Soft shape registration under Lie group frame

we present a method based on a locally affineinvariant constraint for volumetric registration of 3D solid shapes. The core idea of this method is that an affine combination of the given point in 3D solid shapes that are directly connected to the given point, and the corresponding weight of each neighboring point can be obtained by the method of generalized least square. The input of our method is a pair of 3D solid shapes that are represented by a tetrahedral mesh and a voxelized object consisting of a set of voxel cells segmented from Digital Imaging and Communications in Medicine(DICOM) scans. To achieve the registration between the two input DICOM images, we firstly need to do some preprocessing to segment the bones out from the DICOM images and represent the segmented healthy bone and lesion bone using a generic template tetrahedral mesh and a set of voxel cells, respectively. Secondly we apply the standard Iterative Closest Point (ICP) method to briefly align the tetrahedral mesh and the voxelized object. Thirdly we execute a novel registration process that uses as much volumetric information and local geometry information as possible while deforming the generic template tetrahedral mesh of a healthy human bone towards the undelying geometry of the voxel cells. Compared with the previous methods that are based on point or surface, our method requires less auxiliary variables and can better capture the volumetric information of the 3D solid shapes, such as the thickness of the bones. Besides that, using a tetrahedral mesh to represent a solid shape can make the precision of registration greatly improved. Our experimental results demonstrate that the proposed method is robust and is of high registration accuracy.

This paper introduces an improved approach for the volume data registration of human retina. Volume data registration refers to calculating out a near-optimal transformation between two volumes with overlapping region and stitching them together. Iterative closest point (ICP) algorithm is a registration method that deals with registration between points. Classical ICP is time consuming and often traps in local minimum when the overlapping region is not big enough. Optical Coherence Tomography (OCT) volume data are several separate, partially overlapping tiles. To stitch them together is a technology in computer aided diagnosis. In this paper, a new 3D registration algorithm based on improved ICP is presented. First, the Canny edge detector is applied to generate the point cloud set of OCT images. After the detection step, an initial registration method based on the feature points of the point cloud is proposed to determine an initial transformation matrix by using singular value decomposition (SVD) method. Then, an improved ICP method is presented to accomplish fine registration. Corresponding point in the point cloud is weighted to reduce the iteration times of ICP algorithm. Finally, M-estimation is used as the objective function to decrease the impact of outliers. This registration algorithm is used to process human retinal OCT volume pairs that contain an overlapping region of 75ź?ź500ź?ź375 voxels approximately. Then a comparative experiment is conducted on some public-available datasets. The experimental results show that the proposed method outperforms the classical method.

Lie groups are very important objects in algebra, topology and geometry, as it is naturally endowed with two special structures: the algebraic structure of group and the geometric structure of differential manifold. Hence, it is quite meaningful to better know the structure of Lie groups and the actions, and the transformations between them. Two specific theorems about Lie groups and their homomorphism and Lie homomorphism are proved in this paper. First, the fact that there does not exist a surjective Lie group homomorphism between and is introduced. This is done by first having a trial of constructing Lie group surjective homomorphism but fails. Next, this result is further proved by applying topological method that to construct a covering map between and , followed by computing the fundamental group of . Finally, by applying that is a simple group, the fact that there does not exist a surjective group homomorphism between and is demonstrated, together with the statement that is a simple group.

Automatic detection of feather defects using Lie group and fuzzy Fisher criterion for shuttlecock production

As a fundamental process of three-dimensional lidar point cloud data (3D LPCD) processing, numerous registration methods are time consuming and easily fall into local optimum. A 3D LPCD registration method based on the iterative closest point (ICP) algorithm, which is improved by the Gaussian mixture model (GMM) considering corner features, is proposed in this article to address these limitations. The GMM method is used for coarse registration, and the input original 3D LPCD is replaced by corner features extracted by the improved 3D Harris algorithm to improve the efficiency of coarse registration. In addition, a satisfactory initial position between the reference and the moving 3D LPCD is prepared for ICP fine registration by coarse registration; thus, the accuracy of fine registration can be improved. The registration accuracy and efficiency of the new method is proved to be higher than those of four common ICP-based registration methods (3DSC-RANSACICP, 3DSC-SAC-IAICP, FPFH-RANSACICP, and FPFH-SAC-IAICP), and GMM registration methods, and the local optimum problem is effectively addressed.

