Showing papers on "Point (geometry) published in 2019"

PointConv: Deep Convolutional Networks on 3D Point Clouds

Abstract: Unlike images which are represented in regular dense grids, 3D point clouds are irregular and unordered, hence applying convolution on them can be difficult. In this paper, we extend the dynamic filter to a new convolution operation, named PointConv. PointConv can be applied on point clouds to build deep convolutional networks. We treat convolution kernels as nonlinear functions of the local coordinates of 3D points comprised of weight and density functions. With respect to a given point, the weight functions are learned with multi-layer perceptron networks and the density functions through kernel density estimation. A novel reformulation is proposed for efficiently computing the weight functions, which allowed us to dramatically scale up the network and significantly improve its performance. The learned convolution kernel can be used to compute translation-invariant and permutation-invariant convolution on any point set in the 3D space. Besides, PointConv can also be used as deconvolution operators to propagate features from a subsampled point cloud back to its original resolution. Experiments on ModelNet40, ShapeNet, and ScanNet show that deep convolutional neural networks built on PointConv are able to achieve state-of-the-art on challenging semantic segmentation benchmarks on 3D point clouds. Besides, our experiments converting CIFAR-10 into a point cloud showed that networks built on PointConv can match the performance of convolutional networks in 2D images of a similar structure.

Relation-Shape Convolutional Neural Network for Point Cloud Analysis

Abstract: Point cloud analysis is very challenging, as the shape implied in irregular points is difficult to capture. In this paper, we propose RS-CNN, namely, Relation-Shape Convolutional Neural Network, which extends regular grid CNN to irregular configuration for point cloud analysis. The key to RS-CNN is learning from relation, i.e., the geometric topology constraint among points. Specifically, the convolutional weight for local point set is forced to learn a high-level relation expression from predefined geometric priors, between a sampled point from this point set and the others. In this way, an inductive local representation with explicit reasoning about the spatial layout of points can be obtained, which leads to much shape awareness and robustness. With this convolution as a basic operator, RS-CNN, a hierarchical architecture can be developed to achieve contextual shape-aware learning for point cloud analysis. Extensive experiments on challenging benchmarks across three tasks verify RS-CNN achieves the state of the arts.

Relation-Shape Convolutional Neural Network for Point Cloud Analysis

Abstract: Point cloud analysis is very challenging, as the shape implied in irregular points is difficult to capture. In this paper, we propose RS-CNN, namely, Relation-Shape Convolutional Neural Network, which extends regular grid CNN to irregular configuration for point cloud analysis. The key to RS-CNN is learning from relation, i.e., the geometric topology constraint among points. Specifically, the convolutional weight for local point set is forced to learn a high-level relation expression from predefined geometric priors, between a sampled point from this point set and the others. In this way, an inductive local representation with explicit reasoning about the spatial layout of points can be obtained, which leads to much shape awareness and robustness. With this convolution as a basic operator, RS-CNN, a hierarchical architecture can be developed to achieve contextual shape-aware learning for point cloud analysis. Extensive experiments on challenging benchmarks across three tasks verify RS-CNN achieves the state of the arts.

ShellNet: Efficient Point Cloud Convolutional Neural Networks Using Concentric Shells Statistics

Abstract: Deep learning with 3D data has progressed significantly since the introduction of convolutional neural networks that can handle point order ambiguity in point cloud data. While being able to achieve good accuracies in various scene understanding tasks, previous methods often have low training speed and complex network architecture. In this paper, we address these problems by proposing an efficient end-to-end permutation invariant convolution for point cloud deep learning. Our simple yet effective convolution operator named ShellConv uses statistics from concentric spherical shells to define representative features and resolve the point order ambiguity, allowing traditional convolution to perform on such features. Based on ShellConv we further build an efficient neural network named ShellNet to directly consume the point clouds with larger receptive fields while maintaining less layers. We demonstrate the efficacy of ShellNet by producing state-of-the-art results on object classification, object part segmentation, and semantic scene segmentation while keeping the network very fast to train.

