Posts tagged with eigenfaces

Blairthatcher

by and © Aude Oliva & Philippe G. Schyns

A hybrid face presenting Margaret Thatcher (in low spatial frequency) and Tony Blair (in high spatial frequency)

[I]f you … defocus while looking at the pictures, Margaret Thatcher should substitute for Tony Blair ( if this … does not work, step back … until your percepts change).

(Source: cvcl.mit.edu)

Linear combinations of eigenfaces — images like the above — are the cheapest way to store and search photos of faces. Like if you want to computer analyse the faces of everyone at the Superbowl and see if there’s a terrorist.

All of the terrorists’ faces are saved in a database as like {11% Eigenface_1, 6% Eigenface_2, 1% Eigenface_3, …}. So the real face breaks down to just a list of percentages {11%, 6%, 1%, …}.

Articles on Eigenfaces: 1, 2, 3, 4, 5, 6  (sorry to reinforce the google hierarchy)

If you break the faces in the live Superbowl crowd CCTV’s into such a list of percentages, then you can have a computer do a search against known terrorists. (Imagine having the computer search pixel by pixel. It would be like, Hey, the top left of your head looks like the top left of a terrorist’s head! Whoops, that was just the soda machine.)

Final point. There is more than one way to mathematicise a face. Surface normal vectors are one way; the manifold limit of polygons is another; projection is another; and eigenfaces are another. Each way has you conceive of a face differently.

## Eigenstuff

Imagine you have a small collection of things {…}. You take linear combinations (in the spirit of "When Doves Cry inside a Convex Hull") of them, making varied and interesting combinations that explore — even span — a “space”.

Maybe it’s

• the space of all possible faces
• the space of all possible songs
• the space of all possible configurations of a heat engine
• the space of all possible orientations of a sculpture
• the space of all possible paintings
• the space of all possible artistic schools
• the space of all possible states of your desk
• the space of all possible functions ℝ⁴→ℝ
• the space of all possible functions ℝ[0,1]→ℝ[0,1]
• the space of all possible directions in 3-D
• a subset of the space of all possible graphs
• the space of all possible quantum configurations of a molecule
• the space of all possible notes
• the space of all possible ways a week can go

(It’s possible to mathematicise these things in part because of the modern, abstract notion of a “vector”.)

Think about this large space instead of the original small-collection-of-things {…} that generated it. Could a different small-collection-of-things have spanned the exact same space? Surely so.

What about different small-collections {..A..}, {..B..} that could have generated the space? Is there a really simple collection? One whose parts don’t self-interfere. One that’s representative and easy to work with.

If there is such a “canonical basis”, the elements of the basis are called eigenfaces, eigenmodes, eigengraphs, eigenstates, or some other kind of eigenbasis.

Eigenboogers are fundamental to how abstract mathematics made “linear” algebra hugely relevant — to ODE’s, image compression, spectral analysis, crystallography, economics, statistical analysis, geophysics, and teaching artificial intelligence how to spot terrorists at the Superbowl.

Finally, eigenstates and eigenmodes are pretty good for meditating on how the universe is a song, singing itself.