Generalized Radon Transforms

• Generalized Radon transforms are integral operators that extend the classical transform by integrating functions over diverse submanifolds with rich geometric and analytic structures.
• They exhibit sharp mapping and inversion properties under conditions like nondegenerate curvature and the Bolker condition, ensuring stable and artifact-free reconstructions.
• The framework applies to fractal measures, non-Euclidean spaces, and tensor fields, underpinning practical methods in tomography, seismic imaging, and machine learning.

Generalized Radon transforms are a diverse and technically rich class of integral operators that generalize the classical Radon transform by integrating functions (or more general objects such as measures or tensor fields) over a prescribed family of submanifolds, surfaces, or level sets, often with nontrivial geometric, analytic, or algebraic structure. Modern research has expanded the scope of these operators to fractal measures, non-Euclidean spaces, non-linear submanifold families, manifolds with boundary, tensor fields, and connections with microlocal analysis and Fourier integral operators, yielding sharp mapping properties, inversion theorems, stability estimates, and applications in harmonic analysis, tomography, and discrete geometry.

1. Operator Definition and General Framework

The core construction of a generalized Radon transform involves a smooth manifold $M$ (often $\mathbb{R}^n$, a Lie group, or a symmetric space), a parameter space $\Sigma$, and a family of smooth submanifolds $\{H_p\}_{p\in\Sigma}$ of $M$ (typically hypersurfaces or lower-dimensional submanifolds). For a suitable function $f$ on $M$ and possibly a smooth non-vanishing weight $w(p,x)$, the transform acts by

$(R_w f)(p) = \int_{H_p} w(p,x) f(x)\, d\mu_p(x)$

where $d\mu_p$ is the induced measure on $H_p$ (e.g., surface measure). Often, $H_p$ is described as the zero set of a smooth function $\Phi(p,x)$ for fixed $p$.

Variants include transforms over surfaces of revolution, cones, hyperplanes, spheres, or more general level sets, and can integrate fields (functions, measures, tensor fields) with respect to fractal, group-invariant, or weighted measures. Canonical examples arise in (Duan, 2023, Webber et al., 2020, Webber et al., 2023, Agaltsov, 2014), and (Abhishek et al., 4 Feb 2025).

2. Hypotheses, Mapping Properties, and the Bolker Condition

Generalized Radon transforms depend critically on structural hypotheses:

• Ball-size condition for fractal measures: For Borel measures $\mu$ on $\mathbb{R}^d$, $\mu[B(x,r)] \leq Cr^s$ guarantees control over metric entropy and regularity (Duan, 2023).
• Smooth phase and cutoff functions: The defining phase $\Phi(x, y)$ and cutoff $\psi(x, y)$ ensure each integration hypersurface is well-defined and smooth.
• Nondegeneracy of rotational curvature (Monge–Ampère): Full rank of the mixed Hessian matrix, or non-vanishing of the Monge–Ampère determinant $M(\Phi)$, guarantees curvature and prevents degeneracies in mapping properties and inversion (Srivastava et al., 9 Feb 2025, Webber et al., 2020, Webber et al., 2023).
• Bolker condition: The canonical relation associated to the transform must project as an injective immersion onto the data space, which enforces artifact-free reconstructions and FIO structure (Webber et al., 2020, Webber et al., 2023, Homan et al., 2015).
• Sobolev regularity: Mapping properties often hinge on boundedness between Sobolev spaces $L^2 \to H^\alpha$ (Duan, 2023), with $\alpha$ reflecting geometric curvature. Endpoint estimates require threshold conditions on dimensions and regularity, e.g., $s > d - \alpha$ for fractal measures.

3. Main Theorems: Mapping, Inversion, and Sharp Bounds

The analytic theory centers around sharp $L^p$-$L^q$ mapping theorems, inversion formulas, and kernel characterizations:

• Sharp $L^p(\mu) \to L^q(\mu)$ Bounds: On fractal measures with ball-size condition and adequate Sobolev regularity, generalized transforms admit boundedness as operators $A: L^p(\mu) \to L^q(\mu)$ with explicit ranges for $p$ and $q$, which are sharp up to counterexamples on Cantor-type sets (Duan, 2023).
• Inversion and Uniqueness: In various settings (Euclidean, constant curvature, symmetric spaces, tensor tomography), explicit inversion formulas are established, often involving derivative operators, Volterra equations, Mellin–Fourier integrals, or Abel-type inversion. These formulas are dimension-specific and, for tensor fields, require weighted moment transforms to recover all field components (Webber et al., 2023, Agaltsov, 2014, Antipov et al., 2011, Abhishek et al., 4 Feb 2025).
• Support Theorems: Results determine the support of the original object as encoded in the transform, employing Paley–Wiener-type analysis and duality arguments in symmetric spaces (Kuit, 2010, Björklund et al., 9 May 2025).
• Microlocal Regularity: Under analytic and curvature conditions, the transform preserves or enhances regularity, with artifact-free recovery corresponding to the Bolker condition (Homan et al., 2015, Frikel et al., 2015, Webber et al., 2020, Webber et al., 2023).

