Computing Optimal Alignments of Surfaces

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Almost everything we see in the world around us is a two-dimensional surface. Whether with our eyes, with radar or with a laser scanner, we receive data about the geometry of the surface that forms the outer boundary of an object. There is a fundamental need to understand how to analyze this geometric data for purposes of aligning and comparing pairs of surfaces. This project will develop and implements new and rigorous mathematical methods to compare shapes, based on techniques of low-dimensional topology. It will develop new mathematical results and create new computational tools. Applications range from facial recognition to brain mapping to protein classification. The underlying mathematical theory of shape comparison will be extended, and new algorithms to implement the resulting theory will be developed and software made available. Moreover this software will be applied and tested on collections of biological shapes, such as protein surfaces, databases of faces, collections of bones and teeth, and brain cortices.This project will develop new mathematical measures of the distortion based on a variety of measures of the energy needed to stretch one surface over another. Given two surfaces, or shapes, algorithms will be developed to produce an optimal correspondence or alignment between them. The difference or similarity of two shapes will be captured in new distance functions that capture aspects of geometric similarity. The project builds on successful existing approaches used in comparing surfaces that have the topology of a sphere. These methods will be improved and extended to more general surfaces, allowing for comparing surfaces with one, two, or more handles. Past methods based on conformal maps will be extended to allow use of more general surface diffeomorphisms. The project will also produce explicit alignments between partial surfaces, or surfaces with boundary. This partial surface matching allows for matching surfaces when only portions of each have been measured, with other parts obscured. It also allows comparison between pairs of surfaces having different topologies, overcoming a problem faced by current methods. The symmetric distortion energy recently introduced will be extended to these more general contexts. Software implementing these geometric algorithms will be developed and made available. This software will be designed to be usable by scientists in a wide variety disciplines and could lead to breakthroughs in biological and medical understanding.