# Soft Tuy-Completeness for Robust Projection Selection in Cone-Beam CT

arXiv: https://arxiv.org/abs/2605.24023 · HTML: https://mlsanity.com/paper/2605.24023/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 7 of its 47 references that are in the map.
- First posted on arXiv 20 May 2026.

## Builds on (most cited first, 7 of 7)

- [ABC: A Big CAD Model Dataset For Geometric Deep Learning](https://mlsanity.com/paper/1812.06216/) (2018)
- [Learning 3D Gaussians for Extremely Sparse-View Cone-Beam CT Reconstruction](https://mlsanity.com/paper/2407.01090/) (2024)
- [Data Consistent Artifact Reduction for Limited Angle Tomography with Deep Learning Prior](https://mlsanity.com/paper/1908.06792/) (2019)
- [Branch and Bound in Mixed Integer Linear Programming Problems: A Survey of Techniques and Trends](https://mlsanity.com/paper/2111.06257/) (2021)
- [DRACO: Differentiable Reconstruction for Arbitrary CBCT Orbits](https://mlsanity.com/paper/2410.14900/) (2024)
- [Integer Optimization of CT Trajectories using a Discrete Data Completeness Formulation](https://mlsanity.com/paper/2402.10223/) (2024)
- [Application of Gated Recurrent Units for CT Trajectory Optimization](https://mlsanity.com/paper/2405.09333/) (2024)

## Built on by (most cited first, 1 of 1)

- [Gradient-Based Trajectory Optimisation over Continuous Poses for Sparse-View Cone-Beam CT](https://mlsanity.com/paper/2610.09579/) (2026)

## Abstract (arXiv)

This work introduces a continuous soft near-orthogonality score and a resolution-aware saturated coverage objective for projection selection in region-of-interest focused cone-beam CT, grounded in Tuy's completeness theory. Replacing the binary hit-or-miss model of classical Tuy completeness with a graded, differentiable formulation preserves a direct link to achievable feature sizes while enabling both efficient approximate and exact optimisation. We establish that the underlying discrete decision problems are NP-complete via polynomial-time reductions from Set Cover, motivating a submodular greedy algorithm with proven (1-1/e) approximation guarantees and a mixed-integer linear program (MILP) that provides certified optimality bounds. The MILP serves as a quality certificate for the greedy solution rather than a competing optimiser. The primary empirical finding confirms this relationship: across a systematic benchmark spanning six target regions, multiple projection budgets, and four controlled occlusion conditions, the pooled median greedy-to-MILP objective ratio was 0.998, with a substantial fraction of cases certified globally optimal. A binary formulation is included as a diagnostic baseline; it strengthens hard directional completeness but is weaker on the continuous coverage scale. We additionally introduce Effective Spatial Resolution (ESR), a physically interpretable trajectory-level diagnostic that maps directional sampling gaps to achievable feature sizes. ESR correlates reliably with matched reconstruction quality across projection budgets and occlusion levels, providing a practical bridge between the selection stage and the image domain without requiring reconstruction.