Hyperbolic Radon Transform

• Hyperbolic Radon transform is a family of integral transforms that integrate functions along hyperbolic curves or hypersurfaces defined by indefinite quadratic forms.
• It bridges geometric analysis on hyperbolic spaces with explicit inversion formulas, microlocal techniques, and fast algorithms like log-polar FFT.
• Its applications span seismic imaging, medical tomography, and inverse problems, leveraging robust regularization methods to ensure numerical stability.

The hyperbolic Radon transform is a family of integral transforms in which a function is integrated over hyperbolic curves, hypersurfaces, or submanifolds determined by quadratic forms of indefinite signature. This transform arises naturally in inverse problems, mathematical tomography, seismic imaging, and harmonic analysis on spaces of constant negative curvature. In its various instantiations, the hyperbolic Radon transform can be seen as a unifying analytic tool, connecting the geometry of hyperbolic spaces and quadrics to explicit inversion formulas, microlocal structure, and fast algorithms.

1. Foundational Definitions and Settings

Multiple, interrelated definitions of the hyperbolic Radon transform exist, reflecting usage in Euclidean spaces (seismic processing), real constant curvature spaces (hyperbolic & pseudo-Riemannian geometry), and general quadratic forms.

Seismic-Type (Euclidean) Hyperbolic Radon Transform

Given a function $f(x, y)$ on $\mathbb{R}^2$, the (seismic) hyperbolic Radon transform integrates over the hyperbolae

$y^2 = s(x-c)^2 + u$

via

$$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$

This form is foundational for velocity analysis, multiple suppression, and interpolation in reflection seismology (Chihara, 2019).

Quadratic Surface Hyperbolic Radon Transform

Suppose $A$ is a real, symmetric, invertible $n \times n$ matrix of mixed signature. The quadratic Radon transform integrates $f$ over the quadrics

$$Q_{y, s} = \{ x \in \mathbb{R}^n : (x - y)^T A (x - y) = s \}$$

so that

$R_A f(y, s) = \int_{Q_{y, s}} f(x)\,d\sigma(x).$

This generalizes the notion to arbitrary hyperboloids or more general quadric surfaces and holistically describes the microlocal and injectivity properties (Ambartsoumian et al., 12 Feb 2026, Webber et al., 2022).

Hyperbolic Radon in Hyperbolic Space

In real hyperbolic space $H^n$ (hyperboloid model), the totally-geodesic Radon transform integrates a function over all (n-1)-dimensional geodesic hyperplanes or lower-dimensional geodesic subspaces. For $\mathbb{R}^2$0-planes,

$\mathbb{R}^2$1

where $\mathbb{R}^2$2 ranges over all totally-geodesic hyperplanes determined by the Minkowski orthogonality condition $\mathbb{R}^2$3 (Rubin, 2014, Antipov et al., 2011, Rubin, 2021).

2. Analytic, Microlocal, and Geometric Structure

The hyperbolic Radon transform, in its generalized quadratic form, is naturally formulated as a Fourier integral operator (FIO) with explicit phase function and canonical relation. For a smooth function $\mathbb{R}^2$4 and surface parameters (e.g., center $\mathbb{R}^2$5, quadratic form $\mathbb{R}^2$6, offset $\mathbb{R}^2$7), the transform is

$\mathbb{R}^2$8

The microlocal analysis centers on the singularities of the projections of the canonical relation associated with the phase function to the data and image sides.

Bolker Condition

The Bolker condition ensures injectivity and artifact-free invertibility of the Radon transform. It requires the left projection of the canonical relation $\mathbb{R}^2$9 to be an injective immersion. If the union of tangent planes to the acquisition surface $y^2 = s(x-c)^2 + u$0 does not intersect the support of $y^2 = s(x-c)^2 + u$1, the Bolker condition is satisfied, and the normal operator $y^2 = s(x-c)^2 + u$2 is elliptic, yielding stable Sobolev inverses (Webber et al., 2022).

Singularity Types

With indefinite $y^2 = s(x-c)^2 + u$3, the projections can exhibit fold, cusp, or blowdown singularities depending on the geometry of $y^2 = s(x-c)^2 + u$4 and the signature of $y^2 = s(x-c)^2 + u$5 (Ambartsoumian et al., 12 Feb 2026):

• Strictly convex $y^2 = s(x-c)^2 + u$6 and $y^2 = s(x-c)^2 + u$7: only folds;
• $y^2 = s(x-c)^2 + u$8, strictly convex or cylindrical $y^2 = s(x-c)^2 + u$9: presence of cusps or blowdowns;
• Violation of Bolker manifests as “streak” artifacts in reconstructions.

Microlocal Implications

For (data, image) conormal singularities $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$0, visibility in the Radon data is characterized by the right projection of the canonical relation. Regularization, domain truncation, and the geometry of $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$1 play critical roles in the effective microlocal invertibility of the hyperbolic Radon transform (Webber et al., 2022).

3. Inversion Formulas and Analytical Methods

Explicit inversion formulas exist in several models, typically combining Abel-type fractional integrals, back-projection operators, principal value integrals, and FIO calculus.

