Convex Obstacles from Travelling Times

A construction is given for the recovery of a disjoint union of strictly convex smooth planar obstacles from travelling-time information. The obstacles are required to be such that no Euclidean line meets more than two of them.


Introduction
For some n ≥ 1, let K 1 , K 2 , . . . , K n be disjoint closed convex subsets of Euclidean 2-space E 2 ∼ = R 2 , with each boundary ∂K k a C ∞ strictly convex Jordan curve. Let K := ∪ n k=1 K i be contained in the interior of the bounded component B of E 2 \ C, where C ⊂ E 2 is also a strictly convex Jordan curve.
By a geodesic in the closure M of E 2 \ K we mean a piecewise-affine constantspeed curve x : R → M whose junctions are points of reflection on ∂K. The restriction of x to an interval is also called a geodesic, and the set of all geodesics x : [0, 1] → M with x(0), x(1) ∈ C is denoted by X . Then x ∈ X is critical for the length functional calculate K from T , and with a little more effort K can also be reconstructed when n = 2 (see Sect. 4 in [6]). More interestingly 1 , Theorem 1.1 of [10] allows to compute the area of K from T . Importantly, the application of the result in [10] is made possible by the fact (proved in [4]) that the set of points generating trapped trajectories in the exterior of obstacles K considered in this paper has Lebesgue measure zero. Constructing K is equivalent to constructing ∂K, but this seems difficult for n ≥ 3. In the present paper we show how to construct ∂K from T when K is in general position, namely when no line meets more than two connected components of K.
Inverse problems concerning metric rigidity have been studied for a long time in Riemannian geometry: we refer to [8], [1] and their references for more information. In the last 20 years or so similar problems have been considered for scattering by obstacles, where the task is to recover geometric information about an obstacle from its scattering length spectrum [9], or from travelling times of scattering rays in its exterior [6].
In general, an obstacle in the Euclidean space E m ∼ = R m (m ≥ 2) is a compact subset K of R m with a smooth (e.g. C 3 ) boundary ∂K such that Ω K = R m \ K is connected. The scattering rays in Ω K are generalized geodesics (in the sense of Melrose and Sjöstrand [2], [3]) that are unbounded in both directions. Most of these scattering rays are billiard trajectories with finitely many reflection points at ∂K. When K is a finite disjoint union of strictly convex domains, then all scattering rays in Ω K are billiard trajectories, namely geodesics of the type described above.
It turns out that some kinds of obstacles are uniquely recoverable from their travelling times spectra. For example, as mentioned above, this was proved in [5] for obstacles K in R m (m ≥ 3) that are finite disjoint unions of strictly convex bodies with C 3 boundaries. The case m = 2 requires a different proof, given recently in [4].
The set of the so called trapped points (points that generate trajectories with infinitely many reflections) plays a rather important role in various inverse problems in scattering by obstacles, and also in problems on metric rigidity in Riemannian geometry. As an example of M. Livshits shows (see e.g. Figure 1 in [7] or [10]), 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. In dimensions m > 2 examples similar to that of Livshits were given in [7].
The layout of the paper is as folllows.
In §2 we collect some simple observations about linear (non-reflected) geodesics. This leads to the construction of 4(n 2 − n) so-called vacuous arcs β j in T , and then 2(n 2 − n) initial arcs in ∂K. Our plan is to build on the initial arcs, using travelling-time data from reflected rays to construct incremental arcs in ∂K, until eventually 2 the whole of ∂K is found. In §6 we describe an inductive step for constructing incremental arcs from previously determined arcs, and from observations of T . To make the relevant observations we need to understand some of the mathematical structure of T .
The first step towards this understanding is made in §3, 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 §4 to investigate the structure of travelling-time data of nowhere-tangent geodesics. In particular, cusps in so-called telegraphs of T correspond to geodesics that are tangent to ∂K.
The family of all such cusps is studied in §5, where the augmented travellingtime dataT is shown to be the closure of a countable family of disjoint open C ∞ arcsβ j . As described in §6, the property of extendibility can be checked for eachβ j . Whenβ j is extendible it yields an incremental arc in ∂K. Whenβ j is not extendible, a trick using general position replacesβ j by an extendibleβj yielding an incremental arc as previously.

