# Low Dimensional Topology

## July 14, 2010

### The face manifold

Filed under: Misc. — Jesse Johnson @ 11:11 am

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 [1] by Gunnar Carlsson.

1. Very interesting idea, it may have many applications besides social networks.
I wonder what kind of topology you have to define on such a vector space in order to have a manifold with the desired properties on the continuos functions. I think I’m going to read Carlsson’s article next.

Comment by Cristina Valle — July 14, 2010 @ 1:28 pm

2. I think a more appropriate term would be “face space”, but maybe the pun would be lost on most people.

Comment by Ian — July 14, 2010 @ 1:39 pm

3. Wouldn’t it be a manifold? My gut feeling is that this thing is open in that big euclidean space. Small enough changes in a picture of a face should still look like a face, right?

Comment by Pierre — July 15, 2010 @ 5:44 am

4. That’s a good point, Pierre. I was thinking of the face manifold as having positive co-dimension, like an algebraic variety. But if it’s an open set then it would be a manifold. There is a BBC documentary on the human face, hosted by John Cleese, that shows some pictures of faces that aren’t recognizable as faces because of medical conditions that went untreated. I wonder if those would be included in the face manifold (or face space, though I’m not sure I get the pun.)

Comment by Jesse Johnson — July 15, 2010 @ 12:56 pm

5. I have no idea why wordPress put one of my posts (Linear Algebra Survival Guide for Quantum Mechanics — III) as possibly related to this one. Clearly this group knows a lot more math than I ever will. If any of you see anything wrong with any of the posts (there are 9 in the series), please post a comment over there. The point of the series was to make some of the arcane phenomena of linear algebra (the rules for matrix multiplication and the complex inner product, along with why anyone (particularly a physicist ) should be interested in hermitian matrices) seem more natural and plausible.

Luysii

Comment by luysii — July 18, 2010 @ 1:42 pm

6. Really great written post.

Comment by Robert Osborne — April 12, 2020 @ 1:46 am

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