Tensor Field Tomography

• Tensor Field Tomography is the process of reconstructing tensor-valued fields from indirect integral measurements along curves, unifying operator theory with computational methods.
• It addresses challenges such as injectivity, stability, and gauge ambiguities across diverse geometric settings including simple and curved manifolds.
• Applications span medical imaging, materials testing, and seismology, with advanced inversion techniques improving reconstruction accuracy even in high-noise environments.

Tensor Field Tomography is the theory and practice of reconstructing tensor-valued fields from indirect, typically integral measurements along families of curves—most notably geodesics or broken rays—where the measured signal is a (possibly nonlinear) function of the underlying tensor field. Such problems are foundational in areas including medical imaging (diffusion tensor MRI), materials science (stress/strain tomography, elastic modulus recovery), optical and X-ray anisotropy imaging, and seismology. Tensor tomography unifies a range of operator-theoretic, geometric, microlocal, and computational methodologies, confronting injectivity, stability, gauge ambiguities, and efficient inversion in diverse settings from Euclidean domains to curved manifolds and high-noise data regimes.

1. Mathematical Foundations: Geodesic and Broken-Ray Tensor Transforms

At its core, tensor field tomography generalizes the classical X-ray (Radon) transform to symmetric $m$-tensor fields $f$ on a manifold $(M,g)$. The geodesic X-ray transform $I_m$ integrates the contraction $$f_{i_1…i_m}(\gamma(t))\dot\gamma^{i_1}(t)…\dot\gamma^{i_m}(t)$$ along each geodesic $\gamma$:

$$(I_m f)(\gamma) = \int_{0}^{\tau(\gamma)} f_{i_1\dots i_m}(\gamma(t))\,\dot{\gamma}^{i_1}(t)\dots\dot{\gamma}^{i_m}(t)dt.$$

This construction induces a kernel due to the fundamental gauge freedom: any field of the form $f = d p$ (symmetrized covariant derivative of a lower-order tensor $p$) integrates to zero over geodesics; solenoidal injectivity is the statement that these are the only obstructions on a given geometric background (Paternain et al., 2013, Paternain et al., 2011, Ilmavirta et al., 2017, Lehtonen et al., 2017).

Variants include:

• Broken-Ray (V-line) Transforms: The V-line transform generalizes the integration paths to broken rays, modeling single-scattering scenarios. For a symmetric 2-tensor field $f$ in $\mathbb{R}^2$, the longitudinal, transverse, and mixed V-line transforms are defined by integration of contractions such as $\langle f(x), u\otimes u\rangle$ along the two branches starting at each vertex (Ambartsoumian et al., 2023, Ambartsoumian et al., 2024, Bhardwaj, 7 Feb 2025, Mishra et al., 2024).
• Attenuated and Dynamic Transforms: Realistic measurement models incorporate attenuation and refraction, mapping $f$ to boundary observations along geodesics of a Riemannian metric $g=n^2(x)I$ and folding in exponential weights—see attenuated transforms (Vierus et al., 16 Jan 2026, Vierus et al., 2021).

On more general settings (periodic slabs, broken rays with reflecting obstacles, asymptotically hyperbolic surfaces), the nature and structure of the transform's kernel change, but the potential–solenoidal decomposition or its analogs remain central (Ilmavirta et al., 2017, Ilmavirta et al., 2018, Eptaminitakis et al., 5 Oct 2025).

2. Injectivity, Stability, and Gauge Ambiguity

Injectivity results depend critically on geometric assumptions and regularity.

• On simple manifolds (compact, boundary strictly convex, no conjugate points), $I_m$ is solenoidal injective for all $m$ in dimension 2 (Paternain et al., 2011, Paternain et al., 2013), with sharp $L^2 \mapsto H^{1/2}$ stability estimates in nonpositive curvature (Paternain et al., 2020). In higher dimensions, solenoidal injectivity for $m\ge2$ typically requires additional curvature or analyticity hypotheses (Lehtonen et al., 2017, Ilmavirta et al., 2023).
• Gauge ambiguities: On domains with trapped geodesics, product structure, or symmetries (e.g., periodic slabs), the kernel of $I_m$ is no longer exhausted by potential fields: new obstructions arise from fields with special symmetries or boundary conditions (Ilmavirta et al., 2017, Eptaminitakis et al., 5 Oct 2025).
• For broken-ray and V-line transforms, injectivity is sensitive to the geometry of rays and admissibility of the obstacle or branching configuration (Ambartsoumian et al., 2023, Ilmavirta et al., 2018, Ambartsoumian et al., 2024).

