Object Completion using k‐Sparse Optimization

We present a new method for the completion of partial globally‐symmetric 3D objects, based on the detection of partial and approximate symmetries in the incomplete input dataset. In our approach, symmetry detection is formulated as a constrained sparsity maximization problem, which is solved efficiently using a robust RANSAC‐based optimizer. The detected partial symmetries are then reused iteratively, in order to complete the missing parts of the object. A global error relaxation method minimizes the accumulated alignment errors and a non‐rigid registration approach applies local deformations in order to properly handle approximate symmetry. Unlike previous approaches, our method does not rely on the computation of features, it uniformly handles translational, rotational and reflectional symmetries and can provide plausible object completion results, even on challenging cases, where more than half of the target object is missing. We demonstrate our algorithm in the completion of 3D scans with varying levels of partiality and we show the applicability of our approach in the repair and completion of heavily eroded or incomplete cultural heritage objects.     

A method for registration of 3-D shapes

The authors describe a general-purpose, representation-independent method for the accurate and computationally efficient registration of 3-D shapes including free-form curves and surfaces. The method handles the full six degrees of freedom and is based on the iterative closest point (ICP) algorithm, which requires only a procedure to find the closest point on a geometric entity to a given point. The ICP algorithm always converges monotonically to the nearest local minimum of a mean-square distance metric, and the rate of convergence is rapid during the first few iterations. Therefore, given an adequate set of initial rotations and translations for a particular class of objects with a certain level of `shape complexity', one can globally minimize the mean-square distance metric over all six degrees of freedom by testing each initial registration. One important application of this method is to register sensed data from unfixtured rigid objects with an ideal geometric model, prior to shape inspection. Experimental results show the capabilities of the registration algorithm on point sets, curves, and surfaces     

Registering multiview range data to create 3D computer objects

Concerns the problem of range image registration for the purpose of building surface models of 3D objects. The registration task involves finding the translation and rotation parameters which properly align overlapping views of the object so as to reconstruct from these partial surfaces, an integrated surface representation of the object. The registration task is expressed as an optimization problem. We define a function which measures the quality of the alignment between the partial surfaces contained in two range images as produced by a set of motion parameters. This function computes a sum of Euclidean distances from control points on one surfaces to corresponding points on the other. The strength of this approach is in the method used to determine point correspondences. It reverses the rangefinder calibration process, resulting in equations which can be used to directly compute the location of a point in a range image corresponding to an arbitrary point in 3D space. A stochastic optimization technique, very fast simulated reannealing (VFSR), is used to minimize the cost function. Dual-view registration experiments yielded excellent results in very reasonable time. A multiview registration experiment took a long time. A complete surface model was then constructed from the integration of multiple partial views. The effectiveness with which registration of range images can be accomplished makes this method attractive for many practical applications where surface models of 3D objects must be constructed     

Point set registration: coherent point drift

Point set registration is a key component in many computer vision tasks. The goal of point set registration is to assign correspondences between two sets of points and to recover the transformation that maps one point set to the other. Multiple factors, including an unknown nonrigid spatial transformation, large dimensionality of point set, noise, and outliers, make the point set registration a challenging problem. We introduce a probabilistic method, called the Coherent Point Drift (CPD) algorithm, for both rigid and nonrigid point set registration. We consider the alignment of two point sets as a probability density estimation problem. We fit the Gaussian mixture model (GMM) centroids (representing the first point set) to the data (the second point set) by maximizing the likelihood. We force the GMM centroids to move coherently as a group to preserve the topological structure of the point sets. In the rigid case, we impose the coherence constraint by reparameterization of GMM centroid locations with rigid parameters and derive a closed form solution of the maximization step of the EM algorithm in arbitrary dimensions. In the nonrigid case, we impose the coherence constraint by regularizing the displacement field and using the variational calculus to derive the optimal transformation. We also introduce a fast algorithm that reduces the method computation complexity to linear. We test the CPD algorithm for both rigid and nonrigid transformations in the presence of noise, outliers, and missing points, where CPD shows accurate results and outperforms current state-of-the-art methods.     

Simultaneous nonrigid registration of multiple point sets and atlas construction

Group-wise registration of a set of shapes represented by unlabeled point-sets is a challenging problem since, usually this involves solving for point correspondence in a nonrigid motion setting. In this paper, we propose a novel and robust algorithm that is capable of simultaneously computing the mean shape represented by a probability density function from multiple unlabeled point-sets and registering them non-rigidly to this emerging mean shape. This algorithm avoids the correspondence problem by minimizing the Jensen-Shannon (JS) divergence between the point sets. We motivate the use of the JS divergence by pointing out its close relationship to hypothesis testing. We derive the analytic gradient of the cost function in order to efficiently achieve the optimal solution. JS-divergence is symmetric with no bias toward any of the given shapes to be registered and whose mean is being sought. A by product of the registration process is a probabilistic atlas defined as the convex combination of the probability densities of the input point sets being aligned. Our algorithm can be especially useful for creating atlases of various shapes present in images as well as for simultaneously (rigidly or non-rigidly) registering 3D range data sets without having to establish any correspondence. We present experimental results on real and synthetic data.     

