arXiv subjects

Carolin Rossmanith

Publications and source records attributed to Carolin Rossmanith.

3 recordsLinked to original sources

arXiv2018

Phase field approximations of branched transportation problems

In branched transportation problems mass has to be transported from a given initial distribution to a given final distribution, where the cost of the transport is proportional to the transport distance, but subadditive in the transported mass. As a consequence, mass transport is cheaper the more mass is transported together, which leads to the emergence of hierarchically branching transport networks. We here consider transport costs that are piecewise affine in the transported mass with N affine segments, in which case the resulting network can be interpreted as a street network composed of N different types of streets. In two spatial dimensions we propose a phase field approximation of this street network using N phase fields and a function approximating the mass flux through the network. We prove the corresponding $Γ$-convergence and show some numerical simulation results.

math.OC↗

arXiv2016

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

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.

math.CA↗

arXiv2016

Simultaneous Reconstruction and Segmentation for Dynamic SPECT Imaging

This work deals with the reconstruction of dynamic images that incorporate characteristic dynamics in certain subregions, as arising for the kinetics of many tracers in emission tomography (SPECT, PET). We make use of a basis function approach for the unknown tracer concentration by assuming that the region of interest can be divided into subregions with spatially constant concentration curves. Applying a regularized variational framework reminiscent of the Chan-Vese model for image segmentation we simultaneously reconstruct both the labelling functions of the subregions as well as the subconcentrations within each region. Our particular focus is on applications in SPECT with Poisson noise model, resulting in a Kullback-Leibler data fidelity in the variational approach. We present a detailed analysis of the proposed variational model and prove existence of minimizers as well as error estimates. The latter apply to a more general class of problems and generalize existing results in literature since we deal with a nonlinear forward operator and a nonquadratic data fidelity. A computational algorithm based on alternating minimization and splitting techniques is developed for the solution of the problem and tested on appropriately designed synthetic data sets. For those we compare the results to those of standard EM reconstructions and investigate the effects of Poisson noise in the data.

math.NA↗