Point-Based Multi-View Stereo Network

Abstract: We introduce Point-MVSNet, a novel point-based deep framework for multi-view stereo (MVS). Distinct from existing cost volume approaches, our method directly processes the target scene as point clouds. More specifically, our method predicts the depth in a coarse-to-fine manner. We first generate a coarse depth map, convert it into a point cloud and refine the point cloud iteratively by estimating the residual between the depth of the current iteration and that of the ground truth. Our network leverages 3D geometry priors and 2D texture information jointly and effectively by fusing them into a feature-augmented point cloud, and processes the point cloud to estimate the 3D flow for each point. This point-based architecture allows higher accuracy, more computational efficiency and more flexibility than cost-volume-based counterparts. Experimental results show that our approach achieves a significant improvement in reconstruction quality compared with state-of-the-art methods on the DTU and the Tanks and Temples dataset. Our source code and trained models are available at https://github.com/callmeray/PointMVSNet.

A-CNN: Annularly Convolutional Neural Networks on Point Clouds

Abstract: Analyzing the geometric and semantic properties of 3D point clouds through the deep networks is still challenging due to the irregularity and sparsity of samplings of their geometric structures. This paper presents a new method to define and compute convolution directly on 3D point clouds by the proposed annular convolution. This new convolution operator can better capture the local neighborhood geometry of each point by specifying the (regular and dilated) ring-shaped structures and directions in the computation. It can adapt to the geometric variability and scalability at the signal processing level. We apply it to the developed hierarchical neural networks for object classification, part segmentation, and semantic segmentation in large-scale scenes. The extensive experiments and comparisons demonstrate that our approach outperforms the state-of-the-art methods on a variety of standard benchmark datasets (e.g., ModelNet10, ModelNet40, ShapeNet-part, S3DIS, and ScanNet).

PU-GAN: A Point Cloud Upsampling Adversarial Network

Abstract: Point clouds acquired from range scans are often sparse, noisy, and non-uniform. This paper presents a new point cloud upsampling network called PU-GAN, which is formulated based on a generative adversarial network (GAN), to learn a rich variety of point distributions from the latent space and upsample points over patches on object surfaces. To realize a working GAN network, we construct an up-down-up expansion unit in the generator for upsampling point features with error feedback and self-correction, and formulate a self-attention unit to enhance the feature integration. Further, we design a compound loss with adversarial, uniform and reconstruction terms, to encourage the discriminator to learn more latent patterns and enhance the output point distribution uniformity. Qualitative and quantitative evaluations demonstrate the quality of our results over the state-of-the-arts in terms of distribution uniformity, proximity-to-surface, and 3D reconstruction quality.

PU-GAN: a Point Cloud Upsampling Adversarial Network.

Abstract: Point clouds acquired from range scans are often sparse, noisy, and non-uniform. This paper presents a new point cloud upsampling network called PU-GAN, which is formulated based on a generative adversarial network (GAN), to learn a rich variety of point distributions from the latent space and upsample points over patches on object surfaces. To realize a working GAN network, we construct an up-down-up expansion unit in the generator for upsampling point features with error feedback and self-correction, and formulate a self-attention unit to enhance the feature integration. Further, we design a compound loss with adversarial, uniform and reconstruction terms, to encourage the discriminator to learn more latent patterns and enhance the output point distribution uniformity. Qualitative and quantitative evaluations demonstrate the quality of our results over the state-of-the-arts in terms of distribution uniformity, proximity-to-surface, and 3D reconstruction quality.

MeteorNet: Deep Learning on Dynamic 3D Point Cloud Sequences

Abstract: Understanding dynamic 3D environment is crucial for robotic agents and many other applications. We propose a novel neural network architecture called MeteorNet for learning representations for dynamic 3D point cloud sequences. Different from previous work that adopts a grid-based representation and applies 3D or 4D convolutions, our network directly processes point clouds. We propose two ways to construct spatiotemporal neighborhoods for each point in the point cloud sequence. Information from these neighborhoods is aggregated to learn features per point. We benchmark our network on a variety of 3D recognition tasks including action recognition, semantic segmentation and scene flow estimation. MeteorNet shows stronger performance than previous grid-based methods while achieving state-of-the-art performance on Synthia. MeteorNet also outperforms previous baseline methods that are able to process at most two consecutive point clouds. To the best of our knowledge, this is the first work on deep learning for dynamic raw point cloud sequences.

