Hyperbolic Ricci Flow and Its Application in Studying Lateral Ventricle Morphometry

Here we propose a novel method to compute surface hyperbolic parameterization for studying brain morphology with the Ricci flow method. Two surfaces are conformally equivalent if there exists a bijective angle-preserving map between them. The Teichmüller space for surfaces with the same topology is a finite-dimensional manifold, where each point represents a conformal equivalence class, and the conformal map is homotopic to the identity map. A shape index can be defined based on Teichmüller space coordinates, and this shape index is intrinsic and invariant under scaling, translation, rotation, general isometric deformation, and conformal deformation. Using the Ricci flow method, we can conformally map a surface with a negative Euler number to the Poincaré disk and the Teichmüller space coordinates can be computed by geodesic lengths under hyperbolic metric. For lateral ventricular surface registration, we further convert the parameterization to the Klein model where a convex polygon is guaranteed for a multiply connected surface. With the Klein model, diffeomorphisms between lateral ventricular surfaces can be computed with some well known surface registration methods. Compared with prior work, the parameterization does not have any singularities and the intrinsic parameterizations help shape indexing and surface registration. Our preliminary experimental results showed its great promise for analyzing anatomical surface morphology.