1Institute for Mathematics and its Applications, University of
Minnesota, Minneapolis, MN 55455, USA
2Department of Statistics, University of Minnesota, Minneapolis, MN 55455, USA
3Department of Neurology, University of Minnesota, Minneapolis, MN 55455, USA
4Department of Radiology, University of Minnesota, Minneapolis, MN 55455, USA
5Department of Mathematics, Florida State University, Tallahassee, FL 32306, USA
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. Several techniques have been proposed for flattening polygonal representations of surfaces while substantially minimizing metric distortion [1,2], and methods for conformal flattening of polygonal surfaces have also been proposed [3,4]. We describe an efficient method for generating conformal flat maps of triangulated surfaces while minimizing metric distortion within the class of conformal maps. Our method, which controls both angular and metric distortion, involves the solution of a linear system  and a small scale nonlinear minimization. It can be applied to user-defined "patches" or to an entire cortical surface.
Given any pair of surfaces, P and Q, that are both topological disks, and a fixed homeomorphism between their boundaries, there exists a unique harmonic map from P to Q that minimizes the Dirichlet energy and is conformal. We utilize a discretization scheme and optimization-based algorithm for computing the harmonic mapping described in . Since there is always a class of conformal mappings from P to Q related to each other by an automorphism of Q, our algorithm then uses numerical methods to search for that conformal mapping which minimizes the metric distortion of the resulting mesh. Conformal mappings between topological spheres proceed via the mapping for disks using the same technique as . A vertex v and all edges containing it are removed from the input mesh, after that we generate a discrete conformal map from the pruned mesh to the unit disk D; this map is then stereographically projected to the unit sphere S while v is mapped to the "north pole" of S, and the resulting map is again optimized by choosing the S automorphism that minimizes metric distortion.
Results and Conclusions
We created discrete conformal flat maps on the unit sphere from topologically-correct human cerebellar and cerebral cortices; we also flattened selected cortical patches from the cerebellum (Lobes IV and V) and cerebrum (occipital lobe). Mappings obtained for the cerebellar cortex are illustrated in Figure 1. Values for mean angular and normalized metric distortion are compared to those obtained using FreeSurfer  in Table 1. With our conformal method, metric distortion is no worse than that produced by FreeSurfer; moreover, angular information is preserved, and computation time greatly reduced (20 minutes for the spherical flattening of cerebral cortex vs. 10 hours using FreeSurfer).
|Cerebellar Cortex||Cerebral Cortex|
|Our Method||FreeSurfer||Our Method||FreeSurfer|
|Mean angular distortion||4.44°||6.12°||11.51°||23.37°||6.50°||2.66°||8.15°||18.76°|
|Mean normalized metric distortion||40.44%||23.47%||35.53%||46.16%||36.52%||25.37%||21.79%||30.81%|
Figure 1. Conformal flat maps of the human cerebellar cortex. Left panel : Parcellated surface of the cerebellum and its flat map on S; Right panel : Patch from cerebellar Lobes IV and V and its flat maps in a predefined planar region and on D.
1. Fischl B., et al., Neuroimage 9:195-207, 1999.
2. Drury H.M. et al., JCN, 8:1-28, 1996.
3. Hurdal M.K., et al., LNCompSci, 1679:279-286, 1999.
4. Angenent S., et al., TMI 18:700-711, 1999.
5. Eck M., et al., SIGGRAPH95, 173-182.
This work is supported in part by NIH grant MH57180 and NSF grant DMS101339.