Factorized Gradients for Scalable Highly-expressive Parametric Diffeomorphisms

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.