Lidar, or light and range detection, is a method of remote sensing (Tian et al., 2021) that uses light in the form of pulsed lasers to measure distances to objects. It has become an integral technology in various industries, including autonomous vehicles (Elhousni and Huang, 2020), robotics (Mochurad et al., 2023a), and environmental monitoring (Guo et al., 2020).

Lidar systems consist of three main components:

1. Laser transmitter: This component generates short pulses of laser light (usually in the infrared) that are directed at objects in the environment.

Lidar can be:

1. Static Lidar: Used to scan static objects from a fixed position. This is often used in surveying and mapping to create three-dimensional models of the landscape and infrastructure.

However, Lidar has some disadvantages, such as high cost, relatively large size and weight, and sensitivity to weather conditions such as rain or fog. Some of these disadvantages can be compensated for by combining Lidar with other technologies and developing new, more compact and cost-effective Lidar systems.

One of the key applications of lidar is localization, which involves estimating the position and orientation of an object in the environment using data from a lidar sensor (Marck et al., 2013). Localization is extremely important for autonomous vehicles (Lu et al., 2022), where it is necessary to determine the position of the vehicle for safe and efficient operation. Optimization of navigation algorithms and methods can contribute to environmental and economic development, as autonomous vehicles can reduce fuel costs and ensure efficient use of infrastructure (Varsi et al., 2021).

Improving the navigation algorithms of autonomous cars can accelerate the development of smart cities, where autonomous vehicles play an important role in creating integrated and efficient transportation solutions (Phang et al., 2021; Wang et al., 2022).

Lidar’s localization speed is an important factor for real-time applications (Liu et al., 2023), especially for applications such as autonomous driving (Luo et al., 2019) where timely decision making is essential. Traditional localization methods such as the Extended Kalman Filter (EKF) (Zhang, 2019) and the iterative closest point algorithm (ICP) (Zhang et al., 2022), can be computationally expensive and do not meet the requirements of real-time applications (Dabbiru et al., 2020; Shymanskyi et al., 2022).

As it is known (Garland et al., 2008), the CUDA parallel computing platform was developed by NVIDIA, which can significantly accelerate various applications, including Lidar localization. CUDA allows developers to use the massively parallel architecture of modern GPUs, which allows them to process large amounts of Lidar data faster.

The relevance of the conducted research can be considered from the following perspectives:

• Development of autonomous vehicles: With the active growth of the autonomous vehicle industry, the development of new and improvement of existing navigation methods are becoming increasingly relevant. Autonomous vehicles require high accuracy in localization and stable operation of navigation algorithms for safe and efficient movement.

The results of this research can have a positive impact on road safety, cost-effectiveness, environmental sustainability, and transportation accessibility. Additionally, they can contribute to the development of smart cities, integration of transportation systems, and the enhancement of competitiveness for automakers.

The relevance of employing parallel computing in the context of autonomous car navigation becomes evident when considering the following factors:

1. Large Data Volumes (Huang and Cao, 2021): Autonomous vehicles accumulate substantial data from diverse sensors like lidars, radars, and cameras. Swift processing of this data is crucial for appropriate responses to varied situations. Parallel computing enables simultaneous data processing, enhancing the efficiency of the navigation system.

The article analyzes the literature on the topic of the study. This was done with a view to highlighting the main advantages and disadvantages of the current state of the issue under consideration. In the paper (Montañez et al., 2023), the authors employed an extended Kalman filter for the detection of moving objects. The effectiveness of the EKF was assessed using a dataset that includes location information obtained from LiDAR and a radar sensor for an object moving along a trajectory with abrupt changes.

In (Koide et al., 2021) presents an approach that creates a globally consistent 3D map structure based on the loss factor during a real-time GPU-accelerated mapping process. Data is obtained from a 3D Lidar and maps are constructed based on it. The GPU is used to speed up the mapping algorithm during map creation.

