Torus computed tomography for experimental data

Abstract: We implement the torus-based X-ray tomography method introduced by Ilmavirta, Koskela, and Railo in "Torus computed tomography", SIAM J. Appl. Math., 80(4):1947--1976, 2020, for experimental X-ray tomographic data. The numerical implementation is extended to accommodate fan-beam measurements by converting the data to a parallel-beam format and mapping the projection angles to the closed-geodesic directions on the torus. In addition, we consider two extensions of the original framework: the Star TCT method which extends the frequency coverage of the reconstruction, and a numerical implementation of torus backprojection developed by Railo in "Fourier analysis of periodic Radon transforms", J. Fourier Anal. Appl., 26(4):64, 2020, for which we also derive a corresponding regularized formulation. We demonstrate the methods on experimental X-ray data of a walnut and compare them with filtered backprojection. We also introduce a pointwise positivity constraint as a post-processing step, which substantially improves the reconstruction accuracy. The simulated data experiments are revisited using an updated implementation. The results indicate that the proposed extensions improve reconstruction quality and support the applicability of torus-based reconstruction methods to experimental data.

• The paper demonstrates that torus-based reconstruction can process experimental fan-beam X-ray data, with walnut measurements reaching 24.7% relative L2 error for TBP at N=100 versus 22.8% for comparable FBP.
• The paper introduces Star TCT, filtered torus backprojection, and positivity post-processing, while proving convergence for discrete and regularized backprojection methods.
• The paper finds torus methods outperform FBP using the same non-uniform torus angles and avoid directional streaks, but uniformly sampled FBP remains more accurate overall and geometric preprocessing remains computationally expensive.

Overview and motivation

This paper by Salo, Meaney, Koskela, and Railo (2608.16453) demonstrates that the torus-based X-ray tomography framework of Ilmavirta, Koskela, and Railo (Ilmavirta et al., 2019) can be applied to experimentally acquired fan-beam X-ray data. The original Torus CT method reconstructs an image in the Fourier domain on the flat torus $\mathbb{T}^2 = \mathbb{R}^2/\mathbb{Z}^2$, exploiting the fact that each Fourier coefficient $\hat{f}(k)$ is determined by X-ray data along any integer direction $v$ orthogonal to $k$. While the theoretical foundations—injectivity, explicit inversion formulas, stability estimates, and Tikhonov regularization for periodic Radon transforms—were established earlier, the practical applicability to real measurement geometry remained untested. This work closes that gap and introduces three extensions: the Star TCT method for extended frequency recovery, a numerical implementation of torus backprojection (TBP) with a new regularized formulation, and a pointwise positivity post-processing constraint.

Theoretical framework

The method rests on the geodesic X-ray transform $\mathcal{I}$ on $\mathbb{T}^2$, integrating over closed geodesics parametrized by integer directions. By the Fourier slice theorem on the torus, the Fourier transform of the data satisfies $\widehat{\mathcal{I}f}(k,v) = \hat{f}(k)$ when $k \cdot v = 0$ and vanishes otherwise. Consequently, every nonzero Fourier coefficient in a box $\mathbb{Z}_N^2 = [-N,N]^2 \cap \mathbb{Z}^2$ is recoverable from projections in the admissible direction set $Q_N$, consisting of primitive integer vectors orthogonal to some $\hat{f}(k)$0. For $\hat{f}(k)$1, this set contains 3069 unique directions.

The adjoint identity $\hat{f}(k)$2 yields a particularly clean Tikhonov-regularized reconstruction: the minimizer of the Sobolev-space variational problem is simply a filtered version of the adjoint applied to the data, with filter coefficients $\hat{f}(k)$3.

A notable new theoretical contribution is a convergence result for the discrete torus backprojection. Building on the exact inversion formula $\hat{f}(k)$4 over all primitive directions (valid for zero-mean functions), the authors prove that the finite-sum approximation $\hat{f}(k)$5, with mean-value correction, converges to $\hat{f}(k)$6 both distributionally and in $\hat{f}(k)$7 for $\hat{f}(k)$8. They further prove a "filtered TBP" theorem showing that the Tikhonov-regularized solution equals the limit of $\hat{f}(k)$9 convolved with the kernel $v$0, thereby connecting the direct summation approach to the variational regularization theory. This gives TBP the same theoretical guarantees as the Fourier-domain Torus CT.

Methodological extensions

Star TCT exploits the observation that if $v$1, then $v$2 for all integers $v$3, so projection data in direction $v$4 simultaneously determines $v$5 at all multiples. The extended frequency set $v$6 recovers coefficients beyond the standard box without additional measurements; in the experiments, $v$7 with $v$8 doubles the frequency coverage at no data cost.

TBP sums the X-ray projections directly over $v$9 without Fourier analysis or filtering, handling the zero mode via a mean-subtraction correction. Two regularized implementations are provided—a Fourier-domain variant (fFTBP) using FFT-based filtering and a direct convolution variant (cFTBP)—which produce numerically identical results, with fFTBP substantially cheaper.

