## The Image Matching Problem

In the previous example, we saw a vector field representing a specific, finite-dimensional Lie group and then saw specific paths through this group that depend on the initial velocity. While this is an interesting application of matrix groups on its own, we are interested in a much harder problem.

Given a template image and a target image, how can we find the mapping between them? Even more difficult, how can we find the "easiest" (this notion will become clearer) path to deform template into target? These are the very hard problems towards which these workbooks have been developing.

The method we are looking for is a gradient descent based approach to minimizing the cost of paths between two images so as to find the geodesic that connects them. This jargon will become much more intuitively clear in the coming pages.