Volumetric Harmonic Brain Mapping using a Variational Method
Yalin Wang1, Xianfeng Gu2, Tony F. Chan1, Paul M. Thompson3, Shing-Tung Yau4
1Mathematics Department, UCLA
2CISE, University of Florida
3Lab. of Neuro Imaging and Brain Research Institute, UCLA School of Medicine
4Department of Mathematics, Harvard University
{ylwang,chan}@math.ucla.edu, [email protected], [email protected], [email protected]ard.edu
Abstract
Brain surface conformal mapping research has been successful and this motivates our more general investigation
of 3D volumetric brain harmonic mapping. By transforming the full 3D brain volume to a solid sphere, our goal is to
investigate how features map into this canonical 3D coordinate system in the same way as 2D conformal flattening
has helped in analyzing cortical surface geometry. Nonlinear mapping of two brain volumes to a sphere may also
assist with the subsequent nonlinear registration of one brain volume to another. We suggest that 3D harmonic
mapping of brain volumes to a solid sphere can provide a canonical coordinate system for feature identification and
segmentation, as well as anatomical normalization.
We developed two different techniques to tackle the volume mapping problem. The first finds a 3D harmonic
map from a volumetric brain image to a 3D solid sphere and the second uses a sphere carving algorithm to compute
a simplicial decomposition of the volume which is adapted to the surfaces.
We derived the harmonic energy computation equation. We construct a harmonic map in R3with a heat flow
method. First we conformally map the boundary of the 3D volume to a sphere. During this step, we can easily
include other biological constraints (e.g. mappings of superficial sulci). Our method then minimizes the volumetric
harmonic energy while keeping the surface fixed. This mapping is determined by the geometric structure and also
the boundary conditions. Once we fixed the mapping of the brain surfaces, the surface mapping is conformal in our
case, the interior harmonic map consistently maps similar brain structures to canonical locations.
Methods to tetrahedralize the brain for FEM analysis are somewhat rare in the literature, although they are used
occasionally for surgical simulation, or mapping intraoperative brain change. In our current experiments, we apply
the sphere carving algorithm to build a brain-based tetrahedral mesh. The algorithm initial ly constructs a large
sphere that contains all the brain data. Then it keeps removing exterior tetrahedra while maintaining the surface
genus number.
In our experiments, we tested our algorithm on both synthetic and brain volume data. Our synthetic data was a
cube consisting of many tetrahedra. For the brain data, the input image to this algorithm is a binary 3D volumetric
brain image which results from applying a Gaussian mixture tissue classifier to an MRI, in order to classify each
pixel as white matter (in this case, for illustration purposes) and non-white matter. The sphere carving algorithm
then builds a finite element mesh by removing tetrahedra exterior to the binary brain volume. Our experiments show
that the resulting embedding can be used to induce a canonical spherical coordinate system for the brain interior.
Some experimental results are shown in the Figure below.
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