6 research outputs found
An algorithm for optimal transport between a simplex soup and a point cloud
We propose a numerical method to find the optimal transport map between a
measure supported on a lower-dimensional subset of R^d and a finitely supported
measure. More precisely, the source measure is assumed to be supported on a
simplex soup, i.e. on a union of simplices of arbitrary dimension between 2 and
d. As in [Aurenhammer, Hoffman, Aronov, Algorithmica 20 (1), 1998, 61--76] we
recast this optimal transport problem as the resolution of a non-linear system
where one wants to prescribe the quantity of mass in each cell of the so-called
Laguerre diagram. We prove the convergence with linear speed of a damped
Newton's algorithm to solve this non-linear system. The convergence relies on
two conditions: (i) a genericity condition on the point cloud with respect to
the simplex soup and (ii) a (strong) connectedness condition on the support of
the source measure defined on the simplex soup. Finally, we apply our algorithm
in R^3 to compute optimal transport plans between a measure supported on a
triangulation and a discrete measure. We also detail some applications such as
optimal quantization of a probability density over a surface, remeshing or
rigid point set registration on a mesh
Dynamical Optimal Transport on Discrete Surfaces
We propose a technique for interpolating between probability distributions on
discrete surfaces, based on the theory of optimal transport. Unlike previous
attempts that use linear programming, our method is based on a dynamical
formulation of quadratic optimal transport proposed for flat domains by Benamou
and Brenier [2000], adapted to discrete surfaces. Our structure-preserving
construction yields a Riemannian metric on the (finite-dimensional) space of
probability distributions on a discrete surface, which translates the so-called
Otto calculus to discrete language. From a practical perspective, our technique
provides a smooth interpolation between distributions on discrete surfaces with
less diffusion than state-of-the-art algorithms involving entropic
regularization. Beyond interpolation, we show how our discrete notion of
optimal transport extends to other tasks, such as distribution-valued Dirichlet
problems and time integration of gradient flows