Shape registration plays an important role in applications such as 3D object modeling or object recognition. The iterative closest point (ICP) algorithm is widely used for the registration of geometric data. One of its main drawback is its time complexity O(N2), quadratic with the shape size N, which implies long processing time, especially when using high resolution data. Several methods were proposed to accelerate the process. One of the most effective one uses a tree search (k-D tree) to establish closest point relationships and reduces the complexity to O(N logN). This paper reviews several of the existing methods and proposes and analyses a new, even less complex ICP algorithm, that uses a heuristic approach to find the closest points. Based on a local search it permits to reduce the complexity to O(N) and to greatly accelerate the process. A comprehensive analysis and a comparison of the considered algorithm with a tree search method are presented.

The iterative closest point (ICP) algorithm is widely used in three-dimensional (3D) point cloud registration, and it is very stable and robust. However, its biggest drawback is being easily trapped in a local optimal solution, which results in the incorrect registration result. Currently, there is neither a clear effective range to define whether the ICP algorithm will fall into a local optimum nor a study providing a comprehensive evaluation of the ICP algorithm. In this paper, we take the overlap ratio, angle, distance, and noise as the influencing factors of ICP and evaluate the validity, robustness, accuracy, and efficiency of point-to-point and point-to-plane ICP by using four datasets. We first analyze the effective ranges of the two ICP algorithms with respect to overlap ratio, angle, and distance and then propose a universal effective range for the three factors. Next, the effect of Gaussian noise on the validity and accuracy of two ICPs is evaluated. We also analyze the factors influencing ICP accuracy and explain their changing rules. We finally study the effect of different parameters on the efficiency. All results are compared by using point-to-point and point-to-plane ICP algorithms. The results show that the overlap ratio has no effect on validity, but it has a significant influence on accuracy. The angle has a great impact on the validity and efficiency of ICP but has no effect on accuracy. The distance only affects validity, has a limited effect on efficiency and no effect on accuracy. Meanwhile, Gaussian noise has a little effect on the validity. In addition, the general effective range of point-to-point ICP is larger than that of point-to-plane ICP, but the point-to-plane ICP algorithm presents a better efficiency. The point-to-point ICP is more robust to Gaussian noise with respect to validity, while the point-to-plane is more resistant in terms of accuracy.

It has been known since (Pillay, J. Pure Appl. Algebra 53 (1988), no. 3, 239–255)that any group definable in an o $o$ -minimal expansion of the real field can be equipped with a Lie group structure. It is therefore natural to ask when is a Lie group Lie isomorphic to a group definable in such an expansion. Conversano, Starchenko and the first author answered this question in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452) in the case when the group is solvable. This paper answers similar questions in more general contexts. We first give a complete classification in the case when the group is linear. Specifically, a linear Lie group G $G$ is Lie isomorphic to a group definable in an o $o$ -minimal expansion of the reals if and only if its solvable radical has the same property. We then deal with the general case of a connected Lie group, although unfortunately, we cannot achieve a full characterization. Assuming that a Lie group G $G$ has its Levi subgroups with finite center, we prove that in order for G $G$ to be Lie isomorphic to a definable group it is necessary and sufficient that its solvable radical satisfies the conditions given in (Conversano, Onshuus, and Starchenko, J. Inst. Math. Jussieu 17 (2018), no. 2, 441–452).

With the rapid development of binocular reconstruction, fringe projection profilometry, and time of flight, 3D imaging technology has been widely applied in the field of 3D measurement. However, due to limited measurement range and self-occlusion, point cloud registration methods are often used to obtain larger or more complete 3D contours. Although many scholars have proposed various point cloud registration methods, the accuracy and efficiency of point cloud registration still need to be further improved, especially for point clouds with different density or non-rigid transformation. Image registration technology based on image correlation has been developed for many years and has achieved great success in fields such as computer vision, photomechanics, and photogrammetry. Therefore, a simple and direct idea in this paper is to transform the point cloud registration problem into volume image correlation problem. By this, an efficient image registration method based on fast Fourier transform and an inverse compositional Gaussian Newton optimization method that only needs to calculate the Hessian matrix once can be introduced into the point cloud registration field, which can greatly improve the speed and accuracy of point cloud registration. Comparative experiments have shown that our method has doubled the accuracy and efficiency compared to the iterative closest point (ICP) method, and its practicality has also been verified in impeller reconstruction experiments.

