[Paper Review] Determining Generic Point Configurations From Unlabeled Path or Loop Lengths

Ioannis Gkioulekas, Steven J. Gortler+2 more|arXiv (Cornell University)|Sep 12, 2017

Digital Image Processing Techniques36 references3 citations

TL;DR

This paper establishes conditions under which a generic configuration of $ n $ points in $ \mathbb{R}^d $ ($ d \geq 2 $) can be uniquely reconstructed from unlabeled path or loop lengths—i.e., without knowledge of which points were visited or in what order. The key contribution is a sufficient condition—'allows for trilateration'—that guarantees unique recovery up to Euclidean congruence, along with a real-computable algorithm for reconstruction using a new family of algebraic varieties called unsquared measurement varieties.

ABSTRACT

Let $\mathbf{p}$ be a configuration of $n$ points in $\mathbb{R}^d$ for some $n$ and some $d \ge 2$. Each pair of points defines an edge, which has a Euclidean length in the configuration. A path is an ordered sequence of the points, and a loop is a path that has the same endpoints. A path or loop, as a sequence of edges, also has a Euclidean length. In this paper, we study the question of when $\mathbf{p}$ will be uniquely determined (up to an unknowable Euclidean transform) from a given set of path or loop lengths. In particular, we consider the setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing the paths or loops that gave rise to the lengths. Our main result is a condition on the set of paths or loops that is sufficient to guarantee such a unique determination. We also provide an algorithm, under a real computational model, for performing a reconstruction of $\mathbf{p}$ from such unlabeled lengths. To obtain our results, we introduce a new family of algebraic varieties which we call the unsquared measurement varieties. The family is parameterized by the number of points $n$ and the dimension $d$, and our results follow from a complete characterization of the linear automorphisms of these varieties for all $n$ and $d$. The linear automorphisms for the special case of $n = 4$ and $d = 2$ correspond to the so-called Regge symmetries of the tetrahedron.

Motivation & Objective

• To determine when a generic point configuration in $ \mathbb{R}^d $ can be uniquely recovered from unlabeled path or loop length measurements.
• To address the challenge of reconstruction when no combinatorial labels (e.g., sequence of points) accompany the measured lengths.
• To develop a computable algorithm for reconstructing the configuration under a real computational model.
• To characterize the linear automorphisms of unsquared measurement varieties, which are central to the theoretical foundation.
• To generalize prior results on unlabeled edge length recovery to include path and loop lengths, even when only loop measurements are available.

Proposed method

• Introduces a new family of algebraic varieties called 'unsquared measurement varieties' parameterized by $ n $ (number of points) and $ d $ (dimension), which encode the squared distances of paths and loops.
• Uses the Fano variety of 2-planes in these varieties to analyze the structure of possible length measurements and their symmetries.
• Applies trilateration as a core reconstruction strategy: starting from a known triple of points, reconstructs additional points using measured path or loop lengths.
• Employs a real-computable algorithm based on solving systems of polynomial equations derived from the unsquared measurement varieties.
• Leverages the theory of linear automorphisms of these varieties to prove uniqueness, with the $ n=4, d=2 $ case linking to Regge symmetries of the tetrahedron.
• Uses computational algebra systems (e.g., Magma) to verify properties of Fano varieties and singular loci, particularly for small $ n $ and $ d $.

Experimental results

Research questions

• RQ1 Under what conditions can a generic point configuration in $ \mathbb{R}^d $ be uniquely determined from a set of unlabeled path or loop lengths?
• RQ2 Can the absence of labeling information (i.e., no sequence or combinatorial data) be overcome to reconstruct the original configuration?
• RQ3 What is the role of trilateration in enabling reconstruction from unlabeled loop measurements alone?
• RQ4 How do the linear automorphisms of unsquared measurement varieties relate to symmetries such as Regge symmetries in the tetrahedral case?
• RQ5 Is it possible to generalize the rational rank condition for non-singular measurements beyond the 2D case, particularly to 3D?

Key findings

• A configuration of $ n $ generic points in $ \mathbb{R}^d $ ($ d \geq 2 $) is uniquely determined up to Euclidean congruence if the set of measured path or loop lengths 'allows for trilateration'.
• The reconstruction is possible even when only loop lengths are measured and no labeling is available, provided the measurement set satisfies the trilateration condition.
• The unsquared measurement varieties are shown to have a complete characterization of their linear automorphisms, which underpins the uniqueness result.
• For $ n=4 $, $ d=2 $, the linear automorphisms of the unsquared measurement variety correspond exactly to the Regge symmetries of the tetrahedron.
• The Fano variety $ \operatorname{Fano}_2(L_{2,4}) $ is 0-dimensional and equal to the Fano variety of the singular locus, confirming a key structural property.
• The rational rank of the measurement vector $ \mathbf{w} $ must be 6 for non-singular configurations in the $ n=4, d=2 $ case, and this condition ensures full rank in the measurement map.