Highly-Expressive Spaces of Well-Behaved Transformations: Keeping it Simple

We propose novel finite-dimensional spaces of Rⁿ → Rⁿ transformations, n ∈ {1, 2, 3}, derived from (continuously-defined) parametric stationary velocity fields. Particularly, we obtain these transformations, which are diffeomorphisms, by fast and highly-accurate integration of continuous piecewise-affine velocity fields, we also provide an exact solution for n = 1. The simple-yet-highly-expressive proposed representation handles optional constraints (e.g., volume preservation) easily and supports convenient modeling choices and rapid likelihood evaluations (facilitating tractable inference over latent transformations). Its applications include, but are not limited to: unconstrained optimization over monotonic functions, modeling cumulative distribution functions or histograms, time warping, image registration, landmark-based warping, real-time diffeomorphic image editing. Our code is available at https://github.com/freifeld/cpabDiffeo.