  This paper focuses on a similarity measure, known as the Wasserstein
distance, with which to compare images. The Wasserstein distance results from a
partial differential equation (PDE) formulation of Monge's optimal transport
problem. We present an efficient numerical solution method for solving Monge's
problem. To demonstrate the measure's discriminatory power when comparing
images, we use a $1$-Nearest Neighbour ($1$-NN) machine learning algorithm to
illustrate the measure's potential benefits over other more traditional
distance metrics and also the Tangent Space distance, designed to perform
excellently on the well-known MNIST dataset. To our knowledge, the PDE
formulation of the Wasserstein metric has not been presented for dealing with
image comparison, nor has the Wasserstein distance been used within the
$1$-nearest neighbour architecture.
