Spot

Abstract

Optimal transport research has surged in the last decade with wide applications in computer graphics. In most cases, however, it has focused on the special case of the so-called "balanced" optimal transport problem, that is, the problem of optimally matching positive measures of equal total mass. While this approach is suitable for handling probability distributions as their total mass is always equal to one, it precludes other applications manipulating disparate measures. Our paper proposes a fast approach to the optimal transport of constant distributions supported on point sets of different cardinality via one-dimensional slices. This leads to one-dimensional partial assignment problems akin to alignment problems encountered in genomics or text comparison. Contrary to one-dimensional balanced optimal transport that leads to a trivial linear-time algorithm, such partial optimal transport, even in 1-d, has not seen any closed-form solution nor very efficient algorithms to date. We provide the first efficient 1-d partial optimal transport solver. Along with a quasilinear time problem decomposition algorithm, it solves 1-d assignment problems consisting of up to millions of Dirac distributions within fractions of a second in parallel. We handle higher dimensional problems via a slicing approach, and further extend the popular iterative closest point algorithm using optimal transport - an algorithm we call Fast Iterative Sliced Transport. We illustrate our method on computer graphics applications such a color transfer and point cloud registration.

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].…”

“…Later, Rabin et al [RFP14] directly add regularization within the linear program by relaxing the mass preservation constraints. Similar regularization for color transfer can be obtained via unbalanced or partial optimal transport [CPSV18, BC19]. Pitié et al instead propose a variational formulation that enforces the resulting image's gradient to be regularized towards the original image gradients [PKD07] in order to reduce grain.…”

Local color harmony adjustment via color envelope transformation

“…The essence of color migration [7,[10][11][12][13][14][15][16] is the migration of image color distribution features. When the operator adjusts the color of the target image block, in order to make the overall harmony of the image, the other related image blocks need to capture the color style change of the target image block and naturally migrate this change to the related image blocks.…”

“…In this paper, we are concerned with the natural radiation of the operator's local color adjustment effect to the whole image to achieve the harmony of the image theme. Sliced Partial Optimal Transport [7] is a fast optimization algorithm for solving the optimal transport problem by projecting slices on multiple one-dimensional scales to achieve the optimal transport strategy on different base point sets with constant distribution. Based on the sliced local optimal transmission, the operator adjusted image block style radiates to other related image blocks can have high similarity of color distribution between the two parts, while maintaining the content consistency of the target image block before and after adjustment, without changing the image detail texture.…”

Sliced optimal transport on the sphere

“…The idea behind sliced optimal transport has been generalized and transferred to many related problems. There exists sliced variants [8,16] of partial optimal transport [19,27], where only a fraction of mass is transported, and a sliced version [20] of multi-marginal optimal transport [10,12,30], considering the transport between several measures instead of only two. For optimal transport on Riemannian manifolds, sliced Wasserstein distances based on the push-forward of the eigenfunctions of the Laplacian have been proposed in [77].…”