Soft Tuy-Completeness for Robust Projection Selection in Cone-Beam CT
Quick Take
This study presents a soft near-orthogonality score and a resolution-aware objective for projection selection in cone-beam CT, enhancing Tuy's completeness theory. The proposed method achieves a median greedy-to-MILP objective ratio of 0.998 across various benchmarks, confirming its effectiveness in optimizing projection selection while maintaining optimality bounds.
Key Points
- Introduces a continuous soft near-orthogonality score for cone-beam CT projection selection.
- Establishes NP-completeness of discrete decision problems via polynomial-time reductions.
- Achieves a median greedy-to-MILP objective ratio of 0.998 across six target regions.
- Presents Effective Spatial Resolution (ESR) as a diagnostic for feature size mapping.
- MILP serves as a quality certificate for the greedy solution, not as a competing optimizer.
Article Content
From source RSS / original summaryarXiv:2605. 24023v1 Announce Type: new Abstract: 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 $(1-1/\mathrm{e})$ 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.
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