Lili Ju

^{1}Institute for Mathematics and its Applications, University of
Minnesota, Minneapolis, MN 55455, USA

^{2}Department of Statistics, University of Minnesota, Minneapolis,
MN 55455, USA

^{3}Department of Neurology, University of Minnesota, Minneapolis,
MN 55455, USA

^{4}Department of Radiology,
University of Minnesota, Minneapolis, MN 55455, USA

^{5}Department of Mathematics, Florida State University,
Tallahassee, FL 32306, USA

**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. 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 [5] and a small scale nonlinear minimization.
It can be applied to user-defined "patches"
or to an entire cortical surface.

**Methods**

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 [5]. 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 [3]. 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

**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 [1]
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 | |||||

Flat maps | A | B | C | A | A | B | C | A |

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**;

**References**

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.

**Acknowledgements**

This work is supported in part by NIH grant MH57180 and NSF grant DMS101339.