Transformations Based on Continuous Piecewise-Affine Velocity Fields

Oren Freifeld, Søren Hauberg, Kayhan Batmanghelich, Jonn W. Fisher

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We propose novel finite-dimensional spaces of well-behaved transformations. The latter are obtained by (fast and highly-accurate) integration of continuous piecewise-affine velocity fields. The proposed method is simple yet highly expressive, effortlessly handles optional constraints (e.g., volume preservation and/or boundary conditions), and supports convenient modeling choices such as smoothing priors and coarse-to-fine analysis. Importantly, the proposed approach, partly due to its rapid likelihood evaluations and partly due to its other properties, facilitates tractable inference over rich transformation spaces, including using Markov-Chain Monte-Carlo methods. Its applications include, but are not limited to: monotonic regression (more generally, optimization over monotonic functions); modeling cumulative distribution functions or histograms; time-warping; image warping; image registration; real-time diffeomorphic image editing; data augmentation for image classifiers. Our GPU-based code is publicly available.
Original languageEnglish
JournalI E E E Transactions on Pattern Analysis and Machine Intelligence
Issue number12
Pages (from-to)2496-2509
Publication statusPublished - 2017


  • Continuous piecewise-affine velocity fields
  • Diffeomorphisms
  • Tessellations
  • Priors
  • Spatial transformations

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