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Inverting the geodesic ray transform with finite measurements: stability and reconstruction

Yernat M. Assylbekov

arXiv:2609.40015v1math.APcs.NAmath.DGmath.NA

TL;DR

The paper asks whether finitely many local averages of geodesic ray-transform data preserve stability on a fixed finite-dimensional reconstruction space. It proves a fine-mesh stability result and develops convergent iterative reconstructions; reported experiments find that CGM uses substantially fewer iterations than SGD with comparable or smaller errors.

  • Problem

    The paper asks whether sufficiently fine partitions of local averages preserve stability when reconstructing from finitely many geodesic ray-transform measurements.

    Source: §1.1 · p. 1

  • Method

    The paper uses local-average projections and stability estimates to develop steepest-gradient and conjugate-gradient reconstruction methods on piecewise constant spaces.

    Source: §1.3 · p. 2

  • Results

    In all reported experiments, CGM requires substantially fewer iterations than SGD while achieving comparable or smaller reconstruction errors.

    Source: §7 · p. 18

  • Takeaways & Limitations

    The reported reconstructions exhibit artifacts whose appearances depend on the geometry and measurement discretization.

    Source: §7 · p. 18

    Near S(∂M), degenerating influx measure can make basis functions large; adaptive mesh design is relevant numerically, but efficient mesh design is outside the paper’s scope.

    Source: §3 · p. 4

Abstract

from arXiv · show

We study the inversion of the geodesic ray transform from finitely many local averages of its data. Under suitable geometric assumptions, we prove Lipschitz stability on any fixed finite-dimensional reconstruction space when the measurement partition is sufficiently fine. We develop convergent reconstruction algorithms based on Steepest Gradient Descent and Conjugate Gradients and implement them using piecewise constant finite element spaces. Numerical experiments in 2D and 3D illustrate the influence of geometry and measurement discretization on reconstruction quality.

1. Introduction

The paper asks whether finite local averages of geodesic ray-transform data can preserve stability on a prescribed finite-dimensional reconstruction space. It builds on prior full-data and finite-measurement work, develops a stability-based reconstruction approach, and tests its organization through 2D and 3D experiments.

1.1. Statement of the problem.

  • For a fixed finite-dimensional reconstruction space, the paper asks whether sufficiently fine partitions of local averages preserve stability when only finitely many measurements are available.The measurement model averages the transform over a partition of ∂+SM.Source: §1.1 · p. 1

1.2. Previous work.

  • Prior work studies full-data injectivity and stability, reconstruction methods, and sampling or resolution for ray and Radon transforms.The paper specializes the finite-measurement framework of to the geodesic ray transform.Source: §1.2 · p. 2, §1.3 · p. 2

1.3. Approach and main results.

  • The approach uses local-average projection convergence to preserve stability on fine meshes, then derives convergent steepest-gradient and conjugate-gradient methods for piecewise constant spaces.Conjugate gradients terminate within dim W iterations; experiments illustrate effects of geometry and measurement discretization.Source: §1.3 · p. 2

1.4. Structure of the paper.

  • The paper proceeds from transform preliminaries and boundary-space projection construction to stability estimates, reconstruction algorithms, and numerical experiments.It closes with concluding remarks and future research directions.Source: §1.4 · pp. 2–3

2. Preliminaries

The preliminaries set up the measures and L2 spaces used on the manifold, unit sphere bundle, and incoming boundary data space. They extend the geodesic ray transform as a bounded operator on L2.

2.1. Functional Spaces.

  • The preliminaries introduce Riemannian volume, Liouville volume, and the induced boundary measure used to define the paper’s L2 spaces.The spaces are completions of continuous functions with norms induced by the corresponding measures.Source: §2.1 · p. 3, §2.2 · p. 3

2.2. The geodesic ray transform.

  • Under non-trapping and strict-convexity assumptions, the geodesic ray transform extends uniquely to a bounded linear map from L2(M) to L2_µ(∂+SM).The boundedness argument uses Cauchy–Schwarz along geodesics and Santaló’s formula.Source: §2.2 · p. 3
  • The adjoint and Santaló-formula identities ensure the transform’s extension is well defined on L2 equivalence classes.The cited passages introduce the adjoint and state that applying Santaló’s formula yields the identities used for the extension.Source: §2.2 · p. 4

3. Piecewise Constant Finite Element Subspaces on ∂+SM

This section constructs piecewise constant finite-element spaces on the incoming boundary data space and studies their projections. Fine meshes approximate continuous data, while mesh shape near the glancing set matters for numerical stability of basis functions.

3. Piecewise Constant Finite Element Subspaces on ∂+SM

  • A mesh Tρ(∂+SM) is a finite collection of closed cells with non-empty interiors, with size measured using the induced boundary-space metric.The mesh definition also requires its cells to have pairwise intersections of measure zero.Source: §3 · p. 4
  • The finite-dimensional space Vρ(∂+SM) is essentially spanned by cell indicators, and its orthogonal projection gives the best L2_µ approximation from that space.The mesh projection is an orthogonal projection onto the piecewise constant space.Source: §3 · pp. 4–5
  • Because dµ degenerates at S(∂M), basis functions may grow on nearby cells; larger cells there can control their values, making adaptive meshing relevant numerically.Efficient mesh design is outside the paper’s scope.Source: §3 · p. 4
  • For continuous boundary data, projections on sufficiently fine meshes approximate the data uniformly.The proof uses uniform continuity on the compact boundary data space.Source: §3 · p. 5

4. Stability Estimates

The section establishes stability from finitely many local averages on fixed finite-dimensional spaces when the measurement mesh is sufficiently fine. It also extends the estimate to general L2 inputs with an additional approximation-error term.

