TDOA--based localization in two dimensions: the bifurcation curve

In this paper, we complete the study of the geometry of the TDOA map that encodes the noiseless model for the localization of a source from the range differences between three receivers in a plane, by computing the Cartesian equation of the bifurcation curve in terms of the positions of the receivers. From that equation, we can compute its real asymptotic lines. The present manuscript completes the analysis of [Inverse Problems, Vol. 30, Number 3, Pages 035004]. Our result is useful to check if a source belongs or is closed to the bifurcation curve, where the localization in a noisy scenario is ambiguous.Comment: 11 pages, 3 figures, to appear in Fundamenta Informatica

D-Branes on C^3_6 part I: prepotential and GW-invariants

This is the first of a set of papers having the aim to provide a detailed description of brane configurations on a family of noncompact threedimensional Calabi-Yau manifolds. The starting point is the singular manifold C^3/Z_6, which admits five distinct crepant resolutions. Here we apply local mirror symmetry to partially determine the prepotential encoding the GW-invariants of the resolved varieties. It results that such prepotential provides all numbers but the ones corresponding to curves having null intersection with the compact divisor. This is realized by means of a conjecture, due to S. Hosono, so that our results provide a check confirming at least in part the conjecture.Comment: 66 pages, 18 figures, 15 tables; added reference

A comprehensive analysis of the geometry of TDOA maps in localisation problems

In this manuscript we consider the well-established problem of TDOA-based source localization and propose a comprehensive analysis of its solutions for arbitrary sensor measurements and placements. More specifically, we define the TDOA map from the physical space of source locations to the space of range measurements (TDOAs), in the specific case of three receivers in 2D space. We then study the identifiability of the model, giving a complete analytical characterization of the image of this map and its invertibility. This analysis has been conducted in a completely mathematical fashion, using many different tools which make it valid for every sensor configuration. These results are the first step towards the solution of more general problems involving, for example, a larger number of sensors, uncertainty in their placement, or lack of synchronization.Comment: 51 pages (3 appendices of 12 pages), 12 figure

D-Branes on C^3_6. Part I. Prepotential and GW-invariants

This is the first of a set of papers having the aim to provide a detailed description of brane configurations on a family of noncompact threedimensional Calabi-Yau manifolds. The starting point is the singular manifold C^3/Z_6, which admits five distinct crepant resolutions. Here we apply local mirror symmetry to partially determine the prepotential encoding the GW-invariants of the resolved varieties. It results that such prepotential provides all numbers but the ones corresponding to curves having null intersection with the compact divisor. This is realized by means of a conjecture, due to S. Hosono, so that our results provide a check confirming at least in part the conjecture

The algebro-geometric study of range maps

Localizing a radiant source is a widespread problem to many scientific and technological research areas. E.g. localization based on range measurements stays at the core of technologies like radar, sonar and wireless sensors networks. In this manuscript we study in depth the model for source localization based on range measurements obtained from the source signal, from the point of view of algebraic geometry. In the case of three receivers, we find unexpected connections between this problem and the geometry of Kummer's and Cayley's surfaces. Our work gives new insights also on the localization based on range differences.Comment: 38 pages, 18 figure

Source localization and denoising: a perspective from the TDOA space

In this manuscript, we formulate the problem of denoising Time Differences of Arrival (TDOAs) in the TDOA space, i.e. the Euclidean space spanned by TDOA measurements. The method consists of pre-processing the TDOAs with the purpose of reducing the measurement noise. The complete set of TDOAs (i.e., TDOAs computed at all microphone pairs) is known to form a redundant set, which lies on a linear subspace in the TDOA space. Noise, however, prevents TDOAs from lying exactly on this subspace. We therefore show that TDOA denoising can be seen as a projection operation that suppresses the component of the noise that is orthogonal to that linear subspace. We then generalize the projection operator also to the cases where the set of TDOAs is incomplete. We analytically show that this operator improves the localization accuracy, and we further confirm that via simulation.Comment: 25 pages, 9 figure

Growth in a Circular Economy

We present a model of natural resources and growth that stresses the influence of an incomplete circularity of exhaustible natural resources. In particular, we analyze the recycling process and the material balance principle, two fundamental aspects of a circular economy. When market failures arise or complete recycling is not possible for technical reasons, then the equilibrium outcomes in terms of output, consumption, and prices for the material inputs are distorted compared to the socially optimal solution. However, the introduction of a market for waste and a system of subsidies/taxes on virgin and recycled resources enables an internalization of the externalities. The importance of technological progress in order to foster “circularity”, i.e. both to improve resource efficiency in the production process and to enhance the backflow of materials from waste to production, is highlighted

A Geometrical-Statistical Approach to Outlier Removal for TDOA Measurements

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