A few weeks ago, we read two beautifully written papers in our graphics seminar. Both were about measuring the distance between a pair of points. While one deals with points in the interior of a mesh , the other one deals with points on a mesh . I especially liked the discussion we had for the latter, where Misha gave a nice review of Diffusion Distance  in the beginning. I decided I should write it down along with some thoughts before my memory fades.
Some quick math reviews
· Delta Function: a delta function has zero value everywhere, except the origin where it has an impulse response. In 1-D it looks like:
So it is not really a function rather a functional that only makes sense when we consider the integral of its dot-product with some other function. Most importantly, given any function f(x) we have the following property:
In the rest of the article, we will use to denote a delta function that is translated and centered at p. That is: .
· Heat Diffusion on a manifold: (I am going hand-wavy here. More details can be found in .)
, such that
How can we solve it via Fourier Decomposition? (See  for the analogy between the eigenfuntions/eigenvectors of the mesh Laplacian and the Fourier basis.)
Let be eigenpairs of , where
In the discrete case, the solution at time t is formulated as:
Given the initial state , we first compute its frequency coefficients .
Then for any arbitrary time we can predict the state using the above formulation. (Intuitively, this means modulating each frequency component separately by and then recombining them)
Remark: From this interpretation, it becomes clear that Heat Diffusion is a smoothing process that suppresses high-frequency components faster than it suppresses low-frequency components. And when goes to infinity, only the DC term () will survive.
Intuitively, Diffusion Distance between a pair of points can be thought of as following: We create a unit impulse function centered at (imagine a fixed amount of energy concentrated at a point), and we let it diffuse for a period of time t. We create another impulse function centered at , and we also let it diffuse for a period time t. At the end, we look at the difference (measured by L2-norm) between the two distributions. And that is our Diffusion Distance.
Mathematically, this is described by:
and are the distribution functions undergoing Heat Diffusion, with the initial states and .
We use and to denote frequency coefficients of and respectively, i.e., and .
Now, notice the well-known fact that forms an orthonormal basis. This results in two consequences. First, (1) can be worked out to:
Second, finding their coefficients with respect to a certain function is as easy as computing their dot-products with that function. In the case of delta functions, it is even easier, i.e., and
Putting these together, we reach the common definition of the diffusion distance between point and point :
Relationship to Commute-Time Distance and Biharmonic Distance
Commute-Time Distance :
Biharmonic Distance :
 Rustamov, R., Lipman, Y., Funkhouser, T. Interior distance using barycentric coordinates. Computer Graphics Forum (Symposium on Geometry Processing), 28(5), July 2009
 Lipman, Y., Rustamov, R., Funkhouser, T. Biharmonic Distance. ACM Transactions on Graphics 29(3), June 2010.
 GOES, F. D., GOLDENSTEIN, S., AND VELHO, L. 2008. A hierarchical segmentation of articulated bodies. Computer Graphics Forum (Special Issue of Symposium on Geometry Processing) 27, 5, 1349-1356.
 Taubin, G. 1995. A signal processing approach to fair surface design. In ACM SIGGRAPH Conference Proceedings, 351-358.
 FOUSS, F., PIROTTE, A., MICHEL RENDERS, J., AND SAERENS, M. 2006. Random-walk computation of similarities between nodes of a graph, with application to collaborative recommendation. IEEE Transactions on Knowledge and Data Engineering 19, 2007