4. Microlocal Analysis, Canonical Relations, and FIO Paradigm

Sophisticated results use microlocal techniques and Fourier integral operator (FIO) calculus:

• Microlocal correspondence: The transform is an elliptic FIO whose canonical relation links singularities in $f$ to singularities in $Rf$. Bolker’s condition ensures injectivity and absence of artifact directions. Mapping orders are tracked via FIO theory, with Sobolev gains determined by curvature (Webber et al., 2020, Webber et al., 2023, Frikel et al., 2015).
• Visible/Invisible singularities: Only directions satisfying the canonical visibility conditions contribute to reconstructed data. Limited-data reconstructions are characterized via wavefront sets and pseudo-differential operator symbols, and artifact sets are precisely described in terms of microlocal geometry (Frikel et al., 2015, Katsevich, 2021).
• Discrete data resolution: Sampling effects cause smoothing and blurring of singularities, with transition profile and resolution quantifiable via scaling limits and interpolation-theoretic techniques (Katsevich, 2021).

5. Inversion on Measures, Manifolds, and Symmetric Spaces

Advanced frameworks enable generalized Radon theory on non-Euclidean spaces:

• Manifolds and Symmetric Spaces: Definitions and inversion extend to analytic Riemannian manifolds and reductive symmetric spaces, specializing to horospherical and k-plane transforms. The support/inversion theorems involve Eisenstein integrals, spherical harmonics, and representation-theoretic Fourier analysis (Kuit, 2010, Homan et al., 2015, Bernstein et al., 2011).
• Tensor Tomography: For symmetric $m$-tensor fields, generalized transforms include longitudinal and transversal variants, with weighted moments needed for full inversion. Sharp kernel characterizations distinguish solenoidal and potential field components (Abhishek et al., 4 Feb 2025).
• Splines and Representation Embedding: For compact Lie groups, Radon transforms and dual transforms can be inverted approximately via variational splines, and are used to embed representation spaces into function spaces over dynamical systems (Bernstein et al., 2011, Björklund et al., 9 May 2025).
• Transverse Dynamical Systems: Extensions involve Radon-type transforms built over non-homogeneous spaces equipped with group actions and cross-sections, notably yielding Siegel-Radon and Zak transforms with applications in time-frequency analysis and geometry of numbers (Björklund et al., 9 May 2025).

6. Applications, Special Cases, and Operators on Fractals

Generalized Radon transforms have broad technical impact:

• Tomography and Imaging: Spherical, conical, and surface-of-revolution transforms underpin computational tomography (CT, CST, BST, URT), with artifact and stability analysis guiding algorithm design (Webber et al., 2023, Webber et al., 2020).
• Fractal Geometry and Measure Theory: Results on Falconer-type configuration problems leverage multilinear Radon transforms and sharp dimension-dependent kernel estimates to bound measures of distance and simplex sets in fractal sets (Grafakos et al., 2012, Duan, 2023).
• Density Estimation and Machine Learning: Radon and cumulative distribution transforms provide robust, affine-invariant feature extraction for pattern recognition, clustering, and density estimation on manifolds and measure spaces (Beckmann et al., 8 Dec 2025, Webber et al., 2019).
• Lattice Point Counting: Advanced oscillatory integral analysis applies generalized Radon estimates to count lattice points near spheres in Heisenberg groups, with non-isotropic dilation and monogenic curvature overcoming limits of Euclidean bounds (Srivastava et al., 9 Feb 2025).
• Seismic Imaging: Parabolic and hyperbolic Radon transforms recover functions from integrals over fixed-axis graphs with symmetry constraints, admitting explicit inversion formulas even in higher dimensions (Chihara, 2019).

7. Open Problems, Extensions, and Future Directions

Current areas of research involve:

• Extensions of inversion stability to highly singular measures and non-smooth domains.
• Theoretical development for transforms over polynomial and non-polynomial surfaces, or non-orientable manifolds.
• Applications in inverse problems for non-standard, non-Euclidean geometries and in harmonic analysis on Lie groups and dynamical systems.
• Connections to optimal transport and machine learning, especially via Wasserstein metrics and affine-invariant feature mapping (Beckmann et al., 8 Dec 2025).
• Understanding the limits of resolution and artifact generation in highly discretized or noisy reconstructions (Katsevich, 2021).

Recent work suggests broad utility for generalized Radon transforms in both theoretical and applied domains, with progress determined by advances in microlocal analysis, integral geometry, and computational harmonic analysis.