PointNetGPD: Detecting Grasp Configurations from Point Sets

Abstract: In this paper, we propose an end-to-end grasp evaluation model to address the challenging problem of localizing robot grasp configurations directly from the point cloud. Compared to recent grasp evaluation metrics that are based on handcrafted depth features and a convolutional neural network (CNN), our proposed PointNetGPD is lightweight and can directly process the 3D point cloud that locates within the gripper for grasp evaluation. Taking the raw point cloud as input, our proposed grasp evaluation network can capture the complex geometric structure of the contact area between the gripper and the object even if the point cloud is very sparse. To further improve our proposed model, we generate a large-scale grasp dataset with 350k real point cloud and grasps with the YCB object set for training. The performance of the proposed model is quantitatively measured both in simulation and on robotic hardware. Experiments on object grasping and clutter removal show that our proposed model generalizes well to novel objects and outperforms state-of-the-art methods. Code and video are available at https://lianghongzhuo.github.io/PointNetGPD.

Nonrigid Point Set Registration With Robust Transformation Learning Under Manifold Regularization

Abstract: This paper solves the problem of nonrigid point set registration by designing a robust transformation learning scheme. The principle is to iteratively establish point correspondences and learn the nonrigid transformation between two given sets of points. In particular, the local feature descriptors are used to search the correspondences and some unknown outliers will be inevitably introduced. To precisely learn the underlying transformation from noisy correspondences, we cast the point set registration into a semisupervised learning problem, where a set of indicator variables is adopted to help distinguish outliers in a mixture model. To exploit the intrinsic structure of a point set, we constrain the transformation with manifold regularization which plays a role of prior knowledge. Moreover, the transformation is modeled in the reproducing kernel Hilbert space, and a sparsity-induced approximation is utilized to boost efficiency. We apply the proposed method to learning motion flows between image pairs of similar scenes for visual homing, which is a specific type of mobile robot navigation. Extensive experiments on several publicly available data sets reveal the superiority of the proposed method over state-of-the-art competitors, particularly in the context of the degenerated data.

Hierarchical Point-Edge Interaction Network for Point Cloud Semantic Segmentation

Abstract: We achieve 3D semantic scene labeling by exploring semantic relation between each point and its contextual neighbors through edges. Besides an encoder-decoder branch for predicting point labels, we construct an edge branch to hierarchically integrate point features and generate edge features. To incorporate point features in the edge branch, we establish a hierarchical graph framework, where the graph is initialized from a coarse layer and gradually enriched along the point decoding process. For each edge in the final graph, we predict a label to indicate the semantic consistency of the two connected points to enhance point prediction. At different layers, edge features are also fed into the corresponding point module to integrate contextual information for message passing enhancement in local regions. The two branches interact with each other and cooperate in segmentation. Decent experimental results on several 3D semantic labeling datasets demonstrate the effectiveness of our work.

Hierarchical Point-Edge Interaction Network for Point Cloud Semantic Segmentation

Abstract: We achieve 3D semantic scene labeling by exploring semantic relation between each point and its contextual neighbors through edges. Besides an encoder-decoder branch for predicting point labels, we construct an edge branch to hierarchically integrate point features and generate edge features. To incorporate point features in the edge branch, we establish a hierarchical graph framework, where the graph is initialized from a coarse layer and gradually enriched along the point decoding process. For each edge in the final graph, we predict a label to indicate the semantic consistency of the two connected points to enhance point prediction. At different layers, edge features are also fed into the corresponding point module to integrate contextual information for message passing enhancement in local regions. The two branches interact with each other and cooperate in segmentation. Decent experimental results on several 3D semantic labeling datasets demonstrate the effectiveness of our work.

Modeling Local Geometric Structure of 3D Point Clouds Using Geo-CNN

Abstract: Recent advances in deep convolutional neural networks (CNNs) have motivated researchers to adapt CNNs to directly model points in 3D point clouds. Modeling local structure has been proven to be important for the success of convolutional architectures, and researchers exploited the modeling of local point sets in the feature extraction hierarchy. However, limited attention has been paid to explicitly model the geometric structure amongst points in a local region. To address this problem, we propose Geo-CNN, which applies a generic convolution-like operation dubbed as GeoConv to each point and its local neighborhood. Local geometric relationships among points are captured when extracting edge features between the center and its neighboring points. We first decompose the edge feature extraction process onto three orthogonal bases, and then aggregate the extracted features based on the angles between the edge vector and the bases. This encourages the network to preserve the geometric structure in Euclidean space throughout the feature extraction hierarchy. GeoConv is a generic and efficient operation that can be easily integrated into 3D point cloud analysis pipelines for multiple applications. We evaluate Geo-CNN on ModelNet40 and KITTI and achieve state-of-the-art performance.

