# Why is the anisotropic diffusion equivalent to the heat equation when using a constant diffusion coefficient?

The anisotropic diffusion is defined as

$$\frac{\partial I}{\partial t} = \mathrm{div} \left( c(x,y,t) \nabla I \right)= \nabla c \cdot \nabla I + c(x,y,t) \Delta I$$

where $$\operatorname{div}$$ is the divergence operator, $$c(x,y,t)$$ is the diffusion coefficient (which depends on the $$x$$ and $$y$$ coordinates and on the time $$t$$ of the diffusion), $$\nabla I$$ is the gradient of the image and $$\Delta I$$ is the Laplacian of the image and $$I(\cdot ,t):\Omega \rightarrow {\mathbb {R}}$$ is the actual image with domain $$\Omega$$ which is a subset of $$\mathbb{R}^2$$.

Why exactly is the anisotropic diffusion equivalent to the heat equation when using a constant diffusion coefficient?

The heat equation for a function $$I(x, y, t)$$ of three variables $$x$$, $$y$$ (the coordinates) and $$t$$ (the time) is defined as follows
\begin{align} \frac{\partial I}{\partial t} &= \alpha\left(\frac{\partial^2I}{\partial x^2}+\frac{\partial^2I}{\partial y^2}\right) \\ &= \alpha \operatorname{div}(\nabla I) \end{align}
where $$\alpha$$ is a real coefficient called the diffusivity and it thus corresponds to a constant $$c(x, y, t)$$, which, in the case of the anisotropic diffusion equation, can change as a function of $$x$$, $$y$$ and $$t$$.