III: Small: Collaborative Research: Shape Differences in the Biological Sciences

在生物科学的许多领域，发现和量化形状之间的差异是很重要的。这一方法学研究将有助于神经科学和古生物学这两个领域正在进行的研究。在神经科学领域，人们的兴趣主要是追踪阿尔茨海默病的进展和大脑的正常衰老过程，并将MRI扫描中看到的3D形状变化与认知测量和其他变量联系起来。美国国立卫生研究院的研究提供了获取大量此类数据的途径。在古生物学中，关于许多灭绝物种的唯一信息来自化石，对这些化石物种和人类祖先之间关系的估计主要基于形状的异同。目标是将化石形状数据与从现有物种收集的更大的数据集（包括早期研究的形态学和基因组数据）结合起来。这次合作的兴趣在于定义和计算三维生物形状彼此相似的意义；作为具体的例子，他们考虑了灵长类动物骨骼化石的形状和大脑中海马等区域的形状。目前实际的形状差异测量是基于物体表面对应点样本的集合。团队将添加一个连接这些对应点的表面网格，并在其网格副本中通过边缘长度的向量表示形状。两个形状之间的距离就是它们对应的边缘向量之间的欧氏距离。这种表示有一些吸引人的数学性质。通过对网格的一些简单的附加要求，边缘长度向量形成了一个高维欧几里得空间，在这个空间内可以执行标准的统计分析。其次，度量不仅对整个物体的旋转和平移是不变的，而且在很大程度上对物体的一部分相对于其余部分的变换也是不变的。这项研究将包括实验工作，以比较神经生物学和体质人类学中提出的度量和当前的方法。他们还打算研究在输入标本上简化查找和优化相应点和连接点的网格的方法，这是三维数据分析中的一个长期问题。这不仅对促进实验很重要，而且具有良好的实用形状度量将有助于改进该领域的技术。最后，该团队计划研究有趣的相关数学问题，特别是随着采样密度的增加，度量对光滑表面属性的收敛性。他们计划发布两种软件，一种是供从业人员使用的软件，另一种是用相应网格注释的数据集，供其他研究人员使用形状差异的方法。通过这种方式，这项研究将使许多其他分析形状差异的人受益：我们在古生物学和神经科学方面的同事，研究人类、其他动物甚至植物解剖的人，法医科学家等。

Finding and quantifying differences between shapes is important in many areas of the biological sciences. This methodological research will contribute to ongoing research in two areas, neuroscience and paleontology. In neuroscience, the interest is mainly in tracking the progress of Alzheimer's disease and normal aging processes in the brain, and relating the 3D shape changes seen in MRI scans to cognitive measurements and other variables. NIH studies provide access to large amounts of such data. In paleontology, the only information on many extinct species comes from fossils, and estimates of the relationships between these fossil species and human ancestors is based largely on differences and similarities of shape. The goal is to put fossil shape data into the context of much larger sets of data collected from existing species, both morphological and genomic data from earlier research. This collaboration is interested in defining and computing what it means for three-dimensional biological shapes to resemble each other; as specific examples, they consider the shapes of fossil primate bones and of regions in the brain such as the hippocampus. Current practical measures of shape difference are based on sets of corresponding point samples on the object surfaces. The team will add a surface mesh connecting these corresponding points, and represent a shape by the vector of the lengths of the edges in its copy of the mesh. The distance between two shapes is then the Euclidean distance between their corresponding edge vectors. This representation has some attractive mathematical properties. With a few simple additional requirements on the mesh, the edge-length vectors form a high-dimensional Euclidean space, within which standard statistical analyses can be performed. Second, the measure is invariant to rotations and translations not only of the entire object, but to a large extent to transformations of one part of the object with respect to the rest. The research will include experimental work to compare the proposed metric andcurrent methods in both neurobiology and in physical anthropology. They also intend to work on methods to simplify finding and optimizing corresponding points and meshes connecting them on input specimens, a perennial problem in three-dimensional data analysis. Not only is this important to facilitate experiments, but having a good practical shape metric will help improve techniques in this area. Finally, the team plans work on interesting related mathematical problems, specifically the convergence of the metric to property of smooth surfaces as the sampling density is increased There are plans to release both software for the use of practitioners and ensembles of data annotated with corresponding meshes for the use of other researchers into the methodology of shape differences. In this way the research should benefit many others who analyze shape differences: our colleagues in paleontology and neuroscience, people who study the anatomy of humans, other animals and even plants, forensic scientists, and others.