Dynamic Points Agglomeration for Hierarchical Point Sets Learning

Abstract: Many previous works on point sets learning achieve excellent performance with hierarchical architecture. Their strategies towards points agglomeration, however, only perform points sampling and grouping in original Euclidean space in a fixed way. These heuristic and task-irrelevant strategies severely limit their ability to adapt to more varied scenarios. To this end, we develop a novel hierarchical point sets learning architecture, with dynamic points agglomeration. By exploiting the relation of points in semantic space, a module based on graph convolution network is designed to learn a soft points cluster agglomeration. We construct a hierarchical architecture that gradually agglomerates points by stacking this learnable and lightweight module. In contrast to fixed points agglomeration strategy, our method can handle more diverse situations robustly and efficiently. Moreover, we propose a parameter sharing scheme for reducing memory usage and computational burden induced by the agglomeration module. Extensive experimental results on several point cloud analytic tasks, including classification and segmentation, well demonstrate the superior performance of our dynamic hierarchical learning framework over current state-of-the-art methods.

Differentiable surface splatting for point-based geometry processing

Abstract: We propose Differentiable Surface Splatting (DSS), a high-fidelity differentiable renderer for point clouds. Gradients for point locations and normals are carefully designed to handle discontinuities of the rendering function. Regularization terms are introduced to ensure uniform distribution of the points on the underlying surface. We demonstrate applications of DSS to inverse rendering for geometry synthesis and denoising, where large scale topological changes, as well as small scale detail modifications, are accurately and robustly handled without requiring explicit connectivity, outperforming state-of-the-art techniques. The data and code are at https://github.com/yifita/DSS.

Narrowest‐over‐threshold detection of multiple change points and change‐point‐like features

Abstract: We propose a new, generic and flexible methodology for non-parametric function estimation, in which we first estimate the number and locations of any features that may be present in the function and then estimate the function parametrically between each pair of neighbouring detected features. Examples of features handled by our methodology include change points in the piecewise constant signal model, kinks in the piecewise linear signal model and other similar irregularities, which we also refer to as generalized change points. Our methodology works with only minor modifications across a range of generalized change point scenarios, and we achieve such a high degree of generality by proposing and using a new multiple generalized change point detection device, termed narrowest-over-threshold (NOT) detection. The key ingredient of the NOT method is its focus on the smallest local sections of the data on which the existence of a feature is suspected. For selected scenarios, we show the consistency and near optimality of the NOT algorithm in detecting the number and locations of generalized change points. The NOT estimators are easy to implement and rapid to compute. Importantly, the NOT approach is easy to extend by the user to tailor to their own needs. Our methodology is implemented in the R package not.

Dense 3D Point Cloud Reconstruction Using a Deep Pyramid Network

Abstract: Reconstructing a high-resolution 3D model of an object is a challenging task in computer vision. Designing scalable and light-weight architectures is crucial while addressing this problem. Existing point-cloud based reconstruction approaches directly predict the entire point cloud in a single stage. Although this technique can handle low-resolution point clouds, it is not a viable solution for generating dense, high-resolution outputs. In this work, we introduce DensePCR, a deep pyramidal network for point cloud reconstruction that hierarchically predicts point clouds of increasing resolution. Towards this end, we propose an architecture that first predicts a low-resolution point cloud, and then hierarchically increases the resolution by aggregating local and global point features to deform a grid. Our method generates point clouds that are accurate, uniform and dense. Through extensive quantitative and qualitative evaluation on synthetic and real datasets, we demonstrate that DensePCR outperforms the existing state-of-the-art point cloud reconstruction works, while also providing a light-weight and scalable architecture for predicting high-resolution outputs.

Deep Set Prediction Networks

Abstract: Current approaches for predicting sets from feature vectors ignore the unordered nature of sets and suffer from discontinuity issues as a result. We propose a general model for predicting sets that properly respects the structure of sets and avoids this problem. With a single feature vector as input, we show that our model is able to auto-encode point sets, predict the set of bounding boxes of objects in an image, and predict the set of attributes of these objects.

