Linear approximation method for weighted rigid-body registration
Abstract
The Mahalanobis distance of the composite noise in the point data is used to construct an optimization objective function to improve the accuracy of rigid-body registration. The method gives more reliable points and directions larger weights to convert the unweighted registration into a weighted problem. A linear approxi- mation method is used to solve this problem. The translation vector is expressed as a function of the rotation matrix. Constraints on the rotation matrix are eliminated by approximating the rotation matrix with its Lie algebra. The optimal solution is obtained by iteration. Results of three groups of simulations show that this method gives more accurate registration than three other methods and is suitable for applications where high accuracy is required.
The Procrustes criterion is a common measure for the distance between two matricesX andY, and can be interpreted as the sum of squares of the Euclidean distances between their respective column vectors. Often a weighted Procrustes criterion, using, for example, a weighted sum of the squared distances between the column vectors, is called for. This paper describes and analyzes the performance of an algorithm for rotating a matrixX such that the column-weighted Procrustes distance toY is minimized. The problem of rotatingX intoY such that an aggregate measure of Tucker's coefficient of congruence is maximized is also discussed.
. We discuss optimal rotation estimation from two sets of 3-D points in the presence of anisotropic and inhomogeneous noise. We first present a theoretical accuracy bound and then give a method that attains that bound, which can be viewed as describing the reliability of the solution. We also show that an efficient computational scheme can be obtained by using quaternions and applying renormalization. Using real stereo images for 3-D reconstruction, we demonstrate that our method is superior to the least-squares method and confirm the theoretical predictions of our theory by applying the bootstrap procedure. 1 Introduction Determining a rotational relationship between two sets of 3-D points is an important task for 3-D object reconstruction and recognition. For example, if we use stereo vision or range sensing, the 3-D shape can be reconstructed only for visible surfaces. Hence, we need to fuse separately reconstructed surfaces into one object [13]. For this task, we need to determine...
The weighted orthogonal Procrustes problem, an important class of data matching problems in multivariate data analysis, is reconsidered in this paper. It is shown that a steepest descent flow on the manifold of orthogonal matrices can naturally be formulated. This formulation has two important implications: that the weighted orthogonal Procrustes problem can be solved as an initial value problem by any available numerical integrator and that the first order and the second order optimality conditions can also be derived. The proposed approach is illustrated by numerical examples.
This paper describes a new algorithm for estimating the position and orientation of objects. The problem is formulated as an optimization problem using dual number quaternions. The advantage of using this representation is that the method solves for the location estimate by minimizing a single cost function associated with the sum of the orientation and position errors and thus is expected to have a better performance on the estimation, both in accuracy and in speed. Several forms of sensory information can be used by the algorithm. That is, the measured data can be a combination of measured points on an object's surfaces and measured unit direction vectors located on the object. Simulations have been carried out on a Compaq 386/20 computer and the simulation results are analyzed.
We present and validate a novel registration algorithm mapping two data sets, generated from a rigid object, in the presence of Gaussian noise. The proposed method is based on the Unscented Kalman Filter (UKF) algorithm that is generally employed for analyzing nonlinear systems corrupted by additive Gaussian noise. First, we employ our proposed registration algorithm to fit two randomly generated data sets in the presence of isotropic Gaussian noise, when the corresponding points between the two data sets are assumed to be known. Then, we extend the registration method to the case where the data (with known correspondences) is stimulated by anisotropic Gaussian noise. The new registration method not only reliably converges to the correct registration solution, but it also estimates the variance, as a confidence measure, for each of the estimated registration transformation parameters. Furthermore, we employ the proposed registration algorithm for rigid-body, point-based registration where corresponding points between two registering data sets are unknown. The algorithm is tested on point data sets which are garnered from a pelvic cadaver and a scaphoid bone phantom by means of computed tomography (CT) and tracked free-hand ultrasound imaging. The collected 3-D points in the ultrasound frame are registered to the 3-D meshes in the CT frame by using the proposed and the standard Iterative Closest Points (ICP) registration algorithms. Experimental results demonstrate that our proposed method significantly outperforms the ICP registration algorithm in the presence of additive Gaussian noise. It is also shown that the proposed registration algorithm is more robust than the ICP registration algorithm in terms of outliers in data sets and initial misalignment between the two data sets.
The authors present a weighted geometrical feature (WGF)
registration algorithm. Its efficacy is demonstrated by combining points
and a surface. The technique is an extension of Besl and McKay's (1992)
iterative closest point (ICP) algorithm. The authors use the WGF
algorithm to register X-ray computed tomography (CT) and T2-weighted
magnetic resonance (MR) volume head images acquired from eleven patients
that underwent craniotomies in a neurosurgical clinical trial. Each
patient had five external markers attached to transcutaneous posts
screwed into the outer table of the skull. The authors define
registration error as the distance between positions of corresponding
markers that are not used for registration. The CT and MR images are
registered using fiducial paints (marker positions) only, a surface
only, and various weighted combinations of points and a surface. The CT
surface is derived from contours corresponding to the inner surface of
the skull. The MR surface is derived from contours corresponding to the
cerebrospinal fluid (CSF)-dura interface. Registration using points and
a surface is found to be significantly more accurate then registration
using only points or a surface
In this paper, we propose and analyze several methods to estimate a rigid transformation from a set of 3-D matched points or matched frames, which are important features in geometric algorithms. We also develop tools to predict and verify the accuracy of these estimations. The theoretical contributions are: an intrinsic model of noise for transformations based on composition rather than addition; a unified formalism for the estimation of both the rigid transformation and its covariance matrix for points or frames correspondences, and a statistical validation method to verify the error estimation, which applies even when no "ground truth" is available. We analyze and demonstrate on synthetic data that our scheme is well behaved. The practical contribution of the paper is the validation of our transformation estimation method in the case of 3-D medical images, which shows that an accuracy of the registration far below the size of a voxel can be achieved, and in the case of protein substructure matching, where frame features drastically improve both selectivity and complexity. 1.