Linear Geodesics and Vacuous Arcs
From now on let K = ∪ n k=1 K i be an obstacle in E 2 , where K 1 , K 2 , . . . , K n are disjoint closed convex subsets of E 2 with boundaries that are C ∞ strictly convex Jordan curves. As before, assume that K is contained in the interior of the bounded component B of E 2 \ C, where C ⊂ E 2 is also a strictly convex Jordan curve. We also assume that K is in general position. We begin by investigating travelling times of linear geodesics, namely geodesics in X that do not reflect at all. The travelling time data from linear geodesics is where T q 0 is defined as the travelling-time data from geodesics meeting ∂K exactly q times tangentially and nowhere else. By Lemma 1 T q 0 = ∅ for q ≥ 3, 2 Unlike the initial arcs, there are countably many incremental arcs, yielding diminishing additional information from ever-increasing amounts of precisely known data. In practice, insufficient data and limited computing power makes it difficult to carry out more than a few inductive steps, and ∂K is found only approximately.
In the simplest case where n = 1, T 2 0 is empty and ∂K is constructed as the envelope of the line segments [x 0 , nonintersecting bounded open C ∞ arcs β j whose boundaries in∂T 0 comprise T 2 0 which is finite of size 4(n 2 − n).
Proof: For 1 ≤ k = k ≤ n there are 8 directed Euclidean line segments (linear bitangents) tangent to both ∂K k and ∂K k . Each directed linear bitangent is an endpoint of two maximal open arcs of directed line segments that are singly-tangent. The travelling-time data for the linear bitangents is T 2 0 . The travelling-time data for the open arcs β j are the path components of T 1 0 . So n is found from T 1 0 .

Nonlinear Geodesics
In order to construct envelopes of other singly-tangential geodesics, we shall identify the travelling-time data T q of q-times tangential geodesics in X , especially q = 1. Whereas T 1 0 is found by simple inspection of T , some effort is required to isolate T 1 . We first recall some known results about directions of geodesics and travelling-times.
Proof: Because the K k are disjoint and strictly convex, the endpoints of x x0,v0 |[0, 1] are nonconjugate.
Proof: By Lemma 2 and the implicit function theorem, there exist a unique Differentiating with respect tox 0 ∈ U 0 in the direction of δ ∈ R 2 , we find dφx 0 (δ) = − X(x 0 )/ X(x 0 ) , δ , because geodesics are critical for J when variations have fixed endpoints.
The order o(x) of a geodesic x is the number of intersections with ∂K. Write

Arcs and Generators for Nowhere-Tangent Geodesics
Let X q be the space of geodesics x ∈ X that are exactly q-times tangent to ∂K, and set Then T q 0 ⊂ T q and, by Lemma 1, T q = ∅ for q ≥ 3. We have constructed T q x1 . By continuation, the embedding extends uniquely in both directions around C, until just before x is tangent to some ∂K k , which must eventually happen. So T 0 x1 is a countable union of C ∞ embedded arcs α i . Pairwise transversality is proved by contradiction as follows.
x1 is open and dense in T x1 , ∪ q≥1 T q x1 = ∪ i≥1 ∂α i . By continuity, the orders o(α i ) of the xx 0,νi(x0) ∈ X 0 are independent ofx 0 ∈ U i ∩ C. From (1) we obtain, for allx 0 ∈ U i ∩ C, and the arcs α i are similarly bounded. For any i, the closuresᾱ i andᾱ i in T x1 are disjoint for all but finitely many i , where i, i ≥ 1. The generator φ i defines xx 0,νi(x0) ∈ X 0 x1 for everyx 0 ∈ U i ∩ C. For (x 0 , t) ∈ ∂α i , define Proof: We can write t = J(x x0,v0 ) where x x0,v0 ∈ X 1 x1 and v 0 = t. Suppose the last (respectively first) segment of x x0,v0 is not tangent to ∂K. Then, by general position, the first (last) segment is tangent. Perturbing the last segment while maintaining the endpoint x 1 , gives two arcs of nowhere-tangent geodesics, whose initial pointsx 0 lie on the same side of x 0 in C. Along one arc the first (last) segment remains linear and the order decreases by 1. Along the other arc, the first (last) segment breaks into two linear segments, maintaining the order and increasing the travelling time.
Forx 0 near x 0 , the two arcs of geodesics define arcs , Proposition 3 has the Corollary 1. T 1 x1 ∪ T 2 x1 is the closure in T x1 of the set of all points (x 0 , t) ∈ T x1 where T x1 has an isolated cusp.
A C ∞ embedding of T x1 in E 2 is given by (x 0 , t) := x 0 + tν(x 0 ), with ν : C → E 2 some constant-length nonzero outward-pointing normal field. Cusps in T x1 are found by inspecting the telegraph at x 1 , defined as (T x1 ) ⊂ E 2 . Example 1. Figure 1 displays part of (T x1 ) with x 1 = (0.4, 4) with n = 2, and C the circle of radius 4 and centre (0.4, 0). The telegraph is mainly smooth, but different arcs (light-blue, yellow-green and red) meet in cusps, and 6 transversal self-intersections are seen. Cusps (labelled t 1 , t 1 , t 2 , t 2 , t 3 , t 4 ) correspond to tangencies of geodesics ending at x 1 to K 1 or K 2 .
Next we augment T x1 and T to data setsT x1 andT that include initial velocities of geodesics. We first exclude points of intersection of the open arcs α i = α i,x1 in Proposition 2 (these points are reinserted later), by defining α * i = α * i,x1 := α i − ∪ i =i α i . Remark 3. Any α i intersects at most finitely many α i . Because intersections of α i and α i are transversal for i = i , T 0 * x1 := ∪ i∈I α * i,x1 is dense in T 0 x1 := ∪ i∈I α i,x1 , and For (x 0 , t) ∈ T 0 * x1 define u 0 = u x0,t,x1 to be the unit vector −∇φ i (x 0 ) pointing inwards from C. Then set Figure 1: Part of the telegraph in Example 1 To reinsert the excluded points, defineT 0 to be the closure ofT 0 * in