Positivity constraint applies the pointwise projection $k$0 as post-processing. The authors prove it cannot increase the $k$1 error when the ground truth is non-negative, and observe empirically that it benefits Fourier-based methods more than FBP because truncated Fourier series produce Gibbs-type oscillations with large negative excursions.

Simulated-data results

The revised implementation corrects numerical errors present in the original 2020 study and yields substantially improved results. Experiments use $k$2 phantoms (Shepp–Logan, flag, rotated flag) with 2% Gaussian noise, $k$3, and regularization parameters tuned per phantom by minimizing $k$4 error. Representative errors:

Method	Flag, noiseless $k$5	Flag, noisy reg. $k$6	Rot. flag, noisy reg. $k$7
Torus CT	8.49%	17.2%	15.7%
Star TCT	8.01%	17.5%	15.7%
TBP	7.34%	18.4%	16.2%
FBP ($k$8 angles)	26.7%	—	16.4%
FBP (uniform angles)	2.72%	14.1%	11.7%

Three findings stand out. First, in the noiseless setting all torus-based methods outperform FBP computed from the same torus-optimal angle set $k$9—for example, TBP achieves 7.34% $\mathcal{I}$0 error versus 26.7% for FBP on the flag phantom—and FBP exhibits shadow-like streak artifacts under the non-uniform angular distribution of $\mathcal{I}$1 that are absent from torus reconstructions. Second, TBP attains the lowest errors among torus methods on noiseless data but degrades most under noise, while regularized Torus CT and Star TCT remain robust. Third, and importantly as a caveat, FBP with uniformly distributed angles remains the most accurate method overall across all settings. The positivity constraint improves every method, with larger gains for the Fourier-based reconstructions.

Real-data results

The experimental dataset consists of fan-beam measurements of a walnut (2296 × 1200 sinogram, 0.3° angular increments). Since the forward model requires data along geodesic directions $\mathcal{I}$2, the fan-beam sinogram is converted to parallel-beam format via MATLAB's fan2para, and torus directions are approximated by nearest-angle selection. A structural limitation arises here: $\mathcal{I}$3 contains 3069 directions for $\mathcal{I}$4 but only 600 unique projection angles exist, so each measured angle is reused for multiple torus directions. The authors acknowledge this means the reconstruction does not exploit the correct number of independent measurements, though the reuse affects all compared methods equally.

Errors relative to an FBP reference reconstruction decrease monotonically with $\mathcal{I}$5: at $\mathcal{I}$6, TBP reaches $\mathcal{I}$7 versus 25.8% for Torus CT and 24.9% for Star TCT, against 22.8% for FBP with $\mathcal{I}$8 angles. Positivity reduces errors modestly for all methods. Notably, no regularization parameters were found that lower the $\mathcal{I}$9 error on real data—the authors attribute this to interpolation and data-reuse errors dominating over noise, which the Tikhonov scheme does not address. They also note candidly that because the reference itself is an FBP reconstruction, the error metrics may be systematically biased in FBP's favor, so the near-parity of torus methods with FBP should be read conservatively.

Computationally, the sinogram-to-torus mapping dominates: roughly 230 s at $\mathbb{T}^2$0 growing to about 3.8 hours at $\mathbb{T}^2$1 on a desktop CPU, whereas the subsequent reconstructions take only seconds to minutes (TBP fastest at 18–28 s). The implementation is not optimized, and parallelization could reduce these costs.

Limitations and open questions

The paper identifies its principal limitations explicitly. The fan-beam-to-parallel-beam conversion and nearest-angle selection introduce approximation errors that likely dominate the real-data results and are not mitigated by regularization; acquiring data directly along the geodesic angles $\mathbb{T}^2$2 would remove this source of error but has not been attempted. The real-data evaluation rests on a single object and an FBP-derived reference, leaving open whether the favorable comparison persists against ground truth. On simulated data, the question of when torus methods beat FBP with uniform angles—which they do not in any tested configuration—remains unresolved, as does the characterization of phantom classes where the torus-optimal angle structure confers an advantage.

Conclusion

This work establishes that torus-based tomographic reconstruction is viable on experimental fan-beam X-ray data, provides corrected and extended implementations with public code, and contributes new convergence theory for discrete and regularized torus backprojection. The simulated experiments show torus methods clearly superior to FBP restricted to the same non-uniform angles and artifact-free where FBP produces directional streaks, though standard uniformly sampled FBP retains the best overall accuracy. The extensions—Star TCT's free frequency coverage, the computationally efficient filtered TBP, and the positivity post-processing—each yield measurable accuracy improvements. The main obstacles to practical adoption are the computational cost of the sinogram-to-torus mapping and the geometric mismatch between conventional acquisition protocols and closed-geodesic measurement directions.