1. Introduction

2. Related Work

3. Methods

3.1. Dynamic Object Classification

Tracking static features is the key for SLAM systems to maintain localization accuracy in dynamic environments. However, in most scenes, dynamic objects are movabel instead moving, which is the drawback of pure deep learning based methods. The boundary between dynamic and static objects is not clear for deep learning. For instance, books are commonly considered as static objects. But when a person holding a book, the book becomes a dynamic object, which is the same as a car in a parking lot or running in the highway. Therefore, semantic information is not enough to assist robots to detect the moving objects in dynamic environment. In D-VINS, we propose a classification method for obtained semantic labels.

3.1.1. Semantic Label Incremental Updating with Bayes’ Rule

In order to recognize most objects in life, we selected COCO dataset to training the YOLOV5, which contains 80 categories of common objects. Not all objects need to be detected. So we selected 17 most common categories. Firstly, color images are input to the YOLOV5 with TensorRT [30] accelerating to obtain semantic labels of COCO categories. Bounding boxes can locate the approximate region of a dynamic object in an image. The feature points inside the bounding boxes will be given its semantic labels. Since object detection network can only detect semantic information of the current frame and exists missing or incorrect detection problem,D-VINS updates features’ semantic label according to Bayes’ rule to to avoid the error in a certain frame, which transforms the labeling problem into a maximum a posteriori problem.

The $m$th map point in a given world coordinates$W$ is observed by$k$th frame can be written as $P_k^W(x,y,z,1)$. $p_m^k(u,v,1)$ denotes the pixels in camera coordinates that corresponds to the $m$th map point. The projection process of feature points is as follows:

$$s{p}_m^k=KT{P}_k^W.$$

(1)

Where,$s$is the depth of map points and $K$ is the extrinsic matrix. $T$ denotes the transformation matrix from the world coordinates to the observation frame. Denote $l_{true}=\left\{l_i^k|k\in \left(\mathrm{1,2},3,...,N\right),i\in \left(\mathrm{1,2},3,...,\mathrm{M}\right)\right\}$ and $b_{observe}=\left\{b_i^k|k\in \left(\mathrm{1,2},3,...,N\right),i\in \left(\mathrm{1,2},3,...,\mathrm{M}\right)\right\}$ as the ground truth of the semantic labels of the $m$th feature points from the beginning frame to the $k$th frame and the measurements of deep learning bounding boxes. According to Bayes’ rule, there is:

$$P\left(l_{true}|b_{observe}\right)=\frac{P\left(b_{observe}|l_{true}\right)P\left(l_{true}\right)}{P\left(b_{observe}\right)}\propto P\left(b_{observe}|l_{true}\right)P\left(l_{true}\right)$$

(2)

In fact, this is a Maximize a Posterior problem, shown as:

$${\mathit{argmaxl}}_{\mathit{true}}=\mathit{argmax}P\left(l_{true}|b_{observe}\right)=\mathit{argmax}P\left(b_{observe}|l_{true}\right)P\left(l_{true}\right)$$

(3)

When dynamic objects are detected in previous frames, the same result should also be obtained by the current frame. So the semantic labels of feature points are affected by multi-frame in the past. The semantic label probability distribution of the$m$th feature point in$k$th frame is:

$$P\left(l_k^m|b_{1:k}\right)=P\left(l_k^m|l_{k-1}^m,b_{1:k-1}\right)...P\left(l_2^m|l_1^m,b_{1:2}\right)P\left(l_1^m|b_1\right)=\prod _{\mathrm{i}=1}^{\mathrm{k}}P\left(l_i^m|l_{i-1}^m,b_i\right)$$

(4)

When the semantic information of the current frame is obtained, D-VINS will determine whether it is consistent with the previous frames. If the previous semantic label is same as that in the current frame, the detection resut of the current frame is more trusted, and vice versa. The specific algorithm steps are shown in Algorithm 1.

3.1.2. Feature Points Motion State Classification

The feature points classification is divided into two parts. The first part is to classify COCO categories according to the possibility of motion based on life experience. The classification is divided into 1~5 levels. the higher the level the higher the possibility of movement. Those five levels are level I public facilities (traffic light, bench), Level II furniture (chair, sofa, bed), Level III transportation (bicycle, car, motorbike, bus, truck, boat), Level IV sports (football, basketball), people (person) and Level V animals (cat, dog, bird), seen in the Figure 2.

