# Bi-Directional Point Flow Estimation with Multi-Scale Attention for Deformable Lung CT Registration
## 1 Introduction  
Deformable registration of lung CT scans plays a pivotal role in real-world clinical workflows, including respiratory motion compensation [1, 2], longitudinal disease monitoring [3, 4, 5], and radiation therapy planning [6, 7]. Unlike rigid registration methods [8, 9, 10, 11] that assume static global transformations, deformable lung CT registration accounts for the complex non-linear anatomical changes induced by respiration, enabling precise spatial alignment across different breathing phases.  
Traditionally, voxel-based registration has been the dominant paradigm for lung CT registration. These methods optimize the spatial transformation fields using intensity-based similarity metrics, such as mutual information [12, 13] or normalized cross-correlation [14]. While voxel-level registration provides dense deformation estimates, it suffers from critical limitations: (1) high computational and memory costs due to the dense volumetric nature of CT images [15] and (2) sensitivity to the inconsistencies in intensity caused by artifacts, noise, and contrast variability [7, 16].  
Recent advances in geometric deep learning and 3D point cloud representations have enabled a new class of point-based deformable registration, offering computational efficiency and robustness to variations in intensity [17, 18, 19]. These methods typically leveragesparse surface-based representations and geometric feature extraction to estimate dense deformation fields. However, they have not addressed the unique challenges of lung CT registration, where large, non-rigid respiratory deformations and fine-scale anatomical details must be jointly handled.  
Against this backdrop, the recent release of the Lung250M dataset [16] has introduced high-resolution lung CT point clouds across different respiratory phases, enabling foundational studies of point-based registration in lung CT scans. Unlike general registration tasks [17; 18; 19], lung CT registration requires precise modeling of both the global anatomical structure (e.g., inter-lobar coherence) and local correspondence across complex structures such as bronchial trees and vascular bifurcations. In clinical terms, reliable registration of these structures is essential for tracking the disease progression in COPD and measuring the treatment response in pulmonary fibrosis or the accumulating radiation dose across respiratory cycles. These demands necessitate a multi-scale, context-aware framework that can simultaneously capture the long-range consistency and fine-level alignment.  
In this work, we propose a novel point-based deformable lung CT registration framework designed to combine coarse-to-fine flow refinements with bi-directional geometric consistency. Specifically, our approach introduces three key innovations:  
* Geometric keypoint attention at coarse resolutions to enhance global-structure-aware feature matching across widely separated regions;
* Context-aware flow refinement at finer levels via attention-based feature interactions, enabling precise modeling of subtle local deformations;
* A bi-directional registration strategy that jointly estimates the forward and backward flow fields, enforcing transformation consistency and improving the anatomical plausibility.  
By incorporating these components into a unified architecture, the proposed method achieves accurate, stable, and anatomically consistent alignment of lung CT point clouds, significantly improving over the existing point-based baselines [17; 20]. Moreover, our method's use of vascular structures as the keypoints aligns with clinically interpretable anatomical landmarks, offering better transparency and usability than fully voxel-based models.