The geodesic ray transform on Riemannian surfaces with conjugate points
We study the geodesic X-ray transform on compact Riemannian surfaces with conjugate points. Regardless of the type of the conjugate points, we show that we cannot recover the singularities and therefore, this transform is always unstable (ill-posed). We describe the microlocal kernel of and relate it to the conjugate locus. We present numerical examples illustrating the cancellation of singularities. We also show that the attenuated X-ray transform is well posed if the attenuation is positive and there are no more than two conjugate points along each geodesic; but still ill-posed, if there are three or more conjugate points. Those results follow from our analysis of the weighted X-ray transform.
The purpose of this paper is to study the X-ray transform on Riemannian surfaces over geodesics with conjugate points. Let be a fixed directed geodesic on a Riemannian manifold of dimension , and let be a function which support does not contain the endpoints of . We study first the following local problem: what part of the wave front set of can be obtained from knowing the wave front of the (possibly weighted) integrals
of along all (directed) geodesics close enough to ? The analysis can be easily generalized to more general geodesic-like curves as in  or to the even more general case of “regular exponential maps”  as in . For the simplicity of the exposition, we consider the geodesic case only. Since has a Schwartz kernel with singularities of conormal type, could only provide information about near the conormal bundle of . It is well known that if there are no conjugate points on , we can in fact recover near . This goes back to Guillemin [14, 15] for integral transforms satisfying the Bolker condition, see also . The latter allows the use of the clean intersection calculus of Duistermaat and Guillemin  to show that is a pseudo-differential operator (DO), elliptic when . In the geodesic case under consideration, a microlocal study of has been done in [31, 32, 10, 35, 37, 39], and in some of those works, can even be a tensor field.
In the presence of conjugate points, the Bolker condition fails, see also Proposition 3.1. Then is not a DO anymore, and the standard arguments do not apply. Moreover, one can easily construct examples with based on the sphere  where the localized X-ray transform has an infinitely dimensional kernel of distributions singular at or near . Two delta functions of opposite signs placed at two anti-podal points is one such case. In , the second and the third author studied this question assuming that the conjugate points on are of fold type, i.e., has fold type singularities [1, 12]. In dimension , we proved there that there is a loss of derivatives, and we describes the microlocal kernel of modulo that loss. The analysis there was based on the specific properties of the fold conjugate points, and was focused on understanding the structure of as a sum of a DO plus a Fourier Integral Operator (FIO) associated with the conormal bundle of the conjugate locus , see next section. The microlocal structure of in all dimensions was also studied by Sean Holman . In this work, we show that there is a loss of all derivatives, and that this is true for all possible types of conjugate points. Instead of studying , we study itself as an FIO; show that the singularities in the kernel are related by a certain FIO; and describe its canonical relation as a generalization of even if the latter may not be smooth.
Since the answer to the local problem in two dimensions is affirmative when there are no conjugate points, and we show that it is negative when there are, the present paper gives a complete answer to the local problem for ( known near a single ) with the exception of the borderline case when the conjugate points are on .
The global problem, recovery of , and ultimately from known for all (or for a “large” set of) geodesics is different however. Let be two dimensional and non-trapping so that is defined globally. The union of for all directed geodesics in is a double cover of , and for each , there can be possibly resolved by known near each of the two directed geodesics through normal to . We therefore have a system of two equations, and if the weight is not even with respect to the direction, then that system is solvable for a pair of conjugate points, provided that some determinant does not vanish, see (5.1). If the weight is even, then the global problem is equivalent to the local one. Condition (5.1) is naturally satisfied for the geodesic attenuated X-ray transform with positive attenuation. Those two equations however are not enough to resolve the singularities at three points conjugate to each other, and then we still cannot recover . Those results are formulated in Section 5.
Recovery of is directly related to the question of stability of the geodesic X-ray transform (or any other linear map) — given , can we recover in a stable way? We do not study the uniqueness question here but we will just mention that if is real analytic, in many cases, even ones with conjugate points, one can have injectivity based on analytic microlocal arguments. An example of that is a small tubular neighborhood of as above; then the geodesics normal or close to normal to carry enough information to prove injectivity based on analyticity arguments as in [10, 35, 5, 21], for example. Stability however, will be always lost for even weights, for example, if there are conjugate points. The attenuated X-ray transform is stable with one or no conjugate point along each geodesic, and unstable otherwise. The term stable still makes sense even if there is no injectivity; then it indicates that estimate (4.7) holds.
