Fig. 3 The results of the line extraction in one segment. The green lines represent the ground in the segment, and the numerals are the gradients and interceptions of the corresponding lines (a). The pentacles are the length-scales at the input locations, which are processed by an average filter $(a=1)$ (b)
will not be too many seeds in each segment. The small size of the covariance function increases the computational efficiency greatly.
The new seed evaluation is corresponding to the function eval in line 10. Equations 7 and 11 can be used to calculate $K(r_*, r_*$ and $K(r_*, s_p)$ for every test point $(r_*, z_*$.) Equation 10 can then be used to calculate the mean and variance of the output $z_*$ at the test input location $r_*$. In this paper, the rulers (in Eq. 13) proposed by Douillard [^5] are adopted to evaluate the attribute of the input points: the new seeds or outliers. The parameter $t_{\text{model}}$ is the threshold of the variance $V[z]$ of the output $z_*$, while $t_{\text{data}}$ specifies the normalized distance between the real output $z_*$ and the expected mean $\bar{z}$ of the output at the test input $r_*$.
$$ \begin{array}{r l}{V[z]\leq t_{\mathrm{m o d e l}}}&{\frac{|z_{*}-\bar{z}|}{\sqrt{\sigma_{n}^{2}+V[z]}}\leq t_{\mathrm{d a t a}}.}\end{array} $$
For the test point $(r_*, z_*)$, it is classified as the new seed if and only if it satisfies both of the inequalities in Eq. 13. Otherwise, it is an outlier. Starting from the initial seeds (line 5), the seeds are accumulated per iteration until no more new seeds are found (lines 6–11). Figure 4 shows the results of the iterations in the first segment of the scene shown in Fig. 10.
Fig. 4 Cross section of a segment in a polar grid map. The blue points are the likely ground points, while the red circles represent the seeds after per iteration. The point (the pentacle in the bottom-right figure) that is the return of a wire across the road is classified as the outlier
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The final step is the point-wise segmentation, which is corresponding to the function segment in line 13. For segment i, the final seeds, which are found through the iterations above, can be used to model the local ground. For the jth bin $b_j^i$, in segment i, $(r_j^{\min} + r_j^{\max})/2$ is used as the radial coordinate of bin $b_j^i$. Thus, for $r_j^i = (r_j^{\min} + r_j^{\max})/2$, Eq. 10 can be used to predict the mean height of the ground $z_j^i$ in bin $b_j^i$. This means ground height $z_j^i$ of bin $b_j^i$ is defined as the reference ground height $H_{ij}$. For every point $p_k^i$ in the set $P_{b_j^i}' = \{p_k' = (r_k, z_k)| p_k \in P_{b_j^i}\}$, if its relative height $|z_k - H_{ij}|$ is less than $T_r$, $p_k^i$ is classified as a ground point. Otherwise, it is an obstacle point. Since the point-to-point mapping between set $P_{b_j^i}$ and set $P_{b_j^i}'$ is one-to-one, the 3D point $p_k$ and the 2D point $p_k^i$ have the same label. Now, all the labels of the 3D points in the scan can be obtained.
### 3.3 Learning Hyper-Parameters
So far, we have described our method assuming that we had known the hyper-parameters $\theta = \{\sigma_f, a, \sigma_n\}$. However, in practice, they cannot be obtained in advance. In this section, we formulate the regression task in every segment as a supervised learning problem, and focus on how to learn these parameters.
Given a training set $\mathfrak{I} = \{r^m, z^m\}_{m=1}^M$, each segment is a training example of the form $(r^m, z^m)$, where each $r^m = \{r_1^m, \ldots, r_{n_m}^m\}$ is a sequence of $n_m$ radial coordinates in segment m, each $z^m = \{z_1^m, \ldots, z_{n_m}^m\}$ is a corresponding sequence of the heights of the ground, and $n_m$ is the number of points in segment m. In this paper, we assume that these examples are conditionally independent, and try to seek these hyper-parameters by maximizing the pseudo log marginal likelihood [^24]:
$$ \begin{align*}\sum_{m=1}^{M}\log p(z^{m}|r^{m},\theta)&=-\frac{1}{2}\sum_{m=1}^{M}(z^{m})^{T}K_{m}^{-1}z^{m}\\&\quad-\frac{1}{2}\log\left(\prod_{m=1}^{M}|K_{m}|\right)\\&\quad-\frac{\log2\pi}{2}\sum_{m=1}^{M}n_{m},\end{align*} $$
where $K_{m}$ is the covariance matrix for the noisy $z^{m}$ values in segment m, and can be calculated using Eq.7.