Seismic-Type Hyperbolic Radon Inversion (2D)

Given symmetry and vanishing hypotheses (evenness in variables, vanishing on the degenerate set), the principal value integral recovers $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$2 from its hyperbolic Radon data (Chihara, 2019): $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$3

Higher-Dimensional Generalization

For $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$4 variables and arbitrary exponents,

$$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$5

the inversion uses back-projection and fractional derivatives, reducing to standard Radon inverses upon suitable variable changes. The explicit formulas in odd and even dimensions employ principal value integrals and finite projections, respectively (Chihara, 2019).

Inversion for Quadratic and Hyperbolic FIOs

In the convex Bolker-satisfying setting, $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$6 is elliptic, and pseudodifferential inversion is classical. In the fold or cusp case, approximate parametrices can be constructed by localizing away from singular supports (Ambartsoumian et al., 12 Feb 2026, Webber et al., 2022).

Hyperbolic Space/Constant Curvature Inversion

In hyperbolic geometry, inversion is realized via Gegenbauer–Chebyshev fractional integrals, Riemann–Liouville derivatives, or recurrence relations based on spherical means (Rubin, 2014, Antipov et al., 2011). Certain cases admit Helgason-type differentiation-in-parameter (r) formulas and duality with classical Euclidean inversions.

4. Numerical Algorithms, Fast Computation, and Stability

The brute-force evaluation of the hyperbolic Radon transform is computationally cubic in the relevant discretization parameter, posing challenges for large-scale problems.

Log-Polar FFT Methods

By recasting the transform in log-polar coordinates, the fundamentally nonlinear mapping of the hyperbolic summation is reduced to convolution in log-polar space. This allows implementation of the forward (and adjoint) transform using 2D FFTs (Nikitin et al., 2016):

• Data is gridded into log-polar coordinates with cubic spline smoothing;
• FFTs replace direct summation, lowering complexity to $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$7;
• GPU implementations leverage cuFFT and massive parallelism, achieving speed-ups on the order of $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$8 over direct methods at large scale, e.g., $$R_h[f](s, u) = 2 \int_{x \in \mathbb{R},\, s(x-c)^2 + u \geq 0} f(x, \sqrt{s(x-c)^2 + u})\,dx.$$9 (4096) samples.

Stability and Regularization

Numerical inversion requires careful handling of ill-posedness (e.g., derivative amplification in Abel/Volterra equations), especially in the presence of noise. Techniques include regularized finite differences, Tikhonov-type stabilization, and truncation of singular value expansions (Webber, 2023). Increased measurement redundancy, appropriate domain partitioning, and regularization in the reconstruction functional are essential for practical stability.

Discrete Applications

In seismic processing, the hyperbolic Radon transform is central to multiple removal (sparse model fitting under $A$0 penalties), interpolation, and velocity analysis (Nikitin et al., 2016). In ultra-wideband array beamforming, discrete hyperbolic summation focuses near-field (spherical wave) signals and separates them from linear (far-field) events (Ikehara et al., 22 Apr 2026).

5. Generalizations: Constant Curvature and Higher-Rank Transforms

Horospherical and Geodesic Radon Transforms in $A$1

On the real hyperbolic space, the Radon transform extends to integration over totally geodesic subspaces (planes, horospheres, horocycles), with explicit analytic and $A$2-mapping properties. Inversions are available for all lower-dimensional horospherical transforms, generalizing the Euclidean mean-value integrals to hyperbolic geometry via Abel-type operators or Beltrami–Laplace polynomials (Bray et al., 2017, Kumar et al., 2012, Antipov et al., 2011).

Higher-Rank and Dual Transforms

The theory of higher-rank Radon transforms encompasses integration over families of $A$3-geodesic submanifolds mapped onto $A$4-geodesic submanifolds ($A$5), with dual and intertwining structures. Existence, injectivity, and support theorems depend on dimension, symmetries, and model (hyperboloid/Beltrami-Klein/projective analogs), and inversion may be realized through Erdélyi–Kober fractional calculus or Laplace–Beltrami polynomials (Rubin, 2021).

6. Applications and Real-World Contexts

Seismic Imaging

The hyperbolic Radon transform is entrenched in seismic data processing for velocity discrimination, multiple attenuation, and interpolation. It exploits the geometric fact that reflections from subsurface point scatterers follow hyperbolic moveout trajectories in time-offset slices (Chihara, 2019, Nikitin et al., 2016).

Medical and Ultrasound Tomography

In ultrasound reflection tomography, Radon transforms over ellipsoids and hyperboloids with centers on acquisition surfaces are used for imaging in cylindrical and more general geometries. Injectivity, stability, and artifact prediction depend acutely on acquisition surface convexity and associated microlocal geometry (Webber et al., 2022, Webber, 2023).

Harmonic Analysis and Inverse Problems

The hyperbolic Radon and related geodesic transforms enable explicit inversion formulas on spaces of constant curvature, play a role in analytic continuation on symmetric spaces, and underlie wave propagation studies (e.g., SYK model, AdS/dS correspondences) (Stone, 11 Jun 2025).

---

The hyperbolic Radon transform, in its various forms, exhibits rich geometric, functional-analytic, and computational structure. Advancements in fast algorithms, inversion theory, and microlocal analysis continue to drive both foundational understanding and broad applications in inverse problems, signal processing, and integral geometry.