# Bi-Directional Point Flow Estimation with Multi-Scale Attention for Deformable Lung CT Registration
## 3 The Proposed Approach
### The Problem Setup  
Given a pair of lung CT scans, the task of deformable lung CT registration aims to estimate a dense, non-rigid transformation that accurately aligns the anatomical structures between a fixed scan (source) and a moving scan (target). Formally, let \(\mathcal{P}=\{\mathbf{p}_{i}\}_{i=1}^{N}\) and \(\mathcal{Q}=\{\mathbf{q}_{j}\}_{j=1}^{N}\) represent the 3D point clouds extracted from the source and target scans, respectively, where \(N\) denotes the number of sampled points. Our objective is to find a transformation \(\mathcal{F}=\{f_{i}\}_{i=1}^{N}\), mapping each source point \(\mathbf{p}_{i}\in\mathcal{P}\) to its corresponding target location \(\mathbf{q}_{j}\in\mathcal{Q}\).  
We parameterize this transformation as a displacement field, where each point-specific transformation function \(f_{i}:\mathbb{R}^{3}\mapsto\mathbb{R}^{3}\) is defined as  
\[f_{i}(\mathbf{p}_{i})=\mathbf{p}_{i}+\mathbf{v}_{i}, \tag{1}\]  
with \(\mathbf{v}_{i}\in\mathbb{R}^{3}\) being the displacement vector to predict. Given a set of ground-truth correspondences \(\mathcal{C}\) between the source and target points (precomputed using external tools such as corrField [15]), the registration problem is formally expressed as minimizing  
\[\mathcal{F}^{*}=\operatorname*{arg\,min}_{\mathcal{F}}\sum_{(i,j)\in\mathcal{ C}}\|f_{i}(\mathbf{p}_{i})-\mathbf{q}_{j}\|_{2}^{2}. \tag{2}\]  
This optimization encourages the transformed source points \(f_{i}(\mathbf{p}_{i})\) to closely match their ground-truth correspondences \(\mathbf{q}_{j}\), thereby achieving accurate anatomical alignment and maintaining spatial coherence.  
In this work, we propose a learning-based approach to directly estimating the displacement field \(\mathcal{F}\) from the input lung CT point clouds. Inspired by the point-based scene flow framework [17], our architecture incorporates two key innovations. First, we introduce specialized attention mechanisms--geometric keypoint attention at the coarse scale and progressive flow attention at the fine scale--to capture both global structural relationships and local deformation cues better. Second, we employ a bi-directional flow estimation strategy, simultaneously predicting the forward flow (\(\mathcal{F}\), source-to-target) and the backward flow (\(\mathcal{G}=\{g_{j}\}_{j=1}^{N}\), target-to-source). Formally, the backward transformation is defined as  
\[g_{j}(\mathbf{q}_{j})=\mathbf{q}_{j}+\mathbf{u}_{j}, \tag{3}\]  
where \(\mathbf{u}_{j}\in\mathbb{R}^{3}\) denotes the displacement vector for the target points. The backward flow is optimized by minimizing  
\[\mathcal{G}^{*}=\operatorname*{arg\,min}_{\mathcal{G}}\sum_{(i,j)\in\mathcal{ C}}\|\mathbf{p}_{i}-g_{j}(\mathbf{q}_{j})\|_{2}^{2}. \tag{4}\]  
This bi-directional formulation enables the network to leverage mutual consistency between the forward and backward transformations, resulting in more stable, accurate, and anatomically plausible registration outcomes. Figure 1 provides an overview of our network design, and we detail each module in the subsequent sections.