Partial optimal transport for a constant-volume Lagrangian mesh with free boundaries

A survey of Optimal Transport for Computer Graphics and Computer Vision

“…Generalizations of the optimal transport problems have also been investigated. Notably, when the two functions being compared do not have the same mass, this leads to the unbalanced and partial optimal transportation problems [CPSV18, BC19, Lév22], with similar solutions (entropy regularized, sliced, semi‐discrete or dynamical approaches). Similarly, when the two functions live on two different spaces and one only has access to pairwise distances within each of these spaces that need to be matched, the corresponding problem amounts to the Gromov‐Wasserstein problem [Mém11].…”

“…Typical semi‐discrete solvers optimize an energy designed so that, at optimality, the area of each power cell (i.e., the polygonal part of the city going towards its optimal bakery, in the above example) matches the mass of each Dirac (i.e., the amount of bread available at that bakery). This leads to fast, multiscale, implementations in 2‐d [Mér11] and 3‐d [Lév15], or using second order optimization schemes [Lév15, SWS*15, KMT19], and extends to optimal transport on the sphere [CQW*19] or when the masses in the two distributions are different [Lév22]. This is also relatively easy to implement – e.g., as a student project or lab.…”

“…Free‐boundary 3d fluid simulation including viscosity and surface tension, by projecting velocities on divergence‐free velocity fields using semi‐discrete optimal transport, with 500k power cells [Lév22]…”

“…Optimal transportation has been applied to a wide variety of other problems far beyond computer graphics and vision, from economics [Gal18] to operations research. We shall give a few words Figure 16: Free-boundary 3d fluid simulation including viscosity and surface tension, by projecting velocities on divergence-free velocity fields using semi-discrete optimal transport, with 500k power cells [Lév22] on notable applications, either related to the computer graphics and vision communities, or widely adopted by their communities.…”

An efficient semismooth Newton-AMG-based inexact primal-dual algorithm for generalized transport problems

“…In standard optimal transport (2.1), the marginal distributions µ and ν are required to have the same mass. Mathematically, this is quite restrictive and practically, the unbalanced case 1 ⊤ n µ = 1 ⊤ m ν stems from the positive-unlabeled learning [20] and the representation of dynamic meshes for controlling the volume of objects with free boundaries [66].…”