Calculus of Variations and Geometric Measure Theory

A. Brancolini - C. Rossmanith - B. Wirth

Optimal micropatterns in 2D transport networks and their relation to image inpainting

created by brancolin on 26 Sep 2017
modified by brancolini on 14 Nov 2017


Published Paper

Inserted: 26 sep 2017
Last Updated: 14 nov 2017

Journal: Archive for Rational Mechanics and Analysis
Year: 2017
Doi: 10.1007/s00205-017-1192-2

ArXiv: 1601.07402 PDF


We consider two different variational models of transport networks, the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford-Shah image inpainting and techniques developed in that field, we show for a two-dimensional situation that both highly non-convex network optimization tasks can be transformed into a convex variational problem, which may be very useful from analytical and numerical perspectives.

As applications of the convex formulation, we use it to perform numerical simulations (to our knowledge this is the first numerical treatment of urban planning), and we prove the lower bound of an energy scaling law which helps better understand optimal networks and their minimal energies.

Keywords: Optimal transport, Branched transport, Wasserstein distance, Optimal Networks, urban planning, Convex optimization, Irrigation, Image inpainting, micropatterns, energy scaling laws, convex lifting