To set the hyper-parameters by maximizing Eq. 14, we seek the partial derivatives of the pseudo log marginal likelihood with respect to the hyper-parameters. We obtain:
$$ \begin{align*}\frac{\partial}{\partial\theta_{i}}&\left(\sum_{m=1}^{M}\log p(z^{m}|r^{m},\theta)\right)\\&=\frac{1}{2}\sum_{m=1}^{M}tr\left(\left(\alpha_{m}\alpha_{m}^{T}-K_{m}^{-1}\right)\frac{\partial K_{m}}{\partial\theta_{i}}\right),\end{align*} $$
where $tr()$ represents the trace of the matrix, $\alpha_m = K_m^{-1} z_m$, and $\frac{\partial K_m}{\partial \theta_i}$ is a matrix of elementwise derivatives. The specific formulas of $\frac{\partial K_m}{\partial \theta_i}$ are as follows:
$$ \begin{aligned}\frac{\partial K_{m}(r_{i},r_{j})}{\partial\sigma_{f}}&=2\sigma_{f}\left(d_{i}^{2}\right)^{\frac{1}{4}}\left(d_{j}^{2}\right)^{\frac{1}{4}}\left(\frac{d_{i}^{2}+d_{j}^{2}}{2}\right)^{-\frac{1}{2}}\\&\quad\times\exp\left(-\frac{2(r_{i}-r_{j})^{2}}{a^{2}(d_{i}^{2}+d_{j}^{2})}\right)\end{aligned} $$
$$ \begin{align*}\frac{\partial K_{m}(r_{i},r_{j})}{\partial a}&=\sigma_{f}^{2}\big(d_{i}^{2}\big)^{\frac{1}{4}}\big(d_{j}^{2}\big)^{\frac{1}{4}}\left(\frac{d_{i}^{2}+d_{j}^{2}}{2}\right)^{-\frac{1}{2}}\\&\quad\times\exp\left(-\frac{2(r_{i}-r_{j})^{2}}{a^{2}\big(d_{i}^{2}+d_{j}^{2}\big)}\right)\\&\quad\times\left(\frac{4(r_{i}-r_{j})^{2}}{a^{3}\big(d_{i}^{2}+d_{j}^{2}\big)}\right)\end{align*} $$
$$ \frac{\partial K_{m}(r_{i},r_{j})}{\partial\sigma_{n}}=2\sigma_{n}\delta_{ij}, $$
where $d_i = \lg \left( \frac{1}{|g(r_i)|} \right)$. A gradient based optimizer can be adopted to solve the problem.
In this paper, ten labelled scans of street scenes collected in an urban area of Boston [^16] and some manually labelled scans are used to learn the hyper-parameters. Every scan is divided into 180 segments. Some typical segments such as the uphill roads, downhill roads, flat roads, rough roads and so on are chosen to form the training set $\mathfrak{I}$. Figure 5 shows 9 typical segments in the training set $\mathfrak{I}$.
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Fig. 5 9 typical segments chosen from the training set. The blue pluses are training points in the segments. The red lines represent the terrains in the segments with the learnt parameters $\sigma_{f}^{2}=0.0528$, $\sigma_{n}^{2}=0.0012$, a=6.2978)
### 3.4 Segmentation of Obstacle Points Based on Ground
Having labelled all points of a scan, both the obstacles such as vehicles, pedestrians, stumps on the road and the overhanging structures such as tree canopy and wires are classified as obstacles that an ALV cannot travel across. This is a typical under-segmentation problem. However, with the method in this paper, it can be easily solved. Since local path planning of our ALV only depends on a 2.5D grid map, we map the 3D obstacle points in the polar grid map into the grid map with a resolution of 0.2 meter (the reason we use 0.2 meter is that the width of the tyres of our ALV is about 0.2 meter). Symbol $P_{ij}$ is used to represent the set of the obstacle points that are mapped into the grid cell $c_{ij}$.
$$ \begin{aligned}P_{ij}=\left\{p_{k}|i\leq\frac{x_{k}}{0.2} T$, then the points whose relative heights are equal or larger than $z_m$ in $P_{ij}$ are overhanging points. The other points are real obstacle points. The threshold T is chosen in terms of the height of our ALV, $T = 1.8$ m.
