Wang Yalin, Lui Lok Ming, Chan Tony F, Thompson Paul M
Mathematics Department, UCLA, Los Angeles, CA 90095, USA.
Med Image Comput Comput Assist Interv. 2005;8(Pt 2):675-83. doi: 10.1007/11566489_83.
To compare and integrate brain data, data from multiple subjects are typically mapped into a canonical space. One method to do this is to conformally map cortical surfaces to the sphere. It is well known that any genus zero Riemann surface can be conformally mapped to a sphere. Therefore, conformal mapping offers a convenient method to parameterize cortical surfaces without angular distortion, generating an orthogonal grid on the cortex that locally preserves the metric. To compare cortical surfaces more effectively, it is advantageous to adjust the conformal parameterizations to match consistent anatomical features across subjects. This matching of cortical patterns improves the alignment of data across subjects, although it is more challenging to create a consistent conformal (orthogonal) parameterization of anatomy across subjects when landmarks are constrained to lie at specific locations in the spherical parameter space. Here we propose a new method, based on a new energy functional, to optimize the conformal parameterization of cortical surfaces by using landmarks. Experimental results on a dataset of 40 brain hemispheres showed that the landmark mismatch energy can be greatly reduced while effectively preserving conformality. The key advantage of this conformal parameterization approach is that any local adjustments of the mapping to match landmarks do not affect the conformality of the mapping significantly. We also examined how the parameterization changes with different weighting factors. As expected, the landmark matching error can be reduced if it is more heavily penalized, but conformality is progressively reduced.
为了比较和整合大脑数据，通常将多个受试者的数据映射到一个标准空间。一种实现方法是将皮质表面共形映射到球体。众所周知，任何亏格为零的黎曼曲面都可以共形映射到球体。因此，共形映射提供了一种方便的方法来参数化皮质表面而不产生角度失真，在皮质上生成一个局部保持度量的正交网格。为了更有效地比较皮质表面，调整共形参数化以匹配受试者之间一致的解剖特征是有利的。这种皮质模式的匹配改善了受试者间数据的对齐，尽管当地标被限制位于球形参数空间中的特定位置时，要在受试者间创建一致的共形（正交）解剖参数化更具挑战性。在此，我们基于一种新的能量泛函提出一种新方法，通过使用地标来优化皮质表面的共形参数化。在一个包含40个脑半球的数据集上的实验结果表明，地标不匹配能量可以大幅降低，同时有效地保持共形性。这种共形参数化方法的关键优势在于，映射的任何局部调整以匹配地标不会显著影响映射的共形性。我们还研究了参数化如何随不同的加权因子变化。正如预期的那样，如果对地标匹配误差进行更重的惩罚，它可以降低，但共形性会逐渐降低。
