Choice Of Local Coordinate System Research Articles

Cervical spine intervertebral kinematics estimated from inverse kinematics and compared to dynamic X-ray data

Interpretation of shoulder motion across studies has been complicated due to the use of numerous scapular coordinate systems in the literature. Currently, there are no simple means by which to compare scapular kinematics between coordinate system definitions when data from only one coordinate system is known.How do scapular kinematics vary based on the choice of coordinate system and can average rotation matrices be used to accurately convert kinematics between scapular local coordinate systems?Average rotation matrices derived from anatomic landmarks of 51 cadaver scapulae (29 M/22 F; 59 ± 13 yrs; 26R/25 L; 171 ± 11 cm; 70 ± 19 kg; 23.7 ± 5.5 kg/m2) were generated between three common scapular coordinate systems. Absolute angle of rotation was used to determine if anatomical variability within the cadaver population influenced the matrices. To quantify the predictive capability to convert kinematics between the three coordinate systems, the average rotation matrices were applied to scapulothoracic motion data collected from 19 human subjects (10 M/9 F; 43 ± 17 yrs; 19R; 173 ± 9 cm; 71 ± 16 kg; 23.6 ± 4.5 kg/m2) using biplane fluoroscopy. Root mean squared error (RMSE) was used to compare kinematics from an original coordinate system to the kinematics expressed in each alternative coordinate system.The choice of scapular coordinate system resulted in mean differences in scapulothoracic rotation of up to 23°, with overall different shapes and/or magnitudes of the curves. A single average rotation matrix between any two coordinate systems achieved accurate conversion of scapulothoracic kinematics to within 4° of RMSE of the known solution. The average rotation matrices were independent of sex, side, decomposition sequence, and motion.Scapulothoracic kinematic representations vary in shape and magnitude based solely on the choice of local coordinate system. The results of this study enhance interpretability and reproducibility in expressing scapulothoracic motion data between laboratories by providing a simple means to convert data between common coordinate systems. This is necessitated by the variety of available motion analysis techniques and their respective scapular landmark definitions.

Reliable interpretation of scapular kinematics depends on coordinate system definition

BackgroundInterpretation of shoulder motion across studies has been complicated due to the use of numerous scapular coordinate systems in the literature. Currently, there are no simple means by which to compare scapular kinematics between coordinate system definitions when data from only one coordinate system is known. Research questionHow do scapular kinematics vary based on the choice of coordinate system and can average rotation matrices be used to accurately convert kinematics between scapular local coordinate systems? MethodsAverage rotation matrices derived from anatomic landmarks of 51 cadaver scapulae (29 M/22 F; 59 ± 13 yrs; 26R/25 L; 171 ± 11 cm; 70 ± 19 kg; 23.7 ± 5.5 kg/m2) were generated between three common scapular coordinate systems. Absolute angle of rotation was used to determine if anatomical variability within the cadaver population influenced the matrices. To quantify the predictive capability to convert kinematics between the three coordinate systems, the average rotation matrices were applied to scapulothoracic motion data collected from 19 human subjects (10 M/9 F; 43 ± 17 yrs; 19R; 173 ± 9 cm; 71 ± 16 kg; 23.6 ± 4.5 kg/m2) using biplane fluoroscopy. Root mean squared error (RMSE) was used to compare kinematics from an original coordinate system to the kinematics expressed in each alternative coordinate system. ResultsThe choice of scapular coordinate system resulted in mean differences in scapulothoracic rotation of up to 23°, with overall different shapes and/or magnitudes of the curves. A single average rotation matrix between any two coordinate systems achieved accurate conversion of scapulothoracic kinematics to within 4° of RMSE of the known solution. The average rotation matrices were independent of sex, side, decomposition sequence, and motion. SignificanceScapulothoracic kinematic representations vary in shape and magnitude based solely on the choice of local coordinate system. The results of this study enhance interpretability and reproducibility in expressing scapulothoracic motion data between laboratories by providing a simple means to convert data between common coordinate systems. This is necessitated by the variety of available motion analysis techniques and their respective scapular landmark definitions.

An Efficient Method for Forming Parabolic Curves and Surfaces

A new method for the formation of parabolic curves and surfaces is proposed. It does not impose restrictions on the relative positions in space of the sequence of reference points relative to each other, meaning it compares favorably with other prototypes. The disadvantages of the Overhauser and Brever–Anderson methods are noted. The method allows one to effectively form and edit curves and surfaces when changing the coordinates of any given point. This positive effect is achieved due to the appropriate choice of local coordinate systems for the mathematical description of each parabola, which together define a composite interpolation curve or surface. The paper provides a detailed mathematical description of the method of parabolic interpolation of curves and surfaces based on the use of matrix calculations. Analytical descriptions of a composite parabolic curve and its first and second derivatives are given, and continuity analysis of these factors is carried out. For the matrix of points of the defining polyhedron, expressions are presented that describe the corresponding surfaces, as well as the unit normal at any point. The comparative table of the required number of pseudo-codes for calculating the coordinates of one point for constructing a parabolic curve for the three methods is given.

