Structural Brain Mapping
There are three levels of brain mapping: macro (large structures such as lobes or cortices), meso (neuron bundles), and micro (individual neurons). In my work I focus on the mapping of larger cortical surface structures, using MRI data. From raw data, I reconstruct the brain surface, overlaying a triangular mesh on the cortical surface, then use Circle Packing to compute mappings.
Circle Packing
From a triangulated mesh, we identify with each vertex a circle center and with each edge a tangency of two circles. From this, we use software developed by Ken Stephenson at University of Tennessee - Knoxville to map the brain surface to a constant curvature, or flat, surface (Riemann sphere, complex plane, hyperbolic disk). Circle Packing is a quasi-conformal method (as it is discrete rather than continuous), allowing us to investigate how conformal invariants are preserved.
Conformal Invariants
Conformal invariants are properties that are preserved under conformal maps. We wish to investigate how well our quasi-conformal methods preserve the surface. Then we can use these properties to distinguish between healthy and diseased data samples. To do so, we first develop methods for calculating these invariants in the discrete setting.
Conformal Warping
Rather than flat mapping the brain and calculating invariants prior to comparing subjects, we now warp one surface directly to another. Based on methods from Patrice Koehl and Joel Hass, we form adaptations to warp our data, as the existing methods allow for topological spheres, but not topological disks. |
