^{*}

This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution license (http://creativecommons.org/licenses/by/3.0/).

This paper presents a fully-automated method to establish a calibration dataset from on-site scans and recalibrate the intrinsic parameters of a spinning multi-beam 3-D scanner. The proposed method has been tested on a Velodyne HDL-64E S2 LiDAR system, which contains 64 rotating laser rangefinders. By time series analysis, we found that the collected range data have random measurement errors of around ±25 mm. In addition, the layered misalignment of scans among the rangefinders, which is identified as a systematic error, also increases the difficulty to accurately locate planar surfaces. We propose a temporal-spatial range data fusion algorithm, along with a robust RANSAC-based plane detection algorithm to address these issues. Furthermore, we formulate an alternative geometric interpretation of sensory data using linear parameters, which is advantageous for the calibration procedure. The linear representation allows the proposed method to be generalized to any LiDAR system that follows the rotating beam model. We also confirmed in this paper, that given effective calibration datasets, the pre-calibrated factory parameters can be further tuned to achieve significantly improved performance. After the optimization, the systematic error is noticeable lowered, and evaluation shows that the recalibrated parameters outperform the factory parameters with the RMS planar errors reduced by up to 49%.

The need to acquire 3-D information of physical environments has escalated in the last decade. With the rapid advances of Light Detection and Ranging (LiDAR) technology, laser-based active scanning has become a fast, accurate, and popular measurement tool. Stationary terrestrial LiDAR systems in the market such as the Reigl VZ-1000 and Optech ILRIS-3D units, are capable of delivering industrial-level accuracy with errors lower than 10 mm. For the real-time acquisition of panoramic range information, multiple laser rangefinders are attached to a motor, forming a multi-beam LiDAR system. The Velodyne HDL-64E S2, which is equipped with 64 laser emitter-sensor pairs to deliver dynamic panoramic point cloud at 10 Hz within the working range from 0.9 to 120 m, is such a high-definition LiDAR system. Although the LiDAR system was originally designed for the DARPA Grand Challenge and is often used in applications such as mobile navigation and autonomous vehicles ([

In our experiment, we have found that the tested Velodyne HDL-64E S2 achieves an average RMS error of about 2.5 cm. Close examination of the recorded range data has revealed two major problems. The first issue is the layered misalignment of scans on planar surfaces, as shown in

The intrinsic parameters are calibrated by the manufacturer using a plane placed at 25.04 m. The 3-D coordinates of a point becomes less accurate as it moves away from this distance. To address the aforementioned cause of systematic error, we have to further optimize the parameters for a certain range using data acquired in the scanned field. This is known as online or on-site calibration [

There are two major contributions in this paper. First, we formulate an alternative geometric interpretation of sensory data using linear parameters, instead of the nonlinear form specified by the manufacturer. The formulated linearly parameterized form has less correlation and achieves lower RMS error after calibration, as will be shown by the experiments. Second, and the more important contribution, is that we have implemented a framework for automatic on-site calibration using planes that exist within the scene. The calibration process can be easily adapted to other types of LiDAR and has been proven to achieves an RMS error lower than the factory provided calibration parameters.

The rest of this paper is organized as follows: in Section 2, related research is surveyed. InSection 3, the mathematical models of the conversion of range data to 3-D space and the adjustment of parameters are given. In Section 4, we propose a data fusion algorithm and the automatic establishment of calibration dataset. Experimental results are discussed in Section 5, and Section 6 concludes this paper.

There has not been a lot of work published in the literature specific to the calibration of the tested Velodyne LiDAR system since the device is relatively new. In [

The systematic error caused by the inaccurate intrinsic parameters is also examined in [

In [

By referencing previous work, the proposed method also utilizes planar targets for calibration. However, we adopt a linear representation of intrinsic parameters. Moreover, instead of conducting laboratory calibrations, we deploy an automatic mechanism to establish and process calibration data on-site. The proposed method allows the LiDAR system to autonomously adjust its intrinsic parameters while operating online.