Stability is governed by the mapping properties of the normal operator $N=I_m^* I_m$, which is a pseudodifferential operator of order $-1$ and elliptic on solenoidal tensors. Explicit or microlocal Fredholm inverses yield Lipschitz or conditional stability under sharp regularity and coverage conditions (Paternain et al., 2013, Paternain et al., 2020).

3. Explicit Inversion Methods: Analytic and Computational Frameworks

A diversity of analytic and computational inversion strategies are supported across the tensor tomography literature:

• PDE/Energy Methods: The transport equation $X u = -f$ on the unit sphere bundle, where $X$ is the geodesic vector field, underpins energy methods (Pestov identities). For simple manifolds, the Fourier/spherical harmonics decomposition allows mode-by-mode control and reconstruction (Paternain et al., 2011, Ilmavirta et al., 2023, Lehtonen et al., 2017).
• Radon and Generalized Integral Transforms: In flat geometries or for V-lines/Star transforms, inversion proceeds by reducing to the scalar or tensor Radon transform, followed by filtered backprojection, often after algebraic decoupling of components (Ambartsoumian et al., 2023, Bhardwaj, 7 Feb 2025, Mishra et al., 2024). Moment-based methods amplify noise and are ill-suited for high-noise data (Ambartsoumian et al., 2024).
• Elliptic PDEs and ODEs: For certain classes of tensors (e.g., potentials, divergence-free types), reconstructing scalar or vector potentials reduces to solving second-order PDEs (often elliptic, sometimes hyperbolic/parabolic, depending on geometry), with data terms arising from the V-line transforms (Ambartsoumian et al., 2023, Ambartsoumian et al., 2024).
• Adjoint-Based and Iterative Optimization: In complex modalities (attenuated/dynamic transforms, refracting media), the adjoint of the forward operator can be computed by backprojection or solutions to dual transport equations, serving as the basis for iterative regularized reconstruction schemes (e.g., damped Landweber, Nesterov acceleration) (Vierus et al., 16 Jan 2026, Vierus et al., 2021).

Specialized frameworks are needed for three-dimensional or vectorial modalities, including:

• Neural Analysis-By-Synthesis: NeST reconstructs 3D stress tensor fields as neural implicit representations from polarization data, using a differentiable forward model incorporating Jones calculus and jointly optimizing phase unwrapping, tensor field smoothness, and data fidelity (Dave et al., 2024).
• Fourier–Mellin and Angular Harmonic Series: For circular/rotational geometries, tensor field components are recovered via angular Fourier decompositions and Mellin inversion, explicitly solving for the modes of each tensor entry (Bhardwaj, 7 Feb 2025).

4. Experimental and Application Domains

Tensor field tomography methods span a wide array of practical domains:

• Photoelasticity and Stress Tomography: NeST demonstrates non-destructive 3D stress tensor reconstruction from photoelastic birefringence measurements, surpassing 2D-slice classical techniques and linear models, with quantitative error of $6.4\%$ in the reconstructed $\sigma_1-\sigma_2$ field and absolute mean error of $0.15$ MPa (Dave et al., 2024).
• Diffraction Tensor Tomography: Tensorial T2DPC enables volumetric reconstructions of the optical permittivity tensor from polarization-diverse phase-contrast microscopy data, yielding maps of refractive index, birefringence, and principal axis orientation in biological specimens (Xu et al., 2022); scanning electron diffraction tomography reconstructs full 3D strain tensors in materials science, using the transverse ray transform and computational inversion with total variation regularization (Tovey et al., 2020).
• Optical and X-ray Anisotropy Imaging: Multiplexed dielectric tensor tomography reconstructs all six independent entries of the 3D dielectric tensor for optically anisotropic samples, employing off-axis interferometric multiplexing and direct Fourier-domain inversion (Lee et al., 2022). Universal energy-conservation-based frameworks for X-ray dark-field/tensor tomography recover the scattering tensor field using generalized wavefront modulators and analysis of 2D speckle/beam blurring (Lautizi et al., 2024).
• Tomography on Manifolds: Recent results establish injectivity and explicit inversion (or partial range characterization) for tensor X-ray transforms on asymptotically hyperbolic surfaces, periodic slabs, Cartan–Hadamard manifolds, and domains with reflecting obstacles (Eptaminitakis et al., 5 Oct 2025, Ilmavirta et al., 2017, Lehtonen et al., 2017, Ilmavirta et al., 2018).

