Transformations Based on Continuous Piecewise-Affine Velocity Fields

Abstract

We propose novel finite-dimensional spaces of well-behaved ℝn → ℝn 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.

Deep Diffeomorphic Transformer Networks

“…tations of diffeomorphisms (including [17,18]), it is still higher than for an affine spatial transformer. This is due to the added complexity of the transformation.…”

“…Finally, the CUDA implementation from [17,18] lacked TensorFlow interface (as it was not designed for DL architectures) and was also slower than our new implementation. First, here we derived closed-form expressions for the associated matrix exponentials.…”

“…1 Our code is available at: github.com/SkafteNicki/ddtn The CPAB gradient, ∂T θ (x)/∂θ, whose mathematical details can be found in [16], does not have a closed-form expression. Rather, it is given only via the solution of a system of coupled integral equations [18]. As such, TensorFlow's auto-differentiation is inapplicable for computing it.…”

“…Second, we simplified the CPA integration algorithm. Third, we added two additional parallelism levels; while in [17,18] parallelization was done only over different pixels, here we also parallelize over different images as well as the d components of the CPAB gradient. Without these speedups -explained in more detail in [12] -CPAB layers would have been impractical.…”

“…Recently, however, Freifeld et al proposed the CPAB transformations [17,18], which offer a happy medium between expressiveness and efficiency. This makes CPAB transformations a natural choice in a deep-learning context.…”