An algorithm for optimal transport between a simplex soup and a point cloud

Variational Graph Methods for Efficient Point Cloud Sparsification

“…This will enable a comparison with state-of-the-art methods for point cloud sparsification, e.g. the optimal transport based scheme in [MMT17].…”

“…There exists a heuristic method known as Lloyd's algorithm [Llo82] that aims to find barycenters of partitions based on k-means clustering and the related Voronoi cells. More sophisticated methods extend this approach via a variational formulation based on optimal transport and Laguerre cells [De +12 ;MMT17].…”

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

“…Step (7) of the algorithm computes the Newton step vector p, by solving a linear system. Except the boundary term in ∂ 2 K/∂ψ 2 i , this linear system is identical as the one solved in [4]: it corresponds to a Poisson equation discretized with finite elements.…”

“…To do so, we need to convert from the initial representation of the cell, that is, as an intersection of half-spaces" (H-representation), (Π + i ) N v i=1 into an explicit representation of all the vertices and facets of the cell (V-representation). As shown in Figure 7, if the planes are in generic position 7 , each vertex (black) is shared by three planes Π i , Π j , Π k . It is then natural to represent the polytope in dual form (see Figure 7), by a triangulation (T l = {i l , j l , k l }) Nt l=1 shown in red in the Figure (the same representation is exploited by the Boywer-Watson algorithm that we use to compute the Laguerre diagram).…”

“…The approach builds on recent advances in numerical optimal transport, that resulted in Lagrangian schemes for fluid simulation [7] or early Universe reconstruction [4]. In the works cited above, the object fills the entire simulation domain.…”