Total Variation Isoperimetric Profiles
Abstract
Applications such as political redistricting demand quantitative measures of geometric compactness to distinguish between simple and contorted shapes. While the isoperimetric quotient, or ratio of area to perimeter squared, is commonly used in practice, it is sensitive to noisy data and irrelevant geographic features like coastline. These issues are addressed in theory by the isoperimetric profile, which plots the minimum perimeter needed to inscribe regions of different prescribed areas within the boundary of a shape. Efficient algorithms for computing this profile, however, are not known in practice. Hence, in this paper, we propose a convex Eulerian relaxation of the isoperimetric profile using total variation. We prove theoretical properties of our relaxation, showing that it still satisfies an isoperimetric inequality and yields a convex function of the prescribed area. Furthermore, we provide a discretization of the problem, an optimization technique, and experiments demonstrating the value of our relaxation.
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Information & Authors
Information
Published In
SIAM Journal on Applied Algebra and Geometry
Pages: 585 - 613
DOI: 10.1137/18M1215943
ISSN (online): 2470-6566
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 21 September 2018
Accepted: 23 August 2019
Published online: 19 November 2019
Keywords
MSC codes
Authors
Funding Information
Prof. Amar G. Bose Research Grant
Amazon Research Award
Army Research Office https://doi.org/10.13039/100000183 : W911NF-12-R-0011
Massachusetts Institute of Technology https://doi.org/10.13039/100006919
National Science Foundation https://doi.org/10.13039/100000001 : IIS-1838071
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