Conformal Geometry by Miao Jin Xianfeng Gu Ying He & Yalin Wang

Author:Miao Jin, Xianfeng Gu, Ying He & Yalin Wang
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

9.2.3 Hyperbolic Brain Conformal Parameterization with Ricci Flow Method

The Ricci flow is an intrinsic geometric flow that deforms the metric of a Riemannian manifold, such as a 3D surface. The Ricci flow was introduced by Richard Hamilton for general Riemannian manifolds in his seminal work [109] in 1982, and it has gained increasing attention and interest in the engineering field [131, 256, 296, 324, 339]. It plays an important role in the proof of the Poincaré conjecture for three-dimensional manifolds [209–211]. Compared to other conformal parameterization methods used in brain imaging [7, 100, 126, 295, 298], the Ricci flow method can handle surfaces with complicated topologies (boundaries and landmarks) without producing singularities. It also provides a universal and flexible way to compute conformal Riemannian metrics with prescribed Gaussian curvatures. In the discrete case, it is equivalent to optimizing a convex energy. The global optimum exists and is unique, so the computation is stable and efficient.

In conformal geometry, when the Riemannian metric is conformally deformed, curvatures are also changed accordingly. Suppose is changed to , where . Then, the Gaussian curvature will become

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