Given an object a of class pgrid specifying an image and an object b
of class wpp specifiying a more flexible mass distribution at finitely many points,
find the partition of the image (and hence the optimal transport map) that minimizes the total transport cost
for going from a to b.
semidiscrete(a, b, p = 2, method = c("aha"), control = list(), ...)An object describing the optimal transport partition for a and b.
If p=1 an object of class apollonius_diagram having components sites and weights,
as well as (optionally) wasserstein_dist and ret_code (the return code from the call to
semidiscrete1).
If p=2 an objectof class power_diagram having components sites and cells,
as well as (optionally) wasserstein_dist. sites is here a data.frame with columns xi,
eta and w (the weights for the power diagram). cells is a list with as many
2-column matrix components as there are sites, each describing the \(x\)- and \(y\)-coordinates
of the polygonal cell associated with the corresponding site or NULL if the cell of the site is empty.
Plotting methods exist for objects of class apollonius_diagram, power_diagram and
for optimal transport maps represented by either of the two.
an object of class pgrid usually representing an image or the discretization of a measure.
an object of class wpp usually having the same total mass as a.
the power \(\geq 1\) to which the Euclidean distance between points is taken in order to compute costs. Only \(p \in \{1,2\}\) is implemented.
the name of the algorithm to use. Currently only aha is supported.
a named list of parameters for the chosen method or the result of a call to trcontrol. Currently only
the parameters factr and maxit can be set.
currently without effect.
Dominic Schuhmacher dschuhm1@uni-goettingen.de
Björn Bähre bjobae@gmail.com
Valentin Hartmann valentin.hartmann@epfl.ch
This is a wrapper for the functions aha and semidiscrete1. In the former the
Aurenhammer--Hoffmann--Aronov (1998) method for \(p=2\) is implemented in the multiscale variant presented
in Mérigot (2011). In the latter an adapted Aurenhammer--Hoffmann--Aronov method for \(p=1\) is used that
was presented in Hartmann and Schuhmacher (2018).
The present function is automatically called by transport if the
first argument is of class pgrid and the second argument is of class wpp.
F. Aurenhammer, F. Hoffmann and B. Aronov (1998). Minkowski-type theorems and least-squares clustering. Algorithmica 20(1), 61--76.
V. Hartmann and D. Schuhmacher (2017). Semi-discrete optimal transport --- the case p=1. Preprint arXiv:1706.07650
M. Karavelas and M. Yvinec. 2D Apollonius Graphs (Delaunay Graphs of Disks). In CGAL User and Reference Manual. CGAL Editorial Board, 4.12 edition, 2018
Q. Mérigot (2011). A multiscale approach to optimal transport. Computer Graphics Forum 30(5), 1583--1592. tools:::Rd_expr_doi("10.1111/j.1467-8659.2011.02032.x")
Naoaki Okazaki (2010). libLBFGS: a library of Limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS). Version 1.10
plot, transport, aha, semidiscrete1
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