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Research Description

Registration and analysis of neuro-imaging data presents a challenging problem due to the complex folding patterns in the human brain. Specifically, the cortical surface of the brain can be modeled as a highly convoluted 2D surface. Since it is non-flat, the non-euclidean geometry of the cortex needs to be accounted for while performing registration and subsequent signal processing of anatomical and functional signals on the cortex. Techniques from differential geometry offer a powerful set of tools to deal with the convoluted nature of the cortex. We developed a surface parameterization method which computes a 2D coordinate system on the cortex which is then used to compute the surface metric and discretize derivatives in the surface geometry. We presented surface registration techniques to find a point to point correspondence between two surfaces. These can be used to bring surface signals from individual brains to a common surface. Isotropic and anisotropic diffusion filtering methods were formulated for processing of the cortical data. When the surface data is a point-set on the cortex, we proposed a method to quantify its mean and variance with respect to the surface geometry. The registration techniques presented for surface alignment are then extended to volumes to perform full surface and volume registration. This is done by using volumetric harmonic mappings that extend the surface point correspondence to the cortical brain volume. Finally, the volumetric registration is refined by using inverse-consistent linear elastic intensity registration.

Cortical Surface Parameterization

We proposed a parameterization technique for the cortical surface based on $p$-harmonic energy minimization. Angle and area distortion metric were computed to evaluate the performance of this flattening procedure. Finite element method was used for discretization. Our $p$-harmonic method results in a very fast parameterization of high-resolution cortical surfaces and always results in a bijective map.


Cortical Surface Registration

We present a technique that is a generalization of the popular thin-plate spline methods from $R^n$ to a non-Euclidean surface. Our $p$-harmonic mapping method maps each individual cortical hemispheres to the unit squares.  We use the resulting square maps of the cortical hemispheres to assign a coordinate system to the cortex.  We then use these coordinates to compute the metric tensor and Christoffel symbols of the mapping.  In order to register one brain to another, we warp coordinates of one brain with respect to another using sulcal landmarks such that the bending energy is minimized within the true geometry of the surface. This is achieved by solving the resulting variational problem using covariant derivatives and thus making the warping results independent of the coordinate system.

As an improvement over our earlier method, we also proposed an FEM based elastic mapping method that avoids the use of an intermediate surface flattening step for landmark matching. It incorporates the landmark registration into the parameterization method itself. We use the Cauchy-Navier elastic equilibrium equation for performing this matching as explained in the next section. This approach also has the advantage that the computation cost is relatively small and that the resulting alignment is inverse consistent.


Surface Constrained Volumetric Registration

We developed an approach to brain image registration based on harmonic maps that combines surface based and volume based approaches producing a volumetric alignment in which there is also a one-to-one correspondence between points on the two cortical surfaces.

We propose a new method based on harmonic mappings for extending the surface matching to the entire cortical volume, and present a modified intensity alignment to compute the final map. The resulting method, comprising the three steps gives an inverse consistent map  which is capable of aligning both subcortical and sulcal features.


Diffusion Smoothing on Surfaces

Neuroimaging data, such as cortical thickness or neural activation, can often be analyzed more informatively with respect to the cortical surface rather than the entire volume of the brain. This analysis should be carried out in the intrinsic geometry of the surface rather than in the ambient space. We present parameterization based numerical methods for performing isotropic and anisotropic filtering on triangulated surface geometries. In contrast to existing FEM-based methods for triangulated geometries, our approach accounts for the metric of the surface. In order to discretize and numerically compute the isotropic and anisotropic geometric operators, we first parameterize the surface using a $p$-harmonic mapping. We then use this parameterization as our computational space and account for the surface metric while carrying out isotropic and anisotropic filtering. We illustrate these methods in an application to smoothing of mean curvature maps on the cortical surface.
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