## 4 Experimental Evaluation and Analysis
The goal of the experimental evaluation is to demonstrate the usefulness of the ground segmentation algorithm proposed in Section 3.2. In order to evaluate the real-time performance and the accuracy of the algorithm proposed in this paper, two other algorithms proposed in [^5] [^13] are adopted to be compared with. We implement them in C++ as well as the point cloud library (PCL) [^25], and conduct several contrastive experiments both in urban and countryside environments. Our experimental platform is a modified
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Fig. 6 Our ALV used for the experiments in various environments. It is equipped with some cameras, lasers, and positioning sensor for perception
Toyota Land Cruiser, as shown in Fig. 6, equipped with a Velodyne HDL-64E S2, Three SICK LMS 291, and a NovAtel SPAN-CPT GPS-aided inertial navigation system (INS). The Velodyne laser scanner rotates with a speed of 10 HZ. Thus, every scan contains up to 0.13 million points. At the same time, in order to evaluate their accuracy intuitively, a quantitative performance evaluation is conducted on the dataset provided in [^16]. In all the experiments, we use a notebook computer with an Intel P8600 CPU with 2.4 GHz using double cores and 2 GB main memory.
Our algorithm and the algorithm in [^13] are both based on the polar grid map. In the process of constructing the polar grid map of diameter 100 meters, we set the segment size $M = 180$. Every segment is divided into 160 bins, $N = 160$. To simplify computation, the range that the bins that are close to the LIDAR within 20 meters cover is about 0.2 meter along radial direction and the others cover about 0.5 meter along the radial direction. The algorithm in [^5] is based on the three-dimensional grid map, while in this paper, the grid map covers an area of 100 by 100 meters with the horizontal resolution of 0.4 meter and the vertical resolution of 0.2 meter. $^{1}$ The content of every grid cell is the 3D mean of the points in it. In order to increase the computational efficiency, the data compression is performed that the data are decimated by keeping 1 in every 30 of the grid means.
### 4.1 Runtime
As the algorithm is applied to the ALV, the real-time performance of the ground segmentation algorithm is our focus. We have evaluated the proposed algorithm online for numerous scans acquired while autonomously driving in the city of Ordos. The route is in a complex suburban environment that contains some uphill roads, downhill roads, flat roads and so on, as shown in Fig. 7. The algorithms in [^5] [^13] are also implemented off-line using the same scans. Figure 8 shows the results of these algorithms.
Most of the time, with an average of 74.93 ms per scan, our algorithm performs in real time. But, in Interval A, the runtime of every scan is larger than 100 ms. That is because the ALV locates in a large open territory (as shown in Fig. 7) and the ground on the right of ALV is uphill. In these corresponding segments of the polar grid map, the iterative times (lines 6–11 in Algorithm 1) increase, so does the time. The same phenomenon happens to the algorithm in [^5]. From Fig. 8, one can find that our algorithm is slower than the algorithm in [^13] with an average of 35.12 ms per scan, while it is also much faster than the algorithm in [^5] with an average of 678.77 ms per scan. Even though, our algorithm can get a real-time performance, because the refresh rate of the scan is 100 ms per scan.
### 4.2 Accuracy
The dataset provided in [^16] contains ten labelled scans of street scenes collected in an urban area in Boston. Each point has a label such as ground, car, house and so on. We only focus on two kinds of laser points: ground points and obstacle points in this paper. All the non-ground points (the laser points that belong to car, house, person,
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Fig. 7 The global planned map in the city of Ordos acquired from the position sensor (INS)
tree, street sign, fence and other) are gathered into the obstacle class. All ten scans of the laser points are mapped into the polar grid map with diameter of 100 meters and the three-dimensional grid map that covers an area of 100 by 100 meters, respectively. The results of ground segmentation are detailed in Table 1. All the algorithms can get good performances. That is because these scans are collected in the urban streets where the terrains are very flat (some scans where the terrains are rough will be shown in the next subsection). Our algorithm can achieve an accuracy of 0.9767, which is 1.16 % higher than the algorithm in [^13], but 0.17 % lower than the one in [^5]. As known, it can be a disaster for an ALV to misclassify the non-ground points as ground, because this will easily result in a collision with the obstacle. From Table 1, one can see that our algorithm misclassifies the least obstacle points as ground. Thus, our algorithm is the safest one among the three algorithms, to some extent.
