Geometry and Self-Organization

Self-organizing systems typically occur on a geometric domain. The field of computational geometry has developed numerous tools for characterizing local geometry of surfaces and understanding how to extend the Laplacian to these domains. The extension of the Laplacian for these surfaces is called the Laplace-Beltrami Operator. Taking the eigenvectors of this operator over a surface provides a set of basis functions of increasing frequency. In a way, this is analogous to Fourier analysis and spherical harmonics (pictured below).

Interesting, these eigenvectors tend to capture broad features that are semantically meaningful. They tend to capture information about the topology of the surface (or the underlying discrete mesh representing it).

Depth-coded surface (purple is close, yellow is far away)

First eigenvector displayed on the surface

Second eigenvector displayed on the surface

Third eigenvector displayed on the surface

A simple application of the Laplace-Beltrami operator is to use it to model how heat and wave equations might function on non-flat surfaces. These can be generate features for shape matching, segmentation, and much more.

When heat (or molecules) diffuses along a surface, the geodesic distances rather than Euclidean distances are what matter. If we were to place one unit of heat at any location and observe how quickly it dissipates, this gives us the heat kernel. Certain areas dissipate heat more slowly than others. One intriguing consequence this may have is for low-probability, positive feedback-driven processes - they are more likely to initiate in these areas because the molecules are more likely to interact in these local diffusion traps. Something like this has been observed for excitable systems modeled on a torus and for Turing systems on more complex shapes.

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