Singly-Tangent Geodesics
Summarising so far, for x 1 ∈ C, • T x1 is read directly from T , • we have seen how to find arcs α i,x1 and generators φ i for T 0 x1 , andT 0 x1 are obtained using the φ i , •T + :=T 1 ∪T 2 andT 0 are found by varying x 1 .
To distinguishT 1 fromT 2 we need Proposition 4, which is a structural result, analogous to Proposition 2. A geodesic x * ∈ X is said to be bitangent when it has two points of tangency to ∂K. We call x * linear when it has no other points of contact with ∂K.
where the β j are the vacuous arcs in T 1 0 , defined in §2,

for every
4. each (x * 0 , u * 0 , x * 1 , t * ) ∈T 2 is an endpoint of four open arcsβ j ,β j ,β j ,β j , where three of V j , V j , V j , V j are on one side of x * 0 ∈ C, and one is on the other side.
Proof: For (x 0 , u 0 , x 1 , t) ∈T 1 , we have x x0,tu0 ∈ X 1 and x x0,tu0 (1) = x 1 . Now x x0,tu0 is tangent to ∂K at precisely one point. By Lemma 1 this is either the first or last point of contact with ∂K.
If the tangency is first then, perturbing the point of tangency in ∂K gives a small open C ∞ arc around (x 0 , tu 0 , x 1 , t) contained inT 1 . Similarly, if the tangency is last, an open C ∞ arc inT + is given by perturbing the point of tangency in ∂K. So the path componentsβ j ofT 1 inT + are connected smooth 1-dimensional submanifolds of C × S 1 × C × R. They are bounded, nonclosed and, for 1 ≤ j ≤ 4(n 2 − n), can be listed as augmentations of the β j . Then 1. and 2. hold.
0 is tangent to ∂K at both first and last points of contact, and nowhere else. Nearby geodesics in X 1 are obtained by maintaining tangency either at a variable first point of contact, or 3 Including possibly j > 4(n 2 − n). at a variable last point of contact with ∂K. The tangencies at first (respectively last) points of contact generate arcsβ j ,β j (respectivelyβ j ,β j ) inT 1 , separated by (x * 0 , u * 0 , x * 1 , t * ). When the bitangent geodesic x * is linear, there is an open arc V j ⊂ C of initial points of perturbations initially tangent to ∂K k , and another open arc V j ⊂ C of initial points of perturbations initially tangent to ∂K p , as in Figure 2, where x * 0 , V j , V j appear on the right of the illustration. Perturbations whose initial points are in V j (green) and V j (red) have no other points of contact with ∂K. There are also two unlabelled open arcs V j , V j ⊂ C bordered by x * 0 , consisting of initial points of geodesics whose first points of contact are nontangent to ∂K, and whose second points of contact are tangent to ∂K k (green) or ∂K p (red) respectively 4 .
Evidently j = j , because V j and V j are on opposite sides of x * 0 , and similarly j = j , j in Figure 2. Indeed, from the geometry of perturbations of x * , all of j, j , j , j are distinct. In Figure 3 the nonlinear bitangent geodesic x * is tangent to ∂K k and ∂K p at the first and last points of contact respectively. It is not tangent anywhere else to ∂K, but is reflected at other points of contact, as suggested by the illustration. As before, the nonlinear bitangent is perturbed while maintaining tangency either with ∂K k (green) or with ∂K p (red), but now the first and last points of contact remain on ∂K k and ∂K p respectively. The initial points of perturbations tangent to ∂K k sweep out open arcs V j , V j ⊂ C (green) on either side of x * 0 . Initial points of perturbations tangent to ∂K p give the other intervals V j , V j on one side of x * 0 , as indicated by the two red arrows on the left of Figure 3. Again j, j , j , j are distinct. So (x * 0 , u * 0 , x * 1 , t * ) is an endpoint of precisely 4 open arcs and 4. is proved.
Corollary 2.T 1 is the smooth part of the 1-dimensional spaceT + .
So we may write them as o(β j ).
We need the following definitions: • The open arcsβ j ∈B where 1 ≤ j ≤ 4(n 2 − n) are said to be vacuous.
• Denote the envelope of λ j by Λβ j : V j → E 2 .

