scholarly journals The quaternion-based spatial-coordinate and orientation-frame alignment problems

2020 ◽  
Vol 76 (4) ◽  
pp. 432-457
Author(s):  
Andrew J. Hanson
Keyword(s):  
Distance Measure ◽  
Orientation Data ◽  
Quartic Equation ◽  
Spatial Coordinate ◽  
Frame Alignment ◽  
Data Alignment ◽  
Spatial Alignment ◽  
Alignment Problem ◽  
Euclidean Distance Measure ◽  
Point Data

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.

scholarly journals PolyProc: A Modular Processing Pipeline for X-ray Diffraction Tomography

2019 ◽  
Vol 8 (3) ◽  
pp. 388-399 ◽  
Author(s):  
Jiwoong Kang ◽  
Ning Lu ◽  
Issac Loo ◽  
Nancy Senabulya ◽  
Ashwin J. Shahani
Keyword(s):  
Three Dimensional ◽  
Valuable Insight ◽  
Diffraction Tomography ◽  
Computationally Efficient ◽  
X Ray Diffraction ◽  
Orientation Data ◽  
Bulk Specimen ◽  
X Ray ◽  
Data Alignment ◽  
And Performance

Abstract Direct imaging of three-dimensional microstructure via X-ray diffraction-based techniques gives valuable insight into the crystallographic features that influence materials properties and performance. For instance, X-ray diffraction tomography provides information on grain orientation, position, size, and shape in a bulk specimen. As such techniques become more accessible to researchers, demands are placed on processing the datasets that are inherently “noisy,” multi-dimensional, and multimodal. To fulfill this need, we have developed a one-of-a-kind function package, PolyProc, that is compatible with a range of data shapes, from planar sections to time-evolving and three-dimensional orientation data. Our package comprises functions to import, filter, analyze, and visualize the reconstructed grain maps. To accelerate the computations in our pipeline, we harness computationally efficient approaches: for instance, data alignment is done via genetic optimization; grain tracking through the Hungarian method; and feature-to-feature correlation through k-nearest neighbors algorithm. As a proof-of-concept, we test our approach in characterizing the grain texture, topology, and evolution in a polycrystalline Al–Cu alloy undergoing coarsening.

Image Retrieval of Self-Adapt Distance Measure Based on SLLE

2014 ◽  
Vol 989-994 ◽  
pp. 3675-3678
Author(s):  
Xiao Fen Wang ◽  
Hai Na Zhang ◽  
Xiu Rong Qiu ◽  
Jiang Ping Song ◽  
Ke Xin Zhang
Keyword(s):  
Image Retrieval ◽  
High Dimension ◽  
Topological Structure ◽  
Euclidean Distance ◽  
Distance Measure ◽  
Content Based Image Retrieval ◽  
Locally Linear Embedding ◽  
Euclidean Distance Measure ◽  
Linear Embedding ◽  
Dimension Space

Self-adapt distance measure supervised locally linear embedding solves the problem that Euclidean distance measure can not apart from samples in content-based image retrieval. This method uses discriminative distance measure to construct k-NN and effectively keeps its topological structure in high dimension space, meanwhile it broadens interval of samples and strengthens the ability of classifying. Experiment results show the ADM-SLLE date-reducing-dimension method speeds up the image retrieval and acquires high accurate rate in retrieval.

The frame alignment problem in formations of multi-agent systems*

2013 ◽  
Vol 46 (10) ◽  
pp. 262-267
Author(s):  
Carlos H. Caicedo-Núñez ◽  
Sandor M. Veres
Keyword(s):  
Multi Agent Systems ◽  
Agent Systems ◽  
Frame Alignment ◽  
Alignment Problem ◽  
Multi Agent

A convolution-based distance measure for fuzzy singletons and its application in a pattern recognition problem

2020 ◽  
Vol 28 (1) ◽  
pp. 51-63 ◽  
Author(s):  
Rodrigo Naranjo ◽  
Matilde Santos ◽  
Luis Garmendia
Keyword(s):  
Pattern Recognition ◽  
Fuzzy Number ◽  
Euclidean Distance ◽  
Distance Measure ◽  
Distance Measures ◽  
Trading System ◽  
Weighted Distance ◽  
Pattern Recognition Problem ◽  
Euclidean Distance Measure ◽  
Desirable Behavior

A new method to measure the distance between fuzzy singletons (FSNs) is presented. It first fuzzifies a crisp number to a generalized trapezoidal fuzzy number (GTFN) using the Mamdani fuzzification method. It then treats an FSN as an impulse signal and transforms the FSN into a new GTFN by convoluting it with the original GTFN. In so doing, an existing distance measure for GTFNs can be used to measure distance between FSNs. It is shown that the new measure offers a desirable behavior over the Euclidean and weighted distance measures in the following sense: Under the new measure, the distance between two FSNs is larger when they are in different GTFNs, and smaller when they are in the same GTFN. The advantage of the new measure is demonstrated on a fuzzy forecasting trading system over two different real stock markets, which provides better predictions with larger profits than those obtained using the Euclidean distance measure for the same system.

scholarly journals Biases with the Generalized Euclidean Distance measure in disparity analyses with high levels of missing data

Palaeontology ◽  
2019 ◽  
Vol 62 (5) ◽  
pp. 837-849 ◽  
Author(s):  
Oscar E. R. Lehmann ◽  
Martín D. Ezcurra ◽  
Richard J. Butler ◽  
Graeme T. Lloyd
Keyword(s):  
Missing Data ◽  
Euclidean Distance ◽  
Distance Measure ◽  
Euclidean Distance Measure

scholarly journals Clustering of OECD Countries Out of Pocket Health Expenditure Time Series Data

2016 ◽  
Vol 8 (2) ◽  
pp. 23
Author(s):  
Songul Cinaroglu
Keyword(s):  
Euclidean Distance ◽  
Time Series Data ◽  
Distance Measure ◽  
Health Expenditures ◽  
Oecd Countries ◽  
Distance Measures ◽  
Series Data ◽  
Longest Common Subsequences ◽  
Study Results ◽  
Euclidean Distance Measure

<p>Out of pocket health expenditures points out to the payments made by households at the point<br />they receive health services. Frequently these include doctor consultation fees, purchase of<br />medication and hospital bills. In this study hierarchical clustering method was used for<br />classification of 34 countries which are members of OECD (Organization for Economic<br />Cooperation and Development) in terms of out of pocket health expenditures for the years<br />between 1995-2011. Longest common subsequences (LCS), correlation coefficient and<br />Euclidean distance measure was used as a measure of similarity and distance in hierarchical<br />clustering. At the end of the analysis it was found that LCS and Euclidean distance measures<br />were the best for determining clusters. Furthermore, study results led to understand grouping<br />of OECD countries according to health expenditures.</p>

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