Cortical surface flattening using least square conformal mapping with minimal metric distortion

Abstract

Although flattening a cortical surface necessarily introduces metric distortion due to the non-constant Gaussian curvature of the surface, the Riemann mapping theorem states that continuously differentiable surfaces can be mapped without angular distortion. We apply the so-called least-square conformal mapping approach to flatten a patch of the cortical surface onto planar regions and to produce spherical conformal maps of the entire cortex while minimizing metric distortion within the class of conformal maps. Our method, which preserves angular information and controls metric distortion, only involves the solution of a linear system and a nonlinear minimization problem with three parameters and is a very fast approach.