The second part is to classify the feature points into three categories according to the movement in their current frame, which can be divided into absolute stationary points, temporary stationary points and absolute dynamic points. In general, if one object is detected as bench, sofa or potted plant, its feature points are most likely to be a static, which can be involved in the pose estimation and mapping. If the semantic label is animal such as birds, cats or dogs, those features are considered to be dynamic features. Animals usually keep moving and occupy a small area in an image, which have less impact on the SLAM system. As shown in Figure 2, objects of level I and level II are considered as absolute static objects, objects of level III and level IV are temporary static objects and objects of level V are absolute dynamic objects. For temporary static objects, their motion state cannot be determined by prior semantics from deep learning. If such feature points occupy a large area in the image, eliminating all of them will impair the localization accuracy due to insufficient feature points for tracking.

3.2. Features Dynamics Check with IMU Prior and Epipolar Constraint

In order to check current motion state of movable objects, D-VINS calculate dynamic factors to find the absolute dynamic points. For absolute static objects, like furniture, traffic lights and benches, their feature points are kept for all process of SLAM system. For absolute dynamic points like animals, all those feature points are removed and do not participate BA optimization or global pose optimization. For temporary static points, the first part of dynamic factors is calculated by IMU pre-integration, which is utilized as the initial pose of current frame to obtain the reprojection error by projecting the 3D feature points onto the image plane. According to eipipolar geometry, the foundational matrix is calculated to obtain the distance from feature points to their epipolar lines, which is the second part of dynamic factor. If one of the dynamic factors of the feature point is lower than a certain threshold, the point will be labeled as absolute dynamic points.

3.2.1. Dynamic Factor of Reprojection Error Based on IMU Prior Constraint

Conventional visual reprojection projects a feature point from its previous observed frame to the pixels plane of current frame. The reprojection error cannot be calculated if camera pose of current frame is unknown. IMU preintegration provides an initial estimate for current frame’s camera pose, which enables calculating reprojection error to reject dynamic objects with the IMU sensor.

There must be errors between the estimation pose and real camera pose, which means the reprojection points and the observation points are usually not coincident. For map point in world coordinate $\mathit{P}={\left[X_p,Y_p,Z_p,1\right]}^T$, its pixel coordinates projected on the jt frame is ${\mathit{p}}^{C_j}={\left[u^j,v^j,1\right]}^{\mathrm{T}}$. According to Equation (1), the relationship between map points and pixel points according to the camera projection model exists as follows:

$$s\left[\begin{array}{c}{u}^j\\ v^j\\ 1\end{array}\right]=\mathit{K}\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathit{\xi }}^{\wedge }\right)\left[\begin{array}{c}{X}_p\\ Y_p\\ Z_p\\ 1\end{array}\right]$$

(5)

Equation (5) can be written in matrix form ${\mathit{p}}^{{\mathit{C}}_{\mathit{j}}}=\mathit{K}\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathit{\xi }}^{\wedge }\right)\mathit{P}$ and the resisual $r_C$of the reprojection error as follows:

$$r_C={∥{\mathit{p}}^{{\mathit{C}}_{\mathit{j}}}-\frac{1}{s}\mathit{K}\mathrm{e}\mathrm{x}\mathrm{p}\left({\mathit{\xi }}^{\wedge }\right)\mathit{P}∥}_2^2$$

(6)