结合方向包围框改进的三维点云配准算法

为了提升源点云和模板点云在初始相对偏转角度过大时的配准精度,提出了一种结合方向包围框的改进 PointNetLK算法PointNetLK-OBB。该算法用三维点云的方向包围框表示源点云和模板点云的宏观特征,在最近点迭代算法的引导下,对齐源点云和模板点云的方向包围框,并在源点云和模板点云间产生镜面对称效应;根据源点云和模板点云的拟合度探测镜面对称的对称面,得到源点云自身的最佳旋转和平移,完成三维点云配准任务。为了验证算法的有效性,在公开数据集ModelNet40上进行对比实验,实验结果显示,PointNetLK-OBB与PointNetLK相比,提升了源点云和模板点云在初始相对偏转角度过大时的配准精度,对源点云和模板点云间的初始相对位置敏感度降低。创新在于,利用PointNetLK绕开传统点云配准的非凸问题,借助于方向包围框的规整性避开PointNetLK语境下出现的局部最优问题。     

Multiple Solutions in Euler’s Elastic Problem

The paper is devoted to multiple solutions of the classical problem on stationary configurations of an elastic rod on a plane; we describe boundary values for which there are more than two optimal configurations of a rod (optimal elasticae). We define sets of points where three or four optimal elasticae come together with the same value of elastic energy. We study all configurations that can be translated into each other by symmetries, i.e., reflections at the center of the elastica chord and reflections at the middle perpendicular to the elastica chord. For the first symmetry, the ends of the rod are directed in opposite directions, and the corresponding boundary values lie on a disk. For the second symmetry, the boundary values lie on a Möbius strip. As a result, we study both sets numerically and in some cases analytically; in each case, we find sets of points with several optimal configurations of the rod. These points form the currently known part of the reachability set where elasticae lose global optimality.     

深度学习刚性点云配准前沿进展

随着3维采集设备的日渐推广,点云配准在越来越多的领域得到应用。然而,传统方法在低重叠、大量噪声、多异常点和大场景等方面表现不佳,这限制了点云配准在真实场景中的应用。面对传统方法的局限性,结合深度学习技术的点云配准方法开始出现,本文将这种方法称为深度点云配准,并对深度点云配准方法研究进展予以综述。首先,根据有无对应关系对目前的深度学习点云配准方法进行区分,分为无对应关系配准和基于对应关系的点云配准。针对基于对应关系的配准,根据各类方法的主要功能进行详细的分类与总结,其中包括几何特征提取、关键点检测、点对离群值去除、姿态估计和端到端配准,并重点介绍了最新出现的一些方法;针对无对应配准方法,详细介绍了各类方法的特点并对无对应与有对应方法的特点进行了总结。在性能评估中,首先对现有主要的评价指标进行了详细的分类与总结,给出其适用场景。对于真实数据集,给出了特征匹配、点对离群值去除的对比数据,并进行了总结。在合成数据集中,给出了相关方法在部分重叠、实时性和全局配准场景下的对比数据。最后讨论了当前深度点云配准面临的挑战并给出对未来研究方向的展望。     

逐步求精的多视角点云配准方法

针对多视角点云配准问题,本文设计了一个合理的目标函数,便于将多视角配准问题分解成多个双视角配准问题,并考虑了两个要素:1）各帧点云均具有其他所有点云所未覆盖的区域;2）基准帧点云的重要程度高于其他点云.为了求解该目标函数,本文提出了逐步求精的解决策略:根据给定的配准初值构造初始模型,依次取出基准帧以外的每帧点云,利用所提出的双视角配准算法计算该帧点云的配准参数,并修正模型,以便进一步计算后续点云的配准参数.遍历完全部点云构成一次完整的循环,多次循环后可获得精确的多视角配准结果.公开数据集上的实验结果表明,本文所提出的方法能够精确、可靠地实现多视角点云配准.     

On the representations of L-equivalence relations on L-fuzzy sets with applications to locally vague environments

ABSTRACT

L-equivalence relations on L-fuzzy sets, equivalent to L-valued equalities on ordinary sets, are modelling fuzzy similarity of two objects that are partial members of a fuzzily defined set. This paper deals with construction and representation properties of L-equivalence relations on L-fuzzy sets. To this end, we set up a method for constructing them by means of quasi-pseudo-metrics, and then point out how L-equivalence relations on L-fuzzy sets are represented by families of L-fuzzy subsets of L-fuzzy sets. The present results are applied to locally vague environments to elucidate conceptual structures of locally vague environments, and to show how the points in a locally vague environment are described.     