This linear problem is also in the heart of the non-linear problem of recovery a metric or a sound speed (a conformal factor) from the lengths of the geodesics measured at the boundary (boundary rigidity) or from knowledge of the lens relation (lens rigidity), see, e.g., [7, 6, 8, 25, 28, 36, 34, 32, 29]; or from knowledge of the hyperbolic Dirichlet-to-Neumann (DN) map [3, 4, 23, 27, 33, 30, 2]. It is the linearization of the first two ( is a tensor field then); and the lens relation is directly related to the DN map and its canonical relation as an FIO. Although fully non-linear methods for uniqueness (up to isometry) exist, see, e.g., , stability is always derived from stability of the linearization. Very often, see for example , even uniqueness is derived from injectivity and stability of the linearization, see also  for an abstract treatment. Understanding the stability of is therefore fundamental for all those problems.
In seismology, recovery of the jumps between layers, which mathematically are conormal singularities, is actually the main goal. The model then is a linearized map like or a linearized DN map; and the goal is to recover the visible part of the wave front set.
In dimensions , the problem is over-determined. If there is a complete set of geodesics without conjugate points which conormal bundle covers , then can be recovered . In case this is not true or when we have local information only, there can be instability, for example for metrics of product type. In , we formulated a condition for fold conjugate points, under which singularities can still be recovered because then the “artifact” (the term in Theorem 4.3) are of lower order. At present it is not clear however if there are metrics satisfying that condition; and even of they are, the analysis does not cover non-fold conjugate points. Therefore, this problem remains largely open in dimensions .
2. Regular exponential maps and their generic singularities
2.1. Regular exponential maps
2.2. Generic properties of the conjugate locus
We recall here the main result by Warner  about the regular points of the conjugate locus of a fixed point . In fact, Warner considers more general exponential type of maps but we restrict our attention to the geodesic case (see also  for the general case). The tangent conjugate locus of is the set of all vectors so that (the differential of w.r.t. ) is not an isomorphism. We call such vectors conjugate vectors at (called conjugate points in ). The kernel of is denoted by . It is part of that we identify with . By the Gauss lemma, is orthogonal to . The images of the conjugate vectors under the exponential map will be called conjugate points to . The image of under the exponential map will be denoted by and called the conjugate locus of , whoc may not be smooth everywhere. Note that , while . We always work with near a fixed and with near a fixed . Set . Then we are interested in restricted to a small neighborhood of , and in near . Note that may not contain all points near conjugate to along some geodesic; and may not contain even all of those along if the later self-intersects — it contains only those that are of the form with close enough to .
We denote by the set of all conjugate pairs localized as above. In other words, , where runs over a small neighborhood of . Also, we denote by the set , where .
A regular conjugate vector at is defined by the requirement that there exists a neighborhood of , so that any radial ray of contains at most one conjugate vector in . The regular conjugate locus then is an everywhere dense open subset of the conjugate locus and is an embedded -dimensional manifold. The order of a conjugate vector as a singularity of (the dimension of the kernel of the differential) is called an order of the conjugate vector.
In [40, Thm 3.3], Warner characterized the conjugate vectors at a fixed of order at least , and some of those of order , as described below. Note that in , one needs to postulate that remains tangent to at points close to as the latter is not guaranteed by just assuming that it holds at only.
() Fold conjugate vectors. Let be a regular conjugate vector at , and let be one-dimensional and transversal to . Such singularities are known as fold singularities. Then one can find local coordinates near and near so that in those coordinates, is given by
Since the fold condition is stable under small perturbations, as follows directly from the definition, those properties are preserved under a small perturbation of .
() Blowdown of order 1. Let be a regular conjugate vector at and let be one-dimensional. Assume also that is tangent to for all regular conjugate near . We call such singularities blowdown of order 1. Then locally, is represented in suitable coordinates by
Even though we postulated that the tangency condition is stable under perturbations of , it is not stable under a small perturbation of , and the type of the singularity may change then. In some symmetric cases, one can check directly that the type is locally preserved.
() Blowdown of order . Those are regular conjugate vectors in the case where is -dimensional, with . Then in some coordinates, is represented as
In particular, must be tangent to , see also [40, Thm 3.2]. This singularity is unstable under perturbations of , as well. A typical example are the antipodal points on , ; then . Note that in this case, the defining equation of is degenerate, and .
