Radon Slice Method for Data Reconstruction

Updated 15 April 2026

Radon Slice Method is a collection of analytic and computational techniques that transforms high-dimensional data into lower-dimensional slices using the Radon transform and central slice theorem.
It leverages Fourier analysis and support theorems to enable efficient inversion and reconstruction, incorporating algorithms like FFT and advanced interpolation.
Applications include tomography, quantum imaging, and signal processing, with numerical methods optimized for stability and high-resolution data analysis.

The Radon Slice Method refers to a collection of analytic and computational techniques leveraging the structure of the Radon transform and its connection to Fourier analysis, support theorems, integral geometry, and numerical inversion to efficiently process, reconstruct, or analyze high-dimensional data from lower-dimensional slices. The method exploits the central slice (projection-slice) theorem and its generalizations, enabling applications in signal processing, tomography, harmonic analysis, PDEs, quantum imaging, geometric analysis, and computational geometry.

1. Mathematical Foundation of the Radon Slice Method

Fundamentally, the Radon transform of a function $f$ in $\mathbb{R}^n$ , denoted $R[f](\theta, s)$ , is an integral of $f$ over the hyperplane with normal $\theta \in S^{n-1}$ and signed offset $s \in \mathbb{R}$ : $R[f](\theta, s) = \int_{\mathbb{R}^n} f(x) \, \delta(\langle x, \theta \rangle - s) \, dx$ For $n=2$ , the Radon transform reduces to integrating over lines; for higher $n$ , over $(n-1)$ -dimensional hyperplanes. This operation converts the original data into a family of its projections (slices) parameterized by $\mathbb{R}^n$ 0 and $\mathbb{R}^n$ 1.

The central slice theorem states that the 1D Fourier transform (in $\mathbb{R}^n$ 2) of the Radon projection at angle $\mathbb{R}^n$ 3 equals the restriction of the $\mathbb{R}^n$ 4D Fourier transform of $\mathbb{R}^n$ 5 to the radial line in frequency space directed by $\mathbb{R}^n$ 6: $\mathbb{R}^n$ 7 This result, and its multidimensional and algebraic extensions, underlies both theoretical analysis and fast computational schemes across diverse settings (Natroshvili, 2019, Gyongyosi, 2013, Mokhtari et al., 1 May 2025).

2. Analytic and Geometric Generalizations

Beyond basic tomography, the Radon slice methodology has been extended into various geometric frameworks:

Spherical and projective transforms: The spherical slice transform integrates a function on $\mathbb{R}^n$ 8 over the cross-sections by $\mathbb{R}^n$ 9-dimensional planes. Rubin (Rubin, 2021, Rubin, 2014) established that, via stereographic projection, the spherical slice transform can be expressed as a weighted Radon-John transform over $R[f](\theta, s)$ 0-dimensional planes in $R[f](\theta, s)$ 1, allowing import of inversion, support, and kernel results from Euclidean theory.
Clifford analysis and monogenic functions: The Radon and dual Radon slice transforms link monogenic functions (null-solutions to the Dirac operator in $R[f](\theta, s)$ 2) in $R[f](\theta, s)$ 3 to parametric families of (generalized) slice monogenic functions. Explicit expansion theorems, invertibility, and the structure of the function spaces are rigorously analyzed via harmonic and Clifford-algebraic techniques (Colombo et al., 2014).
Minkowski/holographic slicing: In Lorentzian (pseudo-Riemannian) settings, the Radon slice is defined over codimension-1 hyperplanes, crucial for connecting bulk fields (in AdS/dS/Minkowski) to boundary data as in bulk reconstruction and Mellin-space holography (Bhowmick et al., 5 Sep 2025).
Special geometries (cylindrical, planar): The slice approach has been adapted to Radon-type transforms occurring in photoacoustic and other imaging modalities, e.g., cylindrical and planar families of integrating lines or surfaces. Variant slice theorems relate the Fourier-Hankel transforms of data to planes or circles in the original space, yielding inversion and uniqueness (Moon, 2013).

3. Computational Algorithms and Discretization

Efficient numerical realization of Radon slice methods is critical for high-dimensional applications. Key computational strategies include:

Piecewise-polynomial closed-form calculation: For voxelized data in arbitrarily high dimensions, the Radon transform of each axis-aligned cube (voxel) can be computed exactly via a closed-form, piecewise-polynomial expression for the $R[f](\theta, s)$ 4-area of cube-hyperplane intersection. This regularized and slab-averaged variant enables rapid, stable sinogram computation and bypasses high-variance or coarse binning-based approximations (Beinert et al., 13 Mar 2026).
Digital Radon transform (DRT): Fast algorithms (complexity $R[f](\theta, s)$ 5) for the DRT and backprojection enable scalable implementation even for large grids. Slicing on digital grids leverages recursive and block-splitting constructions, as in partitioning angle space into quadrants and establishing exact digital lines (Rim, 2017).
FFT and advanced interpolation: In signal processing and seismic imaging, multidimensional FFTs are employed in conjunction with the generalized Fourier slice theorem (GFST). Polar-to-Cartesian interpolation in frequency space aligns the Radon-domain and data-domain representations for both forward and inverse transforms, dramatically accelerating computations (Mokhtari et al., 1 May 2025).
Sparsity and optimization: Regularized inversion (e.g., $R[f](\theta, s)$ 6 sparsity-promoting ISTA) is seamlessly embedded by treating the Radon slice operator as a fast linear mapping with rapid forward and adjoint computation, enabling high-resolution reconstructions, velocity analysis, and deconvolution (Mokhtari et al., 1 May 2025).