Hyperbolic Ricci Flow and Its Application in Studying Lateral Ventricle Morphometry

Abstract

Abstract.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.

Studying ventricular abnormalities in mild cognitive impairment with hyperbolic Ricci flow and tensor-based morphometry

“…By slicing a cortical surface open along three or more landmark curves, the cortical surface becomes a genus-0 surface with multiple boundaries, which has a negative Euler number. In our prior work (Shi et al, 2012), we illustrated the application of our method on a cortical surface with three landmark curves, which is homotopic to the topological pants. For cortical surfaces with more landmarks, which also have negative Euler numbers, the hyperbolic Ricci flow method is still applicable (Shi et al, 2013c; Shi et al, 2013d), so the remaining processes of the proposed method follow naturally.…”

“…Hyperbolic conformal geometry has an important property that it can induce conformal parameterizations on high-genus surfaces or surfaces with negative Euler numbers and the resulting parameterizations have no singularities (Luo et al, 2008). Motivated by recent advances in hyperbolic conformal geometry based brain imaging research (Shi et al, 2013d; Tsui et al, 2013), including our own work (Shi et al, 2012; Wang et al, 2009b; Wang et al, 2009c), here we propose to use the hyperbolic Ricci flow method to build the canonical parameter domain for ventricular surface registration. The resulting parameterizations are angle-preserving and have no singularity points.…”

Is the study of fungal shape still useful?

“…Other methods used in human medical research rely on surface parameterization using Ricci flow method (explores curvature under the topology of different manifolds), with projection of one of standard models of hyperbolic planes as Poincaré disk and Klein model endowed with hyperbolic metric. 53 As topology of many anatomical features is of higher order genus, novel approaches borrowed from differential geometry explore the possibility to map these entities onto hyperbolic space. A combination of landmarks and projections of the targeted high genus surfaces in hyperbolic plane combined with minimal surfaces was used to compare ear vestibular organs in healthy and impaired individuals.…”