Set Of Spatial Coordinates Research Articles

The quaternion-based spatial-coordinate and orientation-frame alignment problems.

The general problem of finding a global rotation that transforms a given set of spatial coordinates and/or orientation frames (the `test' data) into the best possible alignment with a corresponding set (the `reference' data) is reviewed. For 3D point data, this `orthogonal Procrustes problem' is often phrased in terms of minimizing a root-mean-square deviation (RMSD) corresponding to a Euclidean distance measure relating the two sets of matched coordinates. This article focuses on quaternion eigensystem methods that have been exploited to solve this problem for at least five decades in several different bodies of scientific literature, where they were discovered independently. While numerical methods for the eigenvalue solutions dominate much of this literature, it has long been realized that the quaternion-based RMSD optimization problem can also be solved using exact algebraic expressions based on the form of the quartic equation solution published by Cardano in 1545; focusing on these exact solutions exposes the structure of the entire eigensystem for the traditional 3D spatial-alignment problem. The structure of the less-studied orientation-data context is then explored, investigating how quaternion methods can be extended to solve the corresponding 3D quaternion orientation-frame alignment (QFA) problem, noting the interesting equivalence of this problem to the rotation-averaging problem, which also has been the subject of independent literature threads. The article concludes with a brief discussion of the combined 3D translation-orientation data alignment problem. Appendices are devoted to a tutorial on quaternion frames, a related quaternion technique for extracting quaternions from rotation matrices and a review of quaternion rotation-averaging methods relevant to the orientation-frame alignment problem. The supporting information covers novel extensions of quaternion methods to the 4D Euclidean spatial-coordinate alignment and 4D orientation-frame alignment problems, some miscellaneous topics, and additional details of the quartic algebraic eigenvalue problem.

Topological data analysis quantifies biological nano-structure from single molecule localization microscopy.

MotivationLocalization microscopy data is represented by a set of spatial coordinates, each corresponding to a single detection, that form a point cloud. This can be analyzed either by rendering an image from these coordinates, or by analyzing the point cloud directly. Analysis of this type has focused on clustering detections into distinct groups which produces measurements such as cluster area, but has limited capacity to quantify complex molecular organization and nano-structure.ResultsWe present a segmentation protocol which, through the application of persistence-based clustering, is capable of probing densely packed structures which vary in scale. An increase in segmentation performance over state-of-the-art methods is demonstrated. Moreover we employ persistent homology to move beyond clustering, and quantify the topological structure within data. This provides new information about the preserved shapes formed by molecular architecture. Our methods are flexible and we demonstrate this by applying them to receptor clustering in platelets, nuclear pore components, endocytic proteins and microtubule networks. Both 2D and 3D implementations are provided within RSMLM, an R package for pointillist-based analysis and batch processing of localization microscopy data.Availability and implementationRSMLM has been released under the GNU General Public License v3.0 and is available at https://github.com/JeremyPike/RSMLM. Tutorials for this library implemented as Binder ready Jupyter notebooks are available at https://github.com/JeremyPike/RSMLM-tutorials.Supplementary information Supplementary data are available at Bioinformatics online.

A generalised framework for super-resolution track-weighted imaging

Track-density imaging (TDI) was recently introduced as a method to achieve super-resolution imaging using whole-brain fibre-tracking data (the so called tractogram). A similar approach to achieve super-resolution was later applied for average pathlength mapping (APM). These two methods have in common that the tractogram information is used to create an image with novel contrast and super-resolution properties. In this study, we present a generalised framework for creating super-resolution track-weighted imaging (TWI), where the intensity of the map can be made dependent on any specific property of the streamlines or their set of spatial coordinates. Furthermore, each contrast can be determined by a number of characteristics that are under user control. It is shown that TDI and APM represent specific cases of this generalised framework, and that this framework opens up the possibility of generating a large range of images with novel image contrasts. Finally, it is shown that the same super-resolution principles as those introduced in the original TDI method are also applicable to any of these new images.

Thermodynamic and Differential Entropy under a Change of Variables

The differential Shannon entropy of information theory can change under a change of variables (coordinates), but the thermodynamic entropy of a physical system must be invariant under such a change. This difference is puzzling, because the Shannon and Gibbs entropies have the same functional form. We show that a canonical change of variables can, indeed, alter the spatial component of the thermodynamic entropy just as it alters the differential Shannon entropy. However, there is also a momentum part of the entropy, which turns out to undergo an equal and opposite change when the coordinates are transformed, so that the total thermodynamic entropy remains invariant. We furthermore show how one may correctly write the change in total entropy for an isothermal physical process in any set of spatial coordinates.

Object-Centered Neglect in Patients with Unilateral Neglect: Effects of Left-Right Coordinates of Objects

Abstract When patients with right-sided hemispheric lesions neglect information on the left side, with respect to what set of spatial coordinates is left defined? Two potential reference frames were examined in this study, one where left and right are defined with respect to the midline of the viewer and/or environment (viewer/env-centered) and the other where left and right are defined with respect to the midline of the object (object-centered). By rotating the stimulus 90° clockwise or counterclockwise, and instructing patients with neglect to report the colors appearing around the border of a stimulus, an independent measure was obtained for the number of colors reported from the left and right of the viewer/env- and from the object-based reference frame. Whereas significant object-centered neglect was observed only for upper case asymmetrical letters but not for symmetrical letters nor for drawings of familiar animals or objects, significant viewer/env-based neglect was observed with all the stimulus types. We present an account of the coexistence of neglect in more than one frame of reference and the presence of object-centered neglect under a restricted set of conditions.

Maximum-likelihood estimation of hypocenter with origin time eliminated using nonlinear inversion technique

A new algorithm is applied to inverting arrival time data for hypocenter location. The algorithm incorporates both observed and prior data from a Bayesian point of view. We define marginal probability density function (pdf) to eliminate the origin time from the location problem; the posterior pdf of hypocenter parameters is integrated over the whole range of the origin time. The best estimate of the hypocenter is defined as a set of spatial coordinates which maximizes the marginal pdf. Assuming Gaussian errors in both observed and prior data, we obtain a simple algorithm. Estimation errors of parameters are evaluated by an asymptotic covariance matrix, with which an asymptotic posterior pdf is computed. The algorithm is applied to observed data and is tested. An example of analysis is given for aftershocks of the 1969 Gifuken-chubu earthquake ( M = 6.6) reported by the Japan Meteorological Agency (JMA). The spatial distribution of the aftershocks is supposed to be Gaussian with standard deviation of 15 km. A center of the aftershock distribution, which gives the prior estimates of hypocenters, is also estimated from observed data. Results of the nonlinear inversion of arrival time data are examined in terms of the asymptotic posterior pdf. We found that relocated hypocenters of the aftershocks are concentrated in a narrow region of 2–3 km in width, while the hypocenters previously reported by JMA have a wide distribution of 5–7 km.