We know that the
heat equation diffuses an image
isotropically. This is not good, because it destroys the edges too. Somehow one must find a way for isotropic diffusion, only in a flat region and stop the diffusion near the edges. This is the motivation for Peitro Perona and Jitendra Malik's original formulation of anisotropic diffusion.
The heat equation is $\frac{\partial u}{\partial t}=\Delta u$ i.e. $\frac{\partial u}{\partial t}=\mbox{div}\,(\nabla u)$ with $u(0)=f$.
The idea behind the anisotropic diffusion is to multiply the gradient by a function g that equals 1 when the $\nabla u=0$ i.e. flat regions and goes to 0 as the gradient increases, i.e. near the edges. A typical function that works is $g(x)=\frac{1}{1+(x/b)^2}$. for some constant $b$.
That is $\frac{\partial u}{\partial t}= \mbox{div}\,(g(\Vert\nabla u \Vert^2) \nabla u)$.
Alternatively, this can also be viewed as the gradient descent for the minimization of the energy
$$E(u):=\frac{1}{2}\int_{\Omega} g(\Vert\nabla u\Vert^2).$$
Let us look at some experiments, now.
Here is an experiment with Perona Malik model with
b=0.01, at
t=4.
 |
| Perona Malik at t=4. |
There is hardly any loss of visual quality of the image. Compare it to the original below. Notice there is some loss of quality in her hair.
 |
| The original |
Let us see what happens as we continue to
t=10 below. Ok, the image is a bit too smooth, rather artistic.
 |
| Perona Malik at t=10 |
Comparison with heat equation
Compare these results with the heat equation at
t=4 below. Very smooth, edges are lost. Not good.
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| Heat equation at t=4. |