Based on the geometric interpretation of the range data, a non-linear optimization model can be derived given some observed scene planes. This section first introduces an alternative model for the conversion of range data; then describes the objective function which minimizes plane deviations in terms of quadratic error.

The Velodyne HDL-64E S2 contains 64 laser emitter-sensor pairs which are rigidly attached to a rotating motor, as depicted in _{i}_{i}_{i}_{i}_{i}_{i}

Although it is obvious that the described conversion is not linear, the parameters can actually be interpreted in a linear form by the following steps. First, extract the rotation matrix from the right-hand side of

Let _{θ}

Combining _{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}_{i}

We use the derived linear form (_{i}_{i}

A typical calibration of the geometric sensory parameters is based on the observations of particular objects with known geometry. These objects are known as the calibration targets. Since it is fairly common to find planar objects, such as walls or floors, in both outdoor and indoor environments, plane geometry is ideal for the recalibration of Velodyne LiDAR system in this work.

A 3-D plane that does not pass through the origin can be uniquely defined by a 3-vector _{x}_{y}_{z}_{1}, _{1}, … _{64}, _{64}, _{1}, _{2}, …}, then, for the _{1≤}_{i}_{≤64} _{i}_{i,k}_{i,k}

One may notice that the term

To avoid approaching a trivial solution while maintaining the dependency, we allow the planes to be adjusted within a controllable region. To this end, the parameterization of

Another issue that needs to be dealt with is the fact that the adjustment can be ill-posed under certain conditions. For the _{i}_{it}_{i}_{it}_{i}_{ix}_{iz}_{ix}_{iz}

In this section we describe the establishment and processing of calibration data for the adjustment of parameters. In our work, the calibration data are essentially point-plane correspondences that are extracted from the on-site scans. In order to precisely detect planes, the preprocessing on range data as described in Sections 4.1 and 4.2 are carried out. In Section 4.3 we introduce an automatic plane detection algorithm, which is applied to the point cloud constructed from the preprocessed range data. The methods presented in this section allow the calibration data to be acquired in a fast, yet precise manner.

In the raw measurements, we discovered noticeable temporal instability that causes scattering of range data, as shown in

A spatial-temporal data fusion technique is deployed to integrate raw measurements of multiple spins into more reliable range data. We intend to use continuous range data instead of data acquired from a single spin. The resulting data fusion algorithm is based on the idea of estimating a point using a convex combination of adjacent points. The estimated value at point _{p,q}

Since each laser rangefinder operates at a high speed rate of around 20,000 fires per second, the computation of

In this work the weights are determined by the Gaussian distance ^{−‖}^{p}^{−}^{q}^{‖2}/2^{σ}^{2}. The estimation can be efficiently computed using separable 2-D convolution. To exclude the effect of missing data a normalization term is added to

Since

We apply a derivative-based approach on fused range data to detect points that are continuous and thus likely to be in the calibration planes. The approach works as follows. An

The result is further examined to exclude short segments, which are less likely to contain useful data for the calibration. As a result of removing these outliers, which are the majority of the acquired range data, the detection time of plane is dramatically reduced. Since the segments are detected in the angle-range domain, they may be curved in 3-D space. Despite the curvature, these points are still useful as long as they are planar points.

Finding planes in a point cloud has been a fundamental problem in the field of 3-D modeling. One of the classical solutions is Hough Transform, which searches objects of a particular geometry by means of the accumulator in the parameter domain [

A RANdom SAmpling Consensus (RANSAC) technique that iteratively searches calibration planes in a stochastic manner is developed to locate the calibration targets efficiently. In each round, a point and some of its neighbors are randomly selected. Based on the selected points a best-fitting plane is calculated to minimize residuals in a least-square sense. This plane is then applied to measure its fitness in the global scope with respect to whole point cloud.