In the research (Shreyas Madhav and Rajesh Kanna, 2021), an advanced Lidar 3D SLAM algorithm is introduced for autonomous aerial robots. The alignment process involves the extraction of Fast Point Feature Histogram (FPFH) descriptors, subsequently refined through iterative nearest point registration (NPR). The ultimate trajectory estimation undergoes 3D pose graph optimization to reduce potential overall drift. Simulated results demonstrate a noteworthy 26% decrease in execution time when employing the parallelized algorithm with 4 CPUs compared to its serial counterpart.

In (Jang et al., 2022), the authors detail an algorithm that employs GPU parallel processing to enhance the existing ND map matching process. This optimization resulted in a remarkable 48-fold acceleration while preserving accuracy.

The integration of a semantic image with low-resolution 3D Lidar point clouds and the generation of dense semantic depth maps are addressed in (Lou et al., 2023). Utilizing visual odometry, the method selects functional ORB points with depth information to enhance positional accuracy. During unmanned vehicle positioning, parallel threads are employed to aggregate 3D semantic point clouds.

In the paper (Chiang et al., 2023), the authors leverage Lidar as the primary auxiliary sensor, proposing a Lidar-based simultaneous localization and mapping (SLAM) approach for positioning, navigation, and synchronization. Furthermore, point cloud registration is executed through a three-dimensional normal distribution transform (NDT). The initial Lidar position assumption for Lidar-based SLAM is derived from two sources: one being a differential global navigation satellite system (GNSS) solution, and the other being an inertial navigation system (INS) and an integrated GNSS solution created using an extended Kalman filter with added motion constraints, including zero velocity update and nonholonomic constraint.

An improved NDT algorithm and its FPGA implementation were presented in (Deng et al., 2021). The authors achieved the acceleration of the search operation by using a new data structure called OAVS, which is non-recursive and efficient. The optimized semantic NDT algorithm based on OAVS significantly reduced the number of search operations by eliminating unnecessary queries. Additionally, the proposed streaming FPGA accelerator architecture for SEO-NDT improved real-time performance and ensured energy efficiency. When compared to advanced embedded CPU and GPU processors, the FPGA implementation provided up to 35.85x and 2.44x performance acceleration, respectively.

In (Dong et al., 2021) authors used this method in such a way that it performs all calculations directly on the range images created using 3D LiDAR scans, which avoids explicit processing of the 3D point cloud and quickly selects the poles for each scan.

As indicated in (Mendez Maldonado et al., 2021), the authors developed a hybrid convolutional neural network (CNN) by directly applying a Markovian grid-based localization approach on the GPU. This CNN is capable of simultaneously handling image-based localization and odometry-based probability propagation within a single neural network. The detailed description of the Markovian approach can be found in (Kovtun et al., 2023a).

In (Sun et al., 2020) a new data structure with a spatial partitioning method was presented, which can be successfully built even for large volumes of point clouds. Based on this structure, a KNN search algorithm was developed that works effectively when the distribution of points is uneven. This innovative structure is implemented on both an FPGA accelerator and a GPU.

In the following paper (Xie et al., 2022), introduces a lightweight convolutional neural network (CNN) framework designed for the semantic segmentation of a projection-based LiDAR point cloud. This framework comprises only 1.9 million parameters, marking an 87% reduction compared to leading-edge networks. The evaluation on a GPU revealed a processing time of 38.5 milliseconds per frame and an achieved result of 47.9% mIoU on the Semantic-KITTI dataset. Moreover, the proposed CNN is tailored for FPGAs using the NVDLA architecture, demonstrating a 2.74x speedup compared to a GPU-based implementation and a noteworthy 46x improvement in energy efficiency.

In (Mochurad and Kryvinska, 2021) a parallel parallelization algorithm is proposed to solve the problem of determining the current position of a lidar in 2D based on OpenMP technology. The authors also indicated prospects for further research: 1) optimization of the computing process based on CUDA technology using GPUs; 2) consideration of a more complex spatial domain.

