Mean curvature flow vs. diffusion

Question

Mean curvature flow (MCF) and diffusion-type flows both have smoothing effects on a curve.

I can tell there is a deep connection between the two, as seen in the Merriman-Bence-Osher (MBO) numerical scheme, which uses diffusion to approximate MCF: it takes a characteristic function of a region $\Omega$ and iterates the following

1. apply the heat kernel (i.e. allow the boundary to diffuse), and
2. threshold (to recreate a sharp boundary).

What is the relationship between MCF and diffusive flows and how can one explain this geometrically? Honestly I'm not even sure whether "diffusive flow" is a term, but by that I mean evolution of a curve as described by the heat equation/heat kernel.

Edit: I also read somewhere that Ricci flow on a surface effectively evolves the curvature of the surface with a heat equation. Now, how does Ricci flow come into the picture?

Answer 1

So the best reference for this would probably be Merriman, Bence, and Osher's original paper. In it, they show that the heat equation is locally a mean curvature flow with rate $D\kappa$.

Specifically, consider a region $\Omega$ and its characteristic function $\chi_\Omega$ (let's just call it $\chi$). We can consider the result of a diffusion of $\chi$, and in particular the effect of the diffusion at a point $P$ on the boundary of $\chi$.

The diffusion equation applied to $\chi$ would be $$ \frac{\partial \chi}{\partial t} = D\Delta \chi.$$

Consider the tangent circle to $P$, i.e. if $P$ has local curvature $\kappa = \frac{1}{r}$, the circle with radius $r$.

What is the effect of the diffusion at this point? Rewrite the diffusion equation in the polar coordinates based on the tangent circle to $P$, and we get $$ \frac{\partial \chi}{\partial t} = \frac{D}{r} \frac{\partial \chi}{\partial r} + D\frac{\partial^2 \chi}{\partial r^2} + \frac{D}{r^2}\frac{\partial^2 \chi}{\partial \theta^2}$$

(note this is the diffusion equation LOCAL to the point $P$).

Since $\chi$ is tangent to the circle at $P$, the curve $\chi$ has locally unchanging angle at $P$, i.e. $\frac{\partial^2 \chi}{\partial \theta^2} = 0$.

Thus the diffusion equation at $P$ becomes $$ \frac{\partial \chi}{\partial t} = \frac{D}{r} \frac{\partial \chi}{\partial r} + D\frac{\partial^2 \chi}{\partial r^2} $$

This is an advection-diffusion equation: the term $D\frac{\partial^2 \chi}{\partial r^2}$ acts to diffuse the the boundary of $\chi$, i.e. smoothing the sharp step that the characteristic function has. The term $\frac{D}{r} \frac{\partial \chi}{\partial r}$is the velocity term, which moves the point $P$ (and all points around it) at speed $\frac{D}{r}$. Since the local curvature at $P$ is $\kappa = \frac{1}{r}$, we have that the local change at $P$ simultaneously

1. diffuses out/spreads
2. moves in the direction of its local curvature with rate $D\kappa$.

This is why the MBO scheme alternates diffusion term and thresholding term. The diffusion term takes a MCF step, but the boundary gets diffuse. The thresholding term resharpens the boundary by defining it to be the level set $\chi = \frac{1}{2}$ (which evolves via MCF since it moves exactly with the advective velocity $D\kappa$).

TLDR the diffusion equation applied to the characteristic function $\chi_\Omega$ of a set $\Omega$ causes the boundary $\partial \Omega$ to move according to a mean curvature flow in a small time step, with rate $D\kappa$ where $\kappa$ is the curvature at each point. It only works in a small time step because this analysis relies on local polar coordinates at each point on the boundary.

Answer 2

First: heat flow step smooths the functional.

Second: threshold step redefines the functional to be the indicator function of the 1/2-level set.