A robust and outlier-adaptive method for non-rigid point registration

In this paper, a flexible probabilistic method is introduced for non-rigid point registration, which is motivated by the pioneering research named Coherent Point Drift (CPD). Being different from CPD, our algorithm is robust and outlier-adaptive, which does not need prior information about data such as the appropriate outlier ratio when the point sets are perturbed by outliers. We consider the registration as the alignment of the data (one point set) to a set of Gaussian Mixture Model centroids (the other point set), and initially formulate it as maximizing the likelihood problem, then the problem is solved under Expectation–Maximization (EM) framework. The outlier ratio is also formulated in EM framework and will be updated during the EM iteration. Moreover, we use the volume of the point set region to determine the uniform distribution for modeling the outliers. The resulting registration algorithm exhibits inherent statistical robustness and has an explicit interpretation. The experiments demonstrate that our algorithm outperforms the state-of-the-art method.     

Global Correspondence Optimization for Non‐Rigid Registration of Depth Scans

We present a registration algorithm for pairs of deforming and partial range scans that addresses the challenges of non‐rigid registration within a single non‐linear optimization. Our algorithm simultaneously solves for correspondences between points on source and target scans, confidence weights that measure the reliability of each correspondence and identify non‐overlapping areas, and a warping field that brings the source scan into alignment with the target geometry. The optimization maximizes the region of overlap and the spatial coherence of the deformation while minimizing registration error. All optimization parameters are chosen automatically; hand‐tuning is not necessary. Our method is not restricted to part‐in‐whole matching, but addresses the general problem of partial matching, and requires no explicit prior correspondences or feature points. We evaluate the performance and robustness of our method using scan data acquired by a structured light scanner and compare our method with existing non‐rigid registration algorithms.     

HDRNet: High-Dimensional Regression Network for Point Cloud Registration

Abstract-3D point cloud registration is a crucial topic in the reverse engineering, computer vision and robotics fields. The core of this problem is to estimate a transformation matrix for aligning the source point cloud with a target point cloud. Several learning-based methods have achieved a high performance. However, they are challenged with both partial overlap point clouds and multiscale point clouds, since they use the singular value decomposition (SVD) to find the rotation matrix without fully considering the scale information. Furthermore, previous networks cannot effectively handle the point clouds having large initial rotation angles, which is a common practical case. To address these problems, this paper presents a learning-based point cloud registration network, namely HDRNet, which consists of four stages: local feature extraction, correspondence matrix estimation, feature embedding and fusion and parametric regression. HDRNet is robust to noise and large rotation angles, and can effectively handle the partial overlap and multi-scale point clouds registration. The proposed model is trained on the ModelNet40 dataset, and compared with ICP, SICP, FGR and recent learning-based methods (PCRNet, IDAM, RGMNet and GMCNet) under several settings, including its performance on moving to invisible objects, with higher success rates. To verify the effectiveness and generality of our model, we also further tested our model on the Stanford 3D scanning repository.     

Semi‐isometric Registration of Line Features for Flexible Fitting of Protein Structures

In this paper, we study a registration problem that is motivated by a practical biology problem – fitting protein structures to low‐re solution density maps. We consider registration between two sets of lines features (e.g., helices in the proteins) that have undergone not a single, but multiple isometric transformations (e.g., hinge‐motions). The problem is further complicated by the presence of symmetry in each set. We formulate the problem as a clique‐finding problem in a product graph, and propose a heuristic solution that includes a fast clique‐finding algorithm unique to the structure of this graph. When tested on a suite of real protein structures, the algorithm achieved high accuracy even for very large inputs containing hundreds of helices.     

Registration of 3D Points Using Geometric Algebra and Tensor Voting

We address the problem of finding the correspondences of two point sets in 3D undergoing a rigid transformation. Using these correspondences the motion between the two sets can be computed to perform registration. Our approach is based on the analysis of the rigid motion equations as expressed in the Geometric Algebra framework. Through this analysis it was apparent that this problem could be cast into a problem of finding a certain 3D plane in a different space that satisfies certain geometric constraints. In order to find this plane in a robust way, the Tensor Voting methodology was used. Unlike other common algorithms for point registration (like the Iterated Closest Points algorithm), ours does not require an initialization, works equally well with small and large transformations, it cannot be trapped in “local minima” and works even in the presence of large amounts of outliers. We also show that this algorithm is easily extended to account for multiple motions and certain non-rigid or elastic transformations.     

