Diffusion Equations on Graphs
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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.
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