# On reconstructing n-point configurations from the distribution of distances or areas

@article{Boutin2004OnRN,
title={On reconstructing n-point configurations from the distribution of distances or areas},
author={Mireille Boutin and Gregor Kemper},
journal={ArXiv},
year={2004},
volume={math.AC/0304192}
}
• Published 15 April 2003
• Mathematics, Computer Science
• ArXiv
One way to characterize configurations of points up to congruence is by considering the distribution of all mutual distances between points. This paper deals with the question if point configurations are uniquely determined by this distribution. After giving some counterexamples, we prove that this is the case for the vast majority of configurations. In the second part of the paper, the distribution of areas of sub-triangles is used for characterizing point configurations. Again it turns out… Expand
83 Citations

#### Figures and Topics from this paper

Which Point Configurations Are Determined by the Distribution of their Pairwise Distances?
• Mathematics, Computer Science
• Int. J. Comput. Geom. Appl.
• 2007
This paper focuses on the planar case m = 2 and presents a reconstructibility test with running time O(n11), and the cases of orientation preserving rigid motions (rotations and translations) and scalings are discussed. Expand
On Reconstructing Configurations of Points in ℙ2 from a Joint Distribution of Invariants
• Mathematics, Computer Science
• Applicable Algebra in Engineering, Communication and Computing
• 2004
A set of generators for the invariant field of the combined group Σn×PGL3 is found and a reconstruction principle for point configurations in �’2 from their sub-configurations of five points is obtained. Expand
Labeling Isometric and Almost Isometric n-Point Configurations in R
Consider two n-point configurations X and Y in R with the same distance distribution and distinct distances. We begin with proposing an algorithm that tries to find a bijection ψ : X → Y such that XExpand
On the use of Gromov-Hausdorff Distances for Shape Comparison
These reformulations render these distances more amenable to practical computations without sacrificing theoretical underpinnings, and establish links to several other practical methods proposed in the literature for comparing/matching shapes in precise terms. Expand
Isometries and Equivalences Between Point Configurations, Extended To $\varepsilon$-diffeomorphisms
• Mathematics
• 2017
In this paper, we deal with the Orthogonal Procrustes Problem, in which two point configurations are compared in order to construct a map to optimally align the two sets. This extends this toExpand
Reconstructing Point Sets From Distance Distributions
• Computer Science, Mathematics
• IEEE Transactions on Signal Processing
• 2021
This paper is the first practical approach to solve the large-scale noisy beltway problem where the points lie on a loop using projected gradient descent with a suitable spectral initializer and is robust to noise in the measurements. Expand
Geometrical ambiguity of pair statistics: point configurations.
• Mathematics, Medicine
• Physical review. E, Statistical, nonlinear, and soft matter physics
• 2010
It is shown that pair information is indeed insufficient to uniquely determine the configuration in general, including the reconstruction of atomic structures from experimentally obtained g(2) and a recently proposed "decorrelation" principle. Expand
Distributions of distances and volumes of balls in homogeneous lens spaces
• Mathematics
• 2020
Lens spaces are a family of manifolds that have been a source of many interesting phenomena in topology and differential geometry. Their concrete construction, as quotients of odd-dimensional spheresExpand
On Matching Point Configurations
• 2012
We present an algorithm that verifies if two unlabeled configurations of N points in Rd are or are not an orthogonal transformation of one another, and if applicable, explicitly compute thatExpand
Lossless Representation of Graphs using Distributions
• Computer Science, Mathematics
• ArXiv
• 2007
This paper shows that a large number of graphs are completely determined, up to isomorphism, by the distribution of their sub-triangles, and proposes graph representations in terms of one-dimensional distributions. Expand

#### References

SHOWING 1-10 OF 23 REFERENCES
Algebraic invariants of graphs; a study based on computer exploration
It is shown that some particular sets are not generating, disproving aconjecture of Pouzet related to reconstruction, as well as a lemma of Grigoriev on the invariant ring over digraphs, which provides a very simple minimal generating set of the field ofinvariants. Expand
Invariants ofS4 and the shape of sets of vectors
• Mathematics, Computer Science
• Applicable Algebra in Engineering, Communication and Computing
• 2005
We study a representation ofSn that is related to the shape of sets of vectors in ℝn. We want to determine the invariants of this representation, and obtain a complete description for the case ofS4.
A characteristic free approach to invariant theory
• Mathematics
• 1976
In this paper we treat that portion of classical invariant theory which goes under the name of "first" and "second" fundamental theorem for the classical groups, in a characteristic free way, i.e.,Expand
Computational Invariant Theory
• Computer Science
• 2002
The second edition of this book provides a major update and covers many recent developments in the field of invariant theory, covering fields as disparate as graph theory, coding theory, dynamical systems, and computer vision. Expand
Geometric invariance in computer vision
• Mathematics
• 1992
Part 1 Foundations: algebraic invariants - invariant theory and enumerative combinatorics of young tableaux, Shreeram S. Abhyankar, geometric interpretation of joint conic invariants, Joseph L.Expand
Multiple View Geometry in Computer Vision
• B. Wrobel
• Computer Science
• Künstliche Intell.
• 2001
This book is referred to read because it is an inspiring book to give you more chance to get experiences and also thoughts and it will show the best book collections and completed collections. Expand
The Magma Algebra System I: The User Language
• Computer Science, Mathematics
• J. Symb. Comput.
• 1997
The MAGMA language is presented, the design principles and theoretical background are outlined, and the constructors for structures, maps, and sets are outlined. Expand
Quelques remarques sur les résultats de Tutte concernant le problème de Ulam
© Université de Lyon, 1977, tous droits réservés. L’accès aux archives de la série « Publications du Département de mathématiques de Lyon » implique l’accord avec les conditions généralesExpand
Multiple View Geometry in Computer
• 2001