The researchers in (Mochurad et al., 2023b) introduced a parallel algorithm employing CUDA technologies to establish the 2D position of a lidar through the Particle Filter algorithm. Despite achieving a considerable speedup with this technology, it might appear that their findings challenge the hypothesis presented in our study. Nonetheless, this is not the case, as our investigation focuses on a distinct algorithm, addressing the issue of 3D localization and extending beyond closed-room localization.

In the study (Xu et al., 2022), the use of measurement uncertainty estimation is identified as an effective method for tracking a vehicle, based on LiDAR detectors. The authors propose an extended Kalman filter framework, consisting of two main components: the first is capable of assessing the statistics of measurement noises dependent on the state to detect LiDAR objects, while the second generates multi-hypothesis measurements based on the trajectory of the identified vehicle.

The shift from traditional automobiles to autonomous ones encompasses the integration and enhancement of diverse technologies and computerized algorithms. An integral aspect influencing the efficacy of autonomous vehicles is their localization, along with perception, route planning, and control, where the precision and effectiveness of localization assume a pivotal role in autonomous driving. About (Poulose et al., 2022), the paper underscores the significance of the localization challenge in autonomous vehicles and elucidates its map-based realization employing point cloud matching. The authors introduce a localization system leveraging the Robot Operating System (ROS) in conjunction with Autoware. The empirical findings demonstrate that a map-centric localization system utilizing 3D lidar scanning delivers adequately precise real-time localization for autonomous driving within a university campus setting. The paper provides an exhaustive account of the methodologies for crafting point cloud maps and vehicle localization, along with a systematic guide for implementing a map-based system for autonomous driving.

In cities, there are many regions where the global navigation satellite system does not work, where localization of autonomous driving remains a problem. Various methods have been previously proposed to improve the localization accuracy by using accurate distance measurements obtained from Lidar sensors and for the speed of map construction. This study proposes a parallelized 3D Kalman algorithm using CUDA to accelerate the computational speed of Lidar localization while maintaining the original lidar localization accuracy. Unlike previous papers that parallelize the map construction, this approach parallelizes the Kalman localization algorithm itself.

The aim of this study is to propose a parallel algorithm based on CUDA technology to accelerate lidar localization in 3D space.

The main contribution of this article can be summarized as follows:

1. Anew localization algorithm is proposed that uses the Kalman filter and CUDA technology to accelerate the computational speed of Lidar localization in 3D;

The rest of this article is organized as follows: Chapter 2 describes the proposed parallel localization algorithm, analyzes the computational complexity presents a new theoretical estimate of the speedup based on Ambdahl’s law, and presents the steps of implementing the proposed algorithm using CUDA technology. Chapter 3 describes the environment used for testing and presents the relevant results of numerical experiments. Conclusions and prospects for further research are shown in the last chapter.

The Kalman filter, discussed in (Wo and Biswal, 2023), serves as a powerful recursive tool for estimating the internal state of linear dynamic systems by analyzing a series of noisy measurements. Its application extends across various domains, encompassing engineering, economics, radar, computer vision, and the estimation of structural macroeconomic models. This filter holds significance as a fundamental component in control theory and the development of control systems. In conjunction with the linear-quadratic regulator (LQR), the Kalman filter addresses the challenges posed by linear-quadratic Gaussian (LQG) control problems. Collectively, the Kalman filter, LQR, and LQG controller represent essential solutions to core issues in control theory.

As noted by the authors of (Meng et al., 2023), Kalman filtering is an optimal recursive numerical computing algorithm characterized by the efficiency of program memory use, speed, and suitability for real-time data processing programs.

It is based on a mathematical model of the system and uses the principles of Kalman filtering to combine the predicted state with actual measurements to obtain the best estimate of the system’s state. It has two main steps: prediction and correction. The prediction step uses mathematical models of the system and the previous state estimation to make a forecast of the future state of the system. In the correction step, the predicted state is updated to take into account new measurements that are reduced by noise and incompleteness. The state estimation uses information such as the state vector and covariance matrix to provide an optimal estimate and minimal prediction error. This is achieved by weighting the prediction and new measurements based on their accuracy and uncertainty.