${\mathit{\xi }}^{\wedge }$is the Lie algebra of the $j$th frame in body frame and $\mathit{K}$ is the intrinsic matrix obtained by camera calibration[31]. The camera pose of$j$th frame is obtained by IMU preintegration:

$$\begin{array}{c}{\mathbf{R}}_w^{b_{j-1}}{\mathbf{p}}_{b_j}^w={\mathbf{R}}_w^{b_{j-1}}\left({\mathbf{p}}_{b_{j-1}}^w+{\mathbf{v}}_{b_{j-1}}^w\mathsf{\Delta }\mathrm{t}-\frac{1}{2}{\mathbf{g}}^w{\mathsf{\Delta }\mathrm{t}}^2\right)+{\mathit{\alpha }}_{b_j}^{b_{j-1}}\\ {\mathbf{R}}_w^{b_{j-1}}{\mathbf{v}}_{b_j}^w={\mathbf{R}}_w^{b_{j-1}}\left({\mathbf{v}}_{b_{j-1}}^w-{\mathbf{g}}^w\mathsf{\Delta }\mathrm{t}\right)+{\mathit{\beta }}_{b_j}^{b_{j-1}}\\ {\mathbf{q}}_w^{b_{j-1}}\otimes {\mathbf{q}}_{b_j}^w\&={\mathit{\gamma }}_{b_j}^{b_{j-1}}\end{array}$$

(7)

where, ${\mathit{\alpha }}_{b_j}^{b_{j-1}}$, ${\mathit{\beta }}_{b_j}^{b_{j-1}}$ and ${\mathit{\gamma }}_{b_j}^{b_{j-1}}$ are the pre-integration terms of position, velocity, and pose, which changes the reference frame from the world frame to the local body frame${ b}_{j-1}$; ${\mathbf{p}}_{b_j}^w$, ${\mathbf{v}}_{b_j}^w$ and ${\mathbf{q}}_{b_j}^w$ are system state the state of the jth body frame. From equation (7), the pose of jth frame is obtained. The pixel coordinates in jth frame projected from ith frame is ${\mathit{P}}^{C_j}={\left[u_i^j,v_i^j,1\right]}^T$. And the observation in jth frame is ${\overline{\mathit{P}}}^{C_j}={\left[{\widehat{u}}_i^j,{\widehat{v}}_i^j,1\right]}^T$. By equation (6), the new visual reprojection resisual$r_{project}\left(\mathit{P}\right)$of map point $\mathit{P}$ can be established by camera projection model:

$$\left\{\begin{array}{c}{\mathit{P}}^{C_j}=\pi \left({\mathbf{T}}_b^c{\mathbf{T}}_w^{b_j}{\mathbf{T}}_{b_i}^w{\mathbf{T}}_c^b{P}^{C_j}\right)\\ {\mathit{P}}^{C_i}=\pi \left({\mathbf{T}}_b^c{\mathbf{T}}_w^{b_i}{\mathbf{T}}_{b_j}^w{\mathbf{T}}_c^b{P}^{C_i}\right)\\ r_{project}\left(\mathit{P}\right)={‖{\overline{\mathit{P}}}^{C_j}-{\mathit{P}}^{C_j}‖}_2^2+{‖{\overline{\mathit{P}}}^{C_i}-{\mathit{P}}^{C_i}‖}_2^2\end{array}\right.$$

(8)

where, ${\mathbf{T}}_b^c$ is the transformation matrix from the body frame to camera frame, which is obtained by Kalibr[32]. ${\mathbf{T}}_w^{b_j}$ and ${\mathbf{T}}_{b_i}^w$ represents the transformation matrix between imu frames and the world coordinate.$p_{b_i}^w$ and $p_{b_i}^w$ are translation matrix between body frame and world frame. ${\lambda }_l$ represents the inverse depth of feature point P.$\pi \left(·\right)$represents the pinhole camera projection model.

As shown in the Figure 3, the distance in red denotes the dynamic factor of IMU reprojection error,which is utilized to evaluate how far the object is away from the main optical axis. It shows the observation and projection of the static map point P and the dynamic point P’ in two camera frames. O denotes camera’s optical center. $x_1$ and $x_4$ are feature points matched for the two frames with optical flow. $x_2$ is the feature point projected by the static point P in jth camera frame. $x_3$ is the feature point projected by the dynamic point P’ in j-1th camera frame.

