Dr Stuart Hall discussing Mathematical Image Analysis

In his talk University of Newcastle speaker, Dr Stuart Hall, described how ideas from differential geometry, in particular the Ricci flow, can be useful for image analysis.

In the first part of his talk, he introduced concepts from the theory of parameterised surfaces. Particularly important notions are those of a Riemannian metric I (an inner product on each tangent space) and the second fundamental form II (which measures curvature).

Dr Hall then discussed Gauss’s Theorema Egregium which states that the curvature II depends only on the metric I. As its name suggests, this is a very remarkable observation: if you take a piece of paper and bend it in any way you like (without stretching or tearing, the curvature still vanishes).

Next, Dr Hall introduced the Ricci curvature, which is a geometric flow whereby one roughly speaking evolves the metric proportional to the curvature. The relevant PDE is a diffusion equations which means the regions of concentrated curvature tends to be spread out so that one eventually obtains a surface of constant curvature.

The fact that surfaces carries constant curvature metrics is also known as the Uniformization Theorem. Given an image, viewed as a surface in 3 dimensions, the idea is to use the Ricci flow to map the image onto its canonical domain. In this way one obtains a tool for image registration.

At the end of his talk Dr Stuart Hall speculated on the use of Ricci flow for morphing images.

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