Convex Obstacles from Travelling Times

Lyle Noakes,Luchezar Stoyanov

doi:10.3390/math9192434

Abstract

We consider situations where rays are reflected according to geometrical optics by a set of unknown obstacles. The aim is to recover information about the obstacles from the travelling-time data of the reflected rays using geometrical methods and observations of singularities. Suppose that, for a disjoint union of finitely many strictly convex smooth obstacles in the Euclidean plane, no Euclidean line meets more than two of them. We then give a construction for complete recovery of the obstacles from the travelling times of reflected rays.

Highlights

For some n ≥ 1, let K1, K2, . . . , Kn be disjoint closed convex subsets of Euclidean 2space E2 ∼= R2, with each boundary ∂Kk a C∞ strictly convex Jordan curve
Let K := ∪nk=1Ki be contained in the interior of the bounded component B of E2 \ C, where C ⊂ E2 is a strictly convex Jordan curve
Differentiating with respect to x0 ∈ U0 in the direction of δ ∈ R2, we find dφx0 (δ) = − X(x0)/ X(x0), δ, because geodesics are critical for J when variations have fixed endpoints

Summary

Introduction

For some n ≥ 1, let K1, K2, . . . , Kn be disjoint closed convex subsets of Euclidean 2space E2 ∼= R2, with each boundary ∂Kk a C∞ strictly convex Jordan curve. Livshits (see e.g., Figure 1 in [19] or [4]), in general, the set of trapped points may contain a non-trivial open set In such a case, the obstacle cannot be recovered from travelling times, because of an argument given in [4] due to Livshits based on the reflection properties of planar ellipses. The first step towards this understanding is made, where some simple facts about (typically non-reflected) geodesics are recalled These facts, including a known result for computing initial directions of geodesics, are applied in Section 4 to investigate the structure of travelling-time data of nowhere-tangent geodesics. Most nonlinear geodesics are nowhere-tangent; namely, they lie in Γ0 Their travelling-time data will be used to construct C∞ travelling-time functions φi for Lemma 3, as needed for Propositions 3, 4 of §4. Writing τ(γ) := t for the travelling time (length) of γ ∈ Γ, o(γ)dK ≤ J(γ) ≤ o(γ)diam(C)

Arcs and Generators for Nowhere-Tangent Geodesics

Singly-Tangent Geodesics

Extendible Arcs and the Inductive Step

Conclusions