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Rotation Estimation: A Closed-Form Solution Using Spherical Moments^{ †}

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## Abstract

**:**

## 1. Introduction

## 2. Direct Estimation of Rotation Using Spherical Moments

#### 2.1. Spherical Moments

#### 2.2. Closed-Form Solution of Rotation Estimation

- Define the moment vector $\mathbf{w}$ similarly as for ${\mathbf{w}}_{\mathbf{23}}$. The vector can be built by moment products of two different orders or more. For instance, products such as ${m}_{200}{m}_{220}{m}_{300}$ which combine moments of order 2, 3, and 4 can be also used. the vector $\mathbf{w}$ has to be built from moment products of the same nature and has to also include all of them.
- Using Equation (4), compute the matrices ${\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{x}}$, ${\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{y}}$, and ${\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{z}}$ such that we obtain the following:$$\dot{\mathbf{w}}=\left({\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{x}}\phantom{\rule{0.166667em}{0ex}}\mathbf{w}\right){\omega}_{x}+\left({\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{y}}\phantom{\rule{0.166667em}{0ex}}\mathbf{w}\right){\omega}_{y}+\left({\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{z}}\phantom{\rule{0.166667em}{0ex}}\mathbf{w}\right){\omega}_{z}.$$
- Solve the following system:$$\left[\begin{array}{ccc}{\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{x}}^{\top}& \mathbf{0}& \mathbf{0}\\ {\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{y}}^{\top}& \mathbf{0}& \mathbf{I}\\ {\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{z}}^{\top}& -\mathbf{I}& \mathbf{0}\\ \vdots & \vdots & \vdots \\ \mathbf{0}& \mathbf{0}& {\mathbf{L}}_{{\mathbf{w}}_{\mathbf{23}}/{\omega}_{z}}^{\top}\end{array}\right]\left[\begin{array}{c}{\mathit{\alpha}}_{\mathbf{x}}\\ {\mathit{\alpha}}_{\mathbf{y}}\\ {\mathit{\alpha}}_{\mathbf{z}}\end{array}\right]=\mathbf{0},$$

#### 2.3. Rotation Estimation and Scene Symmetry

## 3. Validation Results

#### 3.1. Simulation Results

- To show the validity of the proposed method to cameras obeying unified model, two different kinds of camera model are used to compute the images. The first corresponds to a simulated fisheye camera with the parameters chosen as focal scaling factors $Fx=Fy=960$ pixels, coordinates of the principal point ${u}_{x}=240$ and ${u}_{y}=320$ pixels, and distortion parameter $\xi =1.6$. The second model corresponds a conventional camera with focal scaling factors $Fx=Fy=600$ pixels and coordinates of the principal point ${u}_{x}=240$ and ${u}_{y}=320$;
- To Validate our approach for large rotational motion;
- To test the effect of translational motion on the accuracy of the estimated rotations.

#### 3.2. Real Experiments

#### 3.3. Discussion

## 4. Conclusions

## Supplementary Materials

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

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**Figure 1.**Results using a fisheye camera model: (

**a**) the reference image and (

**b**) example of rotated images.

**Figure 2.**Simulation results using a fisheye camera model: (

**a**) Estimation error of the rotation and (

**b**) rotation ground truth expressed by Euler angles.

**Figure 3.**Results using conventional camera model: (

**a**) the reference image and (

**b**) example of the rotated images.

**Figure 4.**Simulation results using a perspective camera model: (

**a**) Estimation error of the rotation and (

**b**) rotation ground truth expressed.

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**MDPI and ACS Style**

Hadj-Abdelkader, H.; Tahri, O.; Benseddik, H.-E. Rotation Estimation: A Closed-Form Solution Using Spherical Moments. *Sensors* **2019**, *19*, 4958.
https://doi.org/10.3390/s19224958

**AMA Style**

Hadj-Abdelkader H, Tahri O, Benseddik H-E. Rotation Estimation: A Closed-Form Solution Using Spherical Moments. *Sensors*. 2019; 19(22):4958.
https://doi.org/10.3390/s19224958

**Chicago/Turabian Style**

Hadj-Abdelkader, Hicham, Omar Tahri, and Houssem-Eddine Benseddik. 2019. "Rotation Estimation: A Closed-Form Solution Using Spherical Moments" *Sensors* 19, no. 22: 4958.
https://doi.org/10.3390/s19224958