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

发现和量化形状之间的差异在生物科学的许多领域都很重要。 这种方法学研究将有助于两个领域的研究，神经科学和古生物学。 在神经科学中，主要关注跟踪阿尔茨海默病的进展和大脑中的正常衰老过程，并将MRI扫描中看到的3D形状变化与认知测量和其他变量相关联。 NIH的研究提供了大量此类数据的访问。在古生物学中，许多灭绝物种的唯一信息来自化石，而这些化石物种与人类祖先之间关系的估计主要基于形状的差异和相似性。我们的目标是将化石形状数据纳入从现有物种收集的更大数据集的背景下，包括早期研究的形态学和基因组数据。这项合作对定义和计算三维生物形状彼此相似意味着什么感兴趣;作为具体的例子，他们考虑了灵长类动物骨骼化石和海马体等大脑区域的形状。 目前，形状差异的实际测量是基于对象表面上的对应点样本的集合。 该团队将添加一个连接这些对应点的表面网格，并通过网格副本中的边的长度向量表示形状。两个形状之间的距离是它们对应的边缘向量之间的欧几里得距离。 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，边长向量形成了一个高维欧几里得空间，在该空间内可以执行标准的统计分析。 第二，该测度不仅对整个对象的旋转和平移是不变的，而且在很大程度上对对象的一部分相对于其余部分的变换是不变的。 这项研究将包括实验工作，以比较拟议的度量和目前的方法在神经生物学和体质人类学。 他们还打算研究简化查找和优化输入样本上相应点和连接它们的网格的方法，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，该团队计划在有趣的相关数学问题上开展工作，特别是随着采样密度的增加，度量对光滑表面属性的收敛。计划发布两个软件供从业者使用，并发布用相应网格注释的数据集合，供其他研究人员使用形状差异的方法。 通过这种方式，这项研究应该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领域的同事，研究人类、其他动物甚至植物解剖学的人，法医科学家等等。

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.