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

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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.