采用树表示的全局最优形状匹配算法

提出一种全局优化算法，用于相似不变地在一场景中匹配一个形状。该算法采用支撑树来表示形状，匹配问题被转化成在目标点集中定位这棵树的问题。通过最小化边的空间变换同一个全局空间变换之间的差别，树的每条边的空间变换被强制是一致的。目标函数归结为一个关于边匹配变量的凹二次函数。该函数具有低秩Hessian矩阵，可以通过分支定界法快速地解出。还提出一种新颖的求下界的方案，它可以通过动态规划高效地解出。实验结果表明，所提算法相比主流算法有更好的鲁棒性，特别对于两点集只有部分重叠的情形。     

A symmetry based multiobjective clustering technique for automatic evolution of clusters

In this paper the problem of automatic clustering a data set is posed as solving a multiobjective optimization (MOO) problem, optimizing a set of cluster validity indices simultaneously. The proposed multiobjective clustering technique utilizes a recently developed simulated annealing based multiobjective optimization method as the underlying optimization strategy. Here variable number of cluster centers is encoded in the string. The number of clusters present in different strings varies over a range. The points are assigned to different clusters based on the newly developed point symmetry based distance rather than the existing Euclidean distance. Two cluster validity indices, one based on the Euclidean distance, XB-index, and another recently developed point symmetry distance based cluster validity index, Sym-index, are optimized simultaneously in order to determine the appropriate number of clusters present in a data set. Thus the proposed clustering technique is able to detect both the proper number of clusters and the appropriate partitioning from data sets either having hyperspherical clusters or having point symmetric clusters. A new semi-supervised method is also proposed in the present paper to select a single solution from the final Pareto optimal front of the proposed multiobjective clustering technique. The efficacy of the proposed algorithm is shown for seven artificial data sets and six real-life data sets of varying complexities. Results are also compared with those obtained by another multiobjective clustering technique, MOCK, two single objective genetic algorithm based automatic clustering techniques, VGAPS clustering and GCUK clustering.     

Planar Segment Based Three‐dimensional Point Cloud Registration in Outdoor Environments

We present an odometry‐free three‐dimensional (3D) point cloud registration strategy for outdoor environments based on area attributed planar patches. The approach is split into three steps. The first step is to segment each point cloud into planar segments, utilizing a cached‐octree region growing algorithm, which does not require the 2.5D image‐like structure of organized point clouds. The second step is to calculate the area of each segment based on small local faces inspired by the idea of surface integrals. The third step is to find segment correspondences between overlapping point clouds using a search algorithm, and compute the transformation from determined correspondences. The transformation is searched globally so as to maximize a spherical correlation‐like metric by enumerating solutions derived from potential segment correspondences. The novelty of this step is that only the area and plane parameters of each segment are employed, and no prior pose estimation from other sensors is required. Four datasets have been used to evaluate the proposed approach, three of which are publicly available and one that stems from our custom‐built platform. Based on these datasets, the following evaluations have been done: segmentation speed benchmarking, segment area calculation accuracy and speed benchmarking, processing data acquired by scanners with different fields of view, comparison with the iterative closest point algorithm, robustness with respect to occlusions and partial observations, and registration accuracy compared to ground truth. Experimental results confirm that the approach offers an alternative to state‐of‐the‐art algorithms in plane‐rich environments.     

Fully Automatic Registration of Image Sets on Approximate Geometry

The photorealistic acquisition of 3D objects often requires color information from digital photography to be mapped on the acquired geometry, in order to obtain a textured 3D model. This paper presents a novel fully automatic 2D/3D global registration pipeline consisting of several stages that simultaneously register the input image set on the corresponding 3D object. The first stage exploits Structure From Motion (SFM) on the image set in order to generate a sparse point cloud. During the second stage, this point cloud is aligned to the 3D object using an extension of the 4 Point Congruent Set (4PCS) algorithm for the alignment of range maps. The extension accounts for models with different scales and unknown regions of overlap. In the last processing stage a global refinement algorithm based on mutual information optimizes the color projection of the aligned photos on the 3D object, in order to obtain high quality textures. The proposed registration pipeline is general, capable of dealing with small and big objects of any shape, and robust. We present results from six real cases, evaluating the quality of the final colors mapped onto the 3D object. A comparison with a ground truth dataset is also presented.     

An accurate and fast point-to-plane registration technique

This paper addresses a registration refinement problem and presents an accurate and fast point-to-(tangent) plane technique. Point-to-plane approach is known to be very accurate for registration refinement of partial 3D surfaces. However, the computation complexity for finding the intersection point on a destination surface from a source control point is hindering the algorithm from real-time applications. We introduce a novel point-to-plane registration technique by combining the high-speed advantage of point-to-projection technique. In order to find the intersection point fast and accurately, we forward-project the source point to the destination surface and reproject the projection point to the normal vector of the source point. We show that iterative projections of the projected destination point to the normal vector converge to the intersection point. By assuming the destination surface to be a monotonic function in a new 2D coordinate system, we show contraction mapping properties of our iterative projection technique. Experimental results for several objects are presented for both pair-wise and multi-view registrations.