On sound ranging in some non-proper metric spaces

TL;DR

This paper investigates source localization in non-proper metric spaces, proposing iterative approximation methods and providing implementation details, extending sound ranging techniques beyond traditional metric space assumptions.

Contribution

It introduces iterative approximation algorithms for sound ranging in non-proper metric spaces and analyzes their convergence, including in normed spaces with dense sensors on the unit sphere.

Findings

Approximate solutions can be achieved with arbitrary precision.

Algorithms work in non-proper metric spaces under certain conditions.

Implementation in Julia demonstrates practical applicability.

Abstract

We consider the sound ranging, or source localization, problem --- find the unknown source-point from known moments when the spherical wave of linearly, with time, increasing radius reaches known sensor-points --- in some non-proper metric spaces (closed ball is not always compact). Under certain conditions we approximate the solution to arbitrary precision by the iterative processes with and without a stopping criterion. We also consider this problem in normed spaces with a strictly convex norm when the sensors are dense on the unit sphere. Appended is the implementation of the approximation algorithm in Julia language.