Recent years have witnessed an unprecedented growth in the generation and availability of large scale and complex data. Such growth has fostered the development of new qualitative methods of data analysis. Among the most prominent are the set of methods falling under the rubric of topological methods.
From their inception, topological methods have been gaining in prominence mainly in the studies of scalar data, in particular the topology of scalar functions over manifolds. With the increasing complexity of the available data, there is an interest in extending the topological methods to deal with more complex types of data such as multiple scalar fields and vector fields over manifolds, scalar data over point clouds, as well as scalar fields over high dimensional stratified spaces.
In this dissertation we study the extension of topological methods to the treatment of multiple scalar fields, and scalar fields over a variety of domains ranging from the planar to the stratified.
We first showcase the use of scalar field topological methods to study 2D vector fields over planar domains. We achieve this by transforming the vector field into multiple scalar fields through the use of Hodge decomposition. Hodge decomposition in 2D allows us to transform the vector field into two scalar fields where we can apply scalar field based topological methods. We take advantage of this decomposition to generate topologically-aware visualization of the vector field.
A particular case of multiple scalar functions is a time varying function, where time is treated as a second scalar function.
We study two topological constructs in the context of time varying functions: persistent homology and contour trees. We present an algorithm to incrementally update the persistence pairing of a time-varying function defined on a triangular mesh in O(log n), and contour trees of time-varying functions in O(log n + | lnk(p) ∩ lnk(q) |) where |lnk(p)| is the size of the link of a vertex p in the tetrahedral mesh.
We then examine the relation between multiple scalar functions through the use of the Jacobi set. The Jacobi set attempts to quantify the correlation between two or more scalar fields defined on a common domain. We investigate the computation and simplification of the Jacobi set for point cloud data through the computation of the gradient in eigen space.
Finally we look at recovering spaces of intrinsic dimensions of one (1-strata) from a point cloud embedded in high dimensional space. We make use of the well known Reeb graph to recover the space, and the result is a simple and efficient algorithm that avoids the prohibitive cost of a full space reconstruction using traditional methods.
In summary, we present a variety of methods for exploring the use of topological methods in the study of complex settings involving scalar fields.
We hope in the future to adapt current methods to point clouds sampled from probability distributions with the presence of outliers and background noise.