Geodesic X-ray Transform & Microlocal Analysis

• Geodesic X-ray Transform is an operator that integrates functions, distributions, or tensor fields along geodesics on Riemannian manifolds for applications in tomography and inverse problems.
• Its normal operator decomposes into a pseudodifferential component indicating local action and a Fourier integral operator that captures off-diagonal singularity propagation due to conjugate points.
• The analysis shows that while 2D settings suffer from strong cancellation rendering singularities invisible, higher dimensions enable stable microlocal inversion under a local graph condition.

The geodesic X-ray transform is a central operator in integral geometry and inverse problems, defined as the mapping that associates to a function, distribution, or tensor field its integrals along geodesics of a Riemannian manifold. Originally arising in mathematical physics and tomography, its analysis reveals a deep microlocal structure, particularly in the presence of conjugate points such as fold caustics. The paper of its normal operator—especially in geometric settings where geodesics develop singularities—shows a characteristic decomposition into pseudodifferential and Fourier integral operator components, with profound consequences for invertibility, stability, and the recovery of singularities in the unknowns.

1. Microlocal Decomposition of the Normal Operator

For a general family of geodesics, especially when conjugate points of fold type are present, the normal operator $N = X^* X$ (where $X$ denotes the geodesic X-ray transform and $X^*$ its adjoint) is not a standard elliptic pseudodifferential operator. Instead, it admits a decomposition

$N = A + F$

where:

• $A$ is a pseudodifferential operator (ΨDO) of order $-1$, with principal symbol

$$\sigma_p(A)(x,\xi) = 2\pi \int_{S_x} \delta(\xi(\theta)) [\kappa^*\kappa](x,\theta)\, d\sigma_x(\theta)$$

This part reflects the “local” action of the transform on the function $f$ near $(x,\xi)$, resembling the case with no conjugate points.

• $F$ is a Fourier integral operator (FIO) of order $-n/2$ ($n$ is the manifold’s dimension) whose canonical relation is determined by the geometry of the conjugate locus $\Sigma$ associated to fold singularities in the exponential map. In local coordinates, this canonical relation is explicitly given by

$$\mathcal{C} = \{ (p, \xi; q, \eta)\ :\ (p,q) \in \Sigma,\ \xi = -\eta_i \partial_p \exp^i_p(v),\ \eta \in \operatorname{Coker}(d_v \exp_p(v)),\ \det d_v \exp_p(v) = 0 \}$$

Off-diagonal in nature, $F$ propagates singularities between conjugate points.

This structure is a rigorous manifestation of how the presence of fold caustics prevents the normal operator from being purely elliptic: $A$ reflects propagation along the diagonal in phase space, while $F$ encodes the off-diagonal transfer due to conjugate geometry.

2. Singularities and Cancellation in Two Dimensions

In two dimensions, the interaction of the off-diagonal FIO contributions yields a striking cancellation of singularities. Microlocally, after factoring through conjugate points of fold type, the operator

$\operatorname{Id} - A_2^{-1} F^* A_1^{-1} F$

acts as identity modulo lower order terms, and its principal symbol can vanish. Explicit calculations, for instance in the circular Radon transform, show that the normal operator $R^* R$ can be decomposed as

$R^* R = A_0 + F_+ + F_-$

with $A_0$ a ΨDO and $F_{\pm}$ FIOs with canonical relations mapping $(x, \xi)$ to $(x \pm 2\xi/|\xi|,\xi)$, respectively. The central phenomenon is that the contributions from $F_+$ and $F_-$ can cancel entirely, sometimes to infinite order (as in the fixed-radius circular case). This microlocal cancellation causes singularities of $f$ located at conjugate fold loci to become invisible: even though $f$ is singular, $X f$ may be smooth in a localized region. Therefore, $N$ fails to be microlocally invertible, precluding stable recovery of certain wavefront set directions—a key consequence for inverse problems in dimensions two.

