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Nonlinear Diffusion
These notes summarize the way I present this material, for my benefit. But everything in here is said in more detail, and better, in Weickert’s paper.
Introduction: Motivation for non-standard diffusion
First, let’s begin with some definitions and examples of why we want to use other kinds of diffusion beyond Gaussian smoothing.
We have considered the case of a noisy step edge, and noticed that Gaussian smoothing smoothes across the edge. Gaussian smoothing makes sense away from the edge, but we might want to do something different around the edge. If smoothing is not the same everywhere, we can call this non-homogenous. We want to do this so smoothing is influenced by local image properties.
We might want smoothing to depend on the smoothed image, not the original image. That is, if smoothing eliminates some unimportant structure, that structure should not continue to affect the smoothing. When the smoothing depends on the current, diffused image, though, we get a nonlinear equation for the smoothing, which we call non-linear diffusion.
Finally, we may also want the direction of the flux to depend on image properties. If the intensities are linear (form a ramp) we want the flux to be in the direction in which the ramp is going down, which is the direction of the gradient. This is essentially a 1D problem, with the same direction we had in 1D. If the flux is always in the direction of the gradient, we call the diffusion isotropic. However, near a step edge, we might want the flux to be parallel to the edge, which is perpendicular to the gradient. If the direction of the flux varies in the image, we call it anisotropic.
Non-linear diffusion - Perona-Malik diffusion
We can explain non-homogenous and non-linear diffusion in 1D, so we’ll stick with that, and only move to 2D when we have to. We will use notation consistent with Weickert’s article, so:
f(x) is the density at time 0 (ie the image). u(x,t) is the density at position x and time t.
Normal diffusion is:
\partial u / \partial t = D (\partial^2 u / \partial x^2)
If we stick with isotropic diffusion, we cannot regulate the direction of the diffusion (which is why we can consider the 1D case) we only regulate the amount. It makes sense
to have less diffusion where the image is changing fast, that is, near edge-like things. We can do this with:
\partial u / \partial t = g(x) (\partial^2 u / \partial x^2)
g(x) is a spatially varying diffusion. If g(x) only depends on the initial image, then it is effectively constant and so we get a linear PDE. A natural choice would be a function that decreases with |f’|. One example in use is:
g = 1 / sqrt( 1 + (f’)^2/ \lambda^2)
for some constant \lambda. Notice this decreases from 1 to 0 as f’ grows. When f’ = \lambda, g = ½.
As mentioned above, this produces artifacts, because even after a small structure disappears, it continues to influence the image. It also seems unpleasing, because if two points have the same f’ they will always experience the same amount of smoothing, even if after some smoothing one has a derivative of 0, and the other still has a high derivative. This might happen with a ramp edge with noise wiggles added.
So instead, we make g depend on the current image, not the initial image. This whole approach was introduced by Perona-Malik, who suggested:
g = 1 / (1 + (u’)^2 / \lambda^2)
Instability and artifacts
This works well in practice. Unfortunately, when we analyze Perona-Malik, we find that it is an ill-posed PDE. That means the solutions may not be unique, and they can be unstable (that is, change a lot with small changes in the initial conditions). The way to fix this is with regularization, which means, some kind of smoothing. This is why it works pretty well in practice; discretization acts as a kind of regularization, because it is like approximating the function with a piecewise constant function, which is smoother. This can produce weird results, though, such as worse artifacts when we discretize more. A more principled way to fix this is with
g = 1 / (1 + (u_sigma’)^2 / \lambda^2)
where u_sigma denotes the density smoothed with a Gaussian of sigma.
Anisotropic Diffusion
As mentioned before, we may want to diffuse in different directions in different parts of the image. When there is an edge, we could diffuse orthogonal to it. To do this, we need to start talking about diffusion in 2D.
The gradient is computed on a smaller scale than the intensity variation. If the two are unrelated, so the gradient direction is at a 45 degree angle to the direction of greatest variation, then: If the variation in intensity is almost the same in both directions, smoothing occurs in direction of gradient; as coherence increases, smoothing increases in coherence direction.
Why is it done this way? The direction of flux is never perpendicular to the direction of gradient. In fact it always must have a component in the opposite of the direction of gradient, and no component in the direction of gradient. These properties follow from the fact that the diffusion tensor is positive definite, which is needed to ensure good properties of the diffusion, such as having a unique solution.
We can see the importance of making D positive definite with a simple example. Suppose D had an eigenvector in the direction of the gradient, but this had a negative eigenvalue. This would allow us to construct a diffusion that is the inverse of ordinary diffusion, and that undoes it effects. So, with ordinary diffusion, we can take very different images and smooth them until they are almost constant, and all look nearly the same. If we applied the inverse process, we could start with nearly identical, smooth images, and produce images that look radically different. This means that inverting diffusion is a very unstable process, and can’t be computed effectively.
The Beltrami View
Sochen et al. give another way to formulate anisotropic diffusion, which seems very intuitive. Recall that we can think of ordinary diffusion as the process of replacing each intensity with a weighted average of neighboring intensities, where each neighboring intensity is weighted by the Gaussian of its distance to the pixel. In the Beltrami framework, we do exactly the same thing, except we think of an image as a 2D surface in 3D. Each point, (x,y), with intensity u(x,y) is mapped to the 3D point (x,y,Cu(x,y)). C is a constant, that indicates how to weight intensity vs. spatial coordinates. Then, we take a weighted average, using the geodesic distance in this space, rather than the Euclidean distance in (x,y) space.
Suppose, for example, we have two nearby points on opposite sides of a step edge. The geodesic distance between them will be very big, because it depends on the change in intensity as well as the change in (x,y) coordinates. This means that a point near an edge is updated by the average of points that are mostly on the same side of the edge.
This produces anisotropy in a very natural way. In the direction where there is little variation in intensity, more pixels are nearby in terms of geodesic distance, so more pixels are used in the average.
Notice that the parameter C controls the degree of anisotropy. If C is small, this reduces to Gaussian smoothing. As C grows, the effect of intensity variations is magnified. In the extreme case, as C goes to infinity, only intensity variations matter. Consider the 1D case. Any point on a curve that is not an extrema has a neighborhood of larger intensity
values on one side, and an identical but negative neighborhood on the other side. So the only points affected by this smoothing are the extrema, which move to become less extreme. This evolves piecewise constant regions, where the amount of variation between extrema is constantly reduced. This is called total variation diffusion, because it reduces the total variation of the curve.
Non-linear Diffusion and Segmentation
Note that non-linear diffusion tends to produce a kind of segmentation. We get regions that are piecewise constant, or nearly so, in the course of the diffusion.