Extendible Arcs and the Inductive Step
At the end of §2 the travelling-time data T is used to find 4(n 2 − n) open arcs β j ⊂ T 1 0 . Each of these is augmented, as described in Proposition 4, to a vacuous open arcβ j ⊂T 1 . From the definition in §2 of the conjugatej of j, for 1 ≤ j ≤ 4(n 2 − n), We also obtain C ∞ parameterisations ψ j : V j →β j . More generally (inductively) suppose we have this kind of information where possibly j > 4(n 2 − n).
In precise terms, suppose we are given a C ∞ parameterisation ψ j : V j →β j of some possibly nonvacuous arcβ j ∈B. Here V j ⊂ C is a maximal open arc with the property that, for all x 0 ∈ V j and (x 0 , u 0 , x 1 , t) := ψ j (x 0 ), the first segment of the geodesic x x0,tu0 is tangent to ∂K k . The inductive step extends the open arc Λβ j (V j ) ⊂ ∂K by adjoining another such arc to its clockwise endpoint, as follows.
For x * 0 ∈ C the clockwise terminal limit of x 0 ∈ V j , set By Proposition 4 there are three other open arcsβ j ,β j ,β j ∈B adjacent toβ j ⊂T 1 at (x * 0 , u * 0 , x * 1 , t * ), and the unordered setB j := {β j ,β j ,β j } ⊂ B is found by inspectingT + . In the proof of Proposition 4, the arcsβ j ,β j (respectivelyβ j ,β j ) are generated by geodesics whose first (respectively last) segments are tangent to ∂K. Construct 5 Definition 2.β ∈B * j is an extension ofβ j when the closure Λ j,β of Λβ j (V j ) ∪ Λβ(V ) is a C ∞ strictly convex arc in E 2 . When an extension ofβ j exists, the arcβ j is said to be extendible (otherwise nonextendible).
Proposition 5. Ifβ j is extendible the extensionβ ∈B * j is unique, and Λβ(V ) is an arc in ∂K k . Ifβ j is nonextendible thenβ j is vacuous andβj is extendible.
Proof: By continuity of ψ j , the bitangent x x * 0 ,t * u0 * is tangent to ∂K k at some q := x x * 0 ,t * u0 * (t k ) where 0 < t k < 1, and q is a limit of points of first tangency and first contact with ∂K k . By general position q is either the first point of contact of the bitangent with ∂K or the second point of contact.
If q is the first point of contact thenβ j is extended byβ j whose associated geodesics maintain tangency to ∂K k . Evidently Λβ j (V j ) is an arc in ∂K k . For j * = j or j * = j , and (x 0 ,ũ 0 ,x 1 ,t) ∈β j * near (x * 0 , u * 0 , x * 1 , t * ), the last points of contact of xx 0,tũ0 are tangent to ∂K near q ∈ ∂K k where q = q. By the argument in §3 of [4], the λ(x 0 ,ũ 0 ) are not all tangent to a C ∞ strictly convex arc, namely Λβ j * is not strictly convex, andβ j * does not extendβ j . So the extensionβ =β j is unique.
If alternatively q is the second point of contact, then the first point of tangency is at q := x x * 0 ,t * u0 * (s ) ∈ ∂K k where 0 < t k < t k with k = k. By Lemma 1, q is the first point of contact of the bitangent with ∂K. By Lemma 1, and because q is the second point of tangency, the bitangent is linear with q, q the only points of contact with ∂K. So 1 ≤ j ≤ 4(n 2 − n), and q is the first point of contact of the linear bitangent x x * 1 ,−t * u * 0 with ∂K. Thenβj is extended by requiring tangency to ∂K k of the associated geodesics.
The arc Λβ j (V j ) in ∂K k is therefore extended by an incremental arc Λβ(V ), whereβ is an extension either ofβ j or ofβj. This completes the inductive step. Now the construction of ∂K proceeds as follows. Firstβ j is chosen with 1 ≤ j ≤ 4(n 2 − n), and the inductive step is carried out repeatedly withβ replacing β j after each step, until the incremental arcs Λβ(V ) in ∂K are acceptably 6 small. Then another vacuous arc is used to restart the iterative process. This is repeated until all the vacuous arcs are used. Finally ∂K is the union of the closures of all the arcs (initial and incremental) in ∂K.