3. Microlocal Invertibility in Dimensions Three and Higher

In dimensions three and above, if the canonical relation of $F$ is a local canonical graph (that is, the map $(p,\xi) \mapsto (q,\eta)$ via the caustic locus $\Sigma$ is a local diffeomorphism), then the composed operator

$A_2^{-1} F^* A_1^{-1} F$

is of negative order: $A_2^{-1} F^* A_1^{-1} F \in \Psi^{-\delta}$ with $\delta > 0$. Thus, by composing with parametrices for $A_1$ and $A_2$, the error operator in the inversion gains derivatives, restoring the possibility of stable microlocal inversion. This means that even in the presence of fold conjugate points, if the graph condition holds, all singularities in $f$ can be stably recovered in a microlocal sense—resulting in only mild (typically half-derivative) ill-posedness, in contrast to the catastrophic cancellation in 2D.

4. Examples and Geometric Impact

Several canonical examples showcase the above principles:

Example Type	Geometry and Caustics	Microlocal Behavior
Circular Radon (2D)	Circles as conjugate loci	Infinite order cancellation
Sphere $S^n$	Antipodal points (blow-down)	Infinite-dimensional kernel
Magnetic Geodesics in $\mathbb{R}^3$	Ellipsoidal conjugate locus	Local graph, invertible normal operator
Product Manifolds	Fold in one factor only	Failure of graph condition, partial invertibility

• In the 2D circular transform, the normal operator's off-diagonal terms lead to infinite order cancellation, indicating a maximal loss of microlocal invertibility.
• On $S^n$, conjugate points (antipodal) of blow-down type lead to a large kernel; all odd functions with respect to the antipodal map are invisible.
• For magnetic geodesics in $\mathbb{R}^3$ with constant field, the local graph condition for the FIO is satisfied, leading to invertibility and stable singularity recovery.
• In product manifolds, the graph condition can fail, so microlocal invertibility is retained only in the non-product (i.e., more generic and non-degenerate) directions.

These geometric configurations demonstrate that the type of conjugate points (fold vs. blow-down), the dimension, and the structure of the canonical relation critically determine whether singularities are detectable and stably recoverable.

5. Computation of Principal Symbols and Canonical Relations

The analysis relies on explicit computation of principal symbols and canonical relations associated to both the pseudodifferential and FIO terms. For the ΨDO part: $$\sigma_p(A)(x,\xi) = 2\pi \int_{S_x} \delta(\xi(\theta)) [\kappa^* \kappa](x,\theta)\, d\sigma_x(\theta)$$ which matches the case without conjugate points and expresses the contribution of geodesic directions orthogonal to $\xi$. For the FIO part, the canonical relation is more intricate, encoding the propagation along the conjugate locus, and is written in terms of the geometry of the exponential map at degenerate points where $d_v \exp_p(v)$ is singular. The structure of the canonical relation, and specifically whether it is a local graph, is decisive for invertibility.

6. Implications for Inverse Problems and Tomography

The fine structure elucidated in the microlocal analysis has direct implications for inverse problems in medical imaging (e.g., CT) and seismology:

• In two-dimensional or blow-down settings, singularities of the unknown in directions conormal to caustics may be completely invisible, resulting in nonuniqueness and a failure of stability for the inverse problem.
• In higher dimensions, under appropriate geometric conditions (especially the local graph property for the FIO), stable inversion algorithms are feasible, and all singularities are, in principle, recoverable modulo mild ill-posedness.
• The theory provides concrete model problems—for instance, circular Radon in 2D, magnetic geodesics in 3D, and the paper of fold caustics—which can guide both theoretical understanding and the design of practical inversion algorithms for integral geometric measurements corrupted by conjugate points.

7. Interaction with Layer-Stripping and Weighted Transforms

Although the main results pertain to the unweighted geodesic X-ray transform, the normal operator decomposition framework is essential in the analysis of more complex variable-weighted transforms and their stability in non-simple geometries. The presence of cancellation phenomena and the requirement for microlocal invertibility are key motivators for introducing strategies such as convex foliation, layer-stripping, and weighted inversion, as developed in broader literature on the geodesic X-ray transform. These techniques rely critically on control over the structure of $N$, justifying the emphasis on the precise decomposition and phase space analysis presented in the context of fold caustics.

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This microlocal framework for the geodesic X-ray transform with fold caustics provides a comprehensive lens through which the invertibility and stability properties of integral geometry can be classified, with explicit links to the underlying geometry and dimension. The interplay of pseudodifferential and FIO components in the normal operator captures the full complexity of singularity propagation and cancellation, dictating the feasibility of stable inversion in a rigorous analytic setting (Stefanov et al., 2010).