Rajesh K. Batra

Feature comparisons of 3-D vector fields using earth mover's distance

A method for comparing three-dimensional vector fields constructed from simple critical points is described. This method is a natural extension of previous work (Y. Lavin et al., 1998), which defined a distance metric for comparing two-dimensional fields. The extension to three-dimensions follows the path of our previous work, rethinking the representation of a critical point signature and the distance measure between the points. Since the method relies on topologically based information, problems such as grid matching and vector alignment which often complicate other comparison techniques are avoided. In addition, since only feature information is used to represent, and is therefore stored for each field, a significant amount of compression occurs.

Topology Visualization of the Optical Power Flow through a Novel C-Shaped Nano-Aperture

An ideal visualization tool that has not been used before in studying the optical behavior of near-field apertures is three-dimensional vector field topology. The global view of the vector field structure is deduced by locating singularities (critical points) within the field and augmenting these points with nearby streamlines. We have used for the first time, to the best of our knowledge, three-dimensional topology to analyze the topological differences between a resonant C-shaped nano-aperture and various nonresonant conventional apertures. The topological differences between these apertures are related to the superiority in power throughput of the C-aperture versus conventional round and square sub-wavelength apertures. We demonstrate how topological visualization techniques provide significant insight into the energy enhancement mechanism of the C aperture, and also shed light on critical issues related to the interaction between multiple apertures located in close proximity to each other, which gives rise to cross-talk, for example as a function of distance. Topological techniques allow us to develop design rules for the geometry of these apertures and their desired spot sizes and brightness. The performance of various sub-wavelength apertures can also be compared quantitatively based on their topology. Since topological methods are generically applicable to tensor and vector fields, our approach can be readily extended to provide insight into the broader category of finite-difference-time-domain nano-photonics and nano-science problems.

Topology based methods for quantitative comparisons of vector fields

Computers have enabled the generation and acquisition of large vector and tensor data-sets. Reynolds numbers, and in data acquisition producing large data-set such as weather and ocean current databases. However, techniques for automated searching and comparison of large vector and tensor data-sets have been lacking. Many of the techniques are based on visual comparisons which do not have the quantitative rigor required for automated comparisons. Other methods which use data based comparisons require a common domain, grid alignment, and interpolation often making data- sets acquired from different sources difficult to compare. This paper provides comparison methods that are based on the topology of the vector field. These methods can be used for both 2D and 3D fields, and depending upon the complexity of the field, different levels of comparisons can be utilized to reduce search times. Topology based methods have the added benefits of describing a field that is independent of the grid, and that is based solely on topological components which have a significant compression over the original field.