  In brain imaging, the image acquisition and processing processes themselves
are likely to introduce noise to the images. It is therefore imperative to
reduce the noise while preserving the geometric details of the anatomical
structures for various applications. Traditionally Gaussian kernel smoothing
has been often used in brain image processing and analysis. However, the direct
application of Gaussian kernel smoothing tend to cause various numerical issues
in irregular domains with boundaries. For example, if one uses large bandwidth
in kernel smoothing in a cortical bounded region, the smoothing will blur
signals across boundaries. So in kernel smoothing and regression literature,
various ad-hoc procedures were introduce to remedy the boundary effect.
Diffusion equations have been widely used in brain imaging as a form of noise
reduction. The most natural straightforward way to smooth images in irregular
domains with boundaries is to formulate the problem as boundary value problems
using partial differential equations. Numerous diffusion-based techniques have
been developed in image processing. In this paper, we will overview the basics
of isotropic diffusion equations and explain how to solve them on regular grids
and irregular grids such as graphs.