Generally, it is effective to determining the dynamics of the object by feature points reprojection error. However, this method will fail when the dynamic object is moving along the camera’s optical center ( either toward or away from the camera), which is shown in Figure 3. The reprojection error is close to 0 even P is not a dynamic point. Therefore, we propose additional reprojection error on the previous frame to the conventional visual reprojection in equation (8). Even the point is moving along the optical center, there will be at least one reprojection error is not close to 0. So the two reprojection process are complementary to each other, which minimizes the effect on the dynamic judgment of feature points with the special object motion direction. Then the first part of the dynamic factors ${\lambda }_p$is obtain as below:

$${\lambda }_p=r_{project}\left(P\right)$$

(9)

3.2.1. Dynamic Factor of Epipolar Constraint

Epipolar constraint is a critical property to limit the position of feature points, which is frequently utilized to accelerate the matching process at the front end in various SLAM systems. In D-VINS, the data association between the feature points is obtained by pyramidal iterative Lucas-Kanade optical flow. Then, the seven-point method based on RANSAC is used to calculate the fundamental matrix between two camera frames. The epipolar lines of feature points in the current frame are calculated with the fundamental matrix. The distance from a point to its epipolar line is defined as the second part of dynamic factor. Finally, the distance is used to determine whether the point is dynamic or not. According to the pinhole camera model, the map point P is observed by different camera frames, which is shown in Figure 4. x₁ and x₃ are matched feature points in different frames, and x₂ is the feature point projected to the jth frame by the map point P. The short dashed lines I and I’ are the epipolar lines of the two frames.

${\mathit{x}}_1=\left[u_1,v_1,1\right], {\mathit{x}}_2=[u_2,v_2,1]$ are the homogeneous coordinate forms of the two matched feature points, belonging to the j-1th frame and jth frame, respectively. Then, the epipolar line $I^{’}$of $x_2$ in the jth frame is as follows:

$${\mathit{I}}^{\mathit{’}}=\left[\begin{array}{c}X\\ Y\\ Z\end{array}\right]=\mathit{F}{{\mathit{x}}_1}^T=\mathit{F}\left[\begin{array}{c}{u}_1\\ v_1\\ 1\end{array}\right]$$

(10)

Where $X$,$Y$and$Z$denote the real constants in general form of a straight line(Xu+Yv+Z = 0). F denotes the fundamental matrix. Then, for feature point $x_2$, the epipolar constraint is as follows:

$${\mathit{x}}_2\mathit{F}{{\mathit{x}}_1}^T={\mathit{x}}_2{\mathit{I}}^{\mathit{’}}=\mathbf{0}$$

(11)

In Figure 4, the distance from the point to the epipolar line is marked by the blue line. For the matched feature point $x_i(i=\mathrm{2,3})$ of $x_1$, the residual of epipolar constraint $r_{epipolar}\left(P\right)$ can be described as follows:

$$r_{epipolar}\left(\mathit{P}\right)=\frac{\left|{\mathit{x}}_{\mathit{i}}\mathit{F}{{\mathit{x}}_1}^T\right|}{\sqrt{{‖X‖}^2+{‖Y‖}^2}}$$

(12)

Then the second part of the dynamic factor is obtain as below:

$${\lambda }_e=r_{epipolar}\left(\mathit{P}\right)$$

(13)

For the features of static objects, $r_{epipolar}\left(\mathit{P}\right)$ should be 0 or close to 0. But for the features of dynamic objects, like P’, there is an offset between the real pixel coordinates and its observatoin. But when the feature point moves toward the optical center of j-1 frame, the feature point is still on its epipolar line. So it is hard to determine whether the object is in motion or not. Therefore, when defining whether features are in moving state, it needs to combine the two distances of ${\lambda }_p$and${\lambda }_e$.

The threshold ${\epsilon }_{reproject}$ of the reprojection dynamic factor is set to 4 pixels and ${\epsilon }_{epipolar}$ of the epipolar dynamic factor is set to 3 pixels. If the errors exceeds those thresholds then the feature points are considered as absolute dynamic points and rejected. Then the feature is marked as ADP. So far, the two dynamic factors ${\lambda }_p$ and ${\lambda }_e$ are obtained.

This method enables a more accurate classification of temporary static objects and finds the dynamic feature points. In addition, the feature points of dynamic objects with small movements can be fully utilized by the SLAM system. The specific algorithm steps are shown in Algorithm 2.

4. Experimental Results

5. Discussion

6. Conclusions

References

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