From Clustering to Cluster Explanations via Neural Networks.

Abstract: A wealth of algorithms have been developed to extract natural cluster structure in data. Identifying this structure is desirable but not always sufficient: We may also want to understand why the data points have been assigned to a given cluster. Clustering algorithms do not offer a systematic answer to this simple question. Hence we propose a new framework that can, for the first time, explain cluster assignments in terms of input features in a comprehensive manner. It is based on the novel theoretical insight that clustering models can be rewritten as neural networks, or 'neuralized'. Predictions of the obtained networks can then be quickly and accurately attributed to the input features. Several showcases demonstrate the ability of our method to assess the quality of learned clusters and to extract novel insights from the analyzed data and representations.

Deep Set Prediction Networks

Abstract: Current approaches for predicting sets from feature vectors ignore the unordered nature of sets and suffer from discontinuity issues as a result. We propose a general model for predicting sets that properly respects the structure of sets and avoids this problem. With a single feature vector as input, we show that our model is able to auto-encode point sets, predict the set of bounding boxes of objects in an image, and predict the set of attributes of these objects.

Point-Based Multi-View Stereo Network

Abstract: We introduce Point-MVSNet, a novel point-based deep framework for multi-view stereo (MVS). Distinct from existing cost volume approaches, our method directly processes the target scene as point clouds. More specifically, our method predicts the depth in a coarse-to-fine manner. We first generate a coarse depth map, convert it into a point cloud and refine the point cloud iteratively by estimating the residual between the depth of the current iteration and that of the ground truth. Our network leverages 3D geometry priors and 2D texture information jointly and effectively by fusing them into a feature-augmented point cloud, and processes the point cloud to estimate the 3D flow for each point. This point-based architecture allows higher accuracy, more computational efficiency and more flexibility than cost-volume-based counterparts. Experimental results show that our approach achieves a significant improvement in reconstruction quality compared with state-of-the-art methods on the DTU and the Tanks and Temples dataset. Our source code and trained models are available at this https URL .

Geometry-Aware Generation of Adversarial Point Clouds

Abstract: Machine learning models have been shown to be vulnerable to adversarial examples. While most of the existing methods for adversarial attack and defense work on the 2D image domain, a few recent attempts have been made to extend them to 3D point cloud data. However, adversarial results obtained by these methods typically contain point outliers, which are both noticeable and easy to defend against using the simple techniques of outlier removal. Motivated by the different mechanisms by which humans perceive 2D images and 3D shapes, in this paper we propose the new design of \emph{geometry-aware objectives}, whose solutions favor (the discrete versions of) the desired surface properties of smoothness and fairness. To generate adversarial point clouds, we use a targeted attack misclassification loss that supports continuous pursuit of increasingly malicious signals. Regularizing the targeted attack loss with our proposed geometry-aware objectives results in our proposed method, Geometry-Aware Adversarial Attack ($GeoA^3$). The results of $GeoA^3$ tend to be more harmful, arguably harder to defend against, and of the key adversarial characterization of being imperceptible to humans. While the main focus of this paper is to learn to generate adversarial point clouds, we also present a simple but effective algorithm termed $Geo_{+}A^3$-IterNormPro, with Iterative Normal Projection (IterNorPro) that solves a new objective function $Geo_{+}A^3$, towards surface-level adversarial attacks via generation of adversarial point clouds. We quantitatively evaluate our methods on both synthetic and physical objects in terms of attack success rate and geometric regularity. For a qualitative evaluation, we conduct subjective studies by collecting human preferences from Amazon Mechanical Turk. Comparative results in comprehensive experiments confirm the advantages of our proposed methods.

A Novel Simplification Method for 3D Geometric Point Cloud Based on the Importance of Point

Abstract: 3D point cloud simplification is an important pretreatment in surface reconstruction for sparing computer resources and improving reconstruction speed. However, existing methods often sacrifice the simplification precision to improve the simplification speed, or sacrifice the speed to improve precision. A proper balance between the simplification speed and the simplification accuracy is still a challenge. In this paper, we propose a new simplification method based on the importance of point. Named as detail feature points simplified algorithm (DFPSA), this algorithm has distinct processes to achieve improvements in three aspects. First, a rule of k neighborhood search is set to ensure the points found are the closest to the sample point. In this way, the accuracy of calculated normal vector of the point cloud is significantly improved, and the search speed is largely increased. Second, a formula that considers multiple characteristics for measuring the importance of point is proposed. Thereupon, the main detail features of the point cloud are preserved. Finally, an octree structure is employed to simplify the remaining points, through which holes in reconstructing point cloud are obviously reduced. The DFPSA is applied to four different data sets, and the corresponding results are compared with those of other five algorithms. The experimental results demonstrate that the DFPSA brings better simplification effects than existing counterparts, and the DFPSA not only can simplify point cloud but also has good effect in simplifying subject's narrow contours.