2.3. The 2D case
If , then the order of the conjugate vectors must be one since in the radial direction the derivative of the exponential map is non-zero. Only (F) and () regular conjugate points are possible among the types listed above. For each and , then
|(2.7)||Either is transversal to ; or|
In the first case, is of fold type. The second case is more delicate and depends of the order of contact of with . To be more precise, let be a smooth non-vanishing vector field along so that at each point , is collinear with . Let be a smooth function on with a non-zero differential so that on . Then restricted to , has a zero at , see also  for this and for the definition below.
If this zero is simple, we say that is a simple cusp.
Near a simple cusp, the exponential map has the following normal form 
Regardless of the type of the conjugate vector , the tangent conjugate locus to any is a smooth curve. This follows form  since the order of any conjugate vector is always one. Its image under the exponential map however is locally a smooth curve if is a fold, a point, if is of () type, and a curve with a possible singularity at when the second alternative in (2.7) holds.
3. The geodesic X-ray transform as an FIO
The properties of as an FIO have been studied in , including those of the restricted X-ray transform, in the framework of Guillemin  and Guillemin and Stenberg . We will recall the results in  with some additions.
3.1. The general dimensional case
Assume first . We extend the manifold and the metric to some neighborhood of . We will study restricted to geodesics in some open set of geodesics with endpoints outside . It is clear (and shown below) that is a manifold. The set of all geodesics might not be, if there are trapped ones. We assume first the following
This condition simplifies the exposition. In fact later, we study geodesic manifolds for which the opposite holds, and the general case can be considered as a union of the two. The reason for (3.1) is to guarantee that for any , each element in is conormal to exactly one geodesic in . Let be the points on the geodesics in , in the interior of . We will study acting on distributions supported . In particular, this covers the case of geodesics in some small enough neighborhood of a fixed geodesic as shown in Figure 1.
To parameterize the (directed) geodesics near some , we choose a small oriented hypersurface intersecting transversally. It can be a neighborhood of near or if hits transversely at that particular end. Let be the induced measure in , and let be a smooth unit normal vector field on consistent with the orientation of . Let consist of all with the property that and is not tangent to , and positively oriented, i.e., . Introduce the measure on . Then one can parametrize all geodesics intersecting transversally by their intersection with and the corresponding direction, i.e., by elements in . An important property of is that it introduces a measure on that geodesics set that is invariant under a different choice of by the Liouville Theorem, see e.g., .
We can project onto the unit ball bundle by projecting orthogonally each as a above to . The measure in this representation becomes the induced measure on , and in particular, the factor (which does not even make sense in this representation) disappears. We think of as a chart for the dimensional geodesic manifold (near ). Then (which we identify with ) has a natural symplectic form, that of . We refer to  for an invariant definition of , in fact, they did that in two ways. It also has a natural volume form, which is invariantly defined on . Tangent vectors to can naturally be identified with the Jacobi fields modulo the two trivial ones, and .
We view as the following (continuous) map
The Schwartz kernel of is clearly a delta type of distribution which indicates that it is also a (conormal) Lagrangian distribution, and therefore must be an FIO. To describe it in more detail, we use the double fibration point of view of [11, 16]. Let
be the point-geodesic relation with and being the natural left and right projections. It follows form the analysis below that is smooth of dimension . Then
is a double fibration in the sense of . Note that we switched left and right here compared to some other works in order to get a canonical relation later with complies with the notational convention in . To get the microlocal version of that, following Guillemin , let
be the conormal bundle of . It is a Lagrangian submanifold of , and the associated canonical relation is given by the “twisted” version of :
We then have the microlocal version of the diagram above:
where now and denote the projections indicated above. It is easy to see that their ranges do not include the zero sections.
The weighted X-ray transform then has a Schwartz kernel defined by
One can think of as a function defined on . Then is a distribution conormal to , see [18, section 18.2] in the class . Therefore,
is an FIO of order associated with the Lagrangian , which can be extended to distributions, as well. Its canonical relation is given by above.
Let be as above and fix some local coordinates on ; and let be the dial variable. We denote below by the geodesic issued from , where is unit and has projection on (with a fixed orientation). In those coordinates, the manifold consist of all with the property that for some . We can think of this as a parametric representation
Then the map (3.3) has Jacobian
Here, with and , , and similarly for . The identity operators above are in . Since the bottom right element, the tangent to the geodesic is never zero, we get that the matrix has maximal rank . In particular, this shows that is a smooth manifold of dimension .
