Scientific inquiries generate many different kinds of shapes. Protein backbones are represented as curves, MRI scans give us a three-dimensional representation of the brain, and stress tensors in materials can be represented by point sets in a curved manifold. Because of improvements in data gathering and sensing technologies, we now have large databases of shapes, and have the chance to gain a deeper understanding of scientific processes by performing detailed analysis of these shape collections.
Shape databases are large and complex. Thus, a core challenge when analyzing shape collections is finding succinct descriptions, or synopses, of these collections. Such synopses give us snapshots of the data, describe local and global variations in a population, and provide the building blocks for advanced and rigorous data analysis. However, shapes have complex representations, and the spaces that shapes inhabit are mathematically intricate. Computing compact synopses with provable quality requires new geometric and algorithmic ideas.
In this project, the PI, using experience in the areas of shape matching, data analysis, synopsis data structures, and the geometry of non-Euclidean spaces, will develop a suite of algorithmic techniques for producing synopsis structures on shapes, including generalized means and medians, clusterings, subspace representations, and approximate histograms and distributions. This research lays the foundations for rigorous data analysis in shape spaces. It develops the algorithmic and geometric building blocks that will yield a deeper understanding of these spaces, and in doing so will yield many new ideas for computations in the non-Euclidean spaces that shapes inhabit. More broadly, the computational study of shape as a pathway to deeper scientific inquiry takes advantage of the power of computational methods as well as growing stores of data, and creates new opportunities for the emerging area of data-driven science.