5. V-Line and Broken-Ray Tensor Tomography: Theory and Numerical Results

Theoretical and computational developments in V-line and broken-ray tensor tomography have been substantial:

• Definition and Operators: V-line transforms generalize classical X-ray transforms to broken rays with a vertex, leading to the longitudinal, transverse, and mixed VLTs; the star transform synthesizes data along multiple branches to ensure injectivity and stability (Ambartsoumian et al., 2023, Bhardwaj, 7 Feb 2025, Mishra et al., 2024).
• Explicit Inversion: Inverting VLTs and star transforms is achieved via algebraic reduction to standard Radon inversion or PDE-based recovery, with explicit formulas for potential fields, solenoidal components, and full tensors under injective configurations. Closed-form Radon-based and Fourier–Mellin methods exist, though moment inversion is ill-conditioned and noise-amplifying (Ambartsoumian et al., 2024).

Table: Summary of V-Line Tensor Tomography Operators and Their Inversion

Operator	Definition	Inversion
Longitudinal	$$\mathcal{L}_{u,v}f(x)=\int_0^\infty\langle f(x+tu),u\otimes u\rangle dt+\dots$$	PDE or Radon inversion
Transverse	As above with $u^\perp\otimes u^\perp$	PDE or Radon inversion
Mixed	Weighted combination of $u\otimes u^\perp$ and $v\otimes v^\perp$	Elliptic PDE or moment-based
Star	Linear combination for $m$ directions/weights	Radon + algebraic inversion

Empirical results show that V-line and star-transform reconstructions are robust for smooth phantoms under moderate noise but that high-order moment formulas amplify noise and require regularization. PDE-based inversions are moderately stable; filtered backprojection is effective in the star case with masking outside the known support (Ambartsoumian et al., 2024).

6. Extensions, Challenges, and Frontiers

Outstanding aspects of the field include:

• Manifold and Topological Obstructions: The structure of the kernel and solenoidal injectivity fundamentally depend on the global geometry—e.g., existence of trapped geodesics, boundary topology, or conformal infinity—necessitating the development of decomposition theorems (e.g., transverse traceless–potential–conformal, iterated-tt forms) for range characterization and explicit inversion in settings such as asymptotically hyperbolic surfaces (Eptaminitakis et al., 5 Oct 2025).
• Attenuation, Refraction, Dynamics: Incorporating attenuation, refraction, and time dependence requires models based on inverse problems for transport equations, where viscosity regularization and adjoint-based iterative solvers have shown success (Vierus et al., 16 Jan 2026, Vierus et al., 2021).
• Computational and Data-Theoretic Considerations: The interplay of stability (with respect to noise and incomplete data), computational feasibility (e.g., mesh design, adjoint evaluation), and conditioning (especially for moment or star-type data) shapes both methodology and practical application (Ambartsoumian et al., 2024, Vierus et al., 16 Jan 2026).
• Tensor Tomography in 3D and Anisotropic Media: High-dimensional and anisotropic problems (dielectric, strain/stress, optical, and X-ray anisotropy) drive the evolution of representational and inversion frameworks (from neural implicit networks (Dave et al., 2024) to regularized large-scale optimizations (Tovey et al., 2020, Lee et al., 2022, Lautizi et al., 2024)).

7. Broader Impacts and Future Directions

Tensor field tomography forms the analytical and computational backbone for imaging vectorial and tensorial properties in science and engineering. Theoretical advances—sharp injectivity, stability, and gauge-resolving decompositions—facilitate the development of new modalities (photoelastic, dielectric, dark-field tensor tomography), tighter error guarantees, and more efficient solvers. Open research directions include:

• S-injectivity for higher-rank tensor fields on general geometries,
• Explicit and numerically stable inversion for broken-ray/conical V-lines in 3D,
• Realistic modeling and inversion under strong attenuation, multiple scattering, limited angular coverage,
• Integration of data-driven approaches (neural implicit inversion, learned regularization) with rigorous physics-based models.

References: (Paternain et al., 2011, Paternain et al., 2013, Lehtonen et al., 2017, Ilmavirta et al., 2017, Ilmavirta et al., 2018, Paternain et al., 2020, Tovey et al., 2020, Vierus et al., 2021, Xu et al., 2022, Lee et al., 2022, Ilmavirta et al., 2023, Ambartsoumian et al., 2023, Ambartsoumian et al., 2024, Dave et al., 2024, Lautizi et al., 2024, Mishra et al., 2024, Bhardwaj, 7 Feb 2025, Eptaminitakis et al., 5 Oct 2025, Vierus et al., 16 Jan 2026).