微分同胚为建模复杂空间或时间形变提供了一种灵活且保持拓扑结构的数学工具，广泛应用于多个科学领域。然而，同时实现高表现力和计算效率仍具挑战性。连续分段仿射基（CPAB）变换（Freifeld et al., ICCV 2015; IEEE TPAMI 2017）通过基于选定域剖分的连续速度场对微分同胚族进行参数化，提供了一种有吸引力的解决方案。重要的是，参数向量 $\theta$ 的维度 $d=\dim(\theta)$ 取决于剖分的精细程度，而非数据分辨率。尽管具有这种紧凑的参数化形式，CPAB 优化长期受限于轨迹积分与关于 $\theta$ 的梯度计算之间的紧密耦合。我们提出了 FG-CPAB，一种用于可扩展 CPAB 变换的因子化梯度公式。通过将积分与参数空间投影分离，该方法将梯度计算复杂度从 $\mathcal{O}(dTN)$ 降低至 $\mathcal{O}(TN+dC)$，其中 $T$ 为积分步数，$N$ 为变换点数，$C$ 为剖分单元数。这一重构实现了高达四个数量级的加速（例如在 2D 中约为 $10^4\times$），显著节省了内存（例如在 2D 中峰值 GPU 内存使用量减少约 $500\times$），并大幅提高了表现力。这使得细粒度剖分的 CPAB 首次在 2D 和 3D 中变得实用，从而实现了高表现力的参数化微分同胚。

Diffeomorphisms provide a flexible, topology-preserving mathematical tool for modeling complex spatial or temporal deformations and are used in various scientific fields. However, making these deformations simultaneously highly expressive and computationally efficient remains challenging. Continuous Piecewise-Affine Based (CPAB) transformations (Freifeld et al., ICCV 2015; IEEE TPAMI 2017) offer an attractive solution by parameterizing a family of diffeomorphisms via continuous velocity fields that are piecewise affine with respect to a chosen tessellation of the domain. Importantly, $d=\dim(θ)$, the dimension of the parameter vector $θ$, depends on the fineness of the tessellation rather than on the data resolution. Despite this compact parameterization, CPAB optimization has long been hindered by the tight coupling between trajectory integration and gradient computation with respect to $θ$. We introduce FG-CPAB, a factorized-gradient formulation for scalable CPAB transformations. By separating integration from parameter-space projection, our method reduces gradient computation complexity from $\mathcal{O}(dTN)$ to $\mathcal{O}(TN+dC)$, where $T$ is the number of integration steps, $N$ is the number of transformed points, and $C$ is the number of tessellation cells. This reformulation yields speedups of up to four orders of magnitude (e.g., $\approx 10^4\times$ in 2D), substantial memory savings (e.g., $\approx 500\times$ reduction in peak GPU memory usage in 2D), and dramatically higher expressiveness. This makes fine-tessellation CPAB practical in 2D and 3D for the first time, enabling highly expressive parametric diffeomorphisms.