scanning the laser on the target surface, the six-degree-of-freedom (6-DOF) speed of the target could be reconstructed. Parameters consist of laser geometry, denoted by the x and y co-ordinates on the z = 0 plane, and the laser-direction angles.

**Figure 3.**Experimental setup. Photograph of the system and the experimental condition. MR1~MR4 are the mirrors, and GS is the galvanometer scanner. IR1~IR3 are IR illumination sources and cameras. XI is a monochrome camera. Calibration object was moved manually 1 m away from the system.

**Figure 4.**(

**a**) Calibration object, a white board with ten randomly but asymmetrically placed retroreflective markers. (

**b**) Six motion patterns of the calibration object in its local co-ordinate system, of which the calibration data are composed.

**Figure 5.**Motion reconstruction results of the test set. Specifically, this figure includes the rotational components (top three subfigures) and translational components (bottom three subfigures) of the interframe motion increments in the test set. It was confirmed that the proposed method obviously outperforms the geometric-only method.

**Figure 6.**(

**a**) Used test globe. Retroreflective markers (red-circled) were attached on the globe to track its rotation. (

**b**) Motion-reconstruction results (rotational part).

Motion Pattern | ${\parallel \mathit{\partial}\mathit{\omega}/\mathit{\partial}\mathit{a}\parallel}_{\mathit{\infty}}$ | ${\parallel \mathit{\partial}\mathit{\omega}/\mathit{\partial}\mathit{o}\parallel}_{\mathit{\infty}}$ | ${\parallel \mathit{\partial}\mathit{v}/\mathit{\partial}\mathit{a}\parallel}_{\mathit{\infty}}$ | ${\parallel \mathit{\partial}\mathit{v}/\mathit{\partial}\mathit{o}\parallel}_{\mathit{\infty}}$ |
---|---|---|---|---|

$Subtle$ | 0.0036 | 0.0012 | 0.280 | 0.2201 |

$Rot$ | 0.0088 | 0.0424 | 0.501 | 3.39 |

$Trans$ | 1.31 | 0 | 99.2 | 0 |

Test Set | Rotational (Rad) | Translational (mm) | ||
---|---|---|---|---|

Proposed | Geo | Proposed | Geo | |

$Ro{t}_{x}$ | 0.0139 | 0.0484 | 2.3142 | 9.0864 |

$Ro{t}_{y}$ | 0.0108 | 0.0321 | 2.0543 | 4.9750 |

$Ro{t}_{z}$ | 0.0239 | 0.0321 | 3.8263 | 7.6578 |

$Tran{s}_{x}$ | 0.0101 | 0.0089 | 3.1751 | 3.4187 |

$Tran{s}_{y}$ | 0.0077 | 0.0220 | 1.9044 | 3.8176 |

$Tran{s}_{z}$ | 0.0054 | 0.0075 | 1.6014 | 2.7850 |

$Total$ | 0.0133 | 0.0290 | 2.5977 | 5.7731 |

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**MDPI and ACS Style**

Hu, Y.; Miyashita, L.; Watanabe, Y.; Ishikawa, M.
Visual Calibration for Multiview Laser Doppler Speed Sensing. *Sensors* **2019**, *19*, 582.
https://doi.org/10.3390/s19030582

**AMA Style**

Hu Y, Miyashita L, Watanabe Y, Ishikawa M.
Visual Calibration for Multiview Laser Doppler Speed Sensing. *Sensors*. 2019; 19(3):582.
https://doi.org/10.3390/s19030582

**Chicago/Turabian Style**

Hu, Yunpu, Leo Miyashita, Yoshihiro Watanabe, and Masatoshi Ishikawa.
2019. "Visual Calibration for Multiview Laser Doppler Speed Sensing" *Sensors* 19, no. 3: 582.
https://doi.org/10.3390/s19030582