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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?

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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 $t$.

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