Grant: Discovery Grants Program - Individual

Topology is a branch of mathematics that deals with spatial properties of objects that are preserved under continuous deformations (such as stretching or squeezing an object without tearing or gluing any of its parts). In this branch, we do not ask how big an object is or what is its extension in space but, rather; is it all connected together, or can it be separated into many parts? Does it have any holes or cavities in it? It turns out that these types of properties allow us to design very powerful and flexible methods to distinguish between objects in space and to analyze high dimensional data sets. Many of these methods rely on the extraction of content descriptors capturing global and qualitative properties of the objects and the data sets. In each case, the goal remains to extract features that enhance our understanding of the shapes of objects and the measured data and allow us to represent them in structures that are easy to manipulate for subsequent processing. The main objective of this proposal is to utilize a combination of a theory from topology and the power of computing to solve problems in imaging and other fields where analysis of data sets is a key issue. We will develop topological and algorithmic tools that allow automatic identification, location, and classification of qualitative features of data sets, such as the presence of corners, edges, peaks, pits, passes, and ridge and ravine lines, and their higher dimensional analogues. Once these features are extracted, different techniques are used to analyze relationships between them and represent their configuration in structures such as graphs which allow us to handle this information in an abstract way. We also aim at developing efficient topological shape descriptors for the representation of three-dimensional objects. 3D models are used in the design of objects and components, and are archived to enable reuse and rapid prototyping of new products. The World Wide Web enables access freely or commercially to these digital archives. Thus, it becomes imperative to develop efficient methods for comparison and matching of 3D shapes to enable automatic 3D shape retrieval.