The conormal bundle is at any point of is the space conormal to the range of . Denote by the dual variables of . Then if and only if and
The first condition says that is conormal to the geodesic issued from , which is consistent with the fact that we can only hope to recover conormal singularities at the geodesics involved in the X-ray transform. In particular, we get
What we see immediately is that we have the inclusion above. On the other hand, given , there is a geodesic in normal to by the definition of . Then we can compute and as above, see also the 2D case below. Notice that we have an -dimensional manifold of geodesics normal to ; only one, undirected, if . In the latter case, we may have two directed ones. In particular this implies that the rank of is maximal and equal to but since maps locally to , it has an dimensional kernel, if , and is an isomorphism when .
The next two equations in (3.4) say that the projection of (identified with a vector by the metric) to any non-trivial Jacobi field, at the point , is given. Set
The projection maps the dimensional to the dimensional . We want to find out when this allows us to recover , and therefore, . Assume that we have two different values and of , at which (3.4) holds with the so given with and , respectively. Consider all non-trivial Jacobi fields vanishing at . They form a linear space of dimension . Their projections to at vanish in a trivial way. By (3.4) and by our assumptions, their projections to at must vanish as well. Since is not tangent to the geodesic , we get that at , those Jacobi fields form a subspace of dimension . Therefore, a certain non-trivial linear combination vanishes there, which means that there is a Jacobi field vanishing at both and . Then the corresponding and are conjugate points, as a consequence of our assumption of lack of injectivity.
This argument also proves that is injective as well. Indeed, locally, there are no conjugate points. The problem then can be reduced to showing that is a simple root of the equations (3.4) with given, which follows from the fact that the zeros of Jacobian fields are always simple.
On the other hand, assume that there are no conjugate points along the geodesic . Then any Jacobi field vanishing at would be nonzero for any other . Those (non-trivial) Jacobi fields span an -dimensional linear space as above at any . On the other hand, they are all perpendicular to at . Since , we get a contradiction.
Therefore, is injective if and only if there are no conjugate points along the geodesics in . In particular, it is always locally injective. Moreover, is injective as well.
The X-ray transform is said to satisfy the Bolker condition  if is an injective immersion. Then is a DO of order . It is elliptic at if and only if for at least one with , see, e.g., [10, 35]. Then one can recover singularities conormal to all geodesics over which we integrate by elliptic regularity. The analysis above yields the following.
The Bolker condition is satisfied for if and only if none of the geodesics in has conjugate points.
An indirect indication of the validity of this proposition is the fact that is a DO if and only if there are no conjugate points, as mentioned above. The latter however was proved by analyzing the Schwartz kernel of directly, instead of composing and as FIOs. In the more difficult case of the restricted X-ray transform ( is a submanifold of of the same dimension as , when ), the Bolker condition can be violated even if there are no conjugate points, for examples for the Euclidean metric .
is a Fourier Integral Operator in the class . It satisfies the Bolker condition if and only if the geodesics in have no conjugate points. In the latter case, is a DO of order in .
We also recall the result in  showing that if the conjugate points are of fold type, has a canonical relation constituting of the following non-intersecting canonical relations: the diagonal (a DO part) and , where is the conjugate locus defined as the pairs of all conjugate points, a smooth manifold in that case, see .
3.2. The 2-dimensional case
Assume now . In this case, the three manifolds in the diagram (3.1) have the same dimension, . A natural question is whether and are diffeomorphisms, local or global. The analysis of the -dimensional case answers this already but we will make this more explicit below.
We introduce the scalar Jacobi fields and following  below. Introduce the notation , where for a given vector , we define the covector by , where
Note that has the same length as the vector and is conormal to . The inverse map is then given by . If we think of as a rotation by degrees, then is a rotation by degrees.
With as above (now, a curve transversal to ), every Jacobi vector field along any fixed geodesic close to is a linear combination , where are the initial conditions. As before, . The first of the conditions (3.4) say that for such a point in , we have with some . The last two equations imply , where
The functions and are the projections of the Jacobi fields and to . They solve the scalar Jacobi equation
where is the Gauss curvature, with linearly independent initial conditions. Then
The Bolker condition which we analyzed above says that is an injective immersion and only if there are no conjugate points along the geodesics in . In particular, this is true near if there are no conjugate points on . We will prove this again in this 2D situation. Fix with . We want to see first if the time for which
is unique. Consider the scalar non-zero Jacobi field that vanishes when (3.8) holds. The problem is reduced to showing uniqueness of the solution to with respect to . If there are two solutions however, then they correspond to conjugate points. This proves the injectivity of the projection in this case. The injectivity of differential of follows from the fact that when . Since , we get that is actually a local diffeomorphism. It is global, from to , assuming no conjugate points along any geodesic in .
