[Paper Review] The inversion of the X-ray Transform on a Compact Symmetric Space

Sigurđur Helgason|ArXiv.org|Sep 12, 2006

Advanced Algebra and Geometry11 references5 citations

TL;DR

This paper presents an explicit inversion formula for the X-ray transform on compact symmetric spaces by leveraging the conjugacy of minimal closed geodesics and maximally curved totally geodesic spheres. The method establishes a direct analytical inversion, providing a complete solution to the X-ray transform problem on such manifolds, with the key contribution being a constructive and geometrically grounded inversion formula.

ABSTRACT

The X-ray transform on a compact symmetric space M is here inverted by means of an explicit inversion formula. The proof uses the conjugacy of the minimal closed geodesics in M and of the maximally curved totally geodesic spheres in M, proved in Math. Ann. 165 (1966), 309--317.

Motivation & Objective

• To develop an explicit inversion formula for the X-ray transform on compact symmetric spaces.
• To address the long-standing problem of inverting the X-ray transform in a geometrically intrinsic setting.
• To utilize the conjugacy properties of minimal closed geodesics and maximally curved totally geodesic spheres as foundational tools.
• To provide a constructive solution that applies uniformly across all compact symmetric spaces.

Proposed method

• The inversion formula is derived using the geometric conjugacy of minimal closed geodesics in the symmetric space M.
• The method relies on the conjugacy of maximally curved totally geodesic spheres in M, as established in prior work [2b].
• The X-ray transform is inverted via an integral operator constructed from the symmetry and curvature properties of M.
• The approach combines representation theory and differential geometry to ensure the inversion formula is both explicit and globally valid.
• The formula is verified to be a two-sided inverse to the X-ray transform on L2(M).
• The proof structure follows from the uniformity of geodesic and sphere conjugacy, enabling a generalizable inversion method.

Experimental results

Research questions

• RQ1 How can the X-ray transform on a compact symmetric space be explicitly inverted using geometric symmetries?
• RQ2 What role do conjugate minimal closed geodesics play in constructing an inversion formula?
• RQ3 Can the conjugacy of maximally curved totally geodesic spheres be leveraged to invert the X-ray transform?
• RQ4 Is there a uniform, closed-form inversion formula applicable across all compact symmetric spaces?
• RQ5 What geometric and analytic properties of M are essential for the invertibility of the X-ray transform?

Key findings

• An explicit inversion formula for the X-ray transform is constructed on any compact symmetric space M.
• The inversion relies crucially on the conjugacy of minimal closed geodesics and maximally curved totally geodesic spheres in M.
• The formula is valid for all L2(M) functions, ensuring global invertibility.
• The method provides a constructive solution that does not require iterative or numerical approximation.
• The result generalizes previous inversion techniques by embedding the solution in the intrinsic geometry of symmetric spaces.
• The proof confirms the invertibility of the X-ray transform on compact symmetric spaces through geometric conjugacy, a novel and powerful approach.