The Algebro-geometric Study of Range Maps

The curse of outlier measurements in estimation problems is a well known issue in a variety of fields. Therefore, outlier removal procedures, which enables the identification of spurious measurements within a set, have been developed for many different scenarios and applications. In this paper, we propose a statistically motivated outlier removal algorithm for time differences of arrival (TDOAs), or equivalently range differences (RD), acquired at sensor arrays. The method exploits the TDOA-space formalism and works by only knowing relative sensor positions. As the proposed method is completely independent from the application for which measurements are used, it can be reliably used to identify outliers within a set of TDOA/RD measurements in different fields (e.g. acoustic source localization, sensor synchronization, radar, remote sensing, etc.). The proposed outlier removal algorithm is validated by means of synthetic simulations and real experiments. Index TermsTDOA space, TDOA measurements, range differences, outlier removal.

show abstract

“…This map was thoroughly studied in [19], [21], in the context of planar source localization. Furthermore, in [22] (Remark 10.4) is the proof that…”

Section: The Case N =mentioning

confidence: 99%

“…If G = Σ −1 , (21) and (22) are known in the statistical literature as the Mahalanobis product and norm, respectively.…”

Section: A Statistical Noise Modelmentioning

confidence: 99%

A Geometrical–Statistical Approach to Outlier Removal for TDOA Measurements

Compagnoni

Pini

Canclini

et al. 2017

IEEE Trans. Signal Process.

Self Cite

View full text Add to dashboard Cite

show abstract

“…Remark 4.8. The locus of L 2,4 where the edge lengths of a triangle, (l 12 , l 13 , l 23 ), are held fixed is studied in beautiful detail in [8], where this is shown to be a Kummer surface.…”

Section: Measurement Varietiesmentioning

confidence: 99%

Trilateration using Unlabeled Path or Loop Lengths

Gkioulekas,

Gortler,

Theran

et al. 2020

Preprint

View full text Add to dashboard Cite

Let p be a configuration of n points in R d for some n and some d ≥ 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, which is simply the sum of its Euclidean edge lengths. We are interested in reconstructing p given a set of edge, path and loop lengths. In particular, we consider the unlabeled setting where the lengths are given simply as a set of real numbers, and are not labeled with the combinatorial data describing which paths or loops gave rise to these lengths.In this paper, we study the question of when p will be uniquely determined (up to an unknowable Euclidean transform) from some given set of path or loop lengths through an exhaustive trilateration process. Such a process has been already been used for the simpler problem of unlabeled edge lengths.

show abstract

“…Ho and Chan [11] provided a first development of equations for using TDOA measurements to back out emitter location, coming very close to the polynomial system development given in the next section. The more recent articles [7], [8] provide an advanced description of this TDOA-only setting, based largely on algebraic geometry. Various other approaches have been developed, and at least one [14] has made use of numerical algebraic geometry, though in a rather different context.…”

Section: Introductionmentioning

confidence: 99%

Geolocation with FDOA measurements via polynomial systems and RANSAC

Cameron¹,

Bates²

2018

2018 IEEE Radar Conference (RadarConf18)

View full text Add to dashboard Cite

The problem of geolocation of a transmitter via time difference of arrival (TDOA) and frequency difference of arrival (FDOA) is given as a system of polynomial equations. This allows for the use of homotopy continuation-based methods from numerical algebraic geometry. A novel geolocation algorithm employs numerical algebraic geometry techniques in conjunction with the random sample consensus (RANSAC) method. This is all developed and demonstrated in the setting of only FDOA measurements, without loss of generality. Additionally, the problem formulation as polynomial systems immediately provides lower bounds on the number of receivers or measurements required for the solution set to consist of only isolated points.

show abstract

The Algebro-geometric Study of Range Maps

Cited by 9 publications

References 55 publications

A Geometrical–Statistical Approach to Outlier Removal for TDOA Measurements

A Geometrical–Statistical Approach to Outlier Removal for TDOA Measurements

Trilateration using Unlabeled Path or Loop Lengths

Geolocation with FDOA measurements via polynomial systems and RANSAC

Contact Info

Product

Resources

About