🔢 mathematics

Torus computed tomography for experimental data

This paper implements and extends the torus-based X-ray tomography method for experimental data by adapting it to fan-beam measurements, introducing Star TCT and regularized backprojection techniques, and applying a pointwise positivity constraint to demonstrate significantly improved reconstruction quality on a walnut sample compared to filtered backprojection.

Original authors: Ella Salo, Alexander Meaney, Olli Koskela, Jesse Railo

Published 2026-08-18

📖 6 min read🧠 Deep dive

CC BY 4.0

Original authors: Ella Salo, Alexander Meaney, Olli Koskela, Jesse Railo

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine trying to see inside a solid object without cutting it open. This is the daily challenge of computed tomography, or CT scanning, a technology that uses X-rays to build cross-sectional images of the human body, industrial parts, or even a single walnut. For decades, the standard way to turn these X-ray measurements into a clear picture has been a mathematical technique called filtered backprojection. It works by taking thousands of X-ray beams from different angles and mathematically "smearing" them back across a grid to reconstruct the original shape. While this method is reliable, it struggles when the data is incomplete, noisy, or acquired in a way that doesn't fit the standard grid, often leaving the final image with blurry edges or strange streaks.

A newer approach, known as torus computed tomography, offers a different path. Instead of working directly with the physical shape of the object, this method treats the object as if it were wrapped around a doughnut-shaped surface, technically called a torus. On this surface, the X-ray beams follow straight lines that eventually loop back on themselves, forming closed paths. By analyzing how the X-rays travel along these looping paths, researchers can reconstruct the image by looking at its frequency components—essentially breaking the image down into a pattern of waves rather than building it from pixels. This approach allows for precise control over which details are recovered and offers a way to clean up noise mathematically before the final picture is formed. However, while the theory was sound, it remained unclear whether this elegant mathematical framework could handle the messy, imperfect reality of actual X-ray machines, which often use fan-shaped beams rather than the idealized parallel beams required by the theory.

In a recent study, a team of researchers set out to bridge this gap between theory and practice. They took the torus method and adapted it to work with real-world X-ray data, specifically measurements taken of a walnut. The challenge was significant: the walnut was scanned using a standard fan-beam machine that rotates the object in tiny, equal steps, but the torus method requires data taken along specific, closed-loop directions that do not naturally align with those steps. To solve this, the team developed a way to translate the fan-beam data into a format the torus method could understand. They converted the fan-shaped measurements into parallel beams and then matched each required loop direction to the closest available angle from the scan. This allowed them to apply the torus reconstruction algorithm to a real, physical object for the first time.

The researchers did not stop at simply applying the existing method; they also expanded its capabilities. They introduced a variation called Star TCT, which allows the reconstruction to recover a wider range of image details without needing any extra data, effectively seeing more of the object's internal structure. They also implemented a technique called torus backprojection, which reconstructs the image by summing up the X-ray data directly, offering a different mathematical route to the same goal. To handle the inevitable noise in real scans, they added a regularized version of this backprojection method, which smooths out the image while preserving important details. Finally, they added a simple but powerful post-processing step: a rule that forces any negative values in the image to zero. Since X-ray attenuation—the amount of light the object blocks—cannot be negative in the physical world, this constraint acts as a physical reality check, cleaning up the image further.

When the team tested these methods on the walnut, the results were revealing. The torus-based reconstructions produced clear images that were free from the shadow-like streaks that often plague standard methods when the scanning angles are not perfectly uniform. In fact, the torus images avoided the specific artifacts that appeared in the standard filtered backprojection images, which showed distinct streaks along diagonal and vertical lines. The researchers found that while the standard method using all available angles still produced the most accurate overall image, the torus methods came remarkably close, especially when the positivity constraint was applied. This simple step of removing negative values significantly improved the accuracy of the torus reconstructions, bringing them closer to the quality of the standard method.

The study also revisited previous computer simulations to ensure the new code was working correctly. In these simulated, noise-free environments, the torus methods outperformed the standard approach when using the same limited set of angles, and the new Star TCT variant provided a consistent improvement over the basic torus method. However, the researchers noted that in the presence of noise, the standard method with uniformly distributed angles remained the most accurate overall. The torus backprojection method, while excellent in clean conditions, proved more sensitive to noise, though the regularized versions helped mitigate this issue.

A significant finding of the work was the computational cost. The process of translating the fan-beam data to the torus format is currently the most time-consuming part of the process. For a high-resolution image, this mapping step alone can take nearly four hours on a standard computer, whereas the actual reconstruction happens much faster. Despite this bottleneck, the team demonstrated that the method is viable and that the quality of the images is high enough to be useful. The ability to evaluate the image on any grid without additional cost, thanks to the nature of the torus math, offers a unique flexibility that standard methods do not possess.

Ultimately, this work confirms that the torus computed tomography method is not just a theoretical curiosity but a practical tool that can handle real experimental data. By successfully adapting the method to fan-beam scans and introducing new extensions like Star TCT and positivity constraints, the researchers have shown that this approach can produce high-quality images that avoid the specific artifacts of traditional techniques. While it may not yet replace the standard method in all scenarios, it offers a powerful alternative for situations where data is incomplete or where specific types of image artifacts need to be avoided. The study concludes that with further optimization of the computational steps, torus-based reconstruction could become a robust option for experimental X-ray imaging.

Technical Summary: Torus Computed Tomography for Experimental Data

Problem Statement
Computed tomography (CT) traditionally relies on filtered backprojection (FBP), a method rooted in the Radon transform. While effective for standard acquisition geometries, FBP often yields inadequate results when data is noisy or incomplete. Recent theoretical work has established the geodesic X-ray transform on the flat torus ( $T^2$ ) as a viable alternative framework for reconstruction. However, the original implementation of Torus CT (Ilmavirta et al., 2020) remained largely theoretical or limited to simulated data, with unclear applicability to practical, experimentally acquired X-ray measurements which typically utilize fan-beam geometries and equidistant projection angles rather than the specific closed-geodesic directions required by the torus model.