4. Stability Estimates

  • The stability results assume a compact, non-trapping manifold with convex boundary and Assumption 4.1: no conjugate points or, in dimensions at least three, a smooth strictly convex function.Source: §4 · p. 6
  • The finite-measurement model uses local μ-averages of ray-transform data over cells of the full boundary data space, represented as a piecewise constant function.Each cell average is a bounded linear functional on L2, so pointwise evaluation of Igf is unnecessary.Source: §4 · pp. 6–7
  • For fixed finite-dimensional W, sufficiently fine measurement meshes preserve stability; for general L2 inputs, the estimate includes a W-approximation-error term.The mesh threshold depends on the geometry and W.Source: §4 · p. 7
  • On a fixed finite-dimensional W, boundedness and injectivity of Ig restricted to W suffice for stability; equivalently, ker Ig ∩ W must be {0}.The proof uses compactness of W’s unit sphere.Source: §4 · p. 7

5. Reconstruction Algorithms

The section uses stability of the discretized transform to reconstruct piecewise constant finite element approximations from projected data. Steepest gradient descent converges geometrically, while conjugate gradients reach the exact reconstruction within the dimension of the reconstruction space in exact arithmetic.

5. Reconstruction Algorithms

  • The reconstruction framework uses stability to recover f† from projected ray-transform data and gives convergence proofs adapted to this inverse-problem setting.Source: §5 · p. 9

5.1. W as a piecewise constant FES on M.

  • The reconstruction space Vδ(M) consists of normalized cell indicators on a mesh; they form an orthonormal basis of a finite-dimensional subspace of L2(M).Source: §5.1 · p. 9

5.2. Matrix representations.

  • The discretized measurement operator PρIg|Vδ(M) is represented by a matrix Aρ,δ with N measurement cells and K reconstruction cells.The two coordinate bases are orthonormal.Source: §5.2 · p. 9

5.3. The steepest gradient descent.

  • Steepest gradient descent converges geometrically for sufficiently fine measurement meshes, with its rate determined by the smallest and largest singular values of the discretized transform.If the residual vanishes, the iterate equals f† and the iteration terminates.Source: §5.3 · p. 10

5.4. The conjugate gradient method.

  • Conjugate gradients apply to the least-squares optimization problem, forming each search direction by B-orthogonalizing against earlier directions.The iteration preserves the reconstruction space.Source: §5.4 · pp. 12–13
  • Conjugate gradients reach f† within at most K = dim Vδ(M) iterations in exact arithmetic, and satisfy the same convergence bound as steepest gradient descent.This result holds for a sufficiently fine measurement mesh.Source: §5.4 · p. 12

6. Numerical Experiments

Experiments reconstruct piecewise-constant modified Shepp–Logan/Toft phantoms in 2D and 3D under stereographic, Poincaré, and focusing-lens geometries, using local-average data and SGD or CGM.

6.1. Experimental setup.

  • The tests use conformal metrics on the Euclidean unit ball; stereographic and Poincaré have no conjugate points, while all three geometries are non-trapping with strictly convex boundaries.Geodesic portions inside mesh cells are approximated by sampled-point counts times the time step, with midpoint quadrature used to form sparse matrix entries.Source: §6.1 · p. 14, Calculation of P ∂+SM · p. 15

6.2. 2D setting.

  • The 2D reconstruction uses 12,892 spatial cells and 49,152 measurement cells on uniform Cartesian grids.The spatial grid is 128 × 128; the measurement grid is 256 × 192 in the angular coordinates.Source: §6.2 · p. 16

6.3. 3D setting.

  • The 3D reconstruction uses 137,376 spatial cells and 4,194,304 measurement cells on product grids.The grids are 64 × 64 × 64 in space and 32 × 64 × 32 × 64 in measurement coordinates.Source: §6.3 · p. 17

6.4. Experiments.

  • The stereographic metric gives the smallest reconstruction errors in both dimensions; CGM achieves similar accuracy to SGD while requiring substantially fewer iterations.The comparison is reported for Experiment 1.Source: §6.4 · p. 17
  • In 2D Poincaré, both methods leave a relative error of about 8.3%, despite the large difference in iteration counts; this geometry requires the most iterations in both dimensions.The reconstructions show ring-like artifacts near the center, where the passage reports sparse geodesic sampling.Source: §6.4 · p. 17
  • The focusing-lens reconstruction is more accurate than Poincaré in 2D but has the largest errors in 3D; CGM improves on SGD in both dimensions.The lens experiment lies outside the algorithms’ geometric assumptions; its 2D streaks are consistent with geodesics refocusing near a conjugate point.Source: §6.4 · p. 18

7. Conclusion

Under the stated geometric assumptions, sufficiently fine measurement partitions preserve Lipschitz stability on each fixed finite-dimensional reconstruction space. The experiments show geometry- and discretization-dependent artifacts, motivating further learning-based reconstruction work.

7. Conclusion

  • Sufficiently fine measurement partitions preserve Lipschitz stability on any fixed finite-dimensional reconstruction space under the paper’s geometric assumptions.Across reported experiments, CGM requires substantially fewer iterations than SGD while achieving comparable or smaller reconstruction errors.Source: §7 · p. 18
  • SGD and CGM reconstructions retain artifacts whose appearance depends on the geometry and measurement discretization.The remaining artifacts motivate the author’s ongoing work on deep-learning-based reconstruction.Source: §7 · p. 18
  • Future directions include applying CNNs to CGM reconstructions and unrolling SGD or CGM into learned reconstruction steps, with approximation guarantees and numerical evaluation planned for the end-to-end approach.The CNN direction adapts a reconstruction-and-postprocessing strategy; the unrolled approach incorporates learning into reconstruction steps rather than only the final image.Source: §7 · pp. 18–20
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