We show now that is a diffeomorphism, see (3.5). For , for some , let be the unique, by (3.1), geodesic with the unit so that . Without loss of generality we may assume that the sign above is positive. Let be the point where it hits for the first time, , and let be the projection of the direction at to . Then depend smoothly on as a consequence of the assumption that hits transversely. Thus the inverse of is given by
with the last three functions defined as above. So is a local diffeomorphism. If the opposite to (3.1) holds, then is a double cover.
Combining this with the previous paragraph, we get in particular that given by is a local diffeomorphism.
We summarize those results below. Recall that is an open subset of geodesic and that consist of the interior point on those geodesics, and that is restricted to .
Let . Then under the assumptions in this section,
(a) is an FIO of order associated with the canonical relation given by (3.7), which is a graph of the canonical map described above.
(b) is a local diffeomorphism. It is a global one, from to if and only if there are no conjugate points on the geodesics in .
(c) If there are no conjugate points on the geodesics in , is elliptic at if and only if for such that is collinear with .
Conjugate points destroy the injectivity of , while a violation of condition (3.1) (assumed above) makes -to-.
4. Cancellation of singularities and instability
We are ready now to prove a stronger version of the cancellation of singularities of when . First, we will prove that we have a cancellation of infinite order. Second, the type of the conjugate locus will play no role at all. More precisely, since maps to locally, resolution of singularities without loss of derivatives would mean for any . We proved in  that this is not true if the conjugate points are of fold type, and there is an infinite dimensional space of distributions, for which but , and this is true actually in open conic neighborhoods of those points. This is a loss of at least of derivatives, even if we can recover in another Sobolev space. We show below that we have actually loss of all derivatives, and the type of the conjugate points does not matter.
Assume from now on that is a small neighborhood of some . Then the set has two natural disjoint components, corresponding to the choice of the orientation of the normals to the geodesics. In the representation (3.7), this corresponds to the choice of the sign of . Assume the convention that corresponds to the positive orientation. Then
To understand better what means, observe first that the latter is equivalent to the following. The points and belong to the same geodesic , i.e., , . Next,
In what follows, we drop the dependence on . The Wronskian is independent of and therefore equal to its initial condition . Consider the Jacobi field (as we did above). Then . We have
We therefore proved the following.
if and only if there is a geodesic joining and so that
(a) and are conjugate to each other, along ,
(b) , , , where is the unique non-trivial, up to multiplication by a constant, Jacobi field with .
Of course, (b) is equivalent to saying that , , , where is the unique non-trivial, up to multiplication by a constant, Jacobi field with . In particular, and are conormal to . This generalizes a result in  proved there under the assumption that the conjugate points are of fold type.
From now on, we fix and so that . Let be a small conic neighborhood of , and let be a small conic neighborhood of . Note that the signs of and in Theorem 4.1 above are the opposite if the number of conjugate points between and is even (or zero); and the signs are equal otherwise. Let . By shrinking those neighborhoods a bit, we can assume that . Let , , where, somewhat incorrectly, , if is even, and . Set
Then is a canonical relation itself, and is a diffeomorphism. Also, .
Next theorem extends the corresponding result in  from the case of fold conjugate points (in any dimension) to any type of conjugate points (in two dimensions). It relates the canonical relation directly to the geometry of the conjugate locus.
(a) is a diffeomorphism.
(b) also admits the characterization of Theorem 4.1.
The following theorem describes the microlocal kernel of in this setup.
Let . Then there exists an FIO of order zero with canonical relation with the following property. Let with , , with , small enough. Then
if and only if
The FIO is elliptic if and only if .
Clearly, under the ellipticity assumptions above, we can swap the indices and to obtain
Let be , restricted to distributions with wave front sets in , . Then . We proved above that are FIOs with canonical relations , ; elliptic, if . Then an application of the parametrix to completes the proof. In particular, we get
The theorem implies that we cannot resolve the singularities from the singularities of near .