For an acceptable estimate, which contains a significant number of points within a tolerable deviation ∈, we refine the plane using the Iterative Closest Point (ICP) technique as follows. Firstly, the points that are likely to be in the plane within tolerable deviation are selected. A least-square plane is then calculated subject to these points to replace the initial estimate. The refinement repeats until termination criteria are met that either the update of the plane is significantly small or the number of iterations reaches its limit. Points in the estimated plane are removed from the point cloud, and the new estimate is added to the list of detected planes.

Since the selection of point is done in the global scope, some points contribute to the estimate may be outliers that are far from the majority.

This stochastic process iterates until no more planes can be found or the number of iteration reaches its maximum. The pseudo-code of the algorithm is listed in Algorithm 1 and Algorithm 2. The established point-plane correspondences (see

1

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{i}

_{0},

_{k}

1

_{0}←

_{i}

_{i}

_{i}

_{−1}) <

_{i}

_{i}

_{i}

_{i}

_{−1}‖ is small 5

_{k}

_{i}

The range data are collected in an open space to evaluate the proposed method. The selected site, as shown in

The residual between each point and the corresponding calibration plane is measured to evaluate the performance of parameters. In

Cross validation is also conducted by selecting some subsets of three calibration datasets to perform the recalibration, and evaluating the result using datasets that are not used during the optimization. For an unused dataset, the planes are re-estimated from the point cloud using the optimized intrinsic parameters. The results are given in

To examine the effects of representing the intrinsic parameters in the linear form derived in Section 3.1, we conduct the same experiment using the non-linear representation defined by the manufacturer. The traces of RMS error through first 20 iterations of the optimization process are shown in

Similar to [

As we expected, strong correlations are found between the linear parameters (_{x}_{x}_{y}_{y}_{z}_{z}

The fusion of range measurements reduces the amount of computation. To verify how the reduction of data affects the result of calibration, we conduct the same test, but this time with all calibration datasets established from 894,000 raw measures. The results are compared with the use of fused range data consisting of 377,000 points to study the difference. The RMS error of raw range data is higher than the results listed in

The evaluated errors with and without the fusion of range data is barely distinguishable compared to the errors of factory parameters. The largest differences are 0.08° in angle and 0.164 cm in distance. However, there exists noticeable difference in the running time. The optimization on the fused range data finished in 517 s. In contrast, without the fusion the optimization took 3,132 s to converge. The reduction of data greatly boosts the process of recalibration while achieving a similar result. Yet another observation is that the misalignment of planes is reduced even though it is not explicitly modeled by the objective function.

We present in this work an efficient multi-stage strategy to attain automatic on-site recalibration of the Velodyne HDL-64E S2 system. The proposed method is applicable to both range-angle sensory space as well as Euclidean space. In the first stage, the range data are merged temporally and spatially using a real-time Gaussian-based algorithm. The amount of range data is then reduced by enforcing the continuity constraint. Afterward, we carry out a robust RANSAC-based plane detection algorithm to locate planes in 3-D space. The estimated planes are refined in an ICP manner before the final calibration dataset is established. The intrinsic parameters optimized using the on-site range data achieve an average improvement of 40% over the factory parameters. In the experiment, the plane residuals with a RMS error lower than 1.3 cm is attainable, which is an improvement from previous work. The implementation of the on-site calibration strategy also allows the LiDAR to be automatically calibrated before each acquisition sequence, such that system can use calibration parameters that are more adapted to each individual site for obtaining more accurate results.

A linear form of the parameters is also derived from the range data conversion formula specified by the manufacturer. The parameters represented in the linear form are verified to be less correlated and achieve lower RMS error quicker than the canonical model. The optimization model defined in this paper can be further extended to a LiDAR system that follows the rotating multi-beam model described by the linear parameters. The process is designed to be finished on-site so that one can see and use the improved interpretation of the range data as quickly as possible. For a calibration dataset containing twenty planes with one million points, the whole process (data collection, preprocessing, plane detection, and optimization) can be finished within 10 min on a moderate laptop. Similar to many computer vision applications, we suggest that the calibration process be performed at the beginning of a data acquisition sequence for each different site. For example, calibration is performed once at the beginning for the corridor sequence and the acquired parameters are used throughout the data acquisition for that site. Since the proposed calibration procedure is quick and simple to perform, it is easy to include a calibration before each different range data sequence is taken.

