SCIENCE CHINA Information Sciences, Volume 64 , Issue 1 : 112204(2021) https://doi.org/10.1007/s11432-019-2690-0

## Homography-based camera pose estimation with known gravity direction for UAV navigation

• ReceivedApr 26, 2019
• AcceptedSep 27, 2019
• PublishedDec 14, 2020
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### Abstract

Relative pose estimation has become a fundamental and important problem in visual simultaneous localization and mapping. This paper statistically optimizes the solution for the homography-based relative pose estimation problem. Assuming a known gravity direction and a dominant ground plane, the homography representation in the normalized image plane enables a least squares pose estimation between two views. Furthermore, an iterative estimation method of the camera trajectory is developed for visual odometry. The accuracy and robustness of the proposed algorithm are experimentally tested on synthetic and real data in indoor and outdoor environments. Various metrics confirm the effectiveness of the proposed method in practical applications.

### Acknowledgment

This work was partially supported by National Natural Science Foundation of China (Grant Nos. 61603303, 61803309, 61703343), Natural Science Foundation of Shaanxi Province (Grant No. 2018JQ6070), China Postdoctoral Science Foundation (Grant No. 2018M633574), and Fundamental Research Funds for the Central Universities (Grant Nos. 3102019ZDHKY02, 3102018JCC003).

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• Figure 1

(Color online) Camera coordinate systems aligned with the ground plane (called the ground coordinate system). The detected 3D points lie on the ground plane.

• Figure 2

(Color online) Evaluation of the HLS, 2-point (2pt), and 5-point (5pt) methods during forward ((a) and (c)) and sideways ((b) and (d)) motions with varying image noise.

• Figure 3

(Color online) Evaluation of the HLS, HLS+GN, and 2-point methods during forward ((a)–(d)) and sideways ((e)–(h)) motions with increasing IMU noise from 0$~^{\circ}$ to 1$~^{\circ}$. The image noise is fixed at 0.5 pixels standard deviation.

• Figure 4

(Color online) Accuracy of our method in estimating the homography scale factor. We increase image noise from 0 to 1 shown in (a). Also, (b) and (c) report the effect of pitch/roll noise from 0$~^{\circ}$ to 1$~^{\circ}$, and the image noise is fixed at 0.5 pixels standard deviation.

• Figure 5

(Color online) A few matched feature points in the images taken from the trajectory estimated by the HLS method. (a) The 1st frame; (b) the 400th frame; (c) the 800th frame; (d) the 1200th frame.

• Figure 6

(Color online) Evaluating the trajectory drift errors via the RPE of the HLS method and the 2-point (2pt) method based on the ETH dataset.

• Figure 7

(Color online) Relationship between the estimated and true trajectories in outdoor experiment settings.

• Figure 8

(Color online) Experimental platform (a) and its environment (b): (1) UWB receiver, (2) downward-looking camera, (3) flight controller with IMU, magnetometer, and other components, (4) four UWB anchors, (5) Intel drone, and (6) start point.

• Figure 9

(Color online) Top views of the estimated and the true flight trajectories.

•

Algorithm 1 The proposed HLS method

Require:General corresponding measurements $\bar{X}_{i}$ and $\bar{X}_{j}$, and the known ${\boldsymbol~R}_{i&apos;i}$ and ${\boldsymbol~R}_{j&apos;j}$ at time $i$ and $j$ respectively.

Output: Rotation matrix ${{\boldsymbol~R}}_{ji}$ and translation vector ${{\boldsymbol~t}}_{ji}$.

Pre-rotate each measurement according to (4), and then normalize them to get $\bar{X}_{i&apos;}$ and $\bar{X}_{j&apos;}$;

Calculate the coefficient matrices ${\boldsymbol~A}_{i&apos;}$ and ${\boldsymbol~b}_{i&apos;}$ for all point correspondences according to (17);

Obtain the closed-form solution ${\boldsymbol~H}_{j&apos;i&apos;}$ according to (19);

Acquire the estimation result ${{\boldsymbol~R}}_{ji}$ and ${{\boldsymbol~t}}_{ji}$ according to (15);

Return ${{\boldsymbol~R}}_{ji}$ and ${{\boldsymbol~t}}_{ji}$.

•

Algorithm 2 Iterative camera trajectory estimation

Initialize $d_0$ and set ${\boldsymbol~R}_{00}={\boldsymbol~I},\,{\boldsymbol~t}_{00}=\mathbf{0}$;

$i=0$;

$j=i+1$;

Estimate ${\boldsymbol~R}_{ji}$ and ${\boldsymbol~t}_{ji}$ by Algorithm 1;

Updata camera pose by ${\boldsymbol~R}_{0j}~\leftarrow~{\boldsymbol~R}_{0i}~{\boldsymbol~R}_{ji}^{\rm~T}$ and ${\boldsymbol~t}_{0j}~\leftarrow~{\boldsymbol~t}_{0i}~-~{\boldsymbol~R}_{0i}{\boldsymbol~R}_{ji}^{\rm{T}}{\boldsymbol~t}_{ji}$;

Update $d_j$ according to (22);

Return camera trajectory $\left\{~{\left[{\boldsymbol~R}_{0j}|{\boldsymbol~t}_{0j}\right]}\right\}$;

Iterate time sequence $i~\leftarrow~i+1$;

Repeat from step 3 to step 8.

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