Transport-Based Pattern Theory: A Signal Transformation Approach

In many scientific fields imaging is used to relate a certain physical quantity to other dependent variables. Therefore, images can be considered as a map from a real-world coordinate system to the non-negative measurements being acquired. In this work we describe an approach for simultaneous modeling and inference of such data, using the mathematics of optimal transport. To achieve this, we describe a numerical implementation of the linear optimal transport transform, based on the solution of the Monge-Ampere equation, which uses Brenier's theorem to characterize the solution of the Monge functional as the derivative of a convex potential function. We use our implementation of the transform to compute a curl-free mapping between two images, and show that it is able to match images with lower error that existing methods. Moreover, we provide theoretical justification for properties of the linear optimal transport framework observed in the literature, including a theorem for the linear separation of data classes. Finally, we use our optimal transport method to empirically demonstrate that the linear separability theorem holds, by rendering non-linearly separable data as linearly separable following transform to transport space.