Methodology
This work implements and extends the Torus CT framework to handle experimental data. The core methodology involves three primary technical components:

Data Conversion and Mapping:
- Experimental data is acquired as fan-beam measurements at equally spaced angles. To utilize the Torus CT forward model ( $A_{T^2}$ ), the fan-beam sinogram is first converted to parallel-beam Radon data.
- The parallel-beam projection angles are then mapped to the discrete set of closed-geodesic directions ( $Q_N$ ) on the torus. Since experimental angles do not perfectly align with the required integer vector directions, the method employs a nearest-angle selection strategy.
- The forward operator decomposes closed geodesics into line segments within the fundamental domain $[0, 1)^2$ , computing torus X-ray transforms as sums of standard Radon transform values over these segments.
Reconstruction Algorithms:
- Standard Torus CT: Reconstructs the image by recovering Fourier coefficients $\hat{f}(k)$ from X-ray data projected onto the torus. This relies on the Fourier slice theorem, where data in direction $v$ determines coefficients orthogonal to $v$ .
- Star TCT (Extended Frequency Coverage): An extension that recovers additional Fourier coefficients beyond the standard frequency box $Z_N$ . By exploiting the property that data in direction $v$ determines coefficients for all multiples $mk$ (where $k \perp v$ ), this method expands the recoverable frequency set to $K_{N, \tilde{N}}$ without requiring additional measurements.
- Torus Backprojection (TBP): An alternative spatial-domain reconstruction derived from summing X-ray projections over all primitive integer directions. The paper derives a regularized formulation for TBP, showing that Tikhonov regularization corresponds to convolving the unregularized TBP result with a spatial filtering kernel. Two numerical implementations are provided: a Fourier-domain approach (using FFT) and a direct convolution approach.
Regularization and Post-Processing:
- Tikhonov regularization is applied in the Fourier domain to stabilize the inversion against noise.
- A pointwise positivity constraint ( $f_+ = \max(f, 0)$ ) is introduced as a post-processing step. This leverages the physical prior that X-ray attenuation coefficients are non-negative, aiming to reduce Gibbs-type oscillations and improve accuracy.

Key Contributions
The paper makes the following specific contributions:

Experimental Implementation: Successfully adapts Torus CT to experimental fan-beam X-ray data (specifically a walnut dataset), bridging the gap between theoretical torus geometry and practical acquisition setups.
Algorithmic Extensions: Introduces the Star TCT method to extend frequency recovery and implements Torus Backprojection (TBP) with a derived regularized formulation.
Code Revision: Updates the original numerical implementation from Ilmavirta et al. (2020), correcting simulation errors and optimizing the forward operator.
Positivity Constraint: Demonstrates the efficacy of a simple pointwise positivity constraint as a post-processing step for improving reconstruction quality in attenuation imaging.

Results
The methods were evaluated using both simulated phantoms (Shepp-Logan, Flag, Rotated Flag) and experimental data from a walnut.

Simulated Data:
- The updated implementation yielded substantially improved results compared to the original 2020 study.
- In noiseless settings, all torus-based methods outperformed FBP when using the same set of torus-optimal directions ( $Q_N$ ). FBP with $Q_N$ exhibited shadow-like artifacts due to the non-uniform angular distribution, which were absent in torus-based reconstructions.
- Star TCT provided consistent improvements over standard Torus CT.
- TBP achieved the lowest errors in noiseless settings but was more sensitive to noise.
- In noisy, regularized scenarios, torus-based methods generally produced lower $L^2$ errors than FBP with $Q_N$ for specific phantoms.
- The positivity constraint significantly reduced reconstruction errors for all methods, particularly for Fourier-based approaches where it suppressed Gibbs oscillations.
- FBP using uniformly distributed angles remained the most accurate overall, serving as a strong benchmark.
Experimental Data (Walnut):
- Reconstructions were compared against an FBP reference computed from all available projection angles.
- Torus-based methods produced reconstructions comparable in error to FBP using the same limited set of torus-optimal angles ( $Q_N$ ).
- Crucially, torus-based methods did not exhibit the shadow-like streak artifacts seen in FBP reconstructions using $Q_N$ .
- The positivity constraint reduced errors for both torus and FBP methods but did not alter their relative performance ranking.
- Computational Cost: The dominant bottleneck is the "sinogram-to-torus" mapping step, which scales significantly with the frequency box size $N$ (increasing from ~230s for $N=25$ to ~3.8 hours for $N=100$ ). Once mapped, the actual reconstruction is relatively fast, with TBP being the fastest method.

Significance and Claims
The paper claims that the proposed extensions (Star TCT, TBP, and positivity constraints) improve reconstruction quality and support the applicability of torus-based methods to experimental data. The authors emphasize that the same measurement data can be exploited in multiple ways within the torus framework.

However, the authors maintain a modest stance regarding the experimental results. They acknowledge that the reference FBP reconstruction may introduce a systematic bias in favor of FBP. Furthermore, they note that the approximation errors introduced by converting fan-beam data to parallel-beam and selecting nearest angles likely dominate the error sources in real-data experiments, limiting the effectiveness of Tikhonov regularization in this specific context. The paper concludes that while the methods are viable and artifact-free, further work is needed to characterize the conditions under which they outperform standard FBP and to address the computational bottlenecks of the mapping step.

Technical Summary: Torus Computed Tomography for Experimental Data

More like this