### 4.3 Segmentation in Various Environments
In order to evaluate the usefulness of the proposed algorithm to our ALV, these algorithms are compared by many scans acquired from some urban and countryside environments. Example results are shown in Figs. 9, 10, 11, and 12, which show both the ground segmentation results and the ground surfaces obtained using these algorithms. These scenes include a highway with some
Fig. 8 Runtime of our algorithm (blue) compared to that of [^13] (red) and [^5] (green) for 3200 scans acquired in the city of Ordos
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Table 1 Confusion matrix of the three algorithms
| True\nInferred | Our algorithm | Algorithm in [^5] | Algorithm in [^13] |
| Ground | Non-ground | Ground | Non-ground | Ground | Non-ground |
| Ground | 192251 | 8268 | 196060 | 2570 | 189780 | 10728 |
| Non-ground | 6822 | 439919 | 11479 | 440589 | 11844 | 434901 |
| Accuracy | 0.9767 | 0.9784 | 0.9651 |
traffic, a field road with some road-blocks and two campus roads with a steep slope and a T-junction, respectively. In our implementation, the parameters in our algorithm are set to $\sigma_n^2 = 0.0012$, $\sigma_f^2 = 0.0528$, $a = 6.2978$, $T_{model} = 0.04$, $T_{data} = 3$, $T_r = 0.3$. These parameters are learnt by maximizing the pseudo log marginal likelihood in Section 3.3.
Figures 9–12 show some results of the proposed ground segmentation algorithm in four different scenes compared to those obtained with the algorithms in [^13] and [^5], respectively. Every figure contains two columns: the left column shows the segmentation results of our algorithm, the algorithm in [^5], and the algorithm in [^13], while the other column shows the ground surfaces obtained with the corresponding algorithms. In the left column of the figures, the yellow, red, green points represent the ground, obstacles and overhanging structures, respectively. Since the algorithms in this paper and [^13] are based on the polar grid maps that have been partitioned into many segments, we can only build a local ground model in every segment. The global ground surfaces obtained with our algorithm and the algorithm in [^13] are simply made up by (the combination of) all the local ground models in the segments. $^{2}$
Figure 9 shows a highway scene with a vehicle, a cyclist and a pedestrian. The left column shows the segmentation results of the algorithms, while the right one shows the ground surfaces in the scene obtained using the corresponding algorithms. Our algorithm (as shown in Fig. 9a) and the algorithm in [^5] (as shown in Fig. 9b) can get good performances, while some ground points (the red points in the blue quadrangle) are misclassified as obstacles by the algorithm in [^13] (as shown in Fig. 9c). That is because the vehicle occludes with some ground points, which violates smooth transitions between pairs of successive lines (In [^13], the ground is represented by lines in every segment). The ground surface obtained using the algorithm in [^13] is shown in the right image of Fig. 9c. The flat ground surface in the red quadrangle is misclassified as a bump, which is higher than the ground surface around. Thus, the real ground points in the red quadrangle are misclassified as negative obstacles.
Figure 10 shows a campus scene with an 8.7 % slope. Our algorithm and the algorithm in [^5] classify the slope accurately, as shown in the left column, while the algorithm in [^13] cannot (the red and green points in the blue quadrangle of the left image of Fig. 10c). That is because the root mean squared error of the line fitting for such a steep slope exceeds a predefined threshold. The algorithm in this paper can accumulate the points on the slope as the seeds through the iterations as shown in Fig. 4. Thus, the local ground in every segment can be modelled accurately. The similar iterative processes are implemented in the algorithm in [^5] too.