Riemannian competitive learning for symmetric positive definite matrices clustering

Abstract Symmetric positive definite (SPD) matrices have achieved considerable success in numerous computer vision applications including activity recognition, texture classification, and diffusion tensor imaging. Traditional pattern recognition methods developed in Euclidean space are not suitable for direct processing of SPD matrices because they lie in a Riemannian manifold of negative curvature. In this paper, we draw inspiration from neural network based simple competitive learning, which updates the weight of a neural network in Euclidean space, and propose Riemannian competitive learning which updates the weight in the Riemannian weight space, for SPD manifold clustering. This framework is based on the Fisher-Rao metric which is theoretically very good. We further introduce a conscious competition mechanism to enhance the performance of the RCL algorithm and have developed a robust algorithm termed Riemannian Frequency Sensitive Competitive Learning (rFSCL). Compared with existing methods, the advantages are threefold: (1) rFSCL inherits the online nature of competitive learning, which makes it capable of handling very large data sets; (2) rFSCL inherits the advantage of conscious competitive learning, which indicates that it is less sensitive to the initial values of the cluster centers and that all clusters are fully utilized without the “dead unit” problem associated with many clustering algorithms; and (3) As an intrinsic Riemannian clustering method, rFSCL operates along the geodesic on the manifold in an closed-form analytic manner and is completely independent of the choice of local coordinate systems. Extensive experiments on both synthetic data sets and real data sets show its superior performance to that of other state-of-the-art SPD clustering methods.

Gauge invariant 1PI effective superstring field theory: inclusion of the Ramond sector

We construct off-shell amplitudes in heterotic and type II string theories involving arbitrary combination of Ramond and Neveu-Schwarz sector external states. We also construct the equations of motion of a gauge invariant 1PI effective field theory which reproduces these off-shell amplitudes. Using this construction we prove that the renormalized physical masses do not depend on the choice of local coordinate system and locations of picture changing operators used in defining the off-shell amplitudes. We also use this formalism to examine the conditions under which space-time supersymmetry is unbroken in the quantum theory.

Gauge invariant 1PI effective action for superstring field theory

We construct gauge invariant 1PI effective action for the NS sector of type II and heterotic string field theory. By construction, zero eigenvalues of the kinetic operator of this action determine the renormalized physical masses, and tree level amplitudes computed from this action (after gauge fixing) give the loop corrected S-matrix elements. Using this formalism we can give a simple proof of the result that the renormalized physical masses do not depend on the choice of local coordinate system and locations of picture changing operators used in defining the off-shell amplitude. We also eliminate the need for an infrared regulator in dealing with tadpoles of massless fields.

Large Deformation Analysis of Orthodontic Appliances

The deformations of orthodontic appliances used for space closure are large so that any mathematical analysis will require a nonlinear approach. Existing incremental finite element and finite difference numerical methods suffer from excessive computational effort when analyzing these problems. An accurate segmental technique is proposed to handle these difficulties in an extremely efficient fashion. The segmental technique starts by assuming that an orthodontic appliance is composed of a number of smaller segments, the ends of which undergo small relative rotation. With an appropriate choice of local coordinate system the equilibrium equations for each segment are linearized and solved in a straightforward manner. The segments are then assembled using geometric and force compatibility relations similar to the transfer matrix method. Consequently, the original nonlinear boundary value problem is solved as a sequence of linear initial value problems which converge to the required boundary conditions. As only one segment need be considered at a time, the computations can be performed accurately and efficiently on a PC type computer. Although an iterative solution is used to match the boundary conditions, the time required to solve a given problem ranges from a few seconds to a couple of minutes depending on the initial geometric complexity. The accuracy of the segmental technique is verified by comparison with an exact solution for an initially curved cantilever beam with an end load. In addition, comparisons are made with existing experimental and numerical results as well as with a new set of experimental data. In all cases the segmental technique is in excellent agreement with the results of these other studies.

The Role of the Tangent Mapping in Analyzing Bifurcation Behaviour

AbstractIn the study of solution manifolds of parameter‐dependent nonlinear equations extended systems of equations play an important role, especially for the computation of singular points, such as turning points, bifurcation points, etc. Various extended systems have been proposed in the literature. Here it is shown that a central feature in the construction of extended systems is the tangent map of differential geometry. A theory of extended equations based on the tangent map is presented which also exhibits the close connection with the choice of local coordinate systems. The ideas and results are illustrated with an example of a continuously stirred chemical reactor.

Invariance properties of the multipole expansion

AbstractThe calculations of long‐range interaction energy are often based on multipole expansion. The truncated multipole expansion and interaction energy calculated with it are noninvariant with respect to an arbitrary choice of local coordinate systems. In this paper we show that truncated multipole expansion of form Σ CkR−k is “numerically” independent on a choice of local coordinate systems, if convergence conditions are satisfied.

Invariance properties of the multipole expansion with respect to the choice of the coordinate system

AbstractThe physical interpretation of intermolecular interactions is usually based on the well‐known multipole expansion of the inverse of the interparticle distance. The interaction energy is then interpreted as a sum of terms arising from the interaction of various multipole moments of both systems. It is supposed that the interaction energy calculated via the truncated multipole expansion generally depends on the choice of local coordinate systems through the coordinate dependence of the multipole moments. In this paper we prove that each term of the multipole expansion given in the form ∑k = 1 Ck/Rk is invariant with respect to identical translations and arbitrary rotations of the local coordinate systems. The invariant form of the convergence criterion of the multipole expansion is given and discussed.