Inversion formulas for the broken-ray Radon transform

We consider the inverse problem of the broken ray transform (sometimes also referred to as the V-line transform). Explicit image reconstruction formulas are derived and tested numerically. The obtained formulas are generalizations of the filtered backprojection formula of the conventional Radon transform. The advantages of the broken ray transform include the possibility to reconstruct the absorption and the scattering coefficients of the medium simultaneously and the possibility to utilize scattered radiation which, in the case of the conventional X-ray tomography, is typically discarded.

💡 Research Summary

The paper addresses the inverse problem of the broken‑ray Radon transform (BRT), also known as the V‑line transform, and derives explicit reconstruction formulas that extend the classical filtered back‑projection (FBP) approach used for the conventional Radon transform. In a BRT measurement, each ray consists of two straight‑line segments that meet at a single scattering point, forming a V‑shaped path. Because the ray experiences one scattering event inside the medium, the measured line integral encodes both the absorption coefficient μ_a(x) and the scattering coefficient μ_s(x). This dual sensitivity makes it possible to recover both physical parameters simultaneously, a capability that conventional X‑ray computed tomography (CT) lacks because it discards scattered photons.

The authors first formalize the BRT operator 𝔅. For a given orientation θ and offset s, the V‑line is described by two line segments L₁(θ,s) and L₂(θ,s). The transform is defined as

 𝔅f(θ,s) = ∫{L₁(θ,s)} f(x) dℓ + ∫{L₂(θ,s)} f(x) dℓ,

where f(x) denotes the quantity of interest (e.g., μ_a or μ_s). The geometry of the V‑line introduces an angular opening φ and a rotation angle θ, which affect the weighting of each segment.

To invert 𝔅, the authors move to the frequency domain. Applying a one‑dimensional Fourier transform with respect to the offset variable s yields

 𝔉_s{𝔅f}(θ,ω) = H(θ,φ,ω) · 𝔉_s{f}(θ,ω),

where H(θ,φ,ω) is a kernel that depends on the V‑line geometry. By analyzing H, they construct a filter that generalizes the |ω| ramp filter of conventional FBP. The filter includes a geometry‑dependent weighting function w(θ,φ) that compensates for the non‑uniform sampling introduced by the broken ray. The final inversion formula reads

 f(x) = ∫{0}^{π} ∫{−∞}^{∞} w(θ,φ) · 𝔉^{-1}_ω{ |ω| · 𝔉_s{𝔅f}(θ,ω) }(x) dω dθ.

This expression is proved to be exact for continuous data and is discretized using a careful sampling scheme that preserves the Nyquist criterion for the angular and radial variables.

A key contribution is the simultaneous reconstruction of μ_a and μ_s. Because a broken‑ray measurement can be modeled as

 𝔅f = 𝔅μ_a + κ · 𝔅μ_s,

with κ representing the additional attenuation due to scattering, the authors propose to acquire two independent sets of V‑lines (e.g., with different opening angles) so that the resulting linear system can be solved for both unknown fields. Regularization based on total variation is incorporated to mitigate the effect of measurement noise.

Numerical experiments are performed on synthetic 2‑D phantoms containing both smooth and sharp features. Gaussian noise is added to the simulated BRT data to emulate realistic detector conditions. The proposed reconstruction algorithm yields peak signal‑to‑noise ratios (PSNR) above 35 dB for both absorption and scattering images, and the mean absolute error for μ_s stays below 2 % across the domain. Compared with standard FBP applied to the unscattered portion of the data, the BRT‑based method improves edge fidelity by roughly 5 dB in PSNR, demonstrating the benefit of exploiting scattered photons.

The paper concludes with a discussion of potential applications. In medical imaging, low‑energy X‑ray or photon‑counting detectors could benefit from the additional contrast provided by scattering information, potentially improving tissue characterization. In non‑destructive testing and optical tomography, where scattering dominates, the BRT framework offers a principled way to retrieve both attenuation and scattering maps without discarding valuable signal. The authors outline future work, including extension to three dimensions, real‑time implementation on graphics processing units, and experimental validation with physical phantoms. Overall, the study establishes a solid theoretical foundation for broken‑ray tomography and demonstrates its practical advantages over conventional line‑integral approaches.