Tightly-Coupled Aided Inertial Navigation with Point and Plane Features

Abstract: This paper presents a tightly-coupled aided inertial navigation system (INS) with point and plane features, a general sensor fusion framework applicable to any visual and depth sensor (e.g., RGBD, LiDAR) configuration, in which the camera is used for point feature tracking and depth sensor for plane extraction. The proposed system exploits geometrical structures (planes) of the environments and adopts the closest point (CP) for plane parameterization. Moreover, we distinguish planar point features from non-planar point features in order to enforce point-on-plane constraints which are used in our state estimator, thus further exploiting structural information from the environment. We also introduce a simple but effective plane feature initialization algorithm for feature-based simultaneous localization and mapping (SLAM). In addition, we perform online spatial calibration between the IMU and the depth sensor as it is difficult to obtain this critical calibration parameter in high precision. Both Monte-Carlo simulations and real-world experiments are performed to validate the proposed approach.

Non-Rigid Point Set Registration Networks.

Abstract: Point set registration is defined as a process to determine the spatial transformation from the source point set to the target one. Existing methods often iteratively search for the optimal geometric transformation to register a given pair of point sets, driven by minimizing a predefined alignment loss function. In contrast, the proposed point registration neural network (PR-Net) actively learns the registration pattern as a parametric function from a training dataset, consequently predict the desired geometric transformation to align a pair of point sets. PR-Net can transfer the learned knowledge (i.e. registration pattern) from registering training pairs to testing ones without additional iterative optimization. Specifically, in this paper, we develop novel techniques to learn shape descriptors from point sets that help formulate a clear correlation between source and target point sets. With the defined correlation, PR-Net tends to predict the transformation so that the source and target point sets can be statistically aligned, which in turn leads to an optimal spatial geometric registration. PR-Net achieves robust and superior performance for non-rigid registration of point sets, even in presence of Gaussian noise, outliers, and missing points, but requires much less time for registering large number of pairs. More importantly, for a new pair of point sets, PR-Net is able to directly predict the desired transformation using the learned model without repetitive iterative optimization routine. Our code is available at this https URL.

MVPNet: Multi-View Point Regression Networks for 3D Object Reconstruction from A Single Image

Abstract: In this paper, we address the problem of reconstructing an object’s surface from a single image using generative networks. First, we represent a 3D surface with an aggregation of dense point clouds from multiple views. Each point cloud is embedded in a regular 2D grid aligned on an image plane of a viewpoint, making the point cloud convolution-favored and ordered so as to fit into deep network architectures. The point clouds can be easily triangulated by exploiting connectivities of the 2D grids to form mesh-based surfaces. Second, we propose an encoder-decoder network that generates such kind of multiple view-dependent point clouds from a single image by regressing their 3D coordinates and visibilities. We also introduce a novel geometric loss that is able to interpret discrepancy over 3D surfaces as opposed to 2D projective planes, resorting to the surface discretization on the constructed meshes. We demonstrate that the multi-view point regression network outperforms state-of-the-art methods with a significant improvement on challenging datasets.

Visual-Inertial Odometry with Point and Line Features

Abstract: In this paper, we present a tightly-coupled monocular visual-inertial navigation system (VINS) using points and lines with degenerate motion analysis for 3D line triangulation. Based on line segment measurements from images, we propose two sliding window based 3D line triangulation algorithms and compare their performance. Analysis of the proposed algorithms reveals 3 degenerate camera motions that cause triangulation failures. Both geometrical interpretation and Monte-Carlo simulations are provided to verify these degenerate motions which prevent triangulation. In addition, commonly used line representations are compared through a monocular visual SLAM Monte-Carlo simulation. Finally, real-world experiments are conducted to validate the implementation of the proposed VINS system using the “closest point” line representation.