We construct an infinite dimensional real analytic manifold structure for the space of real analytic mappings from a compact manifold to a locally convex manifold. Here a map is real analytic if it extends to a holomorphic map on some neighbourhood of the complexification of its domain. As is well known the construction turns the group of real analytic diffeomorphisms into a smooth locally convex Lie group. We prove then that the diffeomorphism group is regular in the sense of Milnor. In the inequivalent "convenient setting of calculus" the real analytic diffeomorphisms even form a real analytic Lie group. However, we prove that the Lie group structure on the group of real analytic diffeomorphisms is in general not real analytic in our sense.

We investigate a natural Lie group structure that correlations possess, interpreted as a special case of the quotient manifold structure for correlation matrices. The one‐dimensional setting of the general theory gives rise to additional structure in the form of a natural associative multiplication of correlations making the space of correlations into a Lie group. We explicitly compute left‐invariant vector fields, exponential and logarithmic mappings, and ultimately, a closed‐form distance formula between correlations. The Lie group formalism is then applied to an application in anomalous correlation detection. The advantage of the Lie group method illustrated with time series data of stock prices.

This paper explores the problem of position and speed estimation of a target using satellite-based measurements in a Lie Group Kalman-like filter. The filter employs an intrinsic formulation of a nearly constant velocity model based on the Frenet-Serret frame, embedded in a Lie Group structure. It stands as a more suitable model for the kinematics of a target in space than the usual Euclidean model since it induces a banana-shaped distribution instead of the typical ellipsoidal provided by Gaussian schemes. The paper presents numeric experiments using a real application dataset to evaluate the proposed filtering scheme’s performance against the standard Euclidean representation approach. A differential GNSS solution provided by commercial software furnishes the ground truth. The results indicate a better performance of the proposed Lie Groups filtering scheme.

The survey deals with the investigations reviewed inReferativnyi Zhurnal “Matematika” between 1977–1981. In the survey there are reflected the investigations on the structure of Lie groups and Lie algebras, on their finite-dimensional linear representations and universal enveloping algebras, on the theory of invariants and Lie groups of transformations, and also on continuous and discrete subgroups of Lie groups.

A bounded iterative closest point method for minimally invasive registration of the femur

Surface registration is an inevitable step for solving the correspondence problem in statistical shape modeling and useful for automatic 3D segmentation of objects of a certain shape. Common techniques for a pairwise surface registration rely either on a point-based or a mesh-based representation of the source (moving) and the target (fixed) surface. Iterative closest point (ICP) algorithms operate on unstructured point clouds of surfaces. In the so-called EM-ICP methods, they assume a Gaussian mixture model (GMM) for the registration in order to efficiently estimate the optimal transform and the correspondence probabilities by an iterative expectation-maximization (EM) algorithm. However, point-based techniques cannot preserve the topology of the source surface. Common mesh-based techniques employ different elastic deformations to estimate a parametric model of the optimal registration transform. Both the point-based and the mesh-based schemes can lead to a deformed mesh with flipped or irregular faces. In this paper, we propose a pairwise nonlinear surface registration algorithm that processes both the point clouds and the triangular meshes of surfaces within a unified framework by combing a modified symmetric robust EM-ICP (SR-EM-ICP) algorithm with a mesh optimization mechanism. It improves the registration accuracy and the quality of the deformed mesh in a processing time comparable to the SR-EM-ICP algorithm. These are achieved by an efficient estimation of the inhomogeneous variances of the GMM and the optimal registration transform, reducing the distance between the surfaces and an explicit recovery of the flipped or irregular faces of the deformed mesh in each iteration.