Group Actions and Orbits

Anatomical Orbits (A Preview)

Now, with a basic understanding of group actions and orbits, a mathematical model of anatomy can be defined:

"Anatomy is an orbit under groups of diffeomorphisms." Or put in math language: $$ANATOMY = G \cdot I_{temp} = \left[ I_{temp} \circ \phi, \phi \in G \right]$$. G is a group of diffeomorphisms and $$I_{temp}$$ is a template image. $$G \cdot I_{temp}$$ is the orbit created by the application of every member of G to the original image; $$I_{temp} \circ \phi$$ is the representation for one of the members of G.

We can illustrate this concept by creating a very small sample anatomy, using a stomach as our $$I_{temp}$$. We will choose 19 arbitrary diffeomorphisms to make our G: $$\phi_1,\phi_2,\dots,\phi_{19}$$. One of these diffeomorphisms will be the identity, mapping the original image to itself. The orbit might look like:

anatomical orbit

If we could come up with all the possible diffeomorphisms necessary, we would be able to create the entire anatomy of stomach outlines. Of course, such a task is impossible.