What is m representing in this slide?
It's the number of basis functions used to approximate the original surface (in the box): as you use more and more bases (m), you get a better and better approximation.
So does that mean we are also increasing the number of dimensions? (Because the number of vectors in an orthonormal basis = dimensionality of the space we are working in)
Yes, exactly. When you approximate a function by something like a truncated Fourier series, one way to visualize it is to imagine functions that look more and more like the original one. Another is to imagine that you have a point in a higher and higher dimensional space - the coordinates of this point are of course the coefficients in the basis used for the approximation.
I assume this is what the "will have plenty more to say as course goes on" bullet is for, but will we be discussing filters in regards to signal processing?
The approximation of the dragon reminds me of a cool 3D reconstruction technique that involves dipping the object into water and measuring the changes in the water level to determine the shape of the object. The more times the the object is dipped at different angles, the better the approximation of the shape. After 1000 or so dips, the reconstructed model looks exactly like the original.
Fourier transforms are really neat-- this site lets you experiment with adding different number of sine and cosine waves to approximate a desired graph. An interesting application of Fourier transforms that I've seen has been with Shazam, which identifies music by comparing the Fourier decomposition of the song it needs to identify to its existing database of Fourier decompositions of know songs.
I'm probably getting a little ahead here. On calculating the fourier transform of an image, we get an amplitude map and a phase map. The phase map seems to be (more?) important for visual perception. I wonder if this would also be the case for 3D?
@sickgraph It depends on how you decide to generalize the Fourier transform. For instance, one common approach is to use eigenfunctions of the (real) Laplace-Beltrami operator as your analogues of Fourier modes. In this case there is not a particularly easy way to talk about phase (and yet this approach is perhaps the most common for 3D geometry processing applications). Complex operators may provide other possibilities.