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Diffusion
We’re now going to talk about diffusion processes. This is a physical process in which matter, or heat, for example, spread throughout a medium. This might seem like a non- sequitor with no connection to vision, but diffusion processes are relevant to segmentation for two reasons. First of all, diffusion is a kind of smoothing, and this gives us another set of tools to understand smoothing. For example, if you have a cup of hot coffee and you pour some cold milk into it, what happens to the heat? At first, parts of the cup are hot, and parts are cold. Over time, the heat spreads out, or is smoothed. Second, the result of diffusion can be found by solving a partial differential equation. So discussing diffusion will lead to a discussion of numerical methods for solving PDEs, which are going to be important throughout this course.
Here’s what we want to accomplish today:
Describe diffusion as a PDE.
Describe an explicit method for solving this numerically.
Point out that this explicit method is equivalent to repeated convolution with a small filter.
It turns out that this is equivalent to convolution with a Gaussian. So diffusion low-pass filters a signal. Diffusion and Gaussian convolution and low-pass filtering are all equivalent.
Point out that there are other ways to solve these equations, which have some useful properties (implicit method).
Diffusion as a PDE
One way to think about diffusion of a material in a medium is to imagine the world is divided up into tiny buckets. Each bucket has a discrete number of molecules, say. At every time step, a molecule can stay where it is or jump to a neighboring bucket. We make this simple by assuming it is:
homogenous: This means that the same thing happens everywhere. A particle has the same probability of doing things no matter which bucket it’s in.
isotropic: This means that the particle is equally likely to jump to the left and right.
Some intuitions. If one bucket has a lot of particles, and its neighbor has few, then more particles jump from the first to the second than from the second to the first. So the two buckets tend to become more alike over time (smoothing).
We can take the limit of this model, as our buckets get smaller and our time steps get shorter. It is then natural to express the way in which the concentration of particles change with a differential equation.
C(x,t) is the concentration of particles at position x at time t.
J(x,t) represents the flux of particles. This is the rate at which particles move in the positive direction across the position x at time t.
- J(x,t) = -D \partial C/\partial x.
This means that the flux is inversely proportional to the difference in the concentration of material. If \partial C/\partial x is big, it means the concentration is increasing, so more particles are flowing backwards than forwards. D is the diffusivity. This is a constant which indicates how rapidly particles diffuse in this medium. Note that D is a constant because the diffusion process is homogenous and isotropic. Otherwise, it could be a function of x.
- \partial C/\partial t = - \partial J/ \partial x
Change in concentration is inversely proportional to the change in flux. For example, if flux is constant, no change in concentration. This equation is basically the limit of having a small cell. The difference in flux on the two sides of the cell is the difference in how many particles are entering from the left, and how many are leaving to the right. Sign is negative because flux on the left side of the cell indicates how many particles are entering. If flux is dropping, more particles enter on the left than leave on the right.
Taking the partials of 1 w.r.t x we get: \partial J / \partial x = - D \partial^2 C / \partial x^2. Substituting into 2 we get:
- \partial C/\partial t = D \partial^2 C / \partial x^
This tells us that concentration changes in the direction of the second derivative. This tells us that the concentration gets smoother. For example, at a local maxima, the concentration drops, and at a local minima, the concentration increases.
This diffusion equation is also used in vision to smooth curves.
Numerical solution: Explicit method
We want to compute how C evolves over time. To do this numerically, the main issue is to take derivatives numerically. These will, of course, be an approximation. Taking a Taylor series expansion we find:
C_i+-1 = C_i +- \delta x \partial C/\partial x + ½ \delta x^2 \partial^2 C/\partial x^2 + O(\delta x^3).
this by convolving its initial position with a Gaussian. Or, we’ve seen, we can do it with repeated convolution by a small filter, which is the same thing.
This means that convolving with a Gaussian tells us the solution to the diffusion equation after a fixed amount of time. This is the same as low pass filtering an image. So smoothing, low pass filtering, diffusion, all mean the same thing.