Nevertheless, there are also some differences between our algorithm and the algorithm in [^5]. Comparing the ground surfaces in Fig. 10a and b, one can find that the ground surface in Region A is classified as a flat ground in Fig. 10a while Region A is classified as a slope in Fig. 10b. The main reason is the difference of the distribution of the ground seeds. There are no laser returns in the corresponding Region A (Quad. A) in the left image of Fig. 10b. Our algorithm divides Region A into many segments and gets the local ground models in the segments. Only the seeds, which
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(a) The segmentation result (in the left image) and the ground surface (in the right image) obtained with our algorithm
(b) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^5]
(c) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^13]
Fig. 9 Our segmentation result in (a) of a highway scene, with a vehicle, a cyclist and a pedestrian, compared with the results of the algorithms in [^5] in (b), and [^13] in (c). In the left column of the figure, the ground points, obstacles, and
overhanging structures are colored yellow, red, and green pixels, respectively. The right column shows the ground surfaces obtained with the corresponding algorithms. Figures 10, 11, 12 and 13 are the same
are on the flat ground in these segments, are employed. When the global ground model is calculated using the algorithm in [^5], the seeds both on the flat ground and the slope (the seeds on the slope are found through the iterative processes) can be employed to predict the ground surface in Region A. The mean prediction (in Eq. 10) of the height of the ground surface is a linear combination of the heights of the seeds. Thus, our algorithm classifies Region A as a flat ground, while the algorithm in [^5] classifies it to be a slope.
Figure 11 shows another campus scene with a T-junction. Detecting a long range intersection is very important to local path planning. The
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(a) The segmentation result (in the left image) and the ground surface (in the right image) obtained with our algorithm
(b) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^5]
(c) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^13]
Fig. 10 Our segmentation result in (a) of a campus scene with an 8.7 % slope, compared with the results of the algorithms in [^5] in (b), and [^13] in (c)
algorithms in this paper and [^5] can get good performances (in Fig. 11a and b). That is because these algorithms have seeds on the ground of the intersection. However, the algorithm in [^13] recognizes the intersection as the obstacles as shown in the blue quadrangles of the left image of Fig. 11c, which makes the successive intersection detection impossible [^2]. Compared with the result of the algorithm in [^5], our segmentation result still has some defects. The stair steps are classified as ground in the blue polygons B in the left image of Fig. 11a.
Our algorithm is insensitive to the slowly rising obstacles such as stair steps. Though the height of each stair step is often less than 30 centimeters, it is an obstacle for the ALV in an urban
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(a) The segmentation result (in the left image) and the ground surface (in the right image) obtained with our algorithm
(b) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^5]
(c) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^13]
Fig. 11 Our segmentation result in (a) of a campus scene with a T-junction in front of the ALV, compared with the results of the algorithms in [^5] in (b), and [^13] in (c)
environment. In the segments, which contain the obstacles like the stair steps, the minimum heights in the bins increase gradually. Since there are only a few seeds on the flat ground in each segment, our algorithm may accumulate the points on the stair steps as the seeds through the iterative processes. The stair steps will be misclassified as ground surface. Thus, the height of the ground surface in the segment will increase (e.g., Region B shown in the right image of Fig. 11a). The algorithm in [^5] can get a better result in this situation. That is because it uses many seeds on the global flat ground surface. The points on the stair steps can hardly become the seeds through the iterative process.
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(a) The segmentation result (in the left image) and the ground surface (in the right image) obtained with our algorithm
(b) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^5]
(c) The segmentation result (in the left image) and the ground surface (in the right image) obtained with the algorithm in [^13]
Fig. 12 Our segmentation result in (a) of the countryside scene with some road-blocks, compared with the results of the algorithms in [^5] in (b), and [^13] in (c)
processes in the algorithm in [^5]. The predicted ground surface, which is shown in the right image of Fig. 11b, is very flat, and the stair steps are classified as obstacles accurately.
