A Distance Metric for Finite Sets of Rigid-Body Displacements via the Polar Decomposition

2006 ◽  
Vol 129 (8) ◽  
pp. 883-886 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray ◽  
Jorge Angeles
Keyword(s):  
Singular Value Decomposition ◽  
Rigid Body ◽  
Polar Decomposition ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Coordinate Frames ◽  
Open Research ◽  
Value Decomposition

An open research question is how to define a useful metric on the special Euclidean group SE(n) with respect to: (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances that is useful for the synthesis and analysis of mechanical systems. We discuss a technique for approximating elements of SE(n) with elements of the special orthogonal group SO(n+1). This technique is based on using the singular value decomposition (SVD) and the polar decompositions (PD) of the homogeneous transform representation of the elements of SE(n). The embedding of the elements of SE(n) into SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements. The bi-invariant metric on SO(n+1) is then used to measure the distance between any two displacements. The result is a left invariant PD based metric on SE(n).

Projection Metrics for Rigid-Body Displacements

2005 ◽  
Author(s):  
Pierre M. Larochelle ◽  
Andrew P. Murray
Keyword(s):  
Rigid Body ◽  
Robot Control ◽  
Research Question ◽  
Singular Value ◽  
Invariant Metric ◽  
Euclidean Group ◽  
Robot Calibration ◽  
Coordinate Frames ◽  
Open Research ◽  
Spatial Displacements

An open research question is how to define a useful metric on SE(n) with respect to (1) the choice of coordinate frames and (2) the units used to measure linear and angular distances. We present two techniques for approximating elements of the special Euclidean group SE(n) with elements of the special orthogonal group SO(n+1). These techniques are based on the singular value and polar decompositions (denoted as SVD and PD respectively) of the homogeneous transform representation of the elements of SE(n). The projection of the elements of SE(n) onto SO(n+1) yields hyperdimensional rotations that approximate the rigid-body displacements (hence the term projection metric. A bi-invariant metric on SO(n+1) may then be used to measure the distance between any two spatial displacements. The results are PD and SVD based projection metrics on SE(n). These metrics have applications in motion synthesis, robot calibration. motion interpolation, and hybrid robot control.

A Displacement Metric for Finite Sets of Rigid Body Displacements

2008 ◽  
Author(s):  
Venkatesh Venkataramanujam ◽  
Pierre Larochelle
Keyword(s):  
Euclidean Space ◽  
Rigid Body ◽  
Orthogonal Group ◽  
Polar Decomposition ◽  
Coordinate Frame ◽  
Invariant Metric ◽  
Motion Generation ◽  
Euclidean Group ◽  
Finite Set ◽  
Potential Applications

There are various useful metrics for finding the distance between two points in Euclidean space. Metrics for finding the distance between two rigid body locations in Euclidean space depend on both the coordinate frame and units used. A metric independent of these choices is desirable. This paper presents a metric for a finite set of rigid body displacements. The methodology uses the principal frame (PF) associated with the finite set of displacements and the polar decomposition to map the homogenous transform representation of elements of the special Euclidean group SE(N-1) onto the special orthogonal group SO(N). Once the elements are mapped to SO(N) a bi-invariant metric can then be used. The metric obtained is thus independent of the choice of fixed coordinate frame i.e. it is left invariant. This metric has potential applications in motion synthesis, motion generation and interpolation. Three examples are presented to illustrate the usefulness of this methodology.

Improving TF-IDF with Singular Value Decomposition (SVD) for Feature Extraction on Twitter

2017 ◽  
Author(s):  
Ammar Ismael Kadhim ◽  
Yu-N Cheah ◽  
Inaam Abbas Hieder ◽  
Rawaa Ahmed Ali
Keyword(s):  
Feature Extraction ◽  
Singular Value Decomposition ◽  
Singular Value ◽  
Value Decomposition

Multi-scale singular value decomposition polarization image fusion defogging algorithm and experiment

2020 ◽  
Vol 13 (6) ◽  
pp. 1-10
Author(s):  
ZHOU Wen-zhou ◽  
◽  
FAN Chen ◽  
HU Xiao-ping ◽  
HE Xiao-feng ◽  
...  
Keyword(s):  
Singular Value Decomposition ◽  
Image Fusion ◽  
Singular Value ◽  
Multi Scale ◽  
Polarization Image ◽  
Value Decomposition
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