Kalman filters are constructed upon time-discretized linear dynamical systems, which are represented as Markov chains, as outlined in (Kovtun et al., 2023b). These chains are built on linear operators subject to errors, which may include Gaussian noise. The system’s state is expressed as a vector of real numbers. At each discrete time increment (clock cycle), a linear operator is applied to the state, generating a new state that incorporates some noise and, if available, information from the system control. Subsequently, another linear operator, combined with additional noise, is applied to the true (“hidden”) state to produce the observed outputs. While the Kalman filter shares similarities with the hidden Markov model, a key distinction lies in the fact that the hidden state variables in the Kalman filter assume values in a continuous space, contrasting with the discrete state space of the hidden Markov model. Notably, there exists a robust duality between the equations of the Kalman filter and the hidden Markov model.

• Foresight stage

The Kalman filter model postulates that the actual state at a given time point k is deduced from the state at k − 1 , as illustrated Figure 1: X k p = A X k − 1 + B u k + w k where.

A k is a state transition model applied to the previous state x k − 1 ;

B k is a model of control effects applied to the control vector u k ;

w k is the noise of the process, which is assumed to have a multivariate normal distribution with zero mean and covariance Q k .

The covariance of the predicted P k state is calculated using the following formula: P k p = A P k − 1 A T + Q k   .

• Refinement stage

At a point in time k observation (or measurement) Y k of the present state X k is made in accordance with Y k = C X k m + Z k where C is the observation model that maps the true state space to the observed space, and Z k is the observation noise, assumed to be Gaussian white noise with zero mean and covariance R k .

Next, the covariance of innovations (deviation) is calculated S k which is then used to calculate the optimal Kalman transfer coefficient: S k = H k P k | k − 1 H k T + R k K k = P k | k − 1 H k T S k − 1

After that, we calculate the updated state estimate and its covariance: X k = X k p + K Y   −   H X k p   P k = I   −   K H P k p

The initial state and noise vectors at each cycle { x 0 , w 1 , w k , v 1 . v k } are assumed to be mutually independent.

For the study, a random three-dimensional space of values was generated (Top Streamers on Twitch, 2023) since the object of study is the speed of the algorithm. Random values were generated taking into account the contribution of measurement error and had a nonlinear characteristic.

The tests were conducted on a system with the following characteristics:

In this paper, we developed a parallelized version of the Kalman algorithm in 3D using CUDA to accelerate the computational speed of Lidar localization. Localization using Lidar is relevant for autonomous driving in regions where the global navigation satellite system does not work. The result is a software product that processes object location data faster and can be used for real-time systems, such as autonomous vehicles or car pilot assistance systems, where frequently received data on position and environment need to be processed as quickly as possible. High data rates are optimally suited for CUDA processing.

We tested and measured the efficiency of the sequential Kalman algorithm and the parallel implementation on different sizes of lidar position datasets. It turned out that the use of CUDA is more efficient on larger datasets, from 500 points. On average, using CUDA gives an increase in execution speed of about 2.4 times, while on the optimal set – 3.8 times. Using CUDA on suboptimal sets can lead to losses in execution speed, since due to the peculiarities of the algorithm and acceleration technology, the execution speed is only 80% of the sequential version. This can be explained by the fact that CUDA allows you to calculate many points in parallel at the same time, which reduces the execution time of the algorithm.

In general, the use of CUDA can significantly increase the efficiency of the Kalman algorithm, because in real conditions, it is necessary to constantly process the data coming from the lidar. Nowadays, there are many CUDA-enabled edge devices, which confirms the relevance of the presented algorithm and this study.

Prospects for further research include the possibility of extending the proposed algorithm to the case of 2D LIDAR data and analyzing the proposed algorithm based on SLAM technology (Zhu et al., 2009).