Dissertation
Data-inspired advances in geometric measure theory: Generalized surface and shape metrics
Washington State University
Doctor of Philosophy (PhD), Washington State University
08/2014
DOI:
https://doi.org/10.7273/000005564
Abstract
Author's preferred version is on arXiv. Modern geometric measure theory, developed largely to solve the Plateau problem, has generated a great deal of technical machinery which is unfortunately regarded as inaccessible by outsiders. Consequently, its ideas have not been incorporated into other fields as effectively as possible. Some of these tools (e.g., distance and decompositions in generalized surface space using the flat norm) hold interest from a theoretical perspective but computational infeasibility prevented practical use. Others, like nonasymptotic densities as shape signatures, have been developed independently as useful data analysis tools (e.g., the integral area invariant). Here, geometric measure theory has promise to help close the gaps in our understanding of these ideas. The flat norm measures distance between currents (or generalized surfaces) by decomposing them in a way that is robust to noise. One new result here is that the flat norm can be suitably discretized and approximated on a simplicial complex by means of a simplicial deformation theorem. While not surprising given the classical (cubical) deformation theorem or, indeed, Sullivan's convex cellular deformation theorem (which includes simplicial deformation as a special case), the bounds on the deformation can be made smaller and more practical by focusing on the simplicial case. Computationally, the discretized flat norm can be expressed as a linear programming problem and thus solved in polynomial time. Furthermore, the solution is guaranteed to be integral if the complex satisfies a simple topological condition (absence of relative torsion). This discretized integrality result (with some work) yields a similar statement for the continuous case: the flat norm decomposition of an integral 1-current in the plane can be taken to be integral, something previously unknown for 1-currents which are not boundaries of 2-currents. Nonasymptotic densities (integral area invariants) taken along the boundary of a shape are often enough to reconstruct the shape. This result is easy when the densities are known for arbitrarily small radii but that is not generally possible in practice. When only a single radius is used, variations on reconstruction results (modulo translation and rotation) of polygons and (a dense set of) smooth curves are presented.
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Details
- Title
- Data-inspired advances in geometric measure theory
- Creators
- Sharif N. Ibrahim
- Contributors
- Kevin Royce Vixie (Chair)Bala Krishnamoorthy (Committee Member) - Washington State University, Mathematics and Statistics, Department ofThomas James Asaki (Committee Member) - Washington State University, Mathematics and Statistics, Department of
- Awarding Institution
- Washington State University
- Academic Unit
- Mathematics and Statistics, Department of
- Theses and Dissertations
- Doctor of Philosophy (PhD), Washington State University
- Publisher
- Washington State University
- Number of pages
- 131
- Identifiers
- 99901054231401842
- Language
- English
- Resource Type
- Dissertation