Given a DCEL, how do you identify the unbounded face I have constructed a DCEL using the procedure described in How do I construct a doubly connected edge list given a set of line segments?.

This correctly identifies all faces, however I'm struggling to come up with a way to identify the unbounded face surrounding my graph.

So far my only idea is that by building a polygonal representation of every face, I  could find the face polygon which 'contains' all the others, but this seems kind of messy.
Take any extreme vertex (say among the vertices having the minimum x-coordinate, the one that minimizes y). This vertex is  incident to two edges touching the outer face. Namely the first and the last half-edges leaving this vertex. The first one has the outer face on its right and the last one on its left.