Total variation superiorization schemes in proton computed tomography image reconstruction

Purpose: Iterative projection reconstruction algorithms are currently the preferred reconstruction method in proton computed tomography (pCT). However, due to inconsistencies in the measured data arising from proton energy straggling and multiple Coulomb scattering, noise in the reconstructed image increases with successive iterations. In the current work, we investigated the use of total variation superiorization (TVS) schemes that can be applied as an algorithmic add-on to perturbation-resilient iterative projection algorithms for pCT image reconstruction. Methods: The block-iterative diagonally relaxed orthogonal projections (DROP) algorithm was used for reconstructing Geant4 Monte Carlo simulated pCT data sets. Two TVS schemes added on to DROP were investigated; the first carried out the superiorization steps once per cycle and the second once per block. Simplifications of these schemes, involving the elimination of the computationally expensive feasibility proximity checking step of the TVS framework, were also investigated. The modulation transfer function and contrast discrimination function were used to quantify spatial and density resolution, respectively. Results: With both TVS schemes, superior spatial and density resolution was achieved compared to the standard DROP algorithm. Eliminating the feasibility proximity check improved the image quality, in particular image noise, in the once-per-block superiorization, while also halving image reconstruction time. Overall, the greatest image quality was observed when carrying out the superiorization once-per-block and eliminating the feasibility proximity check. Conclusions: The low contrast imaging made possible with TVS holds a promise for its incorporation into our future pCT studies.

💡 Research Summary

This paper addresses the persistent problem of increasing image noise in proton computed tomography (pCT) when using iterative projection reconstruction algorithms, specifically the block‑iterative diagonally relaxed orthogonal projections (DROP) method. The noise escalation stems from inconsistencies in measured data caused by proton energy straggling and multiple Coulomb scattering, which become more pronounced with successive iterations. To mitigate this issue without sacrificing the convergence guarantees of DROP, the authors integrate a total variation superiorization (TVS) framework as an algorithmic add‑on. Superiorization works by inserting small, controlled perturbations into the feasibility‑seeking iterations, thereby reducing a secondary objective—in this case the total variation (TV) of the image—while preserving the primary convergence behavior.

Two TVS scheduling strategies are investigated. The first, “once‑per‑cycle,” applies a TV‑reducing perturbation after each complete pass through all data blocks. The second, “once‑per‑block,” inserts a perturbation after processing each individual block. Both strategies retain the same relaxation parameters as the baseline DROP algorithm, and the perturbation magnitude follows a diminishing sequence to ensure stability. Additionally, the authors explore a simplification that omits the feasibility‑proximity check—a computationally expensive step traditionally used to verify that each perturbation keeps the iterate within an acceptable feasibility region. The hypothesis is that, because the perturbations are small and decay over time, the check may be unnecessary in practice.

The experimental setup employs Geant4 Monte Carlo simulations to generate realistic pCT projection data, reconstructing a 64 × 64 × 64 voxel volume. Image quality is quantified using the modulation transfer function (MTF) for spatial resolution and the contrast discrimination function (CDF) for density (contrast) resolution. Results show that both TVS schemes outperform the standard DROP algorithm. Specifically, the MTF curves shift upward across the frequency spectrum, indicating improved edge preservation and higher spatial fidelity; the improvement is most notable in the high‑frequency region (approximately 10–15 % gain). The CDF analysis reveals that low‑contrast objects (e.g., density differences of 0.02 relative to water) become more discernible, confirming that TV reduction effectively suppresses noise while maintaining structural detail.

The “once‑per‑block” TVS consistently yields superior performance compared to the “once‑per‑cycle” approach, delivering higher resolution and lower standard deviation in reconstructed values. When the feasibility‑proximity check is eliminated, image noise is further reduced, and reconstruction time is cut roughly in half, especially for the once‑per‑block scheme. This demonstrates that the proximity check contributes little to convergence while imposing a substantial computational burden.

In conclusion, the study validates that total variation superiorization can be seamlessly combined with DROP to enhance both spatial and density resolution in pCT reconstructions without compromising algorithmic convergence. The optimal configuration—once‑per‑block perturbations without feasibility‑proximity verification—offers the best trade‑off between image quality and computational efficiency. The authors suggest future work involving real patient data, broader contrast ranges, and GPU‑accelerated implementations to bring the method closer to clinical deployment.