Centroid

About: Centroid is a research topic. Over the lifetime, 4110 publications have been published within this topic receiving 53637 citations. The topic is also known as: barycenter (geometry) & geometric center of a plane figure.

Generalized procrustes analysis

Abstract: SupposePi(i) (i = 1, 2, ...,m, j = 1, 2, ...,n) give the locations ofmn points inp-dimensional space. Collectively these may be regarded asm configurations, or scalings, each ofn points inp-dimensions. The problem is investigated of translating, rotating, reflecting and scaling them configurations to minimize the goodness-of-fit criterion Σi=1m Σi=1n Δ2(Pj(i)Gi), whereGi is the centroid of them pointsPi(i) (i = 1, 2, ...,m). The rotated positions of each configuration may be regarded as individual analyses with the centroid configuration representing a consensus, and this relationship with individual scaling analysis is discussed. A computational technique is given, the results of which can be summarized in analysis of variance form. The special casem = 2 corresponds to Classical Procrustes analysis but the choice of criterion that fits each configuration to the common centroid configuration avoids difficulties that arise when one set is fitted to the other, regarded as fixed.

Point Set Registration: Coherent Point Drift

Abstract: 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.

Closed-form solution of absolute orientation using orthonormal matrices

Abstract: Finding the relationship between two coordinate systems by using pairs of measurements of the coordinates of a number of points in both systems is a classic photogrammetric task. The solution has applications in stereophotogrammetry and in robotics. We present here a closed-form solution to the least-squares problem for three or more points. Currently, various empirical, graphical, and numerical iterative methods are in use. Derivation of a closed-form solution can be simplified by using unit quaternions to represent rotation, as was shown in an earlier paper [ J. Opt. Soc. Am. A4, 629 ( 1987)]. Since orthonormal matrices are used more widely to represent rotation, we now present a solution in which 3 × 3 matrices are used. Our method requires the computation of the square root of a symmetric matrix. We compare the new result with that obtained by an alternative method in which orthonormality is not directly enforced. In this other method a best-fit linear transformation is found, and then the nearest orthonormal matrix is chosen for the rotation. We note that the best translational offset is the difference between the centroid of the coordinates in one system and the rotated and scaled centroid of the coordinates in the other system. The best scale is equal to the ratio of the root-mean-square deviations of the coordinates in the two systems from their respective centroids. These exact results are to be preferred to approximate methods based on measurements of a few selected points.

Advances in Geometric Morphometrics

Abstract: Geometric morphometrics is the statistical analysis of form based on Cartesian landmark coordinates. After separating shape from overall size, position, and orientation of the landmark configurations, the resulting Procrustes shape coordinates can be used for statistical analysis. Kendall shape space, the mathematical space induced by the shape coordinates, is a metric space that can be approximated locally by a Euclidean tangent space. Thus, notions of distance (similarity) between shapes or of the length and direction of developmental and evolutionary trajectories can be meaningfully assessed in this space. Results of statistical techniques that preserve these convenient properties—such as principal component analysis, multivariate regression, or partial least squares analysis—can be visualized as actual shapes or shape deformations. The Procrustes distance between a shape and its relabeled reflection is a measure of bilateral asymmetry. Shape space can be extended to form space by augmenting the shape coordinates with the natural logarithm of Centroid Size, a measure of size in geometric morphometrics that is uncorrelated with shape for small isotropic landmark variation. The thin-plate spline interpolation function is the standard tool to compute deformation grids and 3D visualizations. It is also central to the estimation of missing landmarks and to the semilandmark algorithm, which permits to include outlines and surfaces in geometric morphometric analysis. The powerful visualization tools of geometric morphometrics and the typically large amount of shape variables give rise to a specific exploratory style of analysis, allowing the identification and quantification of previously unknown shape features.

Deep Hough Voting for 3D Object Detection in Point Clouds

Abstract: Current 3D object detection methods are heavily influenced by 2D detectors. In order to leverage architectures in 2D detectors, they often convert 3D point clouds to regular grids (i.e., to voxel grids or to bird's eye view images), or rely on detection in 2D images to propose 3D boxes. Few works have attempted to directly detect objects in point clouds. In this work, we return to first principles to construct a 3D detection pipeline for point cloud data and as generic as possible. However, due to the sparse nature of the data -- samples from 2D manifolds in 3D space -- we face a major challenge when directly predicting bounding box parameters from scene points: a 3D object centroid can be far from any surface point thus hard to regress accurately in one step. To address the challenge, we propose VoteNet, an end-to-end 3D object detection network based on a synergy of deep point set networks and Hough voting. Our model achieves state-of-the-art 3D detection on two large datasets of real 3D scans, ScanNet and SUN RGB-D with a simple design, compact model size and high efficiency. Remarkably, VoteNet outperforms previous methods by using purely geometric information without relying on color images.