The countryside scene is shown in Fig. 12. There are many road-blocks in front of our ALV which are put into type S on the flat ground to form the curbs of the road. All the three algorithms can separate the road-blocks from the ground, because they all have the abilities to describe a flat road. However, as for a rough road, they have different performances. The algorithm
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Fig. 13 Examples where the performances of our algorithm are poor. The left image shows a campus scene with a car, some persons, some trees and two stair steps (the bottom stair step is 0.15 meter high and the other is 0.2 meter) that are in Quad. C, our algorithm misclassifies the
stair steps as ground, but segments the car, persons, trees and overhanging structures accurately. In the right image, our ALV locates at the intersection of a trunk road and a countryside uphill road. The uphill road is misclassified as obstacles in Quad. D
in [^13] cannot adapt to a rough road, many ground points are classified as obstacles (i.e. the red points in the blue polygon in the left image of Fig. 12c). That is because the rough ground surface makes the root mean square error of the line fitting in the segments large. As there are many seeds on the flat ground, the algorithm in [^5] can get a flat ground surface. It can detect the negative obstacles accurately, while misclassify the slowly rising hillside as obstacles (as shown in the blue quadrangle in the left image of Fig. 12b). Because our algorithm is insensitive to the hillsides, it can classify them as ground accurately.
From the right columns of Figs. 9–12, one can see that the ground surface obtained by the algorithm in [^5] is the smoothest one, the second is our algorithm, and the last is the algorithm in [^13]. In [^5], the algorithm that is based on two-dimensional Gaussian process regression uses the seeds in the whole map to model the global ground. Thus, it keeps the continuity of the ground in the whole map. Our method models the local ground in every segment. It only uses the seeds in each segment. However, less seeds make our algorithm insensitive to the stair steps and hillsides, while less seeds also make our algorithm get a real-time performance. Our algorithm can only keep the continuity in each segment, not between the two adjacent segments. The algorithm in [^13] uses the line fitting method to model the local ground in the segments. Only the relative continuity in each segment can be kept. At the same time, the occlusion and rough terrain will greatly influence the results of the line fitting.
Figure 13 presents some examples where the performances of our algorithm are poor to demonstrate in which environment situations our algorithm is ineffectiveness. In the left image of Fig. 13, our algorithm misclassifies the stair steps as ground (Region C in the left image), because our algorithm divides the polar grid map into many segments. Each segment contains only a few seeds, which makes our algorithm insensitive to the slowly rising obstacles, such as the stair steps. Our algorithm will accumulate the points on the stair steps as seeds to model the local ground surface, and misclassify the stair steps as ground. The other drawback of our algorithm is shown in the right image of Fig. 13. Our ALV locates at the intersection of a trunk road and a countryside uphill road (a 12 % slope), and faces the uphill road. As previously mentioned, we only choose the points whose absolute heights are less than 0.3 meter within a fixed radius 30 meters of the sensor as the initial seeds. However, in this scene, the slope is so steep that there are no initial seeds at all, which makes our algorithm of no avail.
Though our algorithm can be improved further, it had been successfully used in our ALV, which won the champion in the 2011 Chinese Future Challenge that was held in the city of Ordos, China, where the ALV needed to finish the tasks such as avoiding the static obstacles, merging into vehicles, U-turns and so on. With the algorithm
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in this paper and path planning algorithm in [^27], our ALV finished the route of ten kilometers autonomously in the urban environment of Ordos in twenty minutes, without any human intervention.
## 5 Conclusion and Future Work
This paper presents a novel real-time ground segmentation algorithm for an ALV, which combines the advantages of the both algorithms in [^5] and [^13]. The algorithm splits the complex two-dimensional ground segmentation problem into many one-dimensional regression problems to increase the computational efficiency, and keeps the continuity of the ground in every segment. It can get a better accuracy compared with the algorithm in [^13] and the real-time performance compared with the algorithm in [^5]. Our algorithm is useful to the successive object classification and local path planning in real time.
In the future, we will investigate how to keep the continuity between the adjacent segments based on the algorithm in this paper. Also, we will bring in the direction of the normal at each point when choosing the initial seeds, because the normal of the ground is nearly upward in an urban environment. If the normal of a point is not upward, it will not be chosen as an initial seed. This approach may increase the accuracy and robustness of our algorithm, to some extent.
Acknowledgements The work is support in part by National Nature Science Foundations of China under Grant No.61075043 and No. 61375050. Thanks all persons in laboratory who had joined the competition and worked together to won the champion in the third Chinese Future Challenge.
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