The google research blog has an interesting post about something called the “face manifold”. We can think of a digital picture as a vector, where the number of dimensions is three times the number of pixels, and the entries of the vector are the red, green and blue values of each pixel. Within the vector space of all images of a fixed pixel size, there is a hypothetical subset of consisting of all possible pictures of people’s faces. Thought of as a topological space, this subset is called the face manifold (though it seems unlikely that this would actually be a manifold.) Given two faces in this space, a path between them would correspond to a continuous one-parameter family of faces that smoothly transitions (morphs?) from one to the other. Some folks at google have approximated this manifold using public pictures harvested from Orkut (google’s social network platform) and made an application that shows you discrete paths between pictures of your friends. Unfortunately, you need an Orkut account and at least two friends with profile pictures to use it. So I haven’t been able to use it yet (I just signed up for an account five minutes ago.) but from the image on the blog, it looks pretty neat. The idea of the face manifold comes out of a body of much less hypothetical work on the topology of large data sets, which is summarized very nicely in a Bulletin article  by Gunnar Carlsson.
July 14, 2010
The face manifold