In future work, we will incorporate image sensors into the system to study simultaneous calibration of intrinsic and extrinsic parameters (e.g., [

The research is sponsored by National Science Council of Taiwan under grant number NSC 100-2631-H-390-001.

(

Process of the proposed automatic on-site recalibration method.

(

Kernel-based Range Data Fusion with Various σ.

(

Established calibration data containing seven planar surfaces.

The corridor selected to evaluate the proposed method.

Point residuals colour-coded to show calibration planes (

Distribution of point residuals (

Scan of a wall (

Convergence of different parameter representations.

Established calibration datasets.

620,996 | 219,145 | 183,501 | 149,465 | 76% | 8 | |

135,228 | 175,257 | 139,687 | 122,884 | 10% | 6 | |

510,775 | 174,263 | 137,715 | 104,958 | 79% | 6 |

RMS Errors and Improvement After Recalibration.

2.355 cm | 1.323 cm (44%) | 1.209 cm (49%) | 2.014 cm (14%) | 1.285 cm (45%) | |

2.426 cm | 1.532 cm (36%) | 1.776 cm (27%) | 1.363 cm (44%) | 1.491 cm (39%) | |

2.397 cm | 1.733 cm (28%) | 1.457 cm (39%) | 1.307 cm (45%) | 1.412 cm (41%) | |

2.389 cm | 1.493 cm (38%) | 1.482 cm (38%) | 1.663 cm (30%) | 1.386 cm (42%) |

Average correlation matrix and estimated error of linear parameters.

_{x} |
_{y} |
_{z} |
_{x} |
_{y} |
_{z} |
||
---|---|---|---|---|---|---|---|

_{x} |
1.0000 | 0.0735 | 0.0598 | − |
−0.0721 | −0.0545 | 0.0004 cm |

_{v} |
- | 1.0000 | −0.0687 | − |
− |
0.0004 cm | |

_{z} |
- | - | 1.0000 | −0.0349 | − |
− |
0.0003 cm |

_{x} |
- | - | - | 1.0000 | 0.0743 | 0.0368 | 0.1170 cm |

_{v} |
- | - | - | - | 1.0000 | 0.1621 cm | |

_{z} |
- | - | - | - | - | 1.0000 | 0.0769 cm |

Average correlation matrix and estimated error of non-linear parameters.

Δ |
Δ |
Δ |
|||||
---|---|---|---|---|---|---|---|

1.0000 | −0.0679 | 0.0766 | − |
−0.0784 | 0.0641 | 0.0219° | |

- | 1.0000 | − |
0.0509 | − |
0.0231° | ||

- | - | 1.0000 | −0.0780 | − |
0.1765 cm/cm | ||

Δ |
- | - | - | 1.0000 | 0.0747 | −0.0508 | 0.1535 cm |

Δ |
- | - | - | - | 1.0000 | − |
0.1533 cm |

Δ |
- | - | - | - | - | 1.0000 | 0.0004 cm |

Angular and distance plane misalignment with and without data fusion.

0.2062° | 0.6559 cm | 0.1794° | 0.2743 cm | 0.2205° | 0.1129 cm | |

0.2249° | 1.6283 cm | 0.1042° | 1.4106 cm | 0.1520° | 1.4961 cm | |

0.1659° | 0.5163 cm | 0.1803° | 0.2835 cm | 0.0988° | 0.2909 cm | |

0.1257° | 0.7623 cm | 0.1266° | 0.7604 cm | 0.1928° | 0.8136 cm | |

0.1334° | 0.3600 cm | 0.1766° | 0.2037 cm | 0.1449° | 0.1942 cm | |

0.3623° | 0.1985 cm | 0.3169° | 0.0644 cm | 0.2800° | 0.0646 cm | |

0.2031° | 0.6869 cm | 0.1807° | 0.4995 cm | 0.1815° | 0.4954 cm |