Hurdal Monica K, Stephenson Ken
Department of Mathematics, Florida State University, Tallahassee, FL 32306-4510, USA.
Neuroimage. 2004;23 Suppl 1:S119-28. doi: 10.1016/j.neuroimage.2004.07.018.
Cortical flattening algorithms are becoming more widely used to assist in visualizing the convoluted cortical gray matter sheet of the brain. Metric-based approaches are the most common but suffer from high distortions. Conformal, or angle-based algorithms, are supported by a comprehensive mathematical theory. The conformal approach that uses circle packings is versatile in the manipulation and display of results. In addition, it offers some new and interesting metrics that may be useful in neuroscientific analysis and are not available through numerical partial differential equation conformal methods. In this paper, we begin with a brief description of cortical "flat" mapping, from data acquisition to map displays, including a brief review of past flat mapping approaches. We then describe the mathematics of conformal geometry and key elements of conformal mapping. We introduce the mechanics of circle packing and discuss its connections with conformal geometry. Using a triangulated surface representing a cortical hemisphere, we illustrate several manipulations available using circle packing methods and describe the associated "ensemble conformal features" (ECFs). We conclude by discussing current and potential uses of conformal methods in neuroscience and computational anatomy.
皮质扁平化算法正越来越广泛地用于辅助可视化大脑中复杂的皮质灰质层。基于度量的方法最为常见，但存在高度失真的问题。共形或基于角度的算法有全面的数学理论支持。使用圆填充的共形方法在结果的操作和显示方面具有通用性。此外，它还提供了一些新的有趣度量，这些度量可能在神经科学分析中有用，而通过数值偏微分方程共形方法无法获得。在本文中，我们首先简要描述皮质“扁平”映射，从数据采集到映射显示，包括对过去扁平映射方法的简要回顾。然后我们描述共形几何的数学和共形映射的关键要素。我们介绍圆填充的机制，并讨论其与共形几何的联系。使用代表皮质半球的三角化表面，我们展示了使用圆填充方法可进行的几种操作，并描述了相关的“整体共形特征”（ECF）。我们通过讨论共形方法在神经科学和计算解剖学中的当前和潜在用途来得出结论。
