Multiscale Path Metrics for the Analysis of Discrete Geometric Structures
Abstract
The objective of this research is to establish the foundations for a novel mathematical approach describing discrete geometric structures formed by countable (finite or infinite) sets of functions belonging to a given Banach space. The proposed approach is alternative to both spectral graph theory and the theories deriving from manifold approximation techniques (the so-called Òmanifold learningÓ). The key idea is to utilize a new framework for the generation of topological spaces, which can be considered both analytical and algebraic in nature, and completely independent of any preset smoothness assumptions. The discrete topology arising from this new model is surprisingly close to that observed in some Òreal-worldÓ data-sets, and may therefore open the way for the study of discrete geometries observed in various kinds of real data. An important goal of this research is to define guidelines for the choice of novel metric functions which will be best suited for the transcription of global ÒgeometricÓ characteristics of different kinds of data in the presence of non-Gaussian noise. The results obtained by such investigation will provide a more solid foundation to the study of the geometry of data sets produced by sensors operating in the real world, and may also be of use in the understanding of social structure and phenomena based on social network data. The framework will be the basis for reliable methods for measuring distance and similarity between objects found in multi-camera video streams, and for identifying and tracking complex objects (groups of individuals, vehicles, etc.) and characterizing their state. This problem requires by its very nature a high degree of computational agility (needed in order to handle the high data-throughput), and efficient choices of a local metric therefore translate into a significant gain of efficacy of the algorithm.
Document Details
- Document Type
- DoD Grant Award
- Publication Date
- May 22, 2017
- Source ID
- W911NF1610392
Entities
People
- Carlo Tomasi
